values for the central 2 kpc of $\mathrm{NGC\,4321}$ using a multi-line
modelling approach based on cloud-scale multi-line, multi-transition CO
isotopologues. Despite the existing robust $\alpha_{\text{CO}}$ measurement,
we employ the surface density-metallicity-based calibration from Bolatto et
al. (2013) to obtain continuous $\alpha_{\text{CO}}$ values across the full
disc of $\mathrm{NGC\,4321}$ and note that the measurements from Teng et al.
(2023) are on average $0.20\text{\,}\mathrm{dex}$ smaller than the values used
here, but consistent at the 1-sigma level within $0.21\text{\,}\mathrm{dex}$.

###  A.2 Atomic gas – H I 21-cm

We utilise H I 21-cm line observations to measure the atomic gas surface
density. The H I data are from VLA observations associated with HERACLES
(Leroy et al. 2009) mapping several nearby galaxies in H I at $\sim$$$
resolution. We convert the integrated intensity of H I ($W_{\rm 21cm}$) into
atomic gas surface density via (Walter et al. 2008):

$\left(\dfrac{\Sigma_{\text{atom}}}{$\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)=$1.97\text{\times}{10}^{-2}$\left(\dfrac{W_{\rm
21cm}}{$\text{\,}\mathrm{K}\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$}\right)\cos(i),$
(13)

where $\cos(i)$ is accounting for the inclination of the galaxy
($i=$38.5\text{\,}\mathrm{\SIUnitSymbolDegree}$$).

###  A.3 Stellar mass – $3.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$

We use the stellar mass surface density map from Querejeta et al. (2015), who
use $3.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ observations from the Spitzer
Survey of Stellar Structure in Galaxies (S4G; Sheth et al. 2010). The
$3.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ maps are corrected for dust
attenuation using an “Independent Component Analysis” (ICA) method that
separates stars and dust on a pixel-to-pixel basis (for more details, see
Querejeta et al. 2015) We convert the attenuation-corrected
$3.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ map into stellar mass surface
density, $\Sigma_{\star}$, via:

$\left(\dfrac{\Sigma_{\star}}{$\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)=$4.22\text{\times}{10}^{2}$\left(\dfrac{I_{$3.6\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}}{$\text{\,}\mathrm{MJy}\text{\,}{\mathrm{sr}}^{-1}$}\right)\,,$ (14)

which assumes a constant mass-to-light ratio of
$0.6\,$\text{\,}\mathrm{M_{\odot}}$/$\text{\,}\mathrm{L_{\odot}}$$ (Meidt et
al. 2014).

In App. A.4, we use the stellar mass volume density ($\rho_{\star}$) to
compute the dynamical equilibrium pressure of the ISM in the galaxy disc. We
estimate $\rho_{\star}$ from the stellar mass surface density, adopting the
recipes used in Blitz & Rosolowsky (2006); Leroy et al. (2008); Ostriker et
al. (2010); Sun et al. (2020):

$\displaystyle\rho_{\star}=\dfrac{\Sigma_{\star}}{4H_{\star}}=\dfrac{\Sigma_{\star}}{0.54\,R_{\star}}\,,$
(15)

where $H_{\star}$ and $R_{\star}$ are the scale height and radial scale length
of the stellar disc. Above equation assumes an isothermal density profile
along the vertical direction (van der Kruit 1988) and a fixed stellar disc
flattening ratio of $R_{\star}/H_{\star}=7.3$ (Kregel et al. 2002). We adopt
$R_{\star}=$61.1\text{\,}\arcsec$\sim$4.5\text{\,}\mathrm{kpc}$$ from the S4G
photometric decomposition analysis (Salo et al. 2015).

###  A.4 Dynamical equilibrium pressure

We compute the dynamical equilibrium pressure, or ISM pressure
($P_{\text{DE}}$) following the prescription by Sun et al. (2020).
$P_{\text{DE}}$ describes the pressure regulated by the mass content in the
ISM disc and thus provides an important gauge of the local environment of
molecular clouds. The distribution of stars and gas in a galaxy disc can
approximately be described as isothermal fluids in a plane-parallel geometry.
In this prescription the dynamical equilibrium pressure is composed of a
pressure term created by the ISM due to the self-gravity of the ISM disc and a
term due to the gravity of the stars (see e.g. Spitzer 1942), such that:

$\displaystyle P_{\text{DE}}=\dfrac{\pi
G}{2}\Sigma_{\text{gas}}^{2}+\Sigma_{\text{gas}}\sqrt{2G\rho_{\star}}\,\sigma_{\text{gas,z}}\;,$
(16)

where we assume a smooth, single-fluid gas disc, and that all gas shares a
similar velocity dispersion, so that
$\Sigma_{\text{gas}}=\Sigma_{\text{mol}}+\Sigma_{\text{atom}}$ is the total
gas surface density, $\rho_{\star}$ is the stellar mass volume density near
disc midplane and $\sigma_{\text{gas,z}}$ is the velocity dispersion of the
gas perpendicular to the disc.

Sun et al. (2020) proposed a new formalism which takes into account the self-
gravity of the molecular gas at high resolution, i.e.
$100\text{\,}\mathrm{pc}$ scale. In this work, we adopt their formalism, which
is described in the following. To combine the cloud-scale $\Sigma_{\text{mol,
120\,pc}}$ data with the large-scale $\Sigma_{\text{atom}}$ and
$\rho_{\star}$, we split $P_{\text{DE}}$ into two parts: the pressure of the
molecular gas at cloud-scale, $P_{\text{cloud}}$, and the pressure of the
smooth extended atomic gas due to the gravity of all gas (atomic and
molecular) and the stars, $P_{\text{atom}}$. The cloud-scale
$P_{\text{cloud}}$ consists of three terms accounting for the self-gravity of
the molecular gas, the gravity of larger molecular structures and the gravity
of stars:

$\displaystyle P_{\text{cloud}}=$
$\displaystyle\dfrac{3\pi}{8}G\Sigma_{\text{mol,
120\,pc}}^{2}+\dfrac{\pi}{2}G\Sigma_{\text{mol, 120\,pc}}\Sigma_{\text{mol,
260\,pc}}$ $\displaystyle+\dfrac{3\pi}{4}G\rho_{\star}\Sigma_{\text{mol,
120\,pc}}D_{\text{beam}}\;,$ (17)

where $\rho_{\star}$ is computed as described in App. A.3. In Equ. (17),
$\Sigma_{\text{mol, 120\,pc}}$ is given at the cloud-scale resolution (here
$120\text{\,}\mathrm{pc}$) while $\Sigma_{\text{mol, 260\,pc}}$ and
$\rho_{\star}$ describe the distributions of the molecular gas and the stellar
mass density at $260\text{\,}\mathrm{pc}$ scale. In typical units, Equ. (17)
reads:

$\displaystyle\left(\dfrac{P_{\text{cloud}}}{$\text{\,}\mathrm{\textit{k}_{B}}\text{\,}\mathrm{K}\text{\,}{\mathrm{cm}}^{-3}$}\right)=$2.48\text{\times}{10}^{5}$\left(\dfrac{\Sigma_{\text{mol,
120\,pc}}}{${10}^{2}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)^{2}$
(18)
$\displaystyle\quad+\;$3.31\text{\times}{10}^{5}$\left(\dfrac{\Sigma_{\text{mol,
120\,pc}}}{${10}^{2}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)\left(\dfrac{\Sigma_{\text{mol,
260\,pc}}}{${10}^{2}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)$
$\displaystyle\quad+\;$7.01\text{\times}{10}^{4}$\left(\dfrac{\Sigma_{\text{mol,
120\,pc}}}{${10}^{2}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)\left(\dfrac{\rho_{\star}}{${10}^{-1}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-3}$}\right)\left(\dfrac{D_{\text{beam}}}{$150\text{\,}\mathrm{pc}$}\right)\;,$

The large-scale $P_{\text{atom}}$ includes the self-gravity of the atomic gas
and the gravitational interaction of the atomic gas with the (large-scale)
molecular gas and the stars:

$\displaystyle P_{\text{atom}}=\dfrac{\pi G}{2}\Sigma_{\text{atom}}^{2}+\pi
G\Sigma_{\text{atom}}\Sigma_{\text{mol,
260\,pc}}+\Sigma_{\text{atom}}\sqrt{2G\rho_{\star}}\,\sigma_{\text{atom}}\;,$
(19)

where $\sigma_{\text{atom}}$ is the velocity dispersion of the atomic gas
which is fixed at
$\sigma_{\text{atom}}=$10\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$. In
Equ. (19) all quantities are convolved to the large-scale resolution.
Converting to typical units we obtain:

$\displaystyle\left(\dfrac{P_{\text{atom}}}{$\text{\,}\mathrm{\textit{k}_{B}}\text{\,}\mathrm{K}\text{\,}{\mathrm{cm}}^{-3}$}\right)=$3.31\text{\times}{10}^{5}$\left(\dfrac{\Sigma_{\text{atom}}}{${10}^{2}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)^{2}$
(20)
$\displaystyle\quad+\;$6.62\text{\times}{10}^{5}$\left(\dfrac{\Sigma_{\text{atom}}}{${10}^{2}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)\left(\dfrac{\Sigma_{\text{mol,
260\,pc}}}{${10}^{2}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)$
$\displaystyle\quad+\;$1.02\text{\times}{10}^{5}$\left(\dfrac{\Sigma_{\text{atom}}}{${10}^{2}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$}\right)\left(\dfrac{\rho_{\star}}{${10}^{-1}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-3}$}\right)^{1/2}\;,$

Finally, we compute the intensity-weighted average, $\langle P_{\rm
cloud,\,120\,pc}\rangle_{\rm 260\,pc}$, at the large-scale to combine the
molecular gas weight with the large-scale $P_{\text{atom}}$ pixel by pixel:

$\displaystyle\langle P_{\rm DE,\,120\,pc}\rangle_{\rm 260\,pc}=\langle P_{\rm
cloud,\,120\,pc}\rangle_{\rm 260\,pc}+P_{\text{atom}}$ (21)

Figure 10: Comparison of SFR tracer in dense gas scaling relations in the
inner 4 kpc. Top: SFR/HCN, tracing dense gas star formation efficiency, using
three different SFR tracer from left to right, $\text{H}\alpha$ from MUSE,
$21\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from JWST and
$33\text{\,}\mathrm{GHz}$ from VLA. Contours show HCN intensities as in Fig.
3. The dotted ellipse denotes $r_{\rm gal}=$2.75\text{\,}\mathrm{kpc}$$.
Bottom: SFR/HCN, matching the respective above panels, against galactocentric
radius, $r_{\rm gal}$. Blue points indicate detected ($\mathrm{S/N}\geq 3$)
and light blue points denote non-significant ($\mathrm{S/N}<3$) data. The red
hexagon markers show the spectral stacks taken over all data within the bin.
In the middle panel, we show the data obtained from the linear
$21\text{\,}\mathrm{\SIUnitSymbolMicro m}$-to-SFR conversion (Equ. (22)) and
additionally indicate the mean trend inferred from a power law conversion,
i.e. $\mathrm{SFR}\propto L(\mathrm{F2100W})^{1.3}$ (Leroy et al. 2023).
Figure 11: HCN spectroscopic ratios against molecular cloud properties. Top:
HCN/CO and (bottom) SFR/HCN measured at $260\text{\,}\mathrm{pc}$ scale vs
cloud-scale molecular gas properties inferred from $120\text{\,}\mathrm{pc}$
scale CO(2–1) observations. Panels from left to right show molecular cloud
surface density ($\Sigma_{\text{mol}}$), velocity dispersion
($\sigma_{\text{mol}}$) and virial parameter ($\alpha_{\text{vir}}$) on the
$x$-axis, obtained from PHANGS–ALMA as described in Sect. D.1. Similarly to
Fig. 7 and 8, markers indicate the respective environments and line fits as
well as linear regression regimes are determined via MARS. The obtained
thresholds are
$\Sigma_{\text{mol}}=$74\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$$
for SFR/HCN vs. $\Sigma_{\text{mol}}$ and
$\sigma_{\text{mol}}=$8.3\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$ for
SFR/HCN vs. $\sigma_{\text{mol}}$. The linear regression parameters are listed
in Tab. 6. Figure 12: HCN spectroscopic ratio vs stellar mass surface density
similar to Fig. 7 and 8, but using $\Sigma_{\star}$ on the $x$-axis. Stellar
mass is traced via the dust-attenuation corrected
$3.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ emission. The line fit parameters
are listed in Tab. 6. Figure 13: Conversion factor maps. Top: CO(2–1)-to-
CO(1–0) line ratio ($R_{21}$) map computed from CO(2–1) observations
(PHANGS–ALMA) and CO(1–0) observations (ALMA science verification program) at
a common $None$ resolution. Middle: CO-to-$\text{H}_{2}$ conversion factor
($\alpha_{\text{CO}}$) following the prescription from Bolatto et al. (2013)
(Equ. 12) using metallicities (bottom panel) from PHANGS–MUSE optical
recombination lines observations (Williams et al. 2022). Figure 14: Pressure
threshold contours. The map shows the same morphological environment masks as
in Fig. 3. Overlaid are HCN contours in grey and $\langle
P_{\text{DE}}\rangle$ contours at $\langle
P_{\text{DE}}\rangle=$3.8\text{\times}{10}^{5}\text{\,}\mathrm{\textit{k}_{B}}\text{\,}\mathrm{K}\text{\,}{\mathrm{cm}}^{-3}$$
(black) and at $\langle
P_{\text{DE}}\rangle=$1.6\text{\times}{10}^{6}\text{\,}\mathrm{\textit{k}_{B}}\text{\,}\mathrm{K}\text{\,}{\mathrm{cm}}^{-3}$$
(red). The pressure contours represent the threshold values in the pressure
relations discussed in Sect. 4.3. Figure 15: Line ratio trends with
metallicity. Top: HCN/CO(2–1) against metallicity Bottom: HCN/HCO+ against
metallicity. Dark markers indicate significant data ($\rm{S/N}\geq 3$) and
light markers show non-detections ($\rm{S/N}<3$). The red hexagon markers show
the average trends over all data obtained via spectral stacking. We note that
the HCO+ data will be separately published and studied in more detail along
with other dense gas tracers in Neumann et al. in prep. Here, we only show the
HCN/HCO+ variation with metallicity to highlight the flat trends hence
supporting HCN as a tracer of density across the full molecular gas disc of
$\mathrm{NGC\,4321}$. Figure 16: Data product maps compilation. All maps are
convolved to a common resolution of $260\text{\,}\mathrm{pc}$, given by the
native resolution of the HCN data, and sampled to a common, hexagonal pixel
grid at beam size spacing. Figure 17: Similar to to Fig. 4, but using the
galactocentric radius, $r_{\rm gal}$, as the stacking quantity with
$0.5\text{\,}\mathrm{kpc}$ (twice the beam size) bin widths. The panels with
hatched background denote radial bins that are not completely covered by the
field-of-view of the observations and thus not considered for the radial fit
in Fig. 6. Figure 18: Similar to to Fig. 4, but using the dynamical
equilibrium pressure, $\langle P_{\text{DE}}\rangle$, as the stacking quantity
with $0.1\text{\,}\mathrm{dex}$ bin widths. Note that the bins from $\langle
P_{\text{DE}}\rangle=10^{7.0}$ to
$10^{7.4}\,$\text{\,}\mathrm{\textit{k}_{B}}\text{\,}\mathrm{K}\text{\,}{\mathrm{cm}}^{-3}$$
do not contain any spectra and are therefore not shown. Figure 19: Similar to
to Fig. 4, but using the molecular gas surface density, $\Sigma_{\text{mol}}$,
as the stacking quantity with 20 bins from
$\Sigma_{\text{mol}}=3-$353\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$$.
Note that the bin at
$\Sigma_{\text{mol}}=173-$219\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$$
does not contain any spectra and is therefore not shown. Bins with
$\Sigma_{\text{mol}}<$10\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$$
are indicated with hatched backgrounds and are not used for the line fit in
Fig. 9. Table 6: HCN/CO($2-1$) and SFR/HCN correlations with $\Sigma_{\star}$,
$\langle\Sigma_{\text{mol}}\rangle$, $\langle\sigma_{\text{mol}}\rangle$,
$\langle\alpha_{\text{vir}}\rangle$.

x-axis | y-axis | Regime | Slope (stacks) | Slope (los) | Corr. ($p$) | Scatter
---|---|---|---|---|---|---
$\langle\Sigma_{\text{mol}}\rangle$ | HCN/CO(2$-$1) |  | 0.58 | 0.40 (0.04) | 0.43 (0.0) | 0.25
SFR/HCN | $\leq$74\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$$ | 0.00 | -0.75 (0.13) | -0.30 (0.0) | 0.60
$>$74\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$$ | -1.33 | -2.30 (0.45) | -0.53 (0.0) | 0.39
$\langle\sigma_{\text{mol}}\rangle$ | HCN/CO(2$-$1) |  | 0.81 | 0.51 (0.04) | 0.57 (0.0) | 0.23
SFR/HCN | $\leq$8.3\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$ | 0.00 | -1.05 (0.26) | -0.27 (0.0) | 0.52
$>$8.3\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$ | -2.29 | -1.58 (0.16) | -0.60 (0.0) | 0.37
$\langle\alpha_{\text{vir}}\rangle$ | HCN/CO(2$-$1) |  | 0.65 | 0.46 (0.03) | 0.52 (0.0) | 0.21
SFR/HCN |  | -0.84 | -1.11 (0.07) | -0.55 (0.0) | 0.44
$\Sigma_{\star}$ | HCN/CO(2$-$1) |  | 0.52 | 0.44 (0.03) | 0.57 (0.0) | 0.19
SFR/HCN |  | -0.73 | -1.02 (0.06) | -0.50 (0.0) | 0.46

Notes – Linear regression parameters for the respective relations and $x$-axis
regimes presented in Fig. 12 and 11 analogous to Tab. 4. The slopes of the
binned relations (column 4) are either determined by MARS if two distinct
regimes with different linear regression behaviour have been found, or
computed with LinMix if the relation is well described with a single line
according to the MARS algorithm.

##  Appendix B Spectral stacking

We compute spectral stacks using the python package
PyStacker666https://github.com/PhangsTeam/PyStacker presented in Neumann et
al. (2023a). PyStacker uses a high-significance prior (here CO(2–1)) to
determine the velocity field of the line emission. It then uses this velocity
information to shift the spectra of various line to correct for the Doppler
shift, assuming all lines share the same velocity field. Afterwards, we
average the shuffled spectra over larger regions usually resulting in higher-
significance detections. Here, we stack the emission of CO(2–1) (from
PHANGS–ALMA) and HCN(1–0) (this work) via morphological environment (Fig. 4),
galactocentric radius, $r_{\rm gal}$ (Fig. 17), dynamical equilibrium
pressure, $\langle P_{\text{DE}}\rangle$ (Fig. 18), molecular gas surface
density (Fig. 19), $\Sigma_{\text{mol}}$, stellar mass surface density,
$\Sigma_{\star}$, and molecular cloud properties,
$\langle\Sigma_{\text{mol}}\rangle$, $\langle\sigma_{\text{mol}}\rangle$,
$\langle\alpha_{\text{vir}}\rangle$. We use these averaged spectra to compute
integrated intensities, where the velocity-integration window is inferred
based on the average CO(2–1) spectrum using the same masking method as
described in Sect. 3.1. We note that the spectral stacks agree within
$\pm$10\text{\,}\mathrm{\char 37\relax}$$ with binned integrated intensities
computed within the same stacking bins if CO(2–1) is used as a prior to define
the velocity-integration mask.

##  Appendix C SFR tracers

###  C.1 SFR – $21\text{\,}\mathrm{\SIUnitSymbolMicro m}$

We use $21\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (F2100W) emission from
recent JWST–MIRI (Gardner et al. 2023; Wright et al. 2023) observations as a
another probe of SFR, in addition to $\text{H}\alpha$ from PHANGS–MUSE (Sect.
3.4) and $33\text{\,}\mathrm{GHz}$ from VLA (Sect. C.2). These data are part
of the “PHANGS–JWST Treasury Program” (Lee et al. 2023) and have been reduced
via the PHANGS–JWST data reduction pipeline (Williams et al. in prep.).
$\mathrm{NGC\,4321}$ was observed by JWST in June 2023 and we use version 0.9
of the PHANGS–JWST data reduction.

Physically, the strong radiation field from young, massive stars heat up the
surrounding dust, which re-emits at infrared wavelength probed by F2100W.
$21\text{\,}\mathrm{\SIUnitSymbolMicro m}$ point sources correlate well with H
II regions (Hassani et al. 2023) and the F2100W intensity correlates well,
though non-linearly, with extinction-corrected $\text{H}\alpha$ intensity
(Leroy et al. 2023; Belfiore et al. 2023). However, the F2100W also captures
stochastically heated emission from small dust grains that can trace the ISM.
It may thus be both more robust to extinction than $\text{H}\alpha$ and more
subject to contamination by diffuse ISM emission.

To infer $\Sigma_{\text{SFR}}$ from F2100, we use the empirical relation from
Leroy et al. (2023) (their equation 5), which re-scales the
$21\text{\,}\mathrm{\SIUnitSymbolMicro m}$ flux into a
$24\text{\,}\mathrm{\SIUnitSymbolMicro m}$ flux using
$R_{$21\text{\,}\mathrm{\SIUnitSymbolMicro
m}$/$24\text{\,}\mathrm{\SIUnitSymbolMicro m}$}=0.80$ and then converts to SFR
via a linear conversion (e.g. following Kennicutt & Evans 2012), such that the
SFR surface density is given by:

$\left(\dfrac{\Sigma_{\rm
SFR,F2100W}}{$\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{yr}}^{-1}\text{\,}{\mathrm{kpc}}^{-2}$}\right)=$3.7\text{\times}{10}^{-3}$\left(\dfrac{I_{\nu,\rm{F2100W}}}{$\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{sr}}^{-1}$}\right)\cos(i)\;,$
(22)

Thus, we adopt a linear relation between F2100W dust emission and the SFR. We
also contrast this prescription to a power law relation based on Leroy et al.
(2023) which leads to up to a factor of three higher values in the central
$\sim$2\text{\,}\mathrm{kpc}$$ of $\mathrm{NGC\,4321}$.

###  C.2 SFR – $33\text{\,}\mathrm{GHz}$

We note that above SFR tracers might lead to significantly discrepant results
in galaxy centres, where optical recombination lines can become too extinct to
recover robust Balmer decrement corrections and
$21\text{\,}\mathrm{\SIUnitSymbolMicro m}$ emission might be systematically
contaminated by stochastically heated dust grain. Therefore, we use free-free
$33\text{\,}\mathrm{GHz}$ emission from as an additional SFR tracer in the
centre of $\mathrm{NGC\,4321}$. The data are coming from Very Large Array
(VLA) observations of a large sample of galaxies, including
$\mathrm{NGC\,4321}$, at $3\text{\,}\mathrm{GHz}33\text{\,}\mathrm{GHz}$ at
$\sim$$$ resolution (Linden et al. 2020).

At high radio frequencies the ionising flux of young massive stars is directly
proportional to the thermal spectral luminosity. This allows us to trace the
SFR via the thermal part of the $33\text{\,}\mathrm{GHz}$ flux measured by the
VLA following the prescription in Murphy et al. (2012) (their equation 6):

$\displaystyle\left(\dfrac{\Sigma_{\rm
SFR,$33\text{\,}\mathrm{GHz}$}}{$\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{yr}}^{-1}\text{\,}{\mathrm{kpc}}^{-2}$}\right)=$
$\displaystyle\,$5.5\text{\times}{10}^{16}$\,\left(\dfrac{T_{\rm
e}}{${10}^{4}\text{\,}\mathrm{K}$}\right)^{-0.45}\left(\dfrac{\nu}{$\text{\,}\mathrm{GHz}$}\right)^{0.1}\,f_{\rm
thermal}$
$\displaystyle\times\left(\dfrac{I_{\nu,$33\text{\,}\mathrm{GHz}$}}{$\text{\,}\mathrm{erg}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{sr}}^{-1}$}\right)\,\cos(i)\;,$
(23)

where the thermal fraction ($f_{\rm thermal}$) values are taken from Linden et
al. (2020) (their table 4; $\sim$200\text{\,}\mathrm{pc}$$ apertures) and we
adopt an electron temperature of $T_{\rm e}=${10}^{4}\text{\,}\mathrm{K}$$ and
$\nu=$33\text{\,}\mathrm{GHz}$$.

###  C.3 SFR tracer comparison

Throughout this work, we have used Balmer decrement-corrected $\text{H}\alpha$
emission as a tracer of SFR (Sect. 3.4). Despite being overall a robust tracer
of SFR, when corrected for dust attenuation, in the centres of galaxies,
$\text{H}\alpha$ might miss some of the SFR-related $\text{H}\alpha$ emission
due to extremely high attenuation by denser dust thus biasing the SFR estimate
high. Also, in extreme environments, like the centre, $\text{H}\alpha$
emission might be associated with other processes than young stars and thus
bias the SFR high. Therefore, the question is: “How robust are our findings
against the choice of the SFR tracer?”

We use JWST F2100W emission to trace SFR via the hot dust emission that is
less affected by attenuation effects in the centre of the galaxy as long as
these galaxy centres contain sufficient amounts of dust (Sect. C.1). Using the
F2100W-inferred SFR (instead of $\text{H}\alpha$) leads to similar SFR/HCN
values across the disc and in the low-pressure regime for both prescriptions
(linear and power-law conversion) adopted here (see Fig. 10. However, the two
calibrations differ by up to a factor of three in the central kpc of the
galaxy. The change is so severe that using the power-law calibration yields
similar SFRF2100W/HCN values in the centre (median $10^{-1.05}$) as in bar
ends (median $10^{-0.96}$), spiral arms (median $10^{-1.06}$) and interarm
(median $10^{-1.19}$), and statistically the same SFRF2100W/HCN distribution
in the centre compared to the aforementioned environments.

In addition, we compare the two SFR tracers ($\text{H}\alpha$ and F2100W) with
the $33\text{\,}\mathrm{GHz}$-inferred values, being the most robust tracer of
the SFR in the centre of $\mathrm{NGC\,4321}$ (Sect. C.2). We find that
$\text{H}\alpha$ (decrement-corrected) and $33\text{\,}\mathrm{GHz}$ yield
very consistent results across the whole central $1\text{\,}\mathrm{kpc}$
probed by $33\text{\,}\mathrm{GHz}$ emission, agreeing within
$20\text{\,}\mathrm{\char 37\relax}$. F2100W (linear conversion) leads to a
factor of two higher values in the central $260\text{\,}\mathrm{pc}$, but
otherwise consistent values. Therefore, we conclude that for
$\mathrm{NGC\,4321}$ Balmer decrement-corrected $\text{H}\alpha$ emission is
an excellent tracer of the SFR, even in the centre of the galaxy, where
increased dust attenuation could have depreciated $\text{H}\alpha$ as a robust
SFR tracer. Moreover, these results suggest that F2100W is proportional to the
SFR across the full disc of $\mathrm{NGC\,4321}$.

##  Appendix D Scaling relations

###  D.1 Molecular cloud relations

In Fig. 11, we present the relations between HCN/CO and SFR/HCN, respectively
with the properties of molecular gas at GMC scales, i.e. here, at
$120\text{\,}\mathrm{pc}$. The cloud-scale molecular gas properties are
computed following the prescriptions of Sun et al. (2018) and Neumann et al.
(2023b), but adopting the varying $\alpha_{\text{CO}}$ and $R_{21}$
conversions introduced in Sect. A.1.2. We compute the molecular gas surface
density from the CO(2–1) integrated intensity ($W_{\text{CO(2--1)}}$):

$\displaystyle\Sigma_{\text{mol}}=\alpha_{\text{CO}}\,R_{21}^{-1}\,W_{\text{CO(2
--1)}}\,.$ (24)

The molecular gas velocity dispersion is obtained from the CO(2–1) equivalent
line width and corrected for the velocity channel-to-channel correlation:

$\displaystyle\sigma_{\rm measured}=\dfrac{W_{\text{CO(2--
1)}}}{\sqrt{2\pi}T_{\rm peak}}\,,$ (25)
$\displaystyle\sigma_{\text{mol}}=\sqrt{\sigma_{\rm measured}^{2}+\sigma_{\rm
response}^{2}}\,,$ (26)

where $\sigma_{\rm measured}$ assumes a Gaussian line profile and $\sigma_{\rm
response}$ takes into account the instrument channel width and channel-to-
channel correlation. To estimate the virial parameter ($\alpha_{\text{vir}}$),
we assume spherically symmetric clouds with a given density profile
$\rho\propto r^{-1}$, so that $\alpha_{\text{vir}}$ can be computed from the
CO(2–1) data using $\Sigma_{\text{mol}}$ and $\sigma_{\text{mol}}$:

$\displaystyle\alpha_{\text{vir}}=\dfrac{9\,\ln{2}}{\pi\,G}\dfrac{\sigma_{\text{mol}}^{2}}{\Sigma_{\text{mol}}\,D_{\rm
beam}}\propto\dfrac{\sigma_{\text{mol}}^{2}}{\Sigma_{\text{mol}}}\,,$ (27)

where $G$ is the gravitational constant. These molecular gas properties are
computed at $D_{\rm beam}=$120\text{\,}\mathrm{pc}$$ scale and converted to
the HCN resolution, i.e. $260\text{\,}\mathrm{pc}$, via a
$\Sigma_{\text{mol}}$-weighted average, similar to $P_{\text{DE}}$ as
described in Sect. 3.6, hence the notation $\langle
X_{$120\text{\,}\mathrm{pc}$}\rangle_{$260\text{\,}\mathrm{pc}$}$, where $X$
is the quantity to by averaged.

Similar to the relations with radius (Sect. 4.2) and pressure (Sect. 4.3), we
apply the MARS linear regression tool (Sect. 3.8) to the cloud property
relations shown in Fig. 11. We find that HCN/CO correlated positively and with
$\Sigma_{\text{mol}}$, $\sigma_{\text{mol}}$ and $\alpha_{\text{vir}}$ and is
well-described by a single power law over the whole data range that is probed.
Similarly, we observe a negative correlation between SFR/HCN and
$\Sigma_{\text{mol}}$, $\sigma_{\text{mol}}$, $\alpha_{\text{vir}}$. However,
for SFR/HCN vs $\Sigma_{\text{mol}}$ and $\sigma_{\text{mol}}$ we find a
change in the relation (as determined via MARS) at
$\Sigma_{\text{mol}}=$74\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$$
and $\sigma_{\text{mol}}=$8.3\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$,
such that SFR/HCN is roughly constant at low $\Sigma_{\text{mol}}$
($\sigma_{\text{mol}}$) and strongly decreasing with increasing
$\Sigma_{\text{mol}}$ ($\sigma_{\text{mol}}$) at high $\Sigma_{\text{mol}}$
($\sigma_{\text{mol}}$).

###  D.2 Stellar mass relations

Fig. 12 shows the HCN/CO and SFR/HCN scaling relation with the stellar mass
surface density, $\Sigma_{\star}$. We compute $\Sigma_{\star}$ from the
Spitzer $3.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ image as described in
Sect. A.3. Following the same methodology as in Sect. 4.2, 4.3, D.1, we
determine the power law behaviour of the mean trend finding strong
correlations between HCN/CO (positive correlation) and SFR/HCN (negative
correlation) with $\Sigma_{\star}$. For HCN/CO vs. $\Sigma_{\star}$, we
observe a tighter relation ($0.19\text{\,}\mathrm{dex}$ scatter) that is
consistent over all environments and the whole probed data range, spanning two
orders of magnitude in stellar mass surface density
($1\text{\times}{10}^{2}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$
to
$1\text{\times}{10}^{4}\text{\,}\mathrm{M_{\odot}}\text{\,}{\mathrm{pc}}^{-2}$).
The found relation shows that $\Sigma_{\star}$ is a good predictor of HCN/CO
over the whole disc of a nearby galaxy at sub-kpc scales, though most of the
dynamic range of $\Sigma_{\star}$ is covered by only two regions, the centre
and the bar. The relation between SFR/HCN and $\Sigma_{\star}$ shows much
higher scatter ($0.46\text{\,}\mathrm{dex}$) with the bar region being offset
to the main relation by $0.2\text{\,}\mathrm{dex}$, stressing the strongly
suppressed star formation efficiency in the bar (Sect. 5.3). Combined with the
results presented in Sect. D.1, the found relations suggest that the threshold
behaviour in the SFR/HCN vs $P_{\text{DE}}$ relation (Sect. 4.3) is caused by
molecular gas cloud-scale physics rather than larger scale environment.

##  Appendix E Additional figures

Fig. 13 shows the CO(2–1)-to-CO(1–0) line ratio ($R_{\rm 21}$) computed from
the $$4\text{\,}\arcsec$\sim$300\text{\,}\mathrm{pc}$$ resolution CO(1–0) data
from Pan & Kuno (2017) and homogenised CO(2–1) observations form PHANGS–ALMA
(Leroy et al. 2021b). We use the measured $300\text{\,}\mathrm{pc}$-scale
$R_{\rm 21}$ map to infer the cloud-scale ($120\text{\,}\mathrm{pc}$) CO(1–0)
line intensities from the observed CO(2–1) line intensities, which enter the
estimation of the dynamical equilibrium pressure (Sect. 3.6). In Sect. 4.5, we
use the $R_{\rm 21}$ map to convert the $260\text{\,}\mathrm{pc}$-scale
$W_{\text{CO(2--1)}}$ into $W_{\text{CO(1--0)}}$ to compare the HCN-to-CO(1–0)
vs. $\Sigma_{\text{mol}}$ scaling relation with literature findings.

In Fig. 14, we show a map of the morphological environments overlaid with HCN
contours similar to Fig. 3, right panel. In addition, we indicate two loci of
ISM pressure, $P_{\text{DE}}$, matching the pressure thresholds inferred for
the HCN/CO and SFR/HCN vs $P_{\text{DE}}$ scaling relations (Sect. 4.3).

Fig. 15 presents HCN/CO and HCN/HCO+ line ratio trends with metallicity.
Detected sigh line measurements of HCN/CO show no correlation with metallcity
($\rho_{\rm Pearson}=0.12$). The stacked average HCN/CO increases with
metallicity, suggesting that the HCN abundance is more strongly depending on
metallicity than CO, which is expected if nitrogen decreases more sharply with
metallicity than oxygen (Braine et al. 2017, 2023). In this scenario we would
expect a systematic increase of the HCN/HCO+ line ratio with metallicity.
However, we find only a weak correlation between HCN/HCO+ and metallicity
($\rho=0.35$), indicating that metallicity effect play only a minor role
across $\mathrm{NGC\,4321}$, potentially due to the small dynamic range in
metallicity, spanning less than $0.1\text{\,}\mathrm{dex}$ (8.54 to 8.62).

In Fig. 16, we display maps of various data products surface density maps of
the atomic gas, (dense) molecular gas and stellar mas, ISM pressure and SFR
inferred from $\text{H}\alpha$ and $21\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
respectively. In addition, we show the star formation efficiency of the
molecular gas, SFR/CO.