<figure><img src="image_27.png" /><figcaption>**Fig. E.1.** $\chi^2$ converted probability distribution of the 10 000 parameter set of RADMC-3D models for all clumps. The orange point indicates the best-fit model. Blue crosses give the positions of the 30 best-fit models, so there could be overlaps between the different parameter sets due to the binning. For sources G31, G13, G08a, and G19 the results before re-adjusting $T(r)$ based on SED are shown as light blue contours.</figcaption></figure>

gas temperature of 400 K. The CH$_3$CCH abundance is enhanced when the gas temperature reaches 30-40 K due to desorption of CH$_4$ to form CH$_3$CCH (i.e., dissociative recombination of larger hydrocarbons). The CH$_3$CCH abundance is significantly enhanced again when the gas temperature reaches 80-100 K, due to the direct desorption of CH$_3$CCH from grain surfaces. Finally, the models with higher final gas density present a systematically lower CH$_3$CCH abundance since in these high-density models CH$_4$ desorbs at slightly higher temperatures (lower panel of Figure F.1). Similarly, according to the chemical model of Garrod et al. (2017), the abundance of CH$_3$OH experiences two significant enhancements at gas temperatures of 30-40 K and 80-100 K (upper panel of Figure F.1). The chemical modeling include three warm-up models, depending on the timescale for the system to increase from 10-200 K, over $5 \times 10^4$ (fast), $2 \times 10^5$ (medium), and $1 \times 10^6$ (slow) yr (Garrod & Herbst 2006).

To mimic the abundance enhancement in the warm or luke-
warm regions described by these chemical models, we parame-
terized the CH₃OH and CH₃CCH abundance profiles as Equa-
tion 5. Again, we assumed that the A- and E-type CH₃OH have
the same abundance. The parameter ranges of X_in and X_out were
chosen by referencing to N_mol for CH₃CCH and CH₃OH from the
XCLASS/RADEX modeling results (Sections 3.4-3.5) and the
aforementioned chemical models. Figure F.1 shows a compari-
son between the parameter space we explored and the chemical
modeling results.

Currently, only the collisional coefficients of CH₃OH with
para-H₂ are available, although it seems that only in hot shocked
gas is there a significant difference between the thermal rate
coefficients of collisions with ortho- and para-H₂ (Flower et al.
2010). Our non-LTE models were not affected by the uncer-
tain collisional coefficients because we assumed a low ortho-
to para-H₂ ratio (OPR), such that the collisions with ortho-H₂ is
negligible. Chemical models and observations toward early-stage
dense cores indeed indicate an OPR value of ~10⁻³-10⁻² (Flower

et al. 2006, cf. Troscompt et al. 2009), which is well below
the equilibrium value of 3. In the postshock gas, the OPR may
remain low since the short timescale does not allow significant
conversion from para-H₂ to ortho-H₂ (Leurini et al. 2016).

When performing non-LTE modeling of CH<sub>3</sub>CCH, the collisional rates of CH<sub>3</sub>CN (Green 1986) were substituted by those of CH<sub>3</sub>CCH. This is a common approach since CH<sub>3</sub>CCH and CH<sub>3</sub>CN have similar molecular weights and configuration, while collisional rates for CH<sub>3</sub>CCH have not yet been published. The collisional rates of CH<sub>3</sub>OH were quoted from Rabli & Flower (2010) (see Section 3.5).

To create the distributions of physical properties we adopted the sf3dmodels package (Izquierdo et al. 2018) to generate homogeneous grids in Cartesian coordinates, which were then interpolated onto the LIME input format of randomly generated set of points. We used the linear resolution of ~0.015-0.03 pc (depending on the source radius R<sub>clump</sub>, Table 6) as the grid size that corresponds to better than one-fifth the beam size for each source. On each grid we specified a mean gas velocity using a random process to mimic the turbulent velocity field; the direction was uniformly sampled from the 4π solid angle, while the magnitude of the velocity was drawn from a Gaussian distribution with σ =3.5 km s<sup>-1</sup>. We additionally adopted a uniform Doppler broadening of 0.4 km s<sup>-1</sup> (σ<sub>turb</sub> = 0.4 km s<sup>-1</sup>) to accommodate the unresolved (micro-)turbulence velocity. Therefore the intrinsic linewidth for each model is σ<sub>1D</sub> = √(σ<sub>turb</sub><sup>2</sup> + σ<sub>thermal</sub><sup>2</sup>), where σ<sub>thermal</sub> is determined by the assumed gas temperature. This yields linewidths comparable to the observed values (ΔV = 5.3 ± 2.0 km s<sup>-1</sup>).

We postprocessed the output of LIME to match the angular
and velocity resolutions of our observations, and then compared
the annularly and beam averaged synthetic spectra with those
from observed spectral cubes. For each observed clump, the best-
fit model was taken as the one with the lowest $\chi^2$. The parameters
<figure><img src="image_28.png" /><figcaption>**Fig. F.1.** Abundance variations of CH₃OH and CH₃CCH from chemical models (lines) and the explored parameter space (filled) for the LIME models based on ρ<sub>bulk</sub> density model (column A of Table 7). The results from the chemical models are shown as lines (solid, dashed, and dotted): for CH₃OH the abundance profiles for different warm-up timescales are shown (Garrod et al. 2017); for CH₃CCH, abundance profiles of different final collapse densities are shown (Calcutt et al. 2019). The dark green filled region indicates the lower abundance range explored for X<sub>out</sub> by the LIME models, and the light green filled region the upper range explored for X<sub>in</sub>. The vertical gray lines indicate the jump in temperature of 30 and 80 K.</figcaption></figure>

of these best-fit chemical models are summarized in Table 7, column A.

We found that these initial models systematically underes-
timated the intensities of the higher *K* components of CH₃OH
lines in the inner regions for all sources (Figure 13). There are
some sources in which the intensities of the higher *K* compo-
nents were underestimated also at outer radii. This implies that,
in general, the density profiles ρ<sub>bulk</sub> derived from dust continuum
modeling (Appendix E) were not high enough to collisionally
excite the high *K* levels of CH₃OH (e.g., gas may be concentrated
in substructures of higher volume density). This was expected, as
was revealed by the comparison to the ρ<sub>dense</sub> derived by RADEX
modeling: gas densities are 50-200 times higher than that of ρ<sub>bulk</sub>
derived by single-dish dust continuum. Therefore, we updated
the radial density profile in the models according to the RADEX
results ρ<sub>dense</sub> (Section 3.5) following Equation 6. We manually
adjusted the flexible parameters, which are the density scaling
factor f<sub>n</sub> (Equation 6) and f<sub>inc</sub>, to quantify the abundance profile
of X<sub>mol</sub>(r) in Equation 5, in a trial-and-error manner to seek for
better fits to the observational data. Figure F.2 demonstrates how
the line ratios of CH₃OH *K* components vary with gas density
and molecular abundance while keeping a fixed overall molec-
ular column density, as an example using the density model of
G08b.

Our best-fit model parameters for all the clumps are summa-
rized in Table 7, column B. In these results f<sub>r</sub> ranges from 1.5
to 5 in all the sources except G31, indicating RADEX results
only moderately overestimated the gas densities. As mentioned
in Section 3.7, the rather large f<sub>r</sub> in G31 is due to the very high
optical depth of its lower CH<sub>3</sub>OH *K* components in the central
region. In this case the CH<sub>3</sub>OH (5-4) lines do not provide mean-
ingful constraints for the RADEX modeling which was based on
the assumption of moderate optical depth.

When spatially integrating Equation 6 with a spherical sym-
metric assumption, the resulting overall molecular gas masses
(*M*<sub>mod</sub>) considerably exceed those derived based on modeling

dust continuum emission, by integrating ρ<sub>bulk</sub> (see Appendix E). This implies that the dense gas structures traced by CH<sub>3</sub>OH do not have spherically symmetric distributions. Instead, they are local gas concentrations that have small volume filling fac-tors. To reconcile the mass difference, we defined ff<sub>dens</sub> ≡ M<sub>enc</sub>/M<sub>mod</sub>, where M<sub>enc</sub> designates the enclosed molecular gas mass within 0.5 pc radius (a scale encompassing the CH<sub>3</sub>OH emission entirely for all sources) derived from the dust contin-uum models following ρ<sub>bulk</sub> (Equation 2, Appendix E). ff<sub>dens</sub> can be regarded roughly as an upper limit of volume filling factor of the dense gas traced by CH<sub>3</sub>OH. From the radiative transfer modeling point of view, to fit the observed line profiles, in the optically thin limit, varying the dense gas volume does not affect line intensity ratios, while the values of X<sub>mol</sub>(r) can be adjusted accordingly such that the overall molecular column densities are not altered. We discuss the implications from the inferred vol-ume filling factor of dense gas, and the dense gas mass fraction based on comparing ρ<sub>bulk</sub> and ρ<sub>dense</sub> in Section 4.5.

Appendix G: Comparison between observed
spectra and modeling results: Other targets
<figure><img src="image_29.png" /><figcaption>**Fig. F.2.** Line ratios of CH<sub>3</sub>OH (5-4) K components from LIME models based on density profile of G08b (ρ<sub>0</sub> corresponds to the adjusted reference density at 0.1 pc for this source, as in Equation 6). The abundance value corresponds to the outer abundance (X<sub>out</sub>), as in Table 7. The central symbol at the horizontal line of y = 1 corresponds to the best-fit model listed in Table 7 (column B). The comparison of observations with model spectra is presented in Figure 14, top panel.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Fig. G.1. Same as Figure 13, but for other target sources.</figcaption></figure>
<figure><img src="image_31.png" /><figcaption>Fig. G.2. Same as Figure 14, but for other target sources.</figcaption></figure>
<figure><img src="image_32.png" /><figcaption>Fig. G.3. Same as Figure 13, but for other target sources.</figcaption></figure>
<figure><img src="image_33.png" /><figcaption>Fig. G.4. Same as Figure 14, but for other target sources.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>Fig. G.5. Same as Figure 13, but for other target sources.</figcaption></figure>
<figure><img src="image_35.png" /><figcaption>Fig. G.6. Same as Figure 14, but for other target sources.</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>Fig. G.7. Example spectra (archival SMA data, Sect. 2) of thermometer lines and the XCLASS fits of G10.</figcaption></figure>