  Embedded planets are potentially the cause of substructures like gaps and
cavities observed in several protoplanetary disks. Thus, the substructures
observed in the continuum and in line emission encode information about the
presence of planets in the system and how they interact with the natal disk.
The pre-transitional disk around the star PDS 70 is the first case of two young
planets imaged within a dust depleted gap that was likely carved by themselves.
We aim to determine the spatial distribution of the gas and dust components in
the PDS 70 disk. The axisymmetric substructures observed in the resulting
profiles are interpreted in the context of planet-disk interactions. We develop
a thermo-chemical forward model for an axisymmetric disk to explain a subset of
the Atacama Large Millimeter/Submillimeter Array (ALMA) band 6 observations of
three CO isotopologues plus the continuum towards PDS 70. Combining the
inferred gas and dust distributions, the model results in a variable
gas-to-dust ratio profile throughout the disk that spans two orders of
magnitude within the first $130$ au and shows a step gradient towards the outer
disk, which is consistent with the presence of a pressure maxima driven by
planet-disk interactions. We find a gas density drop factor of ${\sim} 19$ at
the location of the planet PDS 70 c with respect to the peak gas density at
$75$ au. Combining this value with literature results on the hydrodynamics of
planet-disk interactions, we find this gas gap depth to be consistent with
independent planet mass estimates from infrared observations. Our findings
point towards gas stirring processes taking place in the common gap due to the
gravitational perturbation of both planets.
