  We present a method of constraining the properties of the $\gamma$-ray
emitting region in flat spectrum radio quasars (FSRQs) in the one-zone proton
synchrotron model, where the $\gamma$-rays are produced by synchrotron
radiation of relativistic protons. We show that for low enough values of the
Doppler factor $\delta$, the emission from the electromagnetic (EM) cascade
which is initiated by the internal absorption of high-energy photons from
photohadronic interactions may exceed the observed $\sim$GeV flux. We use that
effect to derive an absolute lower limit of $\delta$; first, an analytical one,
in the asymptotic limit where the external radiation from the broad line region
(BLR) is negligible, and then a numerical one in the more general case that
includes BLR radiation. As its energy density in the emission region depends on
$\delta$ and the region's distance from the galactic center, we use the EM
cascade to determine a minimum distance for each value of $\delta$. We
complement the EM cascade constraint with one derived from variability
arguments and apply our method to the FSRQ 3C 273. We find that $\delta \gtrsim
18-20$ for $B \lesssim 30$ G and $\sim$day timescale variability; the emission
region is located outside the BLR, namely at $r \gtrsim 10 R_{\rm BLR} \sim 3$
pc; the model requires at pc-scale distances stronger magnetic fields than
those inferred from core shift observations; while the jet power exceeds by at
least one order of magnitude the accretion power. In short, our results
disfavour the proton synchrotron model for the FSRQ 3C 273.
