We show how the interplay between active galactic nuclei (AGN) and merger history determines whether a galaxy quenches star formation at high redshift. We first simulate, in a full cosmological context, a galaxy of total dynamical mass $10^{12}\,M_{\odot}$ at $z=2$. Then we systematically alter the accretion history of the galaxy by minimally changing the linear overdensity in the initial conditions. This "genetic modification" approach allows the generation of three sets of $��$CDM initial conditions leading to maximum merger ratios of 1:10, 1:5 and 2:3 respectively. The changes leave the final halo mass, large scale structure and local environment unchanged, providing a controlled numerical experiment. Interaction between the AGN physics and mergers in the three cases lead respectively to a star-forming, temporarily-quenched and permanently-quenched galaxy. However the differences do not primarily lie in the black hole accretion rates, but in the kinetic effects of the merger: the galaxy is resilient against AGN feedback unless its gaseous disk is first disrupted. Typical accretion rates are comparable in the three cases, falling below $0.1\,M_{\odot}$ yr$^{-1}$, equivalent to around $2\%$ of the Eddington rate or $10^{-3}$ times the pre-quenching star formation rate, in agreement with observations. This low level of black hole accretion can be sustained even when there is insufficient dense cold gas for star formation. Conversely, supernova feedback is too distributed to generate outflows in high-mass systems, and cannot maintain quenching over periods longer than the halo gas cooling time.

Clarifications and added references; accepted for publication in MNRAS