A colleague argues: "Autoregulation makes gene Y repress its own transcription
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Biophysics & Medical Physics · Genetic circuits, regulation and biological information
Forces, Machines and Signals, Ch Genetic circuits and biological pattern
Problem
A colleague argues: "Autoregulation makes gene Y repress its own transcription, so switching gene Y on must take longer to reach its steady level than switching on an unregulated gene X reaching the same level, because Y is fighting itself the whole way up." Test this claim for two genes in a non-dividing bacterium, so growth dilution is negligible and the protein degradation rate is γ=0.1min−1 for both proteins. Gene X is transcribed at a constant rate βX=5nMmin−1. Gene Y is negatively autoregulated with threshold K=50nM and a Hill coefficient so high that we take the limit nH→∞, in which its regulation function becomes a step at K (full-rate production for p<K, none for p>K), with production rate βY=50nMmin−1 below threshold.
((a)) Find the steady-state protein level of each gene.
((b)) Find the half-time to reach that level for each gene.
((c)) State whether the claim is correct, and explain the result in terms of the shapes of the production and loss curves.