Negative autoregulation controls size scaling in confined gene expression reactions

Gene expression via transcription-translation is the most fundamental reaction to sustain biological systems, and complex reactions occur in a small compartment of living cells. There is increasing evidence that physical effects, such as molecular crowding or excluded volume effects of transcriptional-translational machinery, affect the yield of reaction products. On the other hand, transcriptional feedback that controls gene expression during mRNA synthesis is also a vital mechanism that regulates protein synthesis in cells. However, the excluded volume effect of spatial constraints on feedback regulation is not well understood. Here, we study the confinement effect on transcriptional autoregulatory feedbacks of gene expression reactions using a theoretical model. The excluded volume effects between molecules and the membrane interface suppress the gene expression in a small cell-sized compartment. We find that negative feedback regulation at the transcription step mitigates this size-induced gene repression and alters the scaling relation of gene expression level on compartment volume, approaching the regular scaling relation without the steric effect. This recovery of regular size-scaling of gene expression does not appear in positive feedback regulation, suggesting that negative autoregulatory feedback is crucial for maintaining reaction products constant regardless of compartment size in heterogeneous cell populations.