# The Hill‐Type Equation Reveals the Regulatory Principle of Target Protein Expression Led by p53 Pulsing

## TL;DR

This paper shows how the Hill-type equation can predict steady-state protein levels based on mRNA levels during p53 pulsing.

## Contribution

The study demonstrates that the Hill-type equation can equally describe mRNA and protein fold changes under TF pulsing.

## Key findings

- The Hill-type equation accurately predicts both mRNA and protein fold changes for p53 target BAX.
- Amplitude modulation affects protein expression depending on binding affinity.
- Duration and frequency can fine-tune protein expression beyond saturation.

## Abstract

The central dogma indicates the basic direction of gene expression pathways. For activated gene expression, the quantitative relationship between various links from the binding of transcription factors (TFs) to DNA to protein synthesis remains unclear and debated. There is consensus that at a steady state, protein levels are largely determined by the mRNA level. How can we find this steady state? Taking p53 as an example, based on the previously discovered Hill‐type equation that characterizes mRNA expression under p53 pulsing, I proved that the same equation can be used to describe the average steady state of target protein expression. Therefore, at steady state, the average fold changes in mRNA and protein expression under TFs pulsing were the same. This consensus has been successfully demonstrated. For the p53 target gene BAX, the observed fold changes in mRNA and protein expression were 1.40 and 1.28, respectively; the fold changes in mRNA and protein expression calculated using the Hill‐type equation were both 1.35. Therefore, using this equation, we can not only fine‐tune gene expression, but also predict the proteome from the transcriptome. Furthermore, by introducing two quantitative indicators, we can determine the degree of accumulation and stability of protein expression.

The central dogma indicates the direction of gene expression pathways. Taking p53 pulsing as an example, I proved that the average fold changes in mRNA and target protein expression are the same, and the Hill‐type equation can describe such fold changes. This equation reveals the principle of target protein expression. Amplitude modulation is effective with lower binding affinity; however, for higher binding affinity, amplitude modulation quickly reaches saturation. Duration and frequency can tune finely target protein expression beyond saturation.