Hill's function for translational regulation

Question

Transcriptional regulation is generally modeled as a Hill's function (similar to Michaelis-Menten Kinetics):

$$\frac{dm_X}{dt}=\alpha _{m_X}.\frac{R}{K+R} -\beta _{m_X}.m_X$$

Where $m_X$ is the mRNA for some gene-$X$, $R$ is a Regulator $\alpha$ and $\beta$ are formation and degradation rate constants respectively. This equation denotes a saturation kinetics; increasing activator wont cause indefinite increase in transcription. Sounds logical because all promoter sites will be occupied at some point.

In case of a repression the equation looks like:

$$\frac{dm_X}{dt}=\alpha _{m_X}.\frac{K}{K+R} -\beta _{m_X}.m_X$$

I want to model repression of translation using a similar equation. However, the issue is that even though a single mRNA can be saturated by a regulator, increasing mRNA will require more regulators. So effectively the regulator available for a single mRNA molecule will be total regulator ÷ total mRNA.

My question is that whether in such a case the following equation is logical:

$$\frac{dX}{dt}=\alpha _{X}.m_X.\frac{K}{K+R/m_X} -\beta _{X}.X$$

Where $X$ is the protein.

Which means we are taking into consideration the effective concentration of a regulator per mRNA. In other words the Hill's constant $K'$ should scale with mRNA concentration. ($K'=K\times m_X$)

Assumptions:

• Well mixed system
• System at thermodynamic limit
• Amino-acid pool infinite
• Ribosomes infinite

Answer 1

Your logic looks correct to me. Essentially, what you are doing is uniformly distributing the regulator among the available mRNA.

Note that even when using Hill functions to model transcription, the ratio of transcription factor (TF) concentration to the number of TF binding sites must be large - otherwise, you would have to consider binding ratios even at the level of transcription modeling, similarly to what you are doing now for translation. E.g. consider a hypothetical situation where 10 TF molecules compete for binding to 5 different promoter regions, each having 10 binding site repeats - you need to somehow distribute the available 10 TF molecules among 50 target binding sites. Clearly, this is not something accounted for by the standard Hill equation, which would in such case wrongly assume that 10 TF molecules are regulating each construct.

In your specific case, introducing the ratio of the regulator per each mRNA may not even be necessary if this ratio is large, effectively leading to saturation levels anyway. Note that the maximum level that $m_x$ can reach (at steady state) is equal to $max(m_X) = \frac{\alpha_{mX}}{\beta_{mX}}$. If you can ensure this value to be much smaller than your translational regulator concentration, you will get similar results even if you use simply $\frac{K}{K+R}$ for translation modeling.

If the latter assumption cannot be made, then your reasoning makes sense. For each individual mRNA molecule:

$$\text{X produced per mRNA} = \alpha_{X} .\frac{K}{K+R'}$$

where $R'$ is the amount of mRNA bound to this molecule. If $R$ is the total available regulator concentration and uniform binding affinity is assumed, this means that $R'=\frac{R}{m_X}$. Summing this over a total of $m_X$ mRNA and considering degradation yields exactly your final ODE.

Note that another important assumption you are making is that regulator binding/unbinding to/from mRNA is fast compared to translation, and that no cooperative interactions are present.

If you want to verify things further, you could establish system reactions, then derive Hill-free ODE equations according to the law of mass action. You could then compare this model to the one you propose. Furthermore, performing a stochastic simulation might make sense. If you want to go down this road but don't want to implement Gillespie's algorithm on your own, you can use e.g. SGNSim or COPASI.