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reduces the protein's _concentration_). That's represented above by the dashed circle. For simplicity, let's say that it's being reduced at a rate constant \\(\gamma\\) (that letter is a gamma, for anyone who wanted to know). Note that this is not just a rate — it's a _rate constant_, meaning that the actual rate is proportional to the number of molecules. More molecules, more reduction. @@ -55,16 +55,58 @@ $$f(t)=xt=\frac{\beta t}{\gamma}$$ -It's a line — for now. Onwards! +It's a line — for now. + +## Considering Transcription + +Right now, we have protein synthesis as one process — no intermediate steps. In reality, it has two: transcription and translation. The mRNA made in transcription can _also_ be degraded and diluted, just like the proteins made in translation. Let's add another variable to represent mRNA — call it \\(m\\). This can be shown in a diagram: + +![The simplest circuit, now considering both transcription and translation, with mRNA (m) being produced and reduced.](/assets/simplest-circuit-dg2.png) +Credit: CalTech + +The reaction can now be described by two coupled differential equations: + +$$\frac{dm}{dt} = \beta_m - \gamma_mm$$ +$$\frac{dx}{dt} = \beta_pm - \gamma_px$$ + +Now, to find steady-state mRNA and protein concentrations, we set both derivatives to zero and solve, giving us: + +$$m_{ss}=\frac{\beta_m}{\gamma_m}$$ +$$x_{ss}=\frac{\beta_pm_{ss}}{\gamma_p}=\frac{\beta_p\beta_m}{\gamma_p\gamma_m}$$ + +This tells us that steady-state protein concentration, when we consider transcription and translation as separate steps, is proportional to the product of the two synthesis rates and inversely proportional to the product of the two degradation rates. Again, if you think about it, that's pretty intuitive. + +Here's a graph to play with: + +

+ +Onwards! \ No newline at end of file