diff --git a/assets/simplest-circuit-dg2.png b/assets/simplest-circuit-dg2.png new file mode 100644 index 0000000..21f01e8 Binary files /dev/null and b/assets/simplest-circuit-dg2.png differ diff --git a/assets/site.css b/assets/site.css index 7681e9b..e53e6a6 100644 --- a/assets/site.css +++ b/assets/site.css @@ -26,7 +26,6 @@ body { } h1, -h2, p { font-weight: 400; } @@ -52,10 +51,6 @@ h1 { text-align: center; } -h2 { - font-size: 1rem; -} - pre { font-size: 16px; font-family: var(--font-mono); @@ -166,4 +161,5 @@ i.cite { display: block; text-align: center; padding-top: 8px; + font-size: 9pt; } \ No newline at end of file diff --git a/pages/simplest-circuit.md b/pages/simplest-circuit.md index eccdb7b..49abf61 100644 --- a/pages/simplest-circuit.md +++ b/pages/simplest-circuit.md @@ -16,7 +16,7 @@ So, how many proteins there are, with no variables, depends on how many proteins Now, the circuit. This circuit is the simplest possible circuit: a single gene — let's call it \\(x\\) — coding for a single protein \\(p\\) at a rate of \\(\beta\\) molecules per unit time. - + Credit: CalTech However, in real life, proteins aren't just made forever; they're also reduced, through both _active degradation_ (being broken down) and _dilution_ (the cell getting bigger, which 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: + + +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: + +