Simplest circuit

pages/simplest-circuit.md

@@ -2,4 +2,69 @@
 
 <nav-links back="/concepts.html"></nav-links>
 
-# The Simplest Circuit
+# The Simplest Circuit
+
+The key idea for this section is that **steady-state expression levels depend on protein production and removal rates**.
+
+Let's break that down.
+
+First, steady-state conditions mean that all inputs to the system are constant forever. That simplifies things quite a bit.
+
+Expression levels are just the net amount of proteins produced.
+
+So, how many proteins there are, with no variables, depends on how many proteins are being created and how many are being removed. Simple, right?
+
+Now, the circuit. This circuit is the simplest possible circuit: a single gene &mdash; let's call it \\(x\\) &mdash; coding for a single protein \\(p\\) at a rate of \\(\beta\\) molecules per unit time.
+
+![alt text](/assets/simplest-circuit-dg1.png)
+<i class="cite">Credit: CalTech</i>
+
+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 &mdash; it's a _rate constant_, meaning that the actual rate is proportional to the number of molecules. More molecules, more reduction.
+
+There's a differential equation for this:
+$$ \frac{dx}{dt} = \text{production} - (\text{degradation} + \text{dilution}) $$
+Or:
+$$ \frac{dx}{dt} = \beta - \gamma x $$
+Since \\(\gamma\\) counts both degradation and dilution, we can say that:
+$$ \gamma = \gamma_\text{degradation} - \gamma_\text{dilution} $$
+
+Since we're getting into the math, a quick tip: if you (like me) forget what a variable does halfway through the page, just hover over it and it'll tell you what it does. Anyway, back to the show.
+
+To find the net production of the protein under steady state conditions, set the derivative to zero and solve for \\(x\\):
+
+$$0 = \beta - \gamma x$$
+$$-\beta = -\gamma x$$
+$$\frac{-\beta}{-\gamma} = x$$
+$$\frac\beta\gamma = x$$
+
+And we find that **steady-state protein concentration is proportional to the ratio of production and removal rates**. This is another core concept that should be built as intuition.
+
+Since I haven't used the interactive graphing functions I took the bandwidth to import yet, let's graph the general shape of the protein concentration function under the simplest possible conditions (play with the sliders!):
+
+$$f(t)=xt=\frac{\beta t}{\gamma}$$
+
+<div class="graph">
+    <div id="concentration-graph"></div>
+    <div>
+        <label for="beta">β</label>
+        <input type="range" id="beta">
+    </div>
+    <div>
+        <label for="gamma">γ</label>
+        <input type="range" id="gamma">
+    </div>
+</div>
+
+It's a line &mdash; for now. Onwards!
+
+<nav-links back="/concepts.html"></nav-links>
+
+<script>
+    plot('#concentration-graph', (beta, gamma) => `(${beta}x)/${gamma}`, ['#beta', '#gamma'])
+    defineVars([
+        ['γ', 'The rate constant for reduction of protein concentration.'],
+        ['β', 'The rate of protein production, in molecules per unit time.'],
+        ['x', 'The gene in question.'],
+        ['p', 'The protein in question.']
+    ])
+</script>