70 lines
No EOL
3.3 KiB
Markdown
70 lines
No EOL
3.3 KiB
Markdown
<extends template="layouts/base.html" title="The Simplest Circuit"></extends>
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<nav-links back="/concepts.html"></nav-links>
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# The Simplest Circuit
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The key idea for this section is that **steady-state expression levels depend on protein production and removal rates**.
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Let's break that down.
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First, steady-state conditions mean that all inputs to the system are constant forever. That simplifies things quite a bit.
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Expression levels are just the net amount of proteins produced.
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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?
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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.
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<i class="cite">Credit: CalTech</i>
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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.
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There's a differential equation for this:
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$$ \frac{dx}{dt} = \text{production} - (\text{degradation} + \text{dilution}) $$
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Or:
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$$ \frac{dx}{dt} = \beta - \gamma x $$
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Since \\(\gamma\\) counts both degradation and dilution, we can say that:
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$$ \gamma = \gamma_\text{degradation} - \gamma_\text{dilution} $$
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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.
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To find the net production of the protein under steady state conditions, set the derivative to zero and solve for \\(x\\):
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$$0 = \beta - \gamma x$$
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$$-\beta = -\gamma x$$
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$$\frac{-\beta}{-\gamma} = x$$
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$$\frac\beta\gamma = x$$
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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.
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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!):
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$$f(t)=xt=\frac{\beta t}{\gamma}$$
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<div class="graph">
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<div id="concentration-graph"></div>
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<div>
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<label for="beta">β</label>
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<input type="range" id="beta">
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</div>
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<div>
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<label for="gamma">γ</label>
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<input type="range" id="gamma">
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</div>
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</div>
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It's a line — for now. Onwards!
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<nav-links back="/concepts.html"></nav-links>
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<script>
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plot('#concentration-graph', (beta, gamma) => `(${beta}x)/${gamma}`, ['#beta', '#gamma'])
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defineVars([
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['γ', 'The rate constant for reduction of protein concentration.'],
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['β', 'The rate of protein production, in molecules per unit time.'],
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['x', 'The gene in question.'],
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['p', 'The protein in question.']
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])
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</script> |