sources/nuisance-parameter.md

Removing the sample-level nuisance parameter

The factor $\frac{k_s}{\phi_s}$ is the RNA-to-DNA depth ratio of sample $s$ relative to $t = 0$. It is one unknown number per sample, and is shared by every guide. We get rid of it the same way as in the previous tutorial: take a reference.

For the non-targeting reference guides ($K_0 R = 0$, and $m_0$ constant),

$$A_0=\frac{k_s}{{\varphi }_s}A\_0\; =\; \backslash frac\{k\_s\}\{\backslash phi\_s\}$$

Before induction, dCas9 is off, so $R = 0$ and every guide is at steady state with $m_i(0) = \frac{E_{max}}{d + r_i}$. If the guides do not change growth before induction ($r_i = r_0$ at $t = 0$), then $m_i(0) = m_0(0)$ and the ratio of the two normalised ratios is the ratio of late steady states:

$$\frac{A_i}{A_0}=\frac{m_i}{m_0}=\frac{d+r_0}{d+r_i}\cdot \frac{1}{1+K_{i}R}\backslash frac\{A\_i\}\{A\_0\}\; =\; \backslash frac\{m\_i\}\{m\_0\}\; =\; \backslash frac\{d\; +\; r\_0\}\{d\; +\; r\_i\}\; \backslash cdot\; \backslash frac\{1\}\{1\; +\; K\_i\; R\}$$

Plotted against the reference term across late samples, each guide gives a straight line with gradient 1 and intercept $-\log(1 + K_i R)$. The ingredients are:

• Dividing RNA by DNA removes cell number and population growth.
• Dividing by each guide's own $t = 0$ removes per-guide capture efficiency and mRNA stability.
• Dividing by the pooled non-targeting guides removes the per-sample depth factor $\frac{k_s}{\phi_s}$.
• Late timepoints are on the transcript plateau, and the growth term is 1.

What remains is $-\log(1 + K_i R)$. Averaging the left side minus the reference term over the late samples gives this value per guide, $\hat{b}_i$. The most negative $\hat{b}_i$ is the strongest guide, so $-\hat{b}i = \log(1 + K_i R)$ ranks the guides. The transient after induction, the mRNA decay rate $d$, and $E{max}$ are not needed.