# modeling

We model our system mainly by the following assumptions:

1. PAMmer and sgRNA of the target gene are abundant, so all the suvCas9 is bound with them;

$$suvCas9 + sgRNA + PAMmer \to suvCas9::sgRNA::PAMmer$$

2. The total concentration of suvCas9 (located in cytoplasm and nucleus) and the target mRNA (normal and mutated) stay the same;

$$[suvCas9::sgRNA::PAMmer(cytosolic)] + [suvCas9::sgRNA::PAMmer(nucleic)] + [mRNA(canonical)::suvCas9::sgRNA::PAMmer(cytosolic)] + [mRNA(mutated)::suvCas9::sgRNA::PAMmer(cytosolic)] = {c}_{0}$$
$$mRNA(mutated) + mRNA(canonical) = {c}_{R}$$

3. All the related reactions, which are listed below, obey the steady-state assumption, i. e., they are in the equilibrium state;
$$suvCas9::sgRNA::PAMmer(cytosolic) \to suvCas9::sgRNA::PAMmer(nucleic), {K}_{0}$$
$$mRNA(canonical) + suvCas9::sgRNA::PAMmer(cytosolic) \to mRNA(canonical)::suvCas9::sgRNA::PAMmer(cytosolic), {K}_{1}$$
$$mRNA(mutated) + suvCas9::sgRNA::PAMmer(cytosolic) \to mRNA(mutated)::suvCas9::sgRNA::PAMmer(cytosolic), {K}_{2}$$

4. After binding with mRNA, the suvCas9 complex cannot enter the nucleus;
Using the above mentioned assumptions, we can know:
$$[suvCas9::sgRNA::PAMmer(nucleic)] = \frac{{K}_{0}}{1 + {K}_{0} + {K}_{1}[mRNA(canonical)] + {K}_{2}[mRNA(mutated)]} \times {c}_{0}......(1)$$

We also assume:
$$[mRNA(mutated)] = \alpha \times {c}_{R}$$
$$[mRNA(canonical)] = (1- \alpha) \times {c}_{R}$$
$${K}_{1} = t \times {K}_{2}, t >> 1$$
In fact:
$${K}_{0} >> 1$$
So, the equation (1) can be modified to:
$$[suvCas9::sgRNA::PAMmer(nucleic)] / {c}_{0} = \frac{{K}_{0}}{{K}_{2}{c}_{R}(1 - t)\alpha + {K}_{2}{c}_{R}t + {K}_{0}}......(2)$$
The equation (2) is plotted below:

The relation between suvCas9::sgRNA::PAMmer(nucleic) and thymidine kinase(TK) expression is linear in the working condition:
$$[TK] = {k}_{0}[suvCas9::sgRNA::PAMmer(nucleic)]......(3)$$
We assume a sigmoid function between [TK] and probability of cell death:
$$P(cell death) = \frac{1}{1 + e^{-{k}_{1}[TK] + {k}_{2}}}......(4)$$
Integrating the equation (2), (3) and (4), we get:
$$P(cell death) = \frac{1}{1 + e^{{k}_{2} - {c}_{0}{k}_{0}{k}_{1}\frac{{K}_{0}}{{K}_{2}{c}_{R}(1 - t)\alpha + {K}_{2}{c}_{R}t + {K}_{0}}}}$$
It can also be written in the logarithm relative risk format:
$$log(\frac{P(cell death)}{1 - P(cell death)}) = {c}_{0}{k}_{0}{k}_{1}\frac{{K}_{0}}{{K}_{2}{c}_{R}(1 - t)\alpha + {K}_{2}{c}_{R}t + {K}_{0}} - {k}_{2}$$
In this final equation, we can modify several parameters: ${c}_{0}, {k}_{0}$ by the following method:
${c}_{0}$: By changing the suvCas9 promotor and terminator.
${k}_{0}$: By changing the inducible promotor of the TK gene