# Background

In the previous experiments, we conducted a lot of tests on the CRISPR-Cas12a system, which fully proved that the system can be used in the signal amplification process of DNAzyme detection experiments. But the experimental results also show that when using different lengths of ssDNA to trigger the CRISPR-Cas12a reaction, the reaction efficiency is different. In response to this phenomenon, we want to analyze the kinetic process of the CRISPR-Cas12a reaction in order to discover the laws of it.

# CRISPR-Cas12a & Michaelis–Menten Equation

In the previous experimental part, we have introduced how CRISPR-Cas12a is used in our project.

**Fig 2.6.1** CRISPR-Cas12a Reaction in our project

After Trigger DNA enters the reaction system, it will bind to the Cas12a-Probe-gRNA complex and shear the fluorescently labeled probe, resulting in a fluorescent signal. This process takes place in a closed reaction vessel. Various molecules diffuse and collide randomly in the solution. This state conforms to the conditions of use of the Michaelis–Menten kinetics. The progress is as follows:

**Fig 2.6.2** Michaelis–Menten Equation about Cas12a

Michaelis–Menten Equation establish a model which takes the form of an equation describing the rate of enzymatic reactions, by relating reaction rate V0 to the concentration of a substrate [S]. The equation is as follows:

**Fig 2.6.3** Reaction rate V relating to concentration of substrate S

Taking the reciprocal gives Lineweaver-Burk Equation

**Fig 2.6.4** Lineweaver-Burk Equation

where V0 is the reaction velocity, Km is the Michaelis–Menten constant, Vmax is the maximum reaction velocity, and [S] is the substrate concentration.The Lineweaver-Burk Equation can be regarded as a linear function in the form of y=mx+c, while y represents 1/V0, x represents 1/[S]. This means 1/V0 and 1/[S] show a linear relationship.

**Fig 2.6.5** An example of Lineweaver-Burk Plot

# Model Ideas for Our Project

For different product concentration groups, there will be a time-varying reaction product concentration curve (here is fluorescence). In order to solve the production rate of the reaction product, it can be solved by linear regression, that is, for one Time t vector [t1...tn] and a reaction product fluorescence value vector [a1...an]. The relationship between product concentration and time is as follows:

**Fig 2.6.6** The relationship between fluorescence generation and time.

Considering the process of product accumulation, we can clearly find that θ is the rate of fluorescence generation. In this way, by solving for θ, we can get V0. Then we regress the reactant concentration and corresponding reaction rate of each group according to the equation in Fig to solve Km and Vmax.

# Model Result

We designed an experiment in which different lengths of ssDNA were used to trigger the Cas12a system. In this experiment, we used 11 nt, 13 nt, and 15 nt ssDNA at different concentrations as input signals, and measured the fluorescence intensity versus time curve with the help of a microplate reader.

**Fig 2.6.7** Result of 11 nt, 13 nt, 15 nt ssDNA CRISPR-Cas12a
experiment

Then we used the equation in Fig , we can calculate the value of V0.

**Fig 2.6.8** ineweaver-Burk Plot of Cas12a testing

Using the results in the above figure, we finally drew the curve related to V and S. The image is as follows:

**Fig 2.6.9** Michaelis–Menten saturation curve for Cas12a Enzyme with 11
nt, 13nt, 15nt

And the Km can be calculated as follows:

Vmax (sec^-1) | Km (M^-1) | |
---|---|---|

11 nt | 2.52 * 10^-1 | 3.44 * 10^-8 |

13 nt | 5.92 * 10^-2 | 1.16 * 10^-8 |

15 nt | 8.75 * 10^-2 | 1.53 * 10^-8 |

**Table 2.6.10** Vmax and Km value of different ssDNA

# Result analysis

Finally, through calculation, we can find that the Km and Vmax value of 11nt are higher. This can also be seen from the experimental results: a significant increase in fluorescence value can be observed in the curve of 50 nM in the 11nt reaction. This means that our modeling is successful, and the results are in line with experimental expectations. Combining the principle of the Cas12a reaction, we believe that shorter trigger DNA is easier to bind to the Cas12a-gRNA-Probe complex and easier to start the digestion reaction. So 11nt ssDNA has the highest Vmax value in the CRISPR-Cas12a reaction. However, because trigger DNA binds to gRNA according to the principle of base complementary pairing, we can also observe that as the length increases, the Vmax value increases slightly: the 15 nt has higher Vmax value than 13 nt.

# References

1.Chen, Janice S., et al. "CRISPR-Cas12a target binding unleashes indiscriminate single-stranded DNase activity." Science 360.6387 (2018): 436-439.