Our potential inhibitors are expected to
effectively inhibit the binding between Aca1 and *acr*-promoter.
Therefore, in our two-plasmid reporter system for screening the most
effective inhibitors, we expected the following results: in the absence
of an inhibitor, Aca1 can effectively bind to *acr*-promoter,
blocking the expression of eGFP, causing the strength of fluorescence to
decrease; with the presence of inhibitor, the inhibitor binds with Aca1
and disrupts its normal binding with *acr*-promoter, and thus
*acr*-promoter can promote the expression of eGFP and make the
strength of fluorescence in measurement recover to a normal level.

## Model

## Overview

**Introduction**

In order to evaluate the efficiency of our
PDI inhibitor candidates with the fluorescence strength we measured, we
developed a model about the inhibition process of the inhibitor with our
reporter system. In this model, we use a constant named *in vivo*
*K _{I}* to evaluate the

*in vivo*efficiency of inhibitor candidates. (We decided not to develop a model to evaluate the efficiency of our PPI inhibitor candidates because it is obvious that PPI inhibitors are far less effective than PDI inhibitors.)

**Conclusion**

Compared with normal *K _{I}*
that is measured through biochemical approaches, our

*in vivo*

*K*can directly reflect the inhibition efficiency of potential PDI inhibitors. When measuring the exact binding affinity between potential PDI inhibitor and Aca1 protein, it is better to measure the exact K

_{I}_{I}with biochemical approaches. However, because the intracellular local concentration of some molecules may differ from the concentration of those molecules in whole cells, we think that our evaluation of inhibition efficiency through

*in vivo*

*K*is more practical.

_{I}In conclusion, this model can provide us with a way to evaluate the efficiency of certain PDI inhibitors in our reporter system and help us screen effective PDI inhibitors towards Aca1 protein. Also, it is worth mentioning that when our reporter system is applied to screen other effective inhibitors for the interaction between phage specific proteins and their target promoters, our model is still practicable to evaluate the inhibiting efficiency of potential inhibitors.

## Model Design

The PDI inhibitor candidates work on Aca1-dimers instead of monomers and can bind to the DNA binding site of Aca1 proteins, with the binding of one PDI inhibitor is supposed to sufficiently inhibit the normal function of Aca1 dimer. Here we assume that binding of PDI inhibitors can block the dissociation of Aca1 dimer, just like the binding of DNA can stabilize Aca1 dimer. Therefore, the dimerization process of Aca1 is not taken into consideration in this model. Similarly, considering possible cooperation binding of PDI inhibitors (binding of the first PDI inhibitor can effectively inhibit the binding of the second PDI inhibitor), we only take into account the first binding of PDI inhibitor to Aca1 dimer.

Under such conditions, we use our model to
measure a constant, named after *in vivo K _{I}*, to
evaluate the efficiency of PDI inhibitors in bacteria.

In this system, available *acr*-promoter
can promote the expression of eGFP, so we suppose that the concentration
of eGFP is in proportion to the concentration of available
acr-promoters. Also, we have

where *I _{fluorescence}* is the strength of the
fluorescence we measure in the experiments. Therefore, we can get

where *k* is a constant.
Specifically, with the absence of the plasmid expressing Aca1 and the
presence of the plasmid having *egfp* downstream of* acr*-promoter,
we can measure* k* as

(4)

where *I _{1}* is the strength of the fluorescence in
positive control.

We are going to talk about two conditions in this model.

The first condition is the negative control condition in the experiment where no PDI inhibitor is added to the system. Under this condition,

(8)

and

(9)

We define the strength of fluorescence we
measure in this condition as *I _{0}*. Therefore, from
equation (3), we can get

(10)

Taking equation (2), (6), (9), and (10) together, we can get

and therefore, we generate a measurement of the original concentration of Aca1-dimer

(11)

Next, we are going to talk about the
experimental condition where *c(PDI-inhibitor)* of the PDI
inhibitor is added to the reaction system. In an ideal situation, *c(acr-promoter)*
and *c(Aca1-dimer)* in the experiment are strictly maintained to
be the same as those in the negative control. Therefore, with equation
(11), we can build a bridge between those two conditions in order to
roughly measure *K _{I}*, and thus the efficiency of the
PDI inhibitor. With the strength of fluorescence measured in experiment
being

*I*, from equation (2), (3), (6), and (7), we get

(12)

Then, with equation (6) and (11), we can generate [inhibited-Aca1-dimer] as

(13)

Taking equation (1), (4), (5), (12) and (13)
together, *K _{I}* can be measure as

(14)

According to the research by Stanley et al.,
K_{D} has a value of around 50 nM^{1}, which is far
smaller than *kI*, so we can estimate K_{I} as

(15)

By replacing *k* using equation
(4), we can simplify equation (15) as

(16)

where *K _{I}* is in vivo
inhibitory constant of a specific PDI inhibitor,

*c(PDI-inhibitor)*is the concentration of the PDI inhibitor in the system,

*I*is the strength of the fluorescence in the positive control,

_{1}*I*is the strength of the fluorescence in negative control and

_{0}*I*is the strength of the fluorescence in the experimental group.

Clearly, *K _{I}* decreases as

*I*increase. Therefore, increasing in

*I*reflects a higher binding affinity between PDI inhibitor and Aca1 dimer, which indicates a better inhibiting effect of PDI inhibitor. In some cases, the PDI inhibitor candidate may be harmful to

*E.coli*, disrupt the normal expression of GFP in

*E.coli*, and therefore causes

*I*to be smaller than

*I*in the negative control and K

_{0}_{I}to be negative.

## Screening Effective PDI Inhibitor for Aca1

We applied our model to evaluate the
efficiency of different PDI inhibitor candidates and found that 1400-C4
has the smallest *in vivo K _{I}* among those candidates.
Therefore, we concluded that 1400-C4 could have the best inhibition
efficiency and focused on 1400-C4 in the further experiment process.

Table 1.* in vivo*
*K _{I}* of different PDI candidates

compounds |
c(PDI-inhibitor) (μM) |
I_{0} |
I |
I_{1} |
K
_{I }(μM) |
---|---|---|---|---|---|

1388-F10 | 19 | 167401 | 191357 | 1446723 | 115.2079 |

1388-B9 | 19 | 167401 | 192753 | 1446723 | 108.743 |

1401-H11 | 19 | 167401 | 220510.3 | 1446723 | 50.75995 |

1401-G2 | 19 | 167401 | 222787 | 1446723 | 48.58307 |

1402-C10 | 19 | 167401 | 227404.7 | 1446723 | 44.6751 |

1388-D4 | 19 | 167401 | 228061.7 | 1446723 | 44.16743 |

1388-H6 | 19 | 167401 | 237461.3 | 1446723 | 37.94673 |

1405-H10 | 19 | 167401 | 271971.7 | 1446723 | 24.69803 |

1389-B10 | 19 | 167401 | 87026.33 | 1446723 | -37.192 |

1400-C4 | 19 | 167401 | 146922 | 1446723 | -139.539 |

Figure 1. *in vivo* K_{I} of
different PDI candidates

## References

- Stanley, S. Y.
*et al.*Anti-CRISPR-Associated Proteins Are Crucial Repressors of Anti-CRISPR Transcription.*Cell***178**, 1452-1464.e1413, doi:https://doi.org/10.1016/j.cell.2019.07.046 (2019).