Our modelling is based on the following signal transduction network of TOR protein. As we discussed before, the major part of signaling transduction is regulated by the rapamycin-sensitive TORC1 complex either via the Tap42-Sit4/PPA2c or the recently identified Sch9 branches. Nevertheless, Sch9 branch appears to have overlapping functions with cAMP-PKA pathway,and be additionally regulated by proteins not contained in central TOR pathway. Besides, as most of the downstream transription factors are definitely regulated to Tap42-Sit4/PPA2c, we decide to simplify the modelling part to this branch only.

In normal yeast cell, TORC1 (Tor complex 1) with phosphorylated Tor2 is in a functional state, which is able to phosphorylate Tip41 and Tap42 which would bind together when dephosphorylated.

PP2A1 and PP2A2 belong to a PP2A family that has the ability to dephosphorylate other proteins. PP2A1 consists of Pph21/22 and Cdc55/Tpd3, which are catalytic and regulatory subunits respectively. PP2A2 consists of Sit4 and Sap, which are catalytic and regulatory subunits respectively.

However, phosphorylated Tap42 is more likely to bind catalytic subunits of PP2As, like Pph21/22 or Sit4, which means phosphorylated Tap42 suppress the activities of PP2A1 and PP2A2.

PP2As could dephosphorylate some transcription factors, like phosphorylated Rtg1/3, Gcn4, Gln3, etc, as well as phosphorylated Tap42 and Tip41.

The activity of TORC1 can suppress the activities of PP2As.

In our project this year, multiple inhibitors ”FAP”, which are short for furans, acetic acid and phenol could inhibit the activity of TOR protein. When FAP exists in vivo, TORC1 (Tor complex 1) will be dephosphorylated, leading to an inactivated state without the ability to phosphorylate downstream proteins (Notice: The equilibrium constant

can reflect the resistance of Tor2 to FAP to some degree). Another simplification here is that, before we fully understand the mechanism of how FAP inhibit Tor2, we could just treat the interaction between them as complex formation. When Tor2 is bound to FAP, it no longer fulfills the downstream phosphorylation.

As a result, dephosphorylation of Tor2 leads to increased activity of PP2As. Then PP2As turn to be functional again and dephosphorylate a series of transcription factors (Here we use Rtg1/3 as example in the rest modeling part).

Those dephosphorylated transcription factors move into the nucleus (they use to be excluded out of nucleus when phosphorylated), and then activate specific genes. First, dephosphorylated transcription factor could bind with promoter of specific sequences. Here transcription factor Rtg1/3 can activate gene CIT2 by binding its promoter pCIT2. Once bound with transcription factor Rtg1/3, pCIT2 turns into an activated state - pCIT2*. Only activated promoters pCIT2* are able to initiate the transcription process. After transcription, pCIT2* break down into pCIT2, Rtg1/3 and mRNA. These specific mRNAs would complete translation, during which mTOR2p (short for mutant Tor2 protein) would exist in cytoplasm.

For mutant Tor2 protein, it has the identical functions of original Tor2 protein, which means it can phosphorylate Tip41 and Tap42 with the same reaction rate.

However, our mutations give mTor2 protein improved resistance to FAP, which can be demonstrated from the value of reaction rate:

is much more greater than

i As [Tap42p] [Tip41p][Tor1/2p] [PP2A1][PP2A2][RTG1/3p][pCIT2] are treated as factors in signaling transduction, we assume that their total amount would remain unchanged. There only exist different states. For example: the total amount of Tap42 in cytoplasm is unchanged, but Tap42 had two different states, phosphorylated and dephosphorylated. Only certain amount of protein will take part into this gene circuit and transduction loop.

ii Our main mission is about the regulation on transduction of a signal (FAP existing in cytoplasm), thus other complicated mechanism are ignored. Certainly there must be some Tap42 proteins having interactions with other substances, but the small amount of proteins leaked are not taken into our consideration.

iii We assume that mRNA, mutant Tor2 protein and FAP will degrade in a constant rate during normal metabolism in yeast cell.

To demonstrate the above complicated model more clearly, we simplify the original model and devide the whole network into four levels, which could form a feedback loop. After the simplification, it's much easier for reader without much professional knowledge to understand and more convenient to set parameters.

FAP bind with TORC1 to form a nonfunctional polymer, meaning the inhibition by FAP

TORC1 and Rtg1/3 form a nonfunctional polymer, which means that TORC1 suppress the activities of transcription factors in downstream. (This equation combines the equations with PP2As)

Functional transcription factors can activate specific target genes (mutant TORC1 here), and the following processes and equations are same with those in original model.

Follow the assumptions above, we set: i No cross section with TOR pathway is counted in. ii The amount of [TORC1] [Rtg1/3] [pCIT2] is unchanged during the whole process, however, each substance have different existing state. For example, Rtg1/3 has two states, one is functional ([Rtg1/3]) and another one is nonfunctional ([TORC1•Rtg1/3]) iii The mRNA, mutant Tor2 protein and FAP will degrade in a constant rate.

From the Figures above, we know that in the environment without FAP, the amounts of TORC1 and mTORC1 will lead to a steady level after a very short time.
i TORC1: the amount is near 10.037, as we calculate before.
ii mTORC1: the amount is near 0.2872, as we calculate before.
iii Then total amount of functional proteins that could reach "level 2" is 10.037+0.2872=10.3242 .

When FAP are added, the amount of TORC1 is decreased sharply, and then after a period of time, it maintain in a relatively steady level in 48h's fermentation. And the finally amount is about 0.39. From this, we know that FAP greatly inhibits the activity of TORC1.

From the figure above, we found that the amount of mTORC1 is greatly increased after FAP are added in. In previous situation without FAP, the steady amount of mTORC1 is about 0.2872, however, when FAP are added in, after a period of oscillation, the amount of mTORC1 is about 7.9 .

When FAP are added in, the total amount of proteins that have the same function with initial TORC1 can be 0.39+7.9=8.29, of which 95% comes from mTORC1. This substantially proves our gene circuit is able to work in the environment with inhibitors.

From the amount of pCIT2*, we can also draw the same conclusion with that on mTORC1. An interesting thing is that an obvious oscillation appears in the early period.

Taking another look at our "four level structure" mentioned before, we consider this kind of topology as a feedback loop with two nodes, which can account for all the phenomena.

As we know, topology with two nodes wouldn’t give rise to long last oscillation, but a final steady state after a period of fluctuation. In our project, after 2500 seconds' fluctuation, this circuit leads to a steady state.

To evaluate the characteristic of our mutant Tor2 precisely and comprehensively, I conduct further analysis towards mutant Tor2's sensitivity to FAP.

In the simplified model I just show above, the sensitivity of Tor2 to FAP has been greatly decreased fromto . Then questions are raised: what if only three of our four mutant base work? What's the situation if mTor2 with a larger or smaller sensitivity to FAP? Then I try 10 constants ofto see the tendency.
Then we get the following figure from book above:

From the figure above, we discover that When, the amount of mTor2 in cell reach a maximum. In this situation, the sensitivity is deceased only to a certain degree. Then question is raised again: why don't we just choose the mutant Tor2 with?

We discover that as the constant of decreased, the response time decreased, which means that the time for system to reach a steady state. Although system with =10 has a maximum of amount of mutant Tor2, but the response time is longer. The reason for this fact is that mutant Tor2 with a smaller is less sensitive to FAP, so it's easier for system with this kind of Tor2 to reach a steady state.