In our project, AID is the most significant protein. When designing the plasmid construction, we incorporated the Tet-On system as our gene regulation (see Design). In the AID expression model, we qualified how the Tet-On system regulates the expression of AID and predict the eventual concentration of AID in a hybridoma.

The function of AID is mutating DNA sequences. Therefore, we assumed that AID would cross the nuclear pore autonomously after AID is translated in the cytoplasm. However, in our construct design, we eliminated the NES (nuclear export sequence) in the AID. We had concerns about whether it would affect the pattern of AID crossing the nuclear envelope. Therefore, we simply simulated nuclear transport and export, predicting the exchange of the original AID and the NES-deleted AID between the cell nucleus and cytoplasm.

Part 1: gene regulation

An overview of the Tet-On-AID system is shown in Fig. 1 [10]. There are several steps in the system, and the reactions are described according to these steps (rate constants are written as k):

1. Dox­_e (extracellular doxycycline) diffuses into hybridoma through the plasma membrane in the form of Dox­­­_i (intracellular doxycycline)[1]. The parameter deff is a permeability coefficient.





2. Dox_i would bind to rtTA, which initially exists in the cytoplasm, to form Dox_i:rtTA.





3. Doxycycline can be degraded. The rate is assumed to be equal to the distilled water[8].





4. Dox_i-rtTA can either disassociate into Dox_i and rtTA or enter the nucleus, recognizing and binding to the TetO.






5. When Dox_i:rtTA binds to the TetO, transcription can be started and generate mRNA (S_m is referred to as transcription rate).





6. The mRNA results in the translation of the AID protein, and some amount of mRNA can be degraded during the process (S_n is referred to as translation rate, and d_m is degradation rate).






7. Expressed AID can either be degraded (d_n is degradation rate) or diffuse into the nucleus (this will be illustrated in the next session).

Based on the reaction mechanism in equation1.1-1.3, the concentration balances of the transcription factors, doxycycline and rtTA are given in the following ordinary differential equations:

The initial value of [Dox_i] and [Dox_i:rtTA] is set to be zero, while that of the rtTA, which is 1.1 μl/ml, is obtained from the literature[2]. The rate constants and the permeability coefficient are described in the parameter table.

The gene expression regulated by Dox_i:rtTA in equation1.5-1.8 is modeled by the following ODEs:

After modeling the gene expression with ODEs, we wanted to approximate and simplify the generation of mRNA and the expression of AID protein to get the model more easily. Therefore, we incorporated the Hill function to manifest the gene expression (C_N is plasmid copy number):

Part 2: nuclear transport

There are nuclear-localization signal peptides in the AID protein[5]. Protein exchange through nuclear pore is mediated by a receptor-importin, which the small GTPase Ran regulates. Therefore, it will diffuse into the nucleus after it is expressed in the cytoplasm. An overview of the process is shown in Fig. 2 [11].

We built a kinetic model to qualify nuclear-cytoplastic exchange via importin. The model incorporates both nuclear envelope permeability and receptor binding kinetics[12]. A given molecule (here, we refer to AID protein) may pass NPC (nuclear pore complex) autonomously, following its concentration gradient or through receptor-mediated diffusion[6]. The permeabilities relevant to the two modes are p and a (passive and active), respectively. In conclusion, the diffusion rate is related to the permeabilities and the cytoplasmic and nuclear volume (Fick’s law).

We described the kinetic model of the nuclear-cytoplastic exchange in several steps below (Tags C and N refer to cytoplasm and nucleus, respectively. Rate constants are written as k.):

1. In the cytoplasm, a cargo (C_C) AID protein with NLS can bind to importin (I_C), forming a receptor-cargo complex (IC_C).





2. The receptor-cargo complex will diffuse into the nucleus as IC_N.





3. The receptor-cargo complex in the nucleus (IC_N) can dissolve into the cargo (C^N) and importin (I_N).

In addition to receptor-mediated diffusion (a), there would be autonomous passive diffusion (p). This process will be considered in the model.

The original construct AID has NES (nuclear-export signal). Therefore, AID protein, which has diffused into the nucleus, might have a chance to diffuse back to the cytoplasm.

Though the nuclear export process appears to resemble the nuclear transport process mentioned above, we had noticed some differences. Importin is a competitor with the RanGTP, while exportin can bind simultaneously to Ran and protein cargo. The export cargo can be released from the exportin when arriving cytoplasm by hydrolysis of the Ran-associated GTP [13].

We considered two kinetic processes in nuclear export. On the one hand, there is cargo translocation and receptor-mediated diffusion. On the other hand, we investigated the binding kinetics of receptor, exportin, and Ran protein[12]. Besides, like nuclear transport, there are two modes of permeability. Passive transport (p) means the cargo passes the nuclear pore by its concentration gradient. The other is active transport (a), which means the receptor(exportin) mediated diffusion[7].

We described the kinetic model of the nuclear export in several steps below (Tags C and N refer to cytoplasm and nucleus, respectively. Rate constants are written as k.):

1. The exportin in the nucleus (E_N)can bind to RanGTP (R) as the exportin-RanGTP complex (ER_N). However, we noticed from the literature that the equilibrium constant of this reaction is vast (approximately 10⁸) [4]. Thus, we assumed that all the exportins in a nucleus are transferred to ER_N. The rate constants k₇ and k₈ will not be considered in the ODEs.





2. The exportin-RanGTP complex can further bind to a cargo protein (C_N)with NES, forming the exportin-RanGTP-cargo complex (ERC_N).





3. The ERC_N will diffuse into the cytoplasm as ERC_C.





4. In the cytoplasm, GTP hydrolysis on Ran disrupts ERC_C into exportin, RanGDP, and the cargo. We can notice that the hydrolysis is very fast and complete; thus, we can assume that the concentration of ERC_C is approximately zero [14].

In addition to receptor-mediated diffusion (a), there would be autonomous passive diffusion (p). This process will be considered in the model.

In our construct design, we eliminated the C-terminus 16 a.a, which can increase the efficiency of SHM while inhibiting the function of CSR (see Design). However, the NES (nuclear export signal) overlaps the CSR [5]. We thought the removal of the NES might cause changes in diffusion patterns of AID in the nucleus, and even have a negative effect. Therefore, we used the equations above to model the diffusion between the nucleus and cytoplasm of the original AID and our new NES-deleted AID, respectively.

First, based on equations 2.1-2.4 and 3.1-3.5, the concentration balances of the original AID and other NLS and NES system components are given by the following ODEs. The cargo C refers to the original AID protein. Parameter v is the cytoplasm (C) and nucleus (N) volumes. The initial values of IN and IC are set to be 4500nM and 5000nM, respectively.

Second, we assumed that the diffusion of AID without NES would only be subject to NLS and passive transport. Based on equations 2.1-2.4, the concentration balances of AID without NES and other NLS system components are given by the following ODEs. The cargo C refers to the original AID protein. Parameter v is the cytoplasm (C) and nucleus (N) volumes. The initial values of I_N and I_C are set to be 4500nM and 5000nM, respectively [6].

Calculation of translation

The AID is a core protein in our project, and it is barely to find the parameter of AID generation directly from the literature. Therefore, we need to do some simple calculations to estimate.

We used a two amino acids per second elongation rate to calculate. Activation-induced cytidine deaminase (AID) has 188 amino acids (see Design). Therefore, the translation rate is 2/188.    0.011 AID per second

Table of parameters

Parameter	Value	Unit	Reference
deff	3x10-4	1/sec	[1]
k1	0.3	(μl/ml)-1sec-1	[2]
k2	0.03	1/sec	[2]
ddox	10-6	1/sec	[1]
Kd	2.42	none	[2]
CN	5	nM	[3]
Sm	700.87	1/sec	[2]
Sn	0.011	1/sec	assume
dm	0.0075	1/sec	[3]
dn	3.3x10-4	1/sec	[3]
k5	0.03	(nM)-1sec-1	[6]
k6	0.03	1/sec	[6]
p	15	μm3/sec	[6]
a	100	μm3/sec	[6]
k9	0.1	(nM)-1sec-1	[7]
k10	2	1/sec	[7]
vc	4000	μm3	[6]
vn	4000	μm3	[6]

Part 1: the result of gene regulation

After confirming the variables, parameters and equations, we used MATLAB to calculate the ODEs of the Tet-On system.

At first, we set the amount of adding doxycycline is 10 ug/ml. According to equations 1.1-1.3 and their related ODEs, we can get the concentration of the transcription factors in our Tet-On system: doxycycline, rtTA, and doxycycline-rtTA complex. After we added doxycycline, the doxycycline diffuses into the cell (Fig. 4a). Intracellular doxycycline then rapidly binds to rtTA, which already exists in cells (with an initial value of 1.1 μg/ml). As a result, we can see that the amount of rtTA decreases and that of the doxycycline-rtTA complex increase and reach the balance at about 1 hour (Fig. 4b). Eventually, doxycycline-rtTA reaches its maximum of 1.09 μg/ml, while rtTA turns out to be 0.01 μg/ml.

With equations 1.5-1.8 and their corresponding ODEs, which include the use of the Hill function, we can get the amount of generation of the mRNA and the AID protein. The doxycycline-rtTA complex can bind to the promoter and start transcription. The ultimate concentration is 0-145000 nM (Fig. 5a). Next, the mRNA would start the translation of AID. In the end, the AID reaches its balanced concentration of 4.83*10^{^}6 nM (Fig. 5b). We notice that the consuming time of mRNA reaching equilibrium is shorter than that of the AID, which is about 1 hour and 5 hours, respectively.

Then we changed the amount of adding doxycycline. In Fig. 6a, we tested the expression of AID with an extensive concentration of doxycycline, from 0.01 μg/ml to 1000 μg/ml. The concentrations over 10 μg/ml show roughly the same expression value. Moreover, the 100 μg/ml and 1000 μg/ml lines almost overlap. We further changed the value of doxycycline, testing the range from 0.1 to 1 μg/ml with a difference of 0.1 (Fig. 6b).

After the western blot came up(See Proof of Concept), we compared it with the model. The 10 μg/ml result shows full AID expression, while that of 1 μg/ml shows less expression. There is still AID expression when only adding 0.1 μg/ml doxycycline. These results meet the model's outcome (Fig. 6a). However, the AID expression of 1 μg/ml and 0.1 μg/ml seems roughly the same. Sadly, due to time constraints, we were unable to conduct further verifications.

Part 2: the result of nuclear transport

After confirming the variables, parameters ,and equations, we used MATLAB to calculate the ODEs of the nuclear transport system.

We have modeled the original AID and NES-deleted AID with different ODEs systems. At first, we set the cytoplasmic AID concentration is 3000nM. According to the model, we can get the concentration of the two kinds of AID in the nucleus and other system factors, such as the importin, exportin, etc. In Fig. 7, the blue line refers to the system of the original AID, while the yellow line refers to the NES-deleted AID.

Almost all the AID in the cytoplasm immediately binds with the importin (Fig. 7a), forming the importin-AID complex (Fig. 7c). The NES would not affect importin-AID complex transport (Fig. 7c and Fig. 7d), while it slightly lowers the amount of free AID protein in the nucleus (Fig. 7b).

Secondly, we implemented the result of the model of gene regulation. In this model, the AID expression is 4.83*10^{^}6 nM. We changed the amount of AID in the cytoplasm by this numerical value. Then we can get the concentration of the two kinds of AID in the nucleus and other system factors (Fig. 8).

According to the result, we can see that all the yellow and blue lines almost overlap. The deletion of NES has little affection on the transport of the AID to the nucleus. Although the exportin and NES indeed function, the amount of exportin is too little compared to the expressed protein (Fig. 8g), so it can barely affect the diffusion of AID.

Comparing Fig. 7 and 8, we can get some conclusions. If the cargo concentration is less, it is more subject to exportin and diffuses more in active transport. On the other hand, if the cargo concentration is more, it would diffuse more in passive transport. In our design, the ultimately expressed AID protein concentration is very high. Therefore, the deletion of NES would not change the pattern of AID diffusion between the cytoplasm and nucleus.

Our project, AID can aid, aims to find an easier approach for researchers and biotech companies to receive monoclonal antibodies (see Integrated Human Practice). To gain high-affinity antibodies efficiently, our team combines AID, tet-on, and bioreactor to optimize the process of hybridoma cultivation and antibody production. (see Design)

It is obvious that monoclonal antibody plays a significant role in our project. Hence, we decided to build a mathematical model concerning hybridoma culture and monoclonal antibody production. By means of the above model, we can best visualize and complement our experiment, allowing us to predict and construct the optimal condition for those mechanisms.

Initially, we captured the idea of maintaining a high density of hybridoma cells to gain desirable productivity[1]. After the literature review, we realized that hybridoma cultivation deeply depends on the consumption of nutrients and the excretion of toxic byproducts[2].

Glucose and glutamine are significant components utilized by culturing mammalian cells[1]. Likewise, both of them in the culture medium[3] are regarded as nutrients in the model. On the other hand, the lactate and ammonium from fermentation are considered byproducts[1]. Based on the literature, lactate is converted by glucose and glutamine, while ammonium is mainly transformed by glutamine[2]. The four substances mentioned above served as variables in the culturing and producing model. We aimed to gain insight into how nutrients and byproducts affect the experimental outcome throughout the time course.

With the confirmation of the variables, we then searched for the most suitable software for modeling. MATLAB has always been an essential software for assisting researchers in lab procedures. To simulate the experiments, we turned to the MATLAB application, Simbiology, which is well known for its user-friendly interface and all-inclusive functions. Using Simbiology Model Builder, a visual model was constructed and displayed in Fig. 1. The simulation outcome was then provided by Simbiology Model Analyzer.

Summing up, we finally constructed a simbiology model and display it in Fig. 1. The parameter and variables are discribed in the two tables.

To clarify it, we separate the model into two parts.

The fermentation model is our first part. It comprises the kinetics of byproducts and nutrients in cell culture, and the equations are mentioned below (Fig. 2).

The second part is the hybridoma cultivation and monoclonal antibody production model. It continues the previous part by calculating the kinetics of both processes. Through the equations (Fig. 3), we can observe how the mechanisms are affected by the nutrients and byproducts.

For the variables we use, see (Table 1.).

Variables	Initial value	Unit	Symbol
Hybridoma	0.175	gram/liter	H
Glucose	1.72	gram/liter	G
Glutamine	0.258	gram/liter	Q
Lactate	0	gram/liter	L
Ammonium	0	gram/liter	A
Monoclonal antibody	0	gram/liter	Ab
Parameter_u	0.03	1/hour	u

As for the parameters, see (Table 2.).

Parameter	Value	Unit	Reference
m	0.0159	gram/gram/hour	[4]
YA	0.9	gram/gram	[5]
KG	0.863	gram/liter	[6]
KQ	0.00463	gram/liter	[6]
KL	0.061	gram/liter	[6]
krd	0.019	1/hour	[6]
KdL	0.795	gram/liter	[6]
krG	0.042	gram/gram/hour	[6]
KGr	0.029	gram/liter	[6]
krQ	0.00665	gram/gram/hour	[6]
KQr	0.154	gram/liter	[6]
kdQ	0.00315	1/hour	[6]
kgL	0.539	gram/gram	[6]
kQL	0.595	gram/gram	[6]
kA	0.275	liter/gram/hour	[7]
um	0.689	1/hour	[7]
kt	0.00036	1/hour	[8]

According to the experimental protocol, we set the simulation time to 48 hours for monitoring the change after adding culture medium.

Through Simbiology Model Analyzer, we receive some plots indicating the dynamic behavior of the model.

Fig. 4 shows the relationship between nutrients and byproducts. After the supplement with culture medium, it turns out that glucose and glutamine are gradually consumed while lactate and ammonium are produced consequently.

Next, Fig. 5 includes the hybridoma and the former variables. We can tell that hybridoma cells possess unequal growth rates through different periods. In the beginning, hybridoma culture efficiency is skyrocketing owing to sufficient nutrients. However, after the appearance of toxic byproducts, it starts to drop by degree.

As for Fig. 6, it can best represent the positive correlation of monoclonal antibody production rate (Fig. 6a) to hybridoma cell quantity (Fig .6b).

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