Our mathematical model.

Core System Ode model

At the beginning of the project, we wanted to create a more targeted killer t cell. By combining activated cell circuits, it works by simultaneously recognizing two or more surface antigens. Because such a system needs to consider more factors, so we design several simplified models to help us understand and calculate their relationship within the allowable deviation.

We simplified the actual situation, standardized the relevant factors and then built the core system ODE model Hope to be able to confirm in the internal environment of our system can be stable existence and play a practical role.

The target of our modeling is the core cell ttz-synnotch. IC9. Uas-ctx. I2flag. The cell was transfected by NK-92 cell line via lentiviral vector. First, the cell will express our designed synNotch receptor on the surface. The extracellular and intracellular segments are H218 SCFV and Gal4-vp64, respectively.



Such NK cells can recognize surface Antigen HER2-positive breast cancer cells.

The growth of cancer cells in humans generally follows a logistic growth curve, an s shaped growth curve, expressed as:


The instantaneous growth rate is:


The initial number of cancer cells, K for the terminal value (that is, the maximum capacity) , r indicates the speed of tumor progress.

hen NK cells come into contact with cancer cells, they bind to the synNotch receptor, and the intracellular segment of the synNotch receptor, Gal4-VP64, falls off.


If there is no follow-up reaction for a period of time, the COMBINATION WILL SELF-DESTRUCT:


GAL4 can bind to a UAS promoter on another gene to switch on CAR expression and the signaling molecule rIL-2.


In the end, cancer cells identified by CAR and synNotch receptors are cleaved and released by NK cells that bind to them.


Differential equation:


Dimensionless treatment of the parameters:

ke NK cells express synNotch receptors rate 1.0
kg Breast cancer cell growth rate 1.0
[tumor cells]max Maximum number of breast cancer cells 1200
kcombine NK cell and breast cancer cell binding rate 0.22
kkill NK cell killing rate 3.0
δ CAR boot kill constant 0.01
kgenerate Gal 4 initiates downstream expression rate 0.23
σ Gal 4 reaction constant 1.16

After simulation, each cell (material) changes with the time image:


The images show that under ideal conditions, after the core killer cells enter the system, the breast cancer cells stop growing and begin to decrease after a brief response period. At the same time, with the level gradually increased, indicating that the loop works well, killing effect is significant. After a period of time, when breast cancer cells were inhibited at a lower level, the corresponding level of self-reduction, and the expression of killing results can still maintain a certain level for a certain period of time, so as to facilitate detection.

Parameter Sensitivity analysis

We know from the literature that there is a delay of 12-24 hours between the induction of CAR expression on t cells after the recognition of specific antigen by synNotch receptor and the direct activation of CAR expression, and we have reached the same conclusion from experiments. So it's our goal to look at what factors limit the time it takes to take effect, and how the effects are expected to be achieved.


Using the simbiology toolbox in Matlab, we calculated the expression of notch receptor (expression. KF) by NK cells in Vivo The rate at which synNotch receptors bind to breast cancer cells ([ Synnotch ] . Combine) , the rate at which the intracellular segment gal4-VP64 activates downstream gene expression (UAS.generate) , and CAR killing tumor cell rate (kill.kill) as the input parameter Corresponding to the labeled rate of interleukin-2 synthesis (body.FALG.rIL-2) as the output parameter, their limiting effects at different stages of the reaction were compared.

model_3 model_4

According to the simulation results, we can find that the rate of activation of downstream gene expression by intracellular gal4-VP64(UAS.generate) is the first limiting factor of the response efficiency. In addition, the rate at which synNotch receptors bind to breast cancer cells ([ Synnotch ] . Combine ]) also limits the response, although there is a brief crossover This suggests that future improvements in response speed could be directed towards the selection of more efficient and efficient downstream promoters and more efficiently binding receptors at the extracellular level of Synnotch, thus increasing the killing efficiency of core cells.

Biological signal transformation of tumor cells

In addition to killing the tumor cells, we also hope that the modified cells will be able to complete the warning function. It will capture tiny cancer signals, HER2 on the surface of a small number of cancer cells in the early stages, activate the Syn-Notch receptor on NK cells and initiate a downstream response, which is then delivered to 293 cells as a "messenger" of the labelled interleukin-2 factor (FLAG.rIL-2). There, it turns into a fluorescent signal that alerts people and helps kill tumors


We also use ODE45 to solve the differential equation. According to the result of calculation, we find that in the dimensionless model, the low level HER2 signal can be transformed well according to the expectation of experiment Into fluorescence that can be easily detected on the surface of the body.


To split the image, black solid lines represent the moment when t 90 is present, and dotted lines represent the threshold at which the substance is detected. At low HER2 levels, our early warning signal, EGFP, has exceeded the detection threshold and can be observed. It can transform the tumor biological signal which is difficult to be measured into the physical fluorescence signal which is easy to be detected.

At the same time, we have also designed a software for this work, you can get more relevant information in the software interface.