Basic ideas
Half-life and decay rate:
In many occasions the decay rate of a certain macromolecule is measured by halflife. The decay of macromolecules within this model was assumed to a first order reaction $\mathrm{A}->$ products. The decay rate is $\mathrm{k}[\mathrm{A}]$ and the concentration of the reactant will decrease exponentially as $[\mathrm{A}]=[A]_0 * \exp (-\mathrm{kt})$. The time $t_{1 / 2}$ for decrease from $[A]_0$ to $\frac{1}{2}[A]_0$ can be substituted into the equation: $\frac{1}{2}[A]_0=[A]_0 * \exp (-$ $\mathrm{k} t_{1 / 2}$ ). It can be solved for $t_{1 / 2}=\frac{\ln 2}{k}$. The decay rate of a certain molecule, given half-life is $\frac{\ln 2}{t_{1 / 2}}[\mathrm{~A}]$
Chimeric Antigen Receptor (CAR) Expression Model
Introduction
The genetically engineered chimeric antigen receptor gene is introduced and expressed in T cell by transfection. In order to successfully implement the following experiments, the CAR is expected to be sufficiently expressed during cell culture. Therefore, it is important to study the expression pattern of CAR to suggest an optimal culture time. This computational model was built to give a rough picture of CAR expression pattern in a single transfected cell, which was intended to provide some suggestions to CAR-T cell culture on CAR expression perspective.
Model construction
A general pathway obeying central law was used to describe the process of CAR expression (Figure1).
The CAR expression starts with the transcription of pre-mRNA, facilitated by RNA polymerase II. The production of pre-mRNA is described the following equition: $$ \frac{d \text { premRNA }}{d t}=\alpha_{\text {premRNA }} \cdot \text { gene } -\alpha 1 \cdot \text {premRNA }-d_{\text {premRNA }} \cdot \text { premRNA } $$ $\alpha_{\text {premRNA }}$ is the production rate of pre-mRNA specific to the promoter used in the experiment. The amount of gene copies and RNA polymerase II is assumed to be stable inside the cell, therefore, the pre-mRNA production rate was considered to be constant. By integrating the complicated process of pre-mRNA modification, the process of pre-mRNA to mRNA is simplified as a first order reaction and $\alpha 1$ denotes the conversion coefficient. $d_{\text {premRNA }}$ dscribes the speed of pre-mRNA degradation. The transcribed pre-mRNA go through modifications including adding 5' cap, 3' ploy-A tail and splicing inside the nuclear. After modifications, the mature mRNA is exported to the cytoplasm. $$ \frac{d m R N A}{d t}=\alpha 1 \cdot \operatorname{premRNA}-v_{\exp } \cdot m R N A $$ $v_{\text {exp }}$ describe the speed of mRNA transportation trough nuclear membranes.The mature mRNAs in cytoplasm serve as the templates of translation, and can be degraded by various pathways: $$ \frac{d m R N A_{\text {out }}}{d t}=v_{\text {exp }} \cdot m R N A-d_{m R N A_{\text {out }}} \cdot m R N A_{\text {out }} $$ $d_{m R N A A_{o u t}}$ is the average degradation rate of mRNA measured by a previous study. The mRNA was considered to be translated in a constant rate $v_{\text {trans }}$, which is specific to codon sequence. The CAR protein is degraded at an average degradation rate $d_{C A R}$ $$ \frac{d C A R}{d t}=v_{\text {trans }} \cdot m R N A_{\text {out }}-d_{C A R} \cdot C A R $$
Parameters
| Parameter | Meaning | Unit | Value | Reference |
|---|---|---|---|---|
| $\alpha_{\text {premRNA }}$ | premRNA trancription rate | s-1 | 2.4e-2 | (Darzacq et al., 2007) |
| $\alpha 1=\ln 2 / t_{1 / 2}$ | conversion coefficient from premRNA to mRNA | s-1 | 0.693/151.35 | (Zeisel et al., 2011) |
| $d_{\text {premRNA }}=\ln 2 / t_{1 / 2}$ | premRNA degradation rate | s-1 | 0.693/1800 | (Morgado et al., 2012) |
| $v_{\exp }$ | mRNA export rate from nuclear to cytoplasma | s-1 | 2e-3 | (Pokrywka & Goldfarb, 1995) |
| $d_{m R N A_{0 u t}}$ | mRNA degradation rate in cytoplasma | s-1 | 3.3e-3 | (Baudrimont et al., 2017) |
| $\mathcal{v}_{\text {trans }}$ | mRNA translation rate | s-1 | 1.4e-4 | (Reuveni et al., 2011) |
| $d_{C A R}$ | CAR degradation rate | s-1 | 2.778e-6 | (Zhao et al., 2015) |
Table.1: the model parameters for the CAR expression model
Result
The number of successfully transfected plasmids varies from cell to cell, but the pattern of gene expression was not influence, therefore, the input was set as 40 copies per cell as an example.
Based on the result of the simulation, the rate of cytoplasmic mRNA level growth approaches 0 at around 2500 seconds. Meanwhile, the production of CAR significantly slowdown at around 9 to 14 days, with the rate of growth approaches 0 at around 14days, which suggests that the optimal time for transfected CAR gene expression is around 9 days.
Computational model of CAR T-cell immunotherapy
Based on our experiment results and data from previous studies about targeted CAR-T cell transfer that have been applied in clinical trials, like anti-CD19 CAR-T cell therapy, a computational model was built for predicting the therapeutic effect of the DDP4-targeted CAR-T cell. Inspired by a previous mathematical model of CAR-T immunotherapy in preclinical studies of hematological Cancers, cells were divided into three pools and their relationship was presented in Fig.1 (Barros et al., 2021).
Model construction
Senescent fibroblast:
It is assumed that aged fibroblasts cells are generated at a constant rate $\left(r_F\right)$, and they are eliminated by effector CAR-T cells as: $$ \frac{d n_F}{d t}=r_F-\frac{k_E n_F}{K_F+n_F} n_E $$ where $\mathrm{n}_E$ is the number of senescent fibroblasts, $\mathrm{n}_{\bar{E}}$ is the number of activated CAR-T cells, and $k_E$ is the killing rate. The efficacy of eliminating senescent fibroblasts follows the Michaelis-Menten equation with a Michaelis constant $\mathrm{K}_{\mathrm{F}}$, which equals the concentration of the substrate at half the maximum reaction rate, denoting the saturation effect from senescent fibroblasts to the killing efficacy.
Activation of the CAR-T cell.
Only after the activation by the antigen on the aged fibroblast, can the initial nonactivated CAR-T cell be transferred to the activated CAR-T cell and exert its killing effect. The quantitative change of activated CAR-T cells is shown as: $$ \frac{d n_E}{d t}=\frac{r_E n_F}{K_p+n_F} n_E+\frac{r_A n_F}{K_a+n_F} n_M-l_E n_E $$ where $r_E$ is the growth rate of activated CAR-T cells since a few parts of them are memory CAR-T cells and are capable of proliferation, $r_{\mathrm{A}}$ is the activation rate from initial nonactivated CAR-T cell, $\mathrm{n}_{\mathrm{m}}$ is the number of non-activated CAR-T cell. The proliferation and activation rate of the CAR-T cell are influenced by the senescent fibroblast, which are represented by the Michaelis constant $K_p$ and $K_a$, respectively. Lastly, the quantitative change of non-activated CAR-T cells is shown as: $$ \frac{d n_M}{d t}=-\frac{r_A n_F}{K_a+n_F} n_M-l_M n_M $$ where $l_m$ is the apoptosis rate of the non-activated CAR-T cell.
Data collection and parameter selection
| Parameter | Meaning | Unit | Value | Reference |
|---|---|---|---|---|
| $\mathrm{n}_F$ | The number of the senescent fibroblast | 109 | 2000 | |
| $\mathrm{n}_{\mathrm{N}}$ | The number of activated CAR-T cells | 109 | 0 | |
| $\mathrm{n}_{\mathrm{m}}$ | The number of non-activated CAR-T cell | 109 | 16.46 | Bouchkouj et al., 2019 |
| $\mathrm{k}_{\mathrm{E}}$ | Killing rate of activated CAR-T cells | Day-1 | 22.72 | Sahoo et al., 2020 |
| $r_E$ | The growth rate of activated CAR-T cells | Day-1 | 1.62 | Liu et al., 2021 |
| $r_A$ | Activation rate of initial non-activated CAR-T cells | Day-1 | 0.65 | Ribeiro et al., 2002 |
| $r_F$ | The growth rate of senescent fibroblasts | Day-1 | 1-20 | |
| $l_E$ | The apoptosis rate of the activated CAR-T cell | Day-1 | 0.12 | Ribeiro et al., 2002 |
| $l_M$ | The apoptosis rate of the non-activated CAR-T cell | Day-1 | 0.00003 | |
| $K_p$ | Saturation constant to CAR-T cell proliferation | 600 | L. Liu et al., 2021 | |
| $K_F$ | Saturation constant to CAR-T cell killing rate | 6000 | ||
| $K_a$ | Saturation constant to CAR-T cell activation | 1800 |
Table.2: the model parameters for the in vivo simulation
Results
The fibrosis rate varies from patient to patient. Therefore, in the model, the value of the growth rate of senescent ranged from 1 to 20, which served as an example of the real clinical circumstance.
From the simulation result, it was found that DPP4-targeted CAR-T cell could wipe out the senescent fibroblast swiftly and suppress its subsequent increase under different growth rates of senescent fibroblasts, which proved that the therapy was effective for both patients with mild and severe symptoms. It was noted that with the set of faster growth rate of senescent fibroblasts, more activated CAR-T cells would be reserved in vivo. Besides, the remaining amount of non-activated CAR-T cells was only affected by the initial injection amount and time.
All in all, the simulation of the modelling made up the in vivo experiment that has not been conducted, which serves as a good prediction for the effectiveness of the DPP4-targeted CAR-T cell, though the quantitative results provide limited reference since it’s a conceptual model.
References
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