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PASCAR: a multiscale framework to explore the design space of constitutive and inducible CAR T cells

Harshana Rajakaruna, Milie Desai, Jayajit Das  Correspondence email
Published 28 July 2023. DOI: 10.26508/lsa.202302171

Article Figures & Data

Figures

  • Figure 1.
    Figure 1. Multiscale protein abundance structured population dynamic model for CAR T cells (PASCAR) model.

    Single chimeric antigen receptor (CAR) T cells interact with single target cells in the PASCAR model. The strength of CAR T cell signaling depends on the abundance of the CAR–human epidermal growth factor receptor 2 (HER2) complex (C0) in Model NKP or on the abundance of an active complex (CN) formed because of N number of chemical modifications in the CAR–HER2 complex in Model KP. The abundances of C0 (Model NKP) or CN (Model KP) depend on the abundances of CAR (R) and HER2 (H) in single CAR T cell and the target cell, respectively. The rate of lysis (λRH) of target cells because of the cytotoxic response or the proliferation rate (ρRH) of CAR T cells depends on the abundances of C0 (Model NKP) or CN (Model KP) in single CAR T cells. The lysis and the proliferation rates are used to describe the kinetics of populations of CAR T cells and target cells.

  • Figure S1.
    Figure S1. Comparison of the variation of the active chimeric antigen receptor–ligand complex (or CN) with the copy numbers of human epidermal growth factor receptor 2 ligands expressed on target cells for the high- (KD = 17.6 nM) and low-affinity (KD = 210 nM) chimeric antigen receptors where the affinities (kon and koff) are estimated with and without including the diffusion of the receptor and ligand molecules (see details in Supplemental Data 2).
  • Figure 2.
    Figure 2. Protein abundance structured population dynamic model for CAR T cells modeling of cytotoxic and proliferation responses of constitutive chimeric antigen receptor (CAR) T cell against target cells.

    (A) Shows fits for Model NKP to the percentage lysis of target cells 3 d after 10,000 constitutive CAR T cells were incubated with 20,000 target cells. The data and the fits are shown for five different human epidermal growth factor receptor 2 (HER2) expressions with average HER2 abundances at 104.7, 105.2, 105.7, 106.2, 106.9 molecules/cell. The constitutive CAR T cells express either high affinity (KD = 17.6 nM, koff = 9.0 × 10−5 s−1) or low affinity (KD = 210 nM, koff = 6.8 × 10−4 s−1) CARs. The values of percentage lysis at HER2 abundances that are in between the values mentioned above were calculated in the model by interpolating the means and variances of the HER2 distributions (see Figs S11A–D and S12A and B for details). (A, B) Variation of the CAR ligand abundance (or C0) with the copy numbers of HER2 ligands expressed on target cells for the high- and low-affinity CARs in (A). (A, C) Shows fits for Model KP to the percentage lysis data described in (A). (A, D) Variation of the active CAR–ligand complex (or CN) with the copy numbers of HER2 ligands expressed on target cells for the high- and low-affinity CARs in (A). (E) Comparison of the measured means and the variances of CAR abundances at day 3 post co-incubation with the fits obtained from Model KP. The mean HER2 abundances of the target cells used for modeling the co-culture experiments are shown along the x-axis. The means and variances for CAR abundances obtained from Hernandez-Lopez et al (2021) are shown with the green bar. (F) Comparison of the model fits for the percentage lysis, and means and variances of distributions of CAR abundances at day 3 post co-incubation with their experimental counterparts. The variables are made nondimensional by scaling the variables by their respective SDs calculated using the measured values. The goodness of fit is quantified by the correlation coefficient R2 which shows an excellent agreement (R2 = 0.97) between the model and the data. (G) Shows fits (solid lines) for Model KP to the percentage lysis data at day 3 for co-culture experiments at E:T = 1:0.35 (20,000 target cells and 7,000 CAR T cells) with constitutive CAR T cells expressing high-affinity CAR (KD = 1.9 nM, kon = 1.2 × 104 M−1 s−1, koff = 2.2 × 10−5 s−1) at high (WT) and low (+degron) abundances (Hernandez-Lopez et al, 2021) with HER2 expression distributions given for target cells at different mean HER2 levels. The predictions (dashed lines) for percentage lysis at day 3 generated from Model KP for co-culture experiments at E:T = 1:1 (20,000 target cells and 20,000 CAR T cells) are compared with available measurements in Hernandez-Lopez et al (2021) corresponding percentage lysis co-culture are constitutive model fitted (think line) to day 3 CAR expression data, estimating only the initial CAR (high affinity) distribution (mean and variance) for high (WT)- and low (+degoron)-expression CARs, with all other parameters best estimated by the earlier models fitted to high- and low-affinity CAR for constitutive and synthetic notch CAR. (G, H) Shows goodness of the fit for Model KP with the data (percentage lysis, mean, and variances of the CAR abundances at day 3) for the co-culture experiments in (G) for E:T = 1:0.35. The comparison is shown for the nondimensional variables where they are scaled by their SDs calculated using the measured values.

  • Figure S2.
    Figure S2. Fits to chimeric antigen receptor (CAR) expression and percentage lysis data in constitutive CAR T cells for Model NKP.

    (A) Comparison of the model fits for the percentage lysis, and means and variances of distributions of CAR abundances at day 3 post co-incubation with their experimental counterparts. The variables are made nondimensional by scaling the variables by their respective SDs calculated using the measured values. The goodness of fit is quantified by the correlation co-efficient R2 which shows an excellent agreement (R2 = 0.96) between the model and the data. (B) Comparison of the means and the variances of CAR abundances data with NKP model estimations.

  • Figure 3.
    Figure 3. Constitutive chimeric antigen receptor (CAR) T cell subpopulations expressing different CAR abundances show different cytotoxic and proliferative responses.

    (A) Shows lysis rates of target cells expressing mean human epidermal growth factor receptor 2 abundances (=106.2 molecules/cell) where the lysis is mediated by subpopulations of constitutive CAR T cells expressing different CAR abundances. The lysis rates increase with time as the numbers of cells in the CAR T cell subpopulations increase because of cell proliferation. The lysis rates for the CAR T cell subpopulations corresponding to the mean CAR expression (denoted as R-mean) and to the mode of the CAR distribution (denoted as R-peak) at day 3 are marked. The dotted line shows the average of lysis rates across the CAR T cell subpopulations. (B) Shows proliferation rates of CAR T cells as they interact with target cells expressing mean human epidermal growth factor receptor 2 abundances (=106.2 molecules/cell).

  • Figure S3.
    Figure S3. Constitutive chimeric antigen receptor (CAR) T cell subpopulations expressing different CAR abundances show different cytotoxic and proliferative responses.

    (A, C) Shows lysis rates of target cells expressing different mean human epidermal growth factor receptor 2 (HER2) abundances (104.5 versus 106.2 molecules/cell) by subpopulations of CAR T cells expressing different CAR abundances at different times post co-incubation. Results are shown for CARs with high (17.6 nM, top panel) and low (210.0 nM, bottom panel) affinity towards HER2. The dotted line shows the average of lysis rates across the CAR T cell subpopulations. (A, B, C, D) Shows proliferation rates of CAR T cells as they interact with target cells expressing different mean HER2 abundances (104.5 versus 106.2 molecules/cell) for the same assays in (A, C).

  • Figure S4.
    Figure S4. Kinetics of the CAR T cell populations at mean and peak values of CAR abundances.

    (A) Shows the increase in the size of the subpopulations {TR} with time. The “mean” and “peak” indicate the subpopulations corresponding to the mean chimeric antigen receptor (CAR) abundance and the CAR abundance corresponding to the peak of the CAR distribution, respectively, at t = 0. (B) Shows increase in λHR with R for H=H̅. (C) Shows the CAR distributions at days t = 0, 1.0, 2, and 3 d. Note, that the peak of the R distribution remains at intermediate values of R during the kinetics. This behavior can be explained as follows. Consider, TR1(0) and TR2(0) as the sizes of the subpopulations at intermediate (close to the peak) and larger values of R, respectively. If the proliferation rates for the subpopulations are given by g1 and g2, respectively, at t = 0, then assuming exponential growth with constant g1 and g2, the subpopulations’ sizes at a later time t is given by TR1(t)=TR1(0)exp⁡(g1t) and TR2(t)=TR2(0)exp⁡(g2t). Now, to have TR1(t)TR2(t)>1 as we find in (C), the condition TR1(0)TR2(0)>e(g2−g1)t, should hold. In the kinetics considered here, g1,2∝ρR1,2HUH∝R1,2. At t = 0 we find TR10TR20>10 , and g2−g1 ≈ 0.075/day in our model which at t = 1d yields e(g2−g1)t∼1.07. Thus the above condition is satisfied and the peak remains at the intermediate values of R at t>0. In addition, the rate g1 and g2 decreases with time (Fig 3B) which also helps in keeping the above condition intact, and the peak remains at the intermediate values of R for the entire kinetics.

  • Figure 4.
    Figure 4. Protein abundance structured population dynamic model for CAR T cell (PASCAR) modeling of cytotoxic and proliferation responses of constitutive and synthetic notch (synNotch) chimeric antigen receptor (CAR) T cell against target cells.

    (A) Shows fits for Model KP to the percentage lysis of target cells at 3 d after 10,000 synNotch or constitutive CAR T cells were incubated with 20,000 target cells. The data and the fits are shown for five different human epidermal growth factor receptor 2 (HER2) expressions with average HER2 abundances at 104.7, 105.2, 105.7, 106.2, 106.9 molecules/cell. The comparisons of the distributions of CAR abundances between the model and the measured values at day 3 are shown in Fig S13. The synNotch and constitutive CAR T cells express either high affinity (KD = 17.6 nM, koff = 9.0 × 10−5 s−1) or low affinity (KD = 210 nM, koff = 6.8 × 10−4 s−1) CARs. (B) Comparison of the model fits for the percentage lysis, and means and variances of distributions of CAR abundances at day 3 post co-incubation with their experimental counterparts. The variables are made nondimensional by scaling the variables by their respective SDs calculated using the measured values. The goodness of fit is quantified by the correlation co-efficient R2 which shows an excellent agreement (R2 = 0.97) between the model and the data. (C) Comparison of the measured means and the variances of CAR abundances at day 3 post co-incubation with the fits obtained from Model KP. The mean HER2 abundances of the target cells used for modeling the co-culture experiments are shown along the x-axis. The means and variances for CAR abundances obtained from Hernandez-Lopez et al (2021) are shown with green bars. (D) PASCAR model predicted percentage lysis (solid and dashed lines) of target cells at 3 d after 6,000, 12,000, and 15,000 synNotch CAR T cells were incubated with target cells. The data for 10,000 synNotch CAR T cells were used for training the model which are also shown as reference. (E) The predictions in (D) are in good agreement (R2 = 0.98) with the data obtained from Hernandez-Lopez et al (2021). (F) PASCAR model well-predicted percentage lysis (solid line) of target cells at 3 d for a cytotoxic assay at E:T = 0.3 where synNotch CAR T cells expressing high-affinity CAR (KD = 1.9 nM, kon = 1.2 × 104 M−1 s−1, koff = 2.2 × 10−5 s−1) were incubated with target cells. (G) Shows lysis rates of target cells expressing mean HER2 abundances (=106.2 molecules/cell) where the lysis is mediated by subpopulations of synNotch CAR T cells expressing different CAR abundances. (H) Shows proliferation rates of synNotch CAR T cells as they interact with target cells expressing mean HER2 abundances (=106.2 molecules/cell).

  • Figure S5.
    Figure S5. Parameter dependence of the cost function in PASCAR model.

    (A) Correlations between PASCAR model parameters and their significance at α = 0.05 (P ≤ 0.05) level estimated by 1,000 MCMC simulations with replacements. Most parameters are not significantly correlated with Pearson’s correlation coefficients remaining less than 0.5, highest being 0.49. (B) Variations of the cost function (Equation (5)) as pairs of parameters indicated along the axes are varied. The other values are kept fixed at the best estimate values.

  • Figure S6.
    Figure S6. Protein abundance structured population dynamic model for CAR T cells modeling of cytotoxic and proliferation responses of synthetic notch (synNotch) chimeric antigen receptor (CAR) T cell against target cells.

    (A, C) Shows lysis rates of target cells expressing different mean human epidermal growth factor receptor 2 (HER2) abundances (104.5 versus 106.2 molecules/cell) by subpopulations of synNotch CAR T cells expressing different CAR abundances at different times post co-incubation. Results are shown for CARs with high (17.6 nM, top panel) and low (210.0 nM, bottom panel) affinity towards HER2. The dotted line shows the average of lysis rates across the synNotch CAR T cell subpopulations. (A, B, C, D) Shows proliferation rates of synNotch CAR T cells as they interact with target cells expressing different mean HER2 abundances (104.5 versus 106.2 molecules/cell) for the same assays in (A, C).

  • Figure 5.
    Figure 5. Pareto fronts revealing optimal responses of constitutive and synthetic notch (synNotch) chimeric antigen receptor (CAR) T cells against tumor and healthy cells.

    (A) Pareto fronts for constitutive CAR T cells in the plane spanned by % lysis of healthy cells and 1/(% lysis of tumor cells). The Pareto fonts are calculated for different dissociation constants KD = koff/kon where for each KD value, koff and kon = koff/KD are varied to obtain the corresponding front. The Pareto fronts are calculated 5 d after the CAR T cells were incubated with a 1:1 mixture of tumor and healthy cells (Fig S10) in silico (details in the main text) with tumor cell number being 20,000. (B) Variations of % lysis of healthy cells with KD for specific % lysis of tumor cells by constitutive CAR T cells along the Pareto fronts. The % lysis of healthy cells decrease with increasing KD until a certain value (the end points shown on the graph); increasing the KD further decreases % lysis of tumor cells as well (not shown on the graph). (C) Pareto fronts for synNotch CAR T cells for different dissociation constants KD = koff/kon where for each KD value, koff and kon = koff/KD are varied to obtain the corresponding front. The other parameters KH and nH are fixed throughout at 2.42 × 105 molecules/cell and 3.97, respectively. (A) The in silico cytotoxic assay is set up the same way as in (A). (D) Pareto fronts for synNotch CAR T cells for different fixed koff and KD values at (9.0 × 10−5 s−1, 17.6 nM), (5.0 × 10−4 s−1, 3.7 × 104 nM), (5.0 × 10−4 s−1, 1.2 × 105 nM), (7.0 × 10−4 s−1, 1.0 × 106 nM) and (1.5 × 10−3 s−1, 1.1 × 106 nM), where kon = koff/KD. KH and N are varied within (1 × 103, 5 × 107) molecules/cell and (1, 8) on each Pareto front. The symbol size is proportional to N and the shades filling the symbols are proportional to KH. (A) The in silico cytotoxic assays are set up the same way as in (A).

  • Figure S7.
    Figure S7. Variation of the steady state abundances of C0 and CN with KD.

    Shows the variation of C0 and CN with KD where the abundances are computed at mean chimeric antigen receptor = 3,038 and mean human epidermal growth factor receptor 2 abundances for healthy (=104.5) and tumor (=106.2) cells. The koff and kp are set to 9 × 10−5 s−1 and 0.0072. C0 and CN do not change appreciably when chimeric antigen receptor T cells interact with healthy or tumor cells for smaller KD values (<100 nM).

  • Figure S8.
    Figure S8. Pareto fronts revealing optimal responses of constitutive and synthetic notch (synNotch) chimeric antigen receptor (CAR) T cells against tumor and healthy cells.

    (A, B, C) Pareto fronts for constitutive CAR T cells in the plane spanned by % lysis of healthy cells and 1/(% lysis of tumor cells). The Pareto fonts are calculated for different dissociation constants KD = koff/kon, where for each KD value, koff and kon = koff/KD are varied to obtain the corresponding front. The Pareto fronts are calculated 5 d after the CAR T cells were incubated with a 1:4 or 1:1 or 4:1 mixture of healthy and tumor cells in silico (details in the main text) with tumor cell number being 20,000. (D, E, F) Pareto fronts for synNotch CAR T cells for different dissociation constants KD = koff/kon, where for each KD value, koff and kon = koff/KD are varied to obtain the corresponding front. The other parameters KH and nH are fixed throughout at 2.42 × 105 molecules/cell and 3.97, respectively. (A, B, C) The in silico cytotoxic assay is set up the same way as in (A, B, C). (G, H, I) Pareto fronts for synNotch CAR T cells for different fixed koff and KD values at (9.0 × 10−5 s−1, 17.6 nM), (5.0 × 10−4 s−1, 3.7 × 104 nM), (5.0 × 10−4 s−1, 1.2 × 105 nM), (7.0 × 10−4 s−1, 1.0 × 106 nM), and (1.5 × 10−3 s−1, 1.1 × 106 nM), where kon = koff/KD. KH and N are varied within (1 × 103, 5 × 107) molecules/cell and (1, 8) on each Pareto front with healthy to tumor cell ratios varying from 1:4, 1:1, 4:1 with tumor cell number being 20,000. The symbol size is proportional to N and the shades filling the symbols are proportional to KH. (A) The in silico cytotoxic assays are set up the same way as in (A).

  • Figure S9.
    Figure S9. Interpolation of the means and variances of human epidermal growth factor receptor 2 (HER2) abundances expressed by target cells.

    The points show the measured values and the dashed line shows the interpolation. We used the interpolated means and variances to estimate HER2 abundances following log-normal distributions for HER2 expressions not measured in Hernandez-Lopez et al (2021). The estimated HER2 distributions are used in the protein abundance structured population dynamic model for CAR T cell model to evaluate percentage lysis at shown in Figs 2 and 4 in the main text.

  • Figure S10.
    Figure S10. Simulated initial distribution of human epidermal growth factor receptor 2 densities per target cell in a population of mixture of healthy and tumor cells.

    Shows normalized distribution of HER to in a mixture of healthy (104.5 molecules/cell on average) and tumor cells (106.2 molecules/cell on average) at t = 0. The distribution is a superposition of two log-normal distributions with means at 104.5 molecules/cell and 106.2 molecules/cell on average and SD of 0.3.

  • Figure S11.
    Figure S11. Digitized chimeric antigen receptor (CAR) expression data and discretization of the probability distribution function.

    (A) Shows digitized distributions of CAR expressions in synthetic notch CAR T cells obtained from Hernandez-Lopez et al (2021). The units for the abundances are given in terms of AU the fluorescence intensity. (B) Shows the histograms for the CAR distributions used in our modeling. The bin sizes are shown in the figure panels. The AU of fluorescence intensity is converted to the units of molecules/cell based on the calibration of the mean fluorescence intensity to the mean number of CAR abundances performed by Hernandez-Lopez et al (2021). (C) Similar to (A) for constitutive CAR T cells. (D) Similar to (B) for constitutive CAR T cells.

  • Figure S12.
    Figure S12. Digitized human epidermal growth factor receptor 2 (HER2) expression data and discretization of the probability distribution function.

    (A) Shows digitized distributions of HER2 expressions in the target cells obtained from Hernandez-Lopez et al (2021). The units for the abundances are given in terms of AU the fluorescence intensity. (B) Shows the histograms for the HER2 distributions used in our modeling. The bin sizes are shown in the figure panels. The AU of fluorescence intensity is converted to the units of molecules/cell based on the calibration of the mean fluorescence intensity to the mean number of HER2 abundances performed by Hernandez-Lopez et al (2021).

  • Figure S13.
    Figure S13. Distributions of chimeric antigen receptor (CAR) abundances predicted by the protein abundance structured population dynamic model for CAR T cell model at day 3 for co-culture experiments with CAR T cells expressing high- and low-affinity CARs in constitutive and synthetic notch CARs as described in Fig 4 in the main text.

Tables

  • Table 1.

    List of estimated and fixed parameter values.

    ParameterUnitConstitutive + synNotchConstitutive
    λcday−12.09 × 10−8 [1.78 × 10−8, 2.42 × 10−8]1.33 × 10−8 [7.66 × 10−9, 2.05 × 10−8]
    kps−10.0074 [0.006, 0.009]0.0072 [0.0069, 0.0076]
    Nnumber7.1961 [7.1734, 7.2187]6.9031 [6.7617, 7.0459]
    ρcday−15.31 × 10−9 [3.76 × 10−9, 7.12 × 10−9]1.52 × 10−8 [1.47 × 10−8, 1.58 × 10−8]
    μc s.t. μR(consti.)(0)=μcmolecules/cell7.3589 [7.1719, 7.5483]7.2479 [6.9544, 7.5475]
    σR(consti.)(0)molecules2/cell0.4672 [0.4509, 0.4837]0.4841 [0.4748, 0.4935]
    μs s.t. μR(synN)(0)=μsμHnHμHnH+KHnHmolecules/cell6.5367 [6.2714, 6.8075]—
    σR(synN)(0)molecules2/cell1.1257 [0.999, 1.2599]—
    KHmolecules/cell2.46 × 105 [2.46×105, 2.46×105]—
    nHnumber3.8795 [3.7879, 3.9722]—
    Fixed parameters
    koff (high-affinity CAR)s−19.0 × 10−59.0 × 10−5
    kon (high-affinity CAR)nM−1 s−15.1136 × 10−65.1136 × 10−6
    KD (high-affinity CAR)nM17.617.6
    koff (low-affinity CAR)s−16.8 × 10−46.8 × 10−4
    kon (low-affinity CAR)nM−1 s−13.2381 × 10−63.2381 × 10−6
    KD (low-affinity CAR)nM210210

    • The 95% confidence intervals are shown in squared brackets.

Supplementary Materials

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A multiscale framework to explore CAR T cell design space
Harshana Rajakaruna, Milie Desai, Jayajit Das
Life Science Alliance Jul 2023, 6 (10) e202302171; DOI: 10.26508/lsa.202302171
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