## Abstract

We investigated relationships among chimeric TCR (cTCR) expression density, target Ag density, and cTCR triggering to predict lysis of target cells by cTCR^{+} CD8^{+} T human cells as a function of Ag density. Triggering of cTCR and canonical TCR by Ag could be quantified by the same mathematical equation, but cTCR represented a special case in which serial triggering was abrogated. The magnitude of target lysis could be predicted as a function of cTCR triggering, and the predicted minimum cTCR density required for maximal target lysis by CD20-specific cTCR was experimentally tested. cTCR density below ∼20,000 cTCR/cell impaired target lysis, but increasing cTCR expression above this density did not improve target lysis or Ag sensitivity. cTCR downmodulation to densities below this critical minimum by interaction with Ag-expressing targets limited the sequential lysis of targets in a manner that could be predicted based on the number of cTCRs remaining. In contrast, acute inhibition of lysis of primary, intended targets (e.g., leukemic B cells) due to the presence of an excess of secondary targets (e.g., normal B cells) was dependent on the Ag density of the secondary target but occurred at Ag densities insufficient to promote significant cTCR downmodulation, suggesting a role for functional exhaustion rather than insufficient cTCR density. This suggests increasing cTCR density above a critical threshold may enhance sequential lysis of intended targets in isolation, but will not overcome the functional exhaustion of cTCR^{+} T cells encountered in the presence of secondary targets with high Ag density.

Chimeric TCRs (cTCRs), fusion proteins between single-chain Abs specific for surface-expressed tumor associated Ags and components of the TCR signaling complex, have been used to redirect the specificity of T cells toward a number of cancer cell types (1, 2). Expression of these receptors on T cells promotes T cell effector functions, allowing T cells to promote eradication of tumors in murine immunotherapy models. The preclinical antitumor activity of cTCR^{+} T cells in mice has now been well investigated (1, 3–7), and several clinical studies have been performed in patients with cancer (8–11), but relatively little is known about the relationship between cTCR expression density and effector function.

Several studies have examined the relationship between the amount of cTCR expressed and T cell responses to antigenic targets. Analysis of Jurkat T cells expressing cTCR revealed that, for a fixed Ag concentration, an optimal cTCR density existed to achieve maximal IL-2 secretion, with lower cTCR densities inducing submaximal responses and higher densities promoting apoptosis (12). These experiments suggest that cTCR expression density may control both target recognition and cell survival decisions in vivo. In studies with primary human T cells, expression of limiting densities of cTCR rendered the T cells insensitive to targets expressing low Ag density, but permitted efficient lysis of high Ag density targets. These studies in primary cells suggested that a limiting density of cTCR is sufficient to provide a maximal signal if adequately engaged, but that the cTCR could not be fully engaged by targets expressing low densities of Ag (13). Additionally, we have recently shown that cTCR are triggered and downmodulated from the surface of primary T cells in an Ag dose-dependent fashion in response to Ag-expressing targets and that the number of engaged cTCR correlates with degree of T cell degranulation and target lysis (14).

Although these previous studies investigating the role of cTCR expression density and effector function suggested that insufficient cTCR densities may limit T cell function during an initial encounter of a T cell with an Ag-expressing target, these experiments have not examined the extent to which cTCR downmodulation following a previous Ag encounter might impair sequential lysis of targets. We have recently observed that CD20-specific cTCR^{+} T cells demonstrate profoundly impaired lytic activity against CD20^{+} leukemia cells in mice also expressing CD20 on normal B cells (15). This impaired lytic function in part reflected Ag-specific deletion of cTCR^{+} T cells in response to B cells, but may also have partially resulted from acute cTCR downmodulation in response to the overwhelming Ag burden in these animals. Increased cTCR expression density could potentially promote enhanced effector function if cTCR downmodulation limits Ag recognition and target lysis. Alternatively, if functional exhaustion of the lytic capacity of cTCR^{+} T cells occurs in response to excessive numbers of targets, increased cTCR density should not improve target lysis acutely. Thus, understanding the relationship between cTCR expression density and the extent to which cTCR downmodulation impairs sequential target lysis would provide insight into the value of pursuing strategies to increase cTCR expression to enhance T cell function in vivo.

In the current study, we investigated the relationships among cTCR expression density, target Ag density, and cTCR triggering to clarify the manner in which cTCR molecules recognize and respond to target Ag molecules. Using in vitro experimentation and mathematical modeling, we found that the number of cTCR triggered by Ag is constant with respect to cTCR expression density over a wide range of cTCR densities, presumably reflecting the high affinity of single-chain Abs for Ag. Further mathematical modeling demonstrated that triggering of cTCR and canonical TCR by Ag could be described by the same mathematical equation, with cTCR representing a special case in which serial triggering is abrogated. This relationship permitted prediction and testing of the minimum required density of CD20-specific cTCR for induction of maximum target lysis by cTCR^{+} T cells. Downmodulation of TCR below the critical number needed for maximum target lysis was found to impair sequential lysis of targets. However, we found that the intrinsic lytic capacity of the cTCR^{+} T cell, and not insufficient cTCR expression, limits lysis of intended targets (e.g., leukemic B cells) in the presence of an excess of secondary, Ag-expressing targets (e.g., normal B cells). Together, these results suggest that increasing the cTCR expression density may improve sequential lysis of targets by cTCR^{+} T cells that are able to persist and recover lytic capacity, but will not ameliorate the acute inhibition of lysis of primary targets in the presence of an excess of Ag-expressing secondary targets.

## Materials and Methods

### cTCR and Ag density quantitation

cTCR expression density was quantitated using FITC-labeled microbeads and FITC-labeled anti-mouse Fab-specific polyclonal goat Abs as previously described (14). CD20 and CD22 expression densities were quantitated with mouse anti-human mAbs (BD Pharmingen, San Diego, CA): PE anti-CD20 Quantibrite (clone L27), PE anti-CD22 (clone S-HCL-1), and PE-labeled microbeads as previously described (14). Briefly, flow cytometry was performed to determine the mean fluorescence intensity (MFI) of microbeads with a known number of FITC or PE molecules per bead. cTCR or target Ag density was estimated as the number of Ab binding sites per cell from the fluorescence intensity derived by anti-cTCR or anti-CD20/22 Ab staining and interpolation using a standard curve determined from the labeled beads.

### Cell lines and primary cells

The CD20^{+} CD22^{+} Daudi lymphoma and CD20^{−} CD22^{−} Jurkat T cell leukemia cell lines were obtained from the American Type Culture Collection (Manassas, VA). Lymphoblastoid cultured lymphocytes (LCLs) have been described previously (16). Jurkat T cell clones expressing varying Ag densities of CD20 and CD22 have been described previously (14). Human PBMCs were obtained from normal donors with consent. The cells were positively selected for CD8, stimulated with anti-CD3ε mAb, and transduced with retroviral vectors as described previously (14). The Leu16 cTCR is directed against the human CD20 Ag, and the RFB4 cTCR recognizes human CD22 (14, 17). T cells expressing varying levels of cTCRs were obtained by resorting broadly cTCR^{+} T cell populations (obtained by FACS sorting) using immunomagnetic selection with MS columns (Miltenyi Biotec, Auburn, CA), anti-PE microbeads (Miltenyi Biotec), and PE–anti-human-Fcγ Abs (Jackson ImmunoResearch Laboratories, West Grove, PA). Only the T cells expressing the highest level of cTCR were positively selected with this method, leaving a population of cells with lower cTCR expression density. These T cells were expanded in 50 U/ml human rIL-2 for 3–5 d to allow dilution/dissociation of the Abs, followed by use of the T cells in experiments. This approach allowed comparison of the higher and lower density cTCR^{+} populations.

### Chromium release assays

Standard 5-h chromium release assays were performed as described previously (18). For the kinetic analysis performed in Fig. 4*B*, Leu16 cTCR^{+} T cells were incubated at 37°C in a 96-well plate for varying amounts of time with a Jurkat clone expressing ∼ 350,000 CD20 molecules/cell (cold plate). The cTCR^{+} T cells and Jurkat clone were pelleted at 500 × *g* for 3 min after plating to facilitate interaction. Leu16 cTCR^{+} T cells were titrated from 60,000 cells/well down to various E:T ratios relative to the 2000 ^{51}Cr-labeled Daudi used as primary targets; the Jurkat-CD20 clone was plated at 60,000 cells/well in every well and was not titrated (30-fold excess relative to ^{51}Cr-labeled Daudi number). The Leu16 cTCR^{+} T cell/Jurkat-CD20 clone mixture was added after varying times to a plate containing ^{51}Cr-labeled Daudi in 100 μl RPMI 1640 followed by pelleting at 500 × *g* for 3 min. The CD8^{+} T cells were allowed to lyse targets in the ^{51}Cr-labeled hot plate for a total of 5 h per well, after which supernatants were harvested. For the *t* = 0 min condition, Leu16 cTCR^{+} T cells were added to premixed Jurkat-CD20 clone and ^{51}Cr-labeled Daudi plated in the hot plate. For the cold target inhibition assay performed in Fig. 5*A*, Leu16 or RFB4 cTCR^{+} or vector-transduced T cells were added at varying E:T ratios (relative to ^{51}Cr-labeled Daudi) to a plate containing 2,000 ^{51}Cr-labeled Daudi and 60,000 LCLs as cold targets (30-fold excess relative to ^{51}Cr-labeled Daudi number). Supernatants were obtained after 5 h for analysis. For the cold target inhibition assay performed in Fig. 5*B*, the same conditions were employed, but Jurkat clones expressing varying densities of CD20 or CD22 were used as cold targets at a 30-fold excess relative to ^{51}Cr-labeled Daudi. The percentage of target lysis was calculated according to the following formula: % lysis = (experimental [^{51}Cr] release − spontaneous [^{51}Cr] release)/(maximum [^{51}Cr] release − spontaneous [^{51}Cr] release) × 100.

### Flow cytometric target lysis assays (7-aminoactinomycin D assay)

Lysis dose-response curves were generated using 7-aminoactinomycin (7-AAD) target lysis assays as previously described (14). Briefly, cTCR^{+}^{+} CFSE^{+} target cells was quantitated by flow cytometry.

### cTCR downmodulation assays

cTCR downmodulation curves were generated as previously described (14). Briefly, cTCR^{+} T cells and Jurkat target cell clones expressing varying densities of CD20 and CD22 were stained with DDAO and CFSE as for the 7-AAD assay, respectively, and incubated at a 1:1 ratio for 4 h at 37°C. Subsequently, cTCR expression was determined on DDAO^{+} T cells using PE–anti-human-Fcγ Abs compared with T cells incubated with Ag-negative target cells according to the following equation: percent cTCR downmodulation = 100 × (1 − stimulated MFI/unstimulated MFI). The number of cTCRs downmodulated was calculated by multiplying the estimated cTCR expression density by the percentage of cTCRs downmodulated. For the cTCR downmodulation kinetics assay depicted in Fig. 4*A*, DDAO-stained Leu16 cTCR^{+} T cells were incubated at 1 × 10^{5} cells/well with 1 × 10^{5} CFSE-stained Daudi cells and pelleted at 500 × *g* for 3 min in a 96-well plate. The cells were incubated for varying times at 37°C, after which the contents of the well were removed, spun at 500 × *g* at 4°C for 1 min, and immediately resuspended in ice-cold PBS with 2.5 mM EDTA. Subsequently, T cells were stained for cTCR expression using PE–anti-human-Fcγ Abs and analyzed for decreased cTCR expression relative to unstimulated cTCR^{+} T cells. The percent cTCR downmodulation was calculated by the equation described above.

### Curve fitting and calculations

Dose-response curves were calculated according to the equations described below in *Derivation of equations* and *Results* or by variable slope sigmoidal dose-response curves using GraphPad Prism software (version 4.03, GraphPad, San Diego, CA). Closeness of fit of predicted values to the observed data was determined from *R*
^{2} values obtained by the sum of squares method such that *R*
^{2} = 1 − *SS _{Err}
*/

*SS*, with and , where

_{Tot}*y*is an experimentally derived value,

_{i}*f*is a calculated value, and is the average of the experimentally derived values. The value of ρ

_{i}*was obtained with nonlinear regression using an iterative Java-based web application utilizing the method of differential corrections: http://statpages.org/nonlin.html. Values of*

_{TCR,min}*N*and

_{1}*k*for the Viola et al. and Valitutti et al. data sets (19, 20) were determined from plotting (cTCR downmodulated/Ag density)/Ag density using the same method. Alternatively, values of

*N*and

_{1}*k*were determined using this nonlinear regression method to fit Equation 3 (see

*Derivation of equations*) to the observed data sets.

*R*

^{2}values generated by the software were verified by manual calculation. Values for

*K*were obtained in cold-target inhibition assays by fitting one-site binding hyperbolas to [

_{D}^{51}Cr]-release data using GraphPad Prism software (version 4.03, GraphPad), with fixed maximum lysis values based on the assumption of competitive inhibition:inhibition disappears as E:T → ∞. All calculations for equations and predictions were performed using Microsoft Excel 2007 (Microsoft, Redmond, WA). Pearson correlation tests were performed in Fig. 1

*E*and on the data depicted in Fig. 5

*C*using GraphPad Prism software (version 4.03, GraphPad).

### Derivation of equations

If the ratio between cTCR density and the number of target Ag molecules needed to downmodulate half of the cTCR expressed by a T cell (*EC*
_{50}) is constant with respect to cTCR density, then
(1)If cTCRs are not limiting relative to Ag molecules, as seen at low Ag density (ρ*
_{Ag}
*) relative to

*EC*

_{50}(ρ

*) [i.e., ρ*

_{TCR}*< <*

_{Ag}*EC*

_{50}], the following differential equation describes cTCR downmodulation based on a linear model: With solution: where

*T*is the number of cTCRs triggered and downmodulated, ρ

*is Ag density of the targets, and*

_{Ag}*N*

_{1}is the instantaneous cTCR downmodulation in response to one Ag molecule.

As target Ag density becomes larger relative to cTCR density (i.e., ρ*
_{Ag}
* ≥

*EC*

_{50}), the number of cTCR down-modulated per Ag drops due to decreasing availability of cTCR. To account for this, we added a second term to the differential equation to account for this effect based on the Verhulst equation for population growth (21): (i)where

*m*is a fraction representing the maximum number of cTCRs downmodulated at the plateau divided by the cTCR density ρ

_{T}*: (*

_{TCR}*m*=

_{T}*T*

_{max}/ρ

_{TCR}). Separating and integrating yields: (2)Solving for ρ

*when*

_{Ag}*T*= 1/2

*T*(i.e., at ρ

_{max}*=*

_{Ag}*EC*

_{50}), we obtained: (ii)and by substituting: (iii)Canonical TCRs have been shown to be serially triggered by Ag, such that the number of TCRs triggered per Ag molecule is not constant when ρ

*< < ρ*

_{Ag}*, but varies with Ag (19, 20, 22), suggesting that a new expression for the value N*

_{TCR}_{1}was needed that varied as a function of Ag density [i.e.,

*n*=

*f*(ρ

*)] to describe αβ TCR triggering. We used data published by Viola et al. and Valitutti et al. (19, 20), plotting the number of TCRs triggered/Ag molecules offered (N) by the number of Ag molecules offered as done previously (22). On a log-log plot, this relationship appeared linear, suggesting that a power law described the data (Fig. 2*

_{Ag}*C*). Thus, for canonical TCRs: (iv)where

*N*

_{1}is again the instantaneous number of TCR triggered when one molecule of Ag is offered, and

*k*is a serial triggering constant describing the attenuation of serial triggering with increasing Ag density. To obtain an expression for T as a function ρ

*, we multiplied Equation*

_{Ag}*iv*by ρ

_{Ag}_{,}as

*T*=

*N*ρ

*: differentiating with respect to ρ*

_{Ag}*: Incorporating the second term into the differential to account for limiting TCR availability at increasing values of ρ*

_{Ag}*relative to ρ*

_{Ag}*by analogy with Equation*

_{TCR}*i*: (v)integrating yields: (3)As this equation is based on decay of serial triggering by power-law modeling, this suggests that for small values of ρ

*<8 peptide-MHC (pMHC)/cell, TCR downmodulation is described by the power-law behavior described by: The data presented by Viola et al and Valitutti et al. (19, 20) do not extend into this region, however, precluding validation of Equation 3 in this regime. It is therefore possible that the number of cTCRs triggered per Ag molecule may not decrease as sharply with increasing ρ*

_{Ag}*as predicted by a power law for ρ*

_{Ag}*<8, but instead be independent of ρ*

_{Ag}*due to overwhelming abundance of TCRs relative to Ag molecules in this regime, such that doubling ρ*

_{Ag}*would double the number of TCRs downmodulated.*

_{Ag}### ΔT

Equation 2 predicts that cTCR will be downmodulated linearly for ρ*
_{Ag}
* < <

*EC*

_{50}. However, as ρ

*→*

_{Ag}*EC*

_{50}, the number of TCRs downmodulated decreases due to decreasing availability relative to Ag molecules. For a given ρ

*, T cells with greater ρ*

_{Ag}*will be farther away from*

_{TCR}*EC*

_{50}than T cells with lower ρ

*Equation*

_{TCR}*ii*and will therefore downmodulate a greater number of cTCRs according to the following relationship: for a given Ag density, ρ

*, where ρ*

_{Ag}*> ρ*

_{TCR,1}*.*

_{TCR,2}## Results

### Increasing cTCR density increases the number of Ag molecules needed to downmodulate the cTCR population, but does not increase Ag sensitivity of target lysis

We previously noted that increasing cTCR density increased the number of Ag molecules necessary to downmodulate the cTCR population, but the quantitative nature of this relationship remained unclear (14). We therefore compared cTCR downmodulation in response to Ag (as a measure of cTCR engagement) for 10 T cell lines expressing different numbers of cTCR/cell. Plotting cTCR expression density (ρ*
_{TCR}
*) against the number of target Ag molecules needed to downmodulate half of the cTCR population expressed per T cell (

*EC*

_{50}) for the six CD20 and four CD22-specific cTCR

^{+}T cell lines (Leu16 and RFB4, respectively) expressing different numbers of cTCR/cell demonstrated a linear relationship between these parameters (Fig. 1

*A*). This linear relationship suggested that the ratio ρ

*/*

_{TCR}*EC*

_{50}is constant such that: (1)where

*C*is a constant related to the number of cTCRs triggered per Ag molecule. Slopes of 1.67 ± 0.06 and 2.10 ± 0.11 cTCR/Ag were obtained from the Leu16 and RFB4 cTCR plots, respectively, suggesting that this linearity was common to cTCR specific for at least two distinct Ags (Fig. 1

*A*). If the relationship between ρ

*and*

_{TCR}*EC*

_{50}was indeed constant, multiplying the

*EC*

_{50}value obtained in a cTCR downmodulation assay by the constant

*C*should allow the density of cTCR to be predicted such that

*EC*

_{50}*

*C*= ρ

*. Using values of*

_{TCR}*C*= 1.67 ± 0.06 and 2.10 ± 0.11 for Leu16 and RFB4 cTCR, respectively, demonstrated good prediction of ρ

*as a function of*

_{TCR}*EC*

_{50}(

*R*

^{2}= 0.96–0.99; Fig. 1

*B*), supporting the hypothesis of linearity posited by Equation 1. Thus, varying cTCR densities by 15-fold did not alter the number of cTCRs engaged per target Ag, presumably reflecting the high-affinity interaction observed between cTCR and Ag molecules (14). Comparing the six Leu16

^{+}T cell lines with different cTCR densities (Fig. 1

*C*), we noted similar Ag sensitivities (lysis

*EC*

_{50}values) for target lysis (Fig. 1

*D*). Plotting lysis

*EC*

_{50}values versus cTCR density, we found no correlation between these parameters in the range evaluated (Fig. 1

*E*), suggesting that cTCR density was not limiting for the T cell lines (Pearson test;

*p*= 0.47). This result further suggested that the strength of interaction between cTCR and Ag did not vary appreciably in the range tested in these experiments.

### Mathematical modeling suggests that cTCRs demonstrate lower Ag sensitivity compared with canonical TCRs due to the absence of serial triggering

As the relationship between cTCR density and cTCR downmodulation *EC*
_{50} values appeared to be constant, suggesting that the interaction strength between cTCR and Ag was also constant, we hypothesized that the entire cTCR downmodulation curve could be predicted using this constant relationship. This might then permit prediction of T cell killing as a function of Ag density and determination of its impairment by cTCR downmodulation. As target Ag density becomes larger relative to cTCR density (i.e., ρ*
_{Ag}
* ≥

*EC*

_{50}), the number of cTCRs downmodulated per Ag drops, likely due to decreasing availability of cTCR. To account for this, we derived and solved a differential equation based on the Verhulst equation for population growth (21) (see

*Materials and Methods*): (2)where

*T*is the number of cTCR triggered and downmodulated, ρ

_{A}_{g}is Ag density of the targets,

*N*

_{1}is the instantaneous cTCR downmodulation in response to one Ag molecule, and

*m*is a fraction representing the maximum number of cTCR downmodulated at the plateau divided by the cTCR density ρ

_{T}*(*

_{TCR}*m*=

_{T}*T*/ρ

_{max}*). This equation plotted sigmoidal cTCR downmodulation dose-response curves (Fig. 2*

_{TCR}*A*) similar to those depicted in Fig. 1

*C*. To test if this equation accurately described cTCR downmodulation, we determined the value for

*N*

_{1}for the Leu16 cTCR using Equation

*iii*(see

*Materials and Methods*) with a value of

*C*of 1.67 ± 0.06 and

*m*= 0.9 ± 0.05 (obtained from the cell lines depicted in Fig. 1

_{T}*A*). This yielded a value of

*N*

_{1}= 1.04 ± 0.11: close to one cTCR triggered per Ag molecule. A similar value of

*N*

_{1}= 1.36 ± 0.15 was obtained for the RFB4 cTCR with parameters

*C*= 2.1 ± 0.11 and

*m*= 0.94 ± 0.1 (obtained from the cell lines depicted in Fig. 1

_{T}*A*). Using values of ρ

_{TCR}with the constants

*m*and

_{T}*N*

_{1}, entire cTCR downmodulation curves could be generated that closely fit experimentally derived data, supporting the hypothesis of constant cTCR/Ag interaction strength with respect to cTCR density (

*R*

^{2}= 0.86–0.98; Fig. 2

*B*). Similar close fit was found for RFB4 cTCR triggered by CD22 molecules, suggesting that this equation was not limited to the CD20 Ag (not depicted).

As the value of *N*
_{1} was found to be constant and close to 1, this suggested that cTCRs were not being serially triggered by Ag, in contrast to canonical TCRs, which are reported to undergo serial triggering (19, 20). We hypothesized that the expression we derived for cTCR triggering and downmodulation by Ag might represent a special case of the interactions between canonical TCR and pMHC complexes, in which serial triggering was abrogated, possibly due to increased dwell time between the cTCR and Ag molecule (23). To test this hypothesis, we endeavored to determine a relationship between triggering and downmodulation of canonical TCRs and pMHC complexes. Canonical TCRs have been shown to be serially triggered by Ag, such that the number of TCRs triggered per Ag molecule is not constant when ρ*
_{Ag}
* < < ρ

*, but varies with Ag (19, 20, 22), suggesting that a new expression for the value N*

_{TCR}_{1}was needed that varied as a function of Ag density [i.e.,

*n*=

*f*(ρ

*)]. We used data published by Viola et al. and Valitutti et al. (19, 20), plotting the number of TCR triggered/Ag molecule (N) versus the number of Ag molecules per target as done previously (22). On a log-log plot, this relationship appeared linear, suggesting that a power law described the data (Fig. 2*

_{Ag}*C*). Based on the hypothesis that the efficiency of serial triggering decreases as Ag density increases based on a power law, we derived the following equation similar to Equation 2 (see

*Materials and Methods*): (3)where

*N*

_{1}is again the instantaneous number of TCRs triggered when one molecule of Ag is offered, and

*k*is a serial triggering constant describing the attenuation of serial triggering with increasing Ag density. We determined initial values of

*N*

_{1}= 3790 ± 236 and

*k =*0.779 ± 0.024 from Fig. 2

*C*by nonlinear least squares regression (see

*Materials and Methods*). Equation 3 was plotted with the data obtained from the Viola et al and Valitutti et al. (19, 20), with ρ

*set to 30,000 TCR/cell and*

_{TCR}*m*set to 0.9 (both obtained from Ref. 20), which resulted in an

_{T}*R*

^{2}value of 0.769. Error minimization using nonlinear least squares regression demonstrated best fit of Equation 3 to the observed data obtained from Viola et al. (19) with

*N*

_{1}

*=*4630 ± 510 and

*k =*0.785 ± 0.017, yielding an

*R*

^{2}of 0.958 (Fig. 2

*D*). The values of

*N*

_{1}and

*k*obtained from Fig. 2

*C*were close to those generated by direct fitting of Equation 3 to the data, suggesting that modeling the decrease in serial triggering occurring with increasing Ag density using a power law closely models the observed data. Plotting

*N*versus ρ

*for the Valitutti et al. (20) data set yielded*

_{Ag}*N*

_{1}

*=*10879 ± 131 and

*k =*0.891 ± 0.003 (not depicted), whereas Equation 3 demonstrated best fit with

*N*

_{1}

*=*8322 ± 1229 and

*k =*0.727 ± 0.024 (

*R*

^{2}= 0.987; Fig. 2

*D*). This larger discrepancy in predicted values and those obtained by fitting Equation 3 may have resulted from the fewer number of data points in this set.

Comparing Equation 2 with 3, the sole difference was comprised by the exponent *−k* in Equation 3. This suggested that both cTCRs and canonical TCRs respond to similar biological constraints, but that cTCRs do not serially trigger, at *N*
_{1} = 1 and *k* = 0. As previously determined by Viola et al. (19), canonical TCRs triggered by anti-CD3 Abs are not serially triggered, but instead are triggered near a 1:1 ratio with the Ab density on the target cell. We therefore plotted the cTCR downmodulation Equation 2 with the values obtained by Viola et al. (19) when triggering canonical TCRs with anti-CD3 Abs with *N*
_{1} = 0.63 ± 0.09 (obtained from a linear plot of TCR triggered/Ab density), *m _{T}
* = 0.9, and ρ

_{TCR}set to 30,000 TCR/cell (Fig. 2

*D*). Equation 2 demonstrated good fit to the observed data (

*R*

^{2}= 0.832), suggesting that cTCRs behave similarly to a canonical TCR triggered by a high-affinity Ab. This may account for the relatively high target Ag densities required for cTCRs to induce target lysis (14). The Leu16 cTCR required ∼15,000 CD20 molecules/target cell to trigger 10,000 cTCR, whereas the canonical TCR in the Viola et al. study (19) required only ∼100 pMHC molecules (Fig. 2

*D*versus 2

*E*), suggesting that lack of serial engagement of cTCR could decrease the number of TCR triggered and therefore the signal strength generated in response to low Ag densities by two orders of magnitude.

### Target lysis dose-response curves can be predicted for cTCR^{+} T cells based on the number of cTCRs triggered as a function of Ag density

Previous studies with canonical TCRs and cTCRs have suggested that T cells “count” the number of triggered TCRs and induce effector responses in a graded fashion based on the number of TCRs engaged (14, 19). We reasoned that if T cells were indeed counting triggered cTCR, Equation 2 could be modified to predict target lysis curves and be used to predict the minimum number of cTCRs necessary for maximum target lysis induction (ρ*
_{TCR,min}
*). To gain high resolution of cTCR triggering below 25,000 cTCRs, we calculated the number of cTCRs downmodulated for four T cell lines with lines with varying cTCR densities using Equation 2 (Fig. 3

*A*). We noted that for a given value of ρ

*, T cells with higher Ag densities downmodulated a greater number of cTCR, by an amount ΔT (see*

_{Ag}*Materials and Methods*). This occurred as a consequence of Equation 2: T cells with lower cTCR densities downmodulate fewer cTCRs per Ag molecule as ρ

_{Ag}increases as a result of decreasing cTCRs available per Ag. Plotting the percentage of target lysis against the number of cTCRs downmodulated (Fig. 3

*B*), we noted that for a given percentage of target lysis, a greater number of cTCRs were downmodulated by T cells with higher cTCR densities (ΔT; see

*Materials and Methods*). As these four cell lines demonstrated highly similar percentages of target lysis for a given target Ag density (Figs. 1

*F*, 3

*C*), this suggests that not all of the downmodulated cTCRs were actively signaling in T cells with higher cTCR density compared with those with lower cTCR density. We reasoned that a minimum cTCR density existed, ρ

*, such that the cell possessed the exact number of cTCRs necessary to be triggered for maximal target lysis; above this density, additional cTCRs would be triggered, but no additional lysis would be observed, and below this density, maximum potential target lysis would not be achieved at any target Ag density. We therefore hypothesized that as a T cell encountered increasing Ag densities on target cells and triggered increasing numbers of cTCRs, lytic function would increase as the number of triggered cTCRs relative to the minimum number of triggered cTCRs necessary for maximum effector function: (4)where*

_{TCR,min}*L*is the percent lysis achieved for a given number of cTCRs triggered,

*T*, ρ

*is the hypothetical minimum number of cTCRs necessary to induce maximum target lysis, and*

_{TCR,min}*m*is the maximum percent target lysis achieved at the plateau of a target lysis curve by a T cell with ρ

_{L}*≥ ρ*

_{TCR}*. Based on Equation 4, a plot of target lysis versus cTCR downmodulation should be linear for a T cell with ρ*

_{TCR,min}*= ρ*

*, as each increase in cTCR triggering should yield a corresponding increase in target lysis. Indeed, as cTCR expression density decreased, this plot became more linear (Fig. 3*

_{TCR,min}*B*). Error minimization (see

*Materials and Methods*) yielded a value of ρ

*of 20,100 ± 994 cTCR/cell. Plotting Equation 4 with ρ*

_{TCR,min}*= 20,100 cTCR/cell,*

_{TCR,min}*N*

_{1}= 1, and

*m*of 79% (obtained from seven experiments with Leu16

_{L}^{+}T cells;

*m*= 79.3 ± 7.6% lysis) demonstrated close fit to the experimentally observed data (

_{L}*R*

^{2}= 0.90–0.99; Fig. 3

*C*). Thus, Equation 4 could predict target lysis dose-response curves as a function of cTCR triggering with Ag density, minimum cTCR density necessary for maximum lysis, and the maximum lytic potential of the T cell as its parameters, suggesting that T cells count the number of cTCRs triggered and scale effector responses accordingly.

### cTCR downmodulation correlates with impaired sequential lysis of targets by cTCR^{+} T cells

As we predicted that ∼20,000 Leu16 cTCRs were necessary for inducing maximum target lysis by a CD8^{+} T cell, we reasoned that T cells that downmodulated cTCR to densities below this level (i.e., ρ*
_{TCR}
* < ρ

*) would exhibit impaired sequential lysis of targets. To determine if cTCR downmodulation in response to Ag impaired the ability of cTCR*

_{TCR,min}^{+}T cells to perform sequential lysis of targets, we first determined the rate at which cTCR are downmodulated by a high Ag density target to better understand the kinetics of the process. We incubated T cells expressing the Leu16 cTCR with the cell line Daudi, which expresses ∼400,000 CD20 molecules/cell (14), for increasing times and measured the remaining cTCR expression (Fig. 4

*A*). The surface cTCR expression decreased with time and reached maximal downmodulation around 1 h as previously shown for canonical TCR (20, 24). Next, we performed an experiment in which Leu16 cTCR

^{+}T cells were incubated with a Jurkat clone expressing CD20 at ∼350,000 molecules/cell for varying times to downmodulate cTCR at varying E:T ratios, before being added to a second plate containing

^{51}Cr-labeled Daudi lymphoma cells. The T cells were allowed to lyse

^{51}Cr-labeled Daudi for a total of 5 h, after which supernatants were removed for analysis. At the highest E:T ratio, one cTCR

^{+}T cell was present for each Jurkat-CD20 clone. Thus, at this E:T ratio, the ability of the T cells to kill a subsequently introduced Daudi target was analyzed. The lytic capacity of the T cells gradually decreased over time after an instantaneous decrease in apparent lytic activity resulting from the decrease in effective E:T ratio (Fig. 4

*B*). We plotted the lysis achieved at a 30:1 E:T ratio (normalized to maximum at

*t*= 0 min) against the number of cTCRs remaining on the surface of the Leu16 cTCR

^{+}T cells at the time they were incubated with the

^{51}Cr-labeled Daudi targets (Fig. 4

*C*). The lysis values achieved by the T cells decreased as the cTCR density dropped below 2 × 10

^{4}cTCR cells and overlapped with a dose-response curve generated by Leu16 cTCR

^{+}T cells in response to Jurkat target clones expressing varying densities of CD20. This loss of lytic activity could be predicted by plotting normalized lysis obtained by Equation 4 against the number of TCRs triggered, predicted by Equation 2 for a T cell with ρ

_{TCR}equivalent to that used in the assay (

*R*

^{2}= 0.76; Fig. 4

*C*). As T cells have been shown to recycle their lytic capacity within a 5-h period (25), these results suggest that cTCR downmodulation below ρ

*was responsible for observed decrease in lytic capacity in this assay.*

_{TCR,min}### Inhibition of lysis of primary targets by secondary targets results from exhaustion of intrinsic lytic capacity of T cells and not from insufficient cTCR density

We have recently shown that CD20-specific TCR^{+} T cells demonstrate impaired lytic activity against CD20^{+} leukemia (primary targets) if transferred into mice expressing CD20 on normal B cells (secondary targets) (15). This acute decrease in lytic activity could result from cTCR downmodulation, as described above. However, in a setting in which an overwhelming number of nonneoplastic, Ag-expressing cells is present in addition to cancer cells, exhaustion of T cell effector function may occur independently of cTCR downmodulation. To model this second situation, we first tested lysis of ^{51}Cr-labeled Daudi primary targets by CD20-specific Leu16 cTCR^{+} and CD22-specific RFB4 cTCR^{+} T cells (14) in the presence of a 30-fold excess of LCLs (EBV-transformed B cells expressing CD20 and CD22). Lysis of the Daudi cell line by the CD20-specific T cell line was inhibited in a competitive fashion: efficient inhibition of target lysis was observed at low E:T ratios (e.g., ∼40-fold decrease at E:T = 3.3), but the inhibition disappeared as the E:T ratio reached larger values (Fig. 5*A*). In contrast, the CD22-specific T cell line was minimally affected by coincubation with LCL (e.g., ∼1.5-fold decrease in target lysis at E:T = 3.3), as LCLs express insufficient CD22 molecules to induce efficient lysis (Fig. 5*A* and not depicted). This suggested that the magnitude of inhibition of T cell lytic function by the presence of Ag-expressing secondary targets might depend on the Ag density of these targets.

We endeavored to develop an equation to quantify the magnitude of inhibition of T cell lytic effector function by secondary targets, as this would facilitate investigations to determine the dependence of this phenomenon on the target Ag density of the secondary targets. When T cells must interact with both intended and unintended targets, competition between these targets decreases the lysis of the intended targets (Fig. 5*A.1*). Consequently, the E:T ratio necessary to induce a given percentage of target lysis is increased in this setting, providing the appearance of either decreased interaction between the T cells and primary targets or of decreased lytic function per T cell. Both of these processes are likely occurring simultaneously and cannot be differentiated by a cold-target inhibition assay. However, to quantitatively compare the relative magnitude of inhibition of T cell effector function among secondary targets, all that is needed is a measure of how *competitively* these targets are able to inhibit lysis relative to one another. We therefore modeled target lysis curves with an equation similar to the previously described E:T conjugation curves, based on the strength of interaction between T cells and targets and on the observation that cold target inhibition is competitive in nature (Fig. 5*A.1*) (26):
(5)where *L*
_{max} is the percent of target lysis as E:T→ ∞, *R* is the E:T ratio, and *K _{D}
* is a dissociation constant that increases as the strength of interaction between the T cell and target cell decreases. Therefore, assessing the decrease in apparent strength of interaction between T cells and primary targets in the presence of secondary targets (as determined by increasing apparent

*K*values) should provide a quantitative measurement of the ability of a given cell to competitively inhibit the lytic effector function of a T cell. We introduced a parameter termed the “magnitude of inhibition”, Φ =

_{D}*K′*; where

_{D}/K_{D}*K′*is an apparent value obtained in the presence of both primary and secondary targets, whereas

_{D}*K*is the true value obtained in the presence of only one target. Thus, in Fig. 5

_{D}*A.1*,

*K*= 0.75 ± 0.23 for Leu16

_{D}^{+}T cells lysing

^{51}Cr-labeled Daudi in isolation. In contrast,

*K′*= 12 ± 1.9 for Leu16

_{D}^{+}T cells simultaneously lysing both

^{51}Cr-labeled Daudi and unlabeled LCL, demonstrating an apparent decrease in the strength of interactions between the T cells and Daudi cells. These values yielded Φ = 16 ± 5.6, suggesting that the presence of an excess of Ag-expressing secondary targets reduced the apparent interaction between T cells with competent effector function and primary targets by 16-fold.

Having obtained a quantitative measurement of the magnitude of inhibition of lysis by secondary targets, we sought to determine the influence of the Ag density of secondary targets on this parameter. We incubated CD20- or CD22-specific T cells with ^{51}Cr-labeled Daudi in the presence of excess of Jurkat target clones expressing varying densities of CD20 or CD22, respectively, and analyzed primary target lysis after 5 h. If the magnitude of inhibition of lysis of primary targets was dependent on the Ag density of the secondary target, the percentage of primary target lysis should decrease with increasing Ag density on secondary targets. Indeed, primary target lysis by the RFB4 CD22-specific and Leu16 CD20-specific T cells was inhibited in an Ag dose-dependent fashion (Fig. 5*B* and not depicted). We then plotted the magnitude of inhibition of target lysis produced by a target cell clone, Φ, against the Ag density of that clone to compare the Ag responsiveness of this parameter with that of target lysis. Φ demonstrated a sigmoidal curve that overlapped with target lysis curves as a function of Ag density (Fig. 5*C*). The target Ag density required to inhibit lysis by the CD22-specific RFB4 cTCR was ∼4 times greater than that required to inhibit lysis by the CD20-specific Leu16 cTCR. This is consistent with previous findings that the RFB4 cTCR delivers a weaker signal than the Leu16 cTCR due to targeting an epitope distant from the target cell membrane and therefore requires greater target Ag density to induce lysis (Fig. 5*C*) (14). These results suggested that susceptibility of a T cell to inhibition of lytic function by secondary targets depends on both the Ag sensitivity of the cTCR and on the Ag density of the secondary target.

The magnitude of inhibition of T cell lytic capacity achieved by a target cell clone was highly correlated with the efficiency of target lysis induced by that clone [Pearson correlation coefficients: *r* = 0.90 (*p* = 0.001) and *r* = 0.94 (*p* = 0.005) for Leu16 and RFB4, respectively]. This suggested that higher Ag density targets might inhibit lysis to a greater extent by exhausting the lytic capacity of the cells. However, increased cTCR downmodulation in the presence of higher Ag densities might have accounted for the observed diminished lytic function, which could not be excluded from these results. We therefore analyzed the relationship between cTCR downmodulation and inhibition of T cell lytic effector function by secondary targets to assess which of these parameters might exert a greater influence on acute loss of antitumor T cell activity observed in the setting of abundant self-Ag (15). Leu16 cTCR downmodulation in response to Jurkat-CD20 target cell clones expressing varying CD20 densities was measured in concert with analysis of the magnitude of inhibition of lysis of ^{51}Cr-labeled Daudi cells mediated by these same target clones. Both parameters were plotted as a function of the Ag density of the target clones (Fig. 6). Inhibition of T cell lytic function by secondary targets, Φ, reached the half maximal value at Ag densities that induced only minimal cTCR downmodulation, suggesting that exhaustion of lytic function of T cells in the presence of a large number of secondary targets, and not cTCR downmodulation, was responsible for the inhibition of lysis of primary targets in this setting.

## Discussion

Although a substantial body of literature exists investigating Ag recognition by canonical αβ TCR (27–31), few studies have addressed Ag recognition by cTCRs (12–14). In this study, we performed a quantitative analysis of the relationships among cTCR expression density, target molecule expression density, cTCR triggering, and target lysis. This quantitative analysis allowed mathematical modeling of these processes, leading to iterative rounds of experimentation and modeling. By deriving differential equations based on empirical observations, we found that triggering of cTCR and canonical TCR could be described by the same mathematical expression, with cTCR representing a special case in which serial triggering was absent. Previous modeling and experimentation has suggested that canonical TCRs possessing very high affinity for antigenic pMHC and demonstrating low TCR:pMHC dissociation rates exhibit decreased serial triggering and weaker total signaling per pMHC molecule (23, 32). However, at high Ag densities, TCRs possessing low dissociation rates for antigenic pMHC have been shown to exhibit no decrease in maximal effector function, as serial triggering was not necessary for signal amplification at these densities (33). Similarly, CD4 T cell clones required ∼100-fold greater number of high-affinity anti-CD3–stimulating ligands compared with lower-affinity pMHC to trigger the ∼15,000 TCRs necessary for maximal cytokine production, demonstrating that high-affinity interactions do not necessarily limit maximal effector function, but increase the number of target ligands required to achieve it (19). These findings suggested that loss of serial triggering resulting from TCR-ligand interactions exhibiting low dissociation rates could limit the number of TCRs triggered at low Ag density by two orders of magnitude. Equation 3 was derived based on empirical observation of canonical αβ TCR triggering by pMHC, but was found to also describe αβ TCR triggering by high-affinity anti-CD3 mAb and cTCR triggering by Ag if *N*
_{1} was set to a value near one and the serial triggering constant *k* was set to zero, as seen with cTCR. Together, these results suggest that cTCRs are triggered by Ags similar to αβ TCR triggered by anti-CD3 mAb and explain the relatively high number of Ag molecules required to stimulate target lysis. However, evaluating the number of cTCRs required to reach maximal target lysis (Fig. 3) and cytokine production (not depicted) and comparing that with published values for αβ TCR requirements (19) illustrates that T cells require a similar number of canonical or cTCRs to be triggered for maximal function, namely ∼20,000 molecules per cell, suggesting that cTCRs do not necessarily deliver a weaker signal than an αβ TCRs.

Analyzing cTCR downmodulation in the context of target lysis by T cells, we found that sequential lysis of targets was impaired once cTCR expression density fell below the minimum value required to maximally induce target lysis. The degree of this impairment could be predicted using equations based on a constant value for the number of cTCRs triggered/Ag molecules expressed on the target cell, suggesting that this relationship remained constant for ρ_{TCR} < ρ_{TCR,min}. However, a previous study demonstrated that at low cTCR densities, T cells could lyse only high Ag density targets, demonstrating immunoselectivity (13). This suggests that for ρ_{TCR} < ρ_{TCR,min}, cTCR^{+} T cells may exhibit an apparent decrease in the avidity of the cTCR for Ag molecules, resulting in decreased cTCRs triggered for a given Ag concentration. As low densities of cTCRs are rapidly downmodulated by targets expressing low Ag densities, thereby impairing sequential target lysis, achieving immunoselectivity between normal and transformed tissues by limiting cTCR expression densities is likely not a viable strategy for tumor immunotherapy.

cTCR targeting epitopes of CD22 located far from the target cell membrane deliver weaker signals per triggered cTCR and consequently require greater Ag densities to induce target lysis (14). As a result of this lower Ag sensitivity and the lower expression of CD22 on normal versus malignant B cell lymphoma cell lines, T cells expressing the CD22-specific RFB4 cTCR mediated selective lysis of lymphoma cell lines while sparing normal B cells. In the current study, we noted that the capacity of secondary targets to inhibit the lysis of primary targets was dependent on the Ag density of the secondary target and on the Ag sensitivity of the cTCR (Fig. 5*C*). The CD20-specific Leu16 cTCR induced maximal lysis at lower Ag densities, but also was maximally inhibited by secondary targets at lower Ag densities than was the CD22-specific RFB4 cTCR. This suggests that T cells expressing relatively low sensitivity TCR might demonstrate an advantage by retaining lytic function following encounter with tissues expressing low densities of tumor-associated Ags. However, the RFB4 cTCR delivers a weaker signal than the Leu16 cTCR, requiring a greater number of cTCRs to induce maximal lysis (ρ_{TCR,min}).This suggests that a far greater expression density of RFB4 cTCR would be required to prevent cTCR downmodulation to below ρ_{TCR,min} following encounter with an Ag^{+} tumor cell, possibly impairing sequential target lysis. This constraint may also limit sequential lysis induced by low-affinity canonical TCRs targeting partially agonistic altered peptide ligands, which have been shown to induce weaker signals than full agonists (34, 35). If high cTCR expression density could be maintained, cTCR^{+} T cells with a diminished sensitivity to target Ags might exhibit a survival and antileukemic advantage compared with high-sensitivity cTCR^{+} T cells in mice bearing a large burden of secondary target cells expressing low target Ag density relative to the primary target cells.

Our recent in vivo studies treating leukemia in mice that express CD20 on both normal B cells and leukemic cells demonstrate rapid loss of CD20-specific T cell effector function in vivo (15). cTCR^{+} T cells underwent exponential decay following infusion, reaching nearly undetectable numbers after 5 d, suggesting that activation-induced cell death limited antileukemic activity of the T cells. However, prior to deletion, loss of effector function in this setting resulting from cTCR downmodulation or exhaustion of T cell lytic capacity in the presence of large number of targets could limit antileukemic activity. Although cTCR downmodulation by targets expressing high Ag density impaired sequential lysis of targets (Fig. 4*C*), inhibition of lysis of primary targets in the presence of an excess of secondary targets reached half-maximal values at target Ag densities insufficient to produce significant cTCR downmodulation (Fig. 6). This suggests that exhaustion of lytic effector function in the presence of an excess of secondary targets acutely limits T cell function independently of cTCR downmodulation. This might result from the ability of CD8^{+} T cells to induce lysis of multiple targets simultaneously by directing lytic granules to spatially distinct T cell-target contact points (36), leading to maximal degranulation at high target densities (14). Therefore, at least three factors limit the activity of cTCR^{+} T cells when transferred into an Ag burdened environment: cTCR downmodulation limiting sequential lysis, exhaustion of T cell effector function, and, eventually, T cell deletion via activation-induced cell death. Interventions to improve immunotherapy using cTCR^{+} T cells should be aimed at alleviating these constraints.

Finally, we have previously found that normal human B cells expressing 2 × 10^{5} CD20 molecules are sufficient to maximally downmodulate Leu16 cTCR on T cells expressing ~10^{5} cTCR/cell (14). Equation 2 predicts that T cells expressing 0.5–1 × 10^{6} cTCR/cell would downmodulate only a small fraction of their cTCRs in response to an encounter with an individual B cell, allowing further cTCRs to be triggered in response to subsequent targets. This might enhance sequential lysis of targets. However, T cells overexpressing cTCR might undergo prolonged signaling in response to multiple targets. Sustained TCR signaling has been associated with activation-induced cell death (37), suggesting that TCR downmodulation may be necessary to dampen ongoing TCR signaling in the presence of large Ag burdens. Overexpression of cTCR might therefore hasten Ag-induced T cell death (12), limiting sequential lysis of targets. Additionally, increasing the number of cTCRs available to be triggered to maintain signaling in response to multiple targets would only improve sequential lysis of targets if the T cells remain viable and recycle lytic granules, after becoming acutely functionally exhausted in response to a large Ag burden. It is therefore likely that a secondary intervention will be necessary to improve T cell survival in the setting of chronic Ag-induced signaling in addition to increasing cTCR expression density, such as provision of exogenous cytokines (8, 38) or inclusion of costimulatory domains in cTCR (39–41).

## Acknowledgments

We thank Brian N. Lundstrom and Kai J. Miller for assistance with preparation of the manuscript.

**Disclosures** M.C.J. received licensing agreement revenues from his COH CD20-CAR construct patent and has ownership interests in a start-up company covering the CD20-CAR.

## Footnotes

This research was supported by National Institutes of Health Grants R21 CA 117131 (to O.W.P.), R01 CA18029 (to P.D.G.), and R01 CA33084 (to P.D.G.), grants from the Lymphoma Research Foundation (to O.W.P.) and the Leukemia & Lymphoma Society (LLS 7040 to O.W.P. and P.D.G.), and gifts from David and Patricia Giuliani, Mary and Geary Britton-Simmons, the Hext Family Foundation, and James and Sherry Raisbeck (to O.W.P.). S.E.J. is supported by a stipend from the National Institutes of Health Medical Scientist Training Program, a Poncin Award, a Cancer Research Institute training grant, and an Achievement Rewards for College Scientists grant.

Abbreviations used in this paper:

- 7-AAD
- 7-aminoactinomycin D
- cTCR
- chimeric TCR
- LCL
- lymphoblastoid cultured lymphocyte
- MFI
- mean fluorescence intensity
- pMHC
- peptide-MHC.

- Received November 19, 2009.
- Accepted February 15, 2010.

- Copyright © 2010 by The American Association of Immunologists, Inc.