A mathematical model of natural killer cell activity


  • Anna Scherbakova,

    1. Department of Hematology and Oncology, University of Tartu, Tartu 51014, Estonia
    2. Department of Immunotherapy, Competence Centre for Cancer Research, Tallinn 12618, Estonia
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  • Helen Lust,

    1. Department of Hematology and Oncology, University of Tartu, Tartu 51014, Estonia
    2. Department of Immunotherapy, Competence Centre for Cancer Research, Tallinn 12618, Estonia
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  • Hele Everaus,

    1. Department of Hematology and Oncology, University of Tartu, Tartu 51014, Estonia
    2. Department of Immunotherapy, Competence Centre for Cancer Research, Tallinn 12618, Estonia
    3. Hematology and Bone Marrow Transplantation Unit, Department of Hematology and Oncology, Tartu University Hospital, Tartu 50417, Estonia
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  • Alar Aints

    Corresponding author
    1. Department of Hematology and Oncology, University of Tartu, Tartu 51014, Estonia
    2. Department of Immunotherapy, Competence Centre for Cancer Research, Tallinn 12618, Estonia
    3. Cell Therapy Laboratory, Department of Hematology and Oncology, Tartu University Hospital, Tartu 50417, Estonia
    • Cell Therapy Laboratory, Tartu University Hospital, Raja 31, 50417 Tartu, Estonia
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Natural killer (NK) cells are capable of lysing their target cells with the help of perforin. The application of these cells for immunotherapy requires the estimation of their potency for the purpose of validation and batch-to-batch comparison. Cytotoxicity measurements have been carried out at only a few effector target ratios, therefore, allowing only semiquantitative assessment at best. By using a novel approach of varying the effector target ratio continuously and careful analysis of the experimental data after the reactions, we have achieved a precision necessary for constructing a mathematical model of cytotoxic reaction. Curve-fitting to experimental data indicates that NK cell cytotoxicity follows the law of mass action and fits the model of a single ligand–receptor interaction. The method allows to use the value of half-maximal lysis to describe the potency of cytotoxic NK cells numerically. © 2013 International Society for Advancement of Cytometry

Natural killer cells (NK cells) are large granular lymphocytes that derive from bone marrow and comprise 5–15% of all peripheral blood mononuclear cells (PBMC). NK cells are defined as CD3−CD19−CD56+ and NKp46+ cells.

NK cell-mediated cytotoxicity occurs primarily through the perforin/granzyme-dependent pathway (1), although NK cells can also kill target cells by death receptor ligands, such as Fas ligand and tumor necrosis factor related apoptosis inducing ligand (2). Their effector functions are regulated by homeostatic balance between activating and inhibitory signals through a wide repertoire of receptors on their surface.

The ability of NK cells to kill tumor cells has provided the rational basis for their exploitation in novel immunotherapy approaches, primarily in the treatment of acute myeloid leukemias (AML). Studies have shown that reconstituting NK cells may be important in infection, graft-vs.-host disease and graft-vs.-leukemia effects (3). Pioneering results of Ruggeri et al. (4) have reported that allogeneic NK cells can mediate antileukemic effects against AML after haploidentical hematopoietic stem-cell transplantation, with Killer Immunoglobulin-like Receptor (KIR) ligand incompatibility in the graft-vs.-host direction. Inhibitory receptors, such as long-tailed KIR and CD94/NKG2A, have been recognized as important determinants of NK cell activity in hematopoietic stem-cell transplantation (5). Different combinations of activating and inhibitory receptors on NK cells can reduce graft-vs.-host disease, promote engraftment, and provide graft-vs.-tumor responses. The use of KIR ligand incompatibility produces significant graft-vs.-leukemia effect in patients with AML at high risk of relapse (6).

Since 1968, quantitation of cell-mediated cytotoxicity is performed by radioactive chromium 51chromium-release assay (CRA). In this assay, target cells are incubated with the Na2 51CrO4 solution, which passively diffuses into the cells and interacts with intracellular proteins. 51Cr-labelled target cells are coincubated with effector cells for 4–16 h, then the amount of 51Cr released from the lysed cells into supernatant is measured by beta or gamma counter. The amount of radioactivity released from the target cells is directly proportional to the number of target cells killed by NK cells.

Nevertheless, CRA has some disadvantages:

  • 1Health risks associated with radioactivity and financial expenditure, because 51Cr has short half-life and should be regularly replaced.
  • 2Can be used only for labeling of those target cells that spontaneously bind the amount of 51Cr sufficient for detection.
  • 3Sensitivity of CRA depends on background—results cannot be interpreted, if spontaneous 51Cr release is too high.
  • 4Variability of CRA results is relatively high.
  • 5Results are uniparametric and cell death is not quantified at a single-cell level.

Considering the disadvantages of radioactive methods, flow cytometry based methods were developed as an alternative to CRA. We have used a carboxyfluorescin succinimidyl ester (CFSE) and 7-amino actinomycin D (7-AAD) based method similar to the one described by Lecoeur et al (7), except that we used CFSE to label target cells and 7-AAD as a second label for permeabilized target cells to achieve unambiguous identification of target cell populations.

Freshly isolated and activated NK cells have been used for immunotherapy of renal cell carcinoma and AML with significant success, which was dependent on KIR ligand mismatch and in vivo expansion of NK cells (8). The discovery of NK cell in vitro expansion in CellGro stem cell growth medium (9) has opened the possibility to contemplate NK cell immunotherapy using activated in vitro expanded NK cells in quantities unobtainable from a single donor. The beneficial effect of cell quantity has been demonstrated by the recent success of tumor-specific T-cell immunotherapy (10).

A prerequisite to using any therapeutic preparation is demonstration of potency in quantifiable terms.

We have refined cytometry-based cytotoxicity assay using double labeling for target cell identification and established a mathematical model of NK cell activity.

Materials and Methods

Cell Lines and Culture

PMBC were obtained from peripheral blood of healthy donors from Tartu University Hematology and Oncology Hospital Hematology and Bone Marrow Transplantation department. Donors were previously informed according to Tartu University Human Studies Committee of Ethics (151/101; 21.08.2006). Donor PBMCs were isolated by density-gradient centrifugation using Ficoll-Paque PLUS lymphopreparation kit (Amersham Biosciences).

Cytotoxic activity of NK cells was tested against six cell lines: (1) K562 (HLA-C negative)—human erythroleukemia cell line (American Type Culture Collection); (2) Namalwa (HLA-C*07)—human lymphoblastoid cell line derived from Burkitt lymphoma (American Type Culture Collection); (3) four human B-lymphoblastoid cell lines: HLA negative LCL 721.221-wt and three LCL transfected with different HLA-C group alleles [LCL 721.221-Cw*0401 (IHW03096), 721.221-Cw*1202 (IHW03093), and 721.221-Cw*1403 (IHW03094)]. HLA Cw*0401 allele bears group 2 epitope and HLA-Cw*1202 and HLA-Cw*1403 alleles have group 1 epitopes. These cell lines were kindly provided by Dr. Marcel Tilanus of Utrecht University.

PBMC and all tumor cell lines were incubated at 37°C in 5% CO2 atmosphere.

PBMC were cultured in six well plates (Falcon BD, Le Point De Claix, France) at 1 × 106 cell/ml. Culture medium contained CellGro SCGM (CellGenix GmbH, Freiburg, Germany) with 5% human serum (Tartu University Hospital), IL-2 (Proleukin, Chiron), 1000 U/ml, and OKT3 (mouse monoclonal antihuman anti CD3 antibody) (Ortho Biotech, Raritan, NJ), 10 ng/ml for three weeks according to Carlens et al (9). Unlike Carlens, antibody OKT3 was added during the whole culturing period. Antibiotics were not added.

All tumor cell lines were cultured in 50 ml tissue culture flasks (BD labware Europe, Maylan Cedex, France) in 5–6 ml RPMI 1640 medium (Invitrogen Gibco, Grand island, NY) containing 10% of fetal calf serum (Invitrogen Gibco, Grand Island, NY) at density of 250,000–500,000 cell/ml.

PBMC Immunophenotyping

Immunophenotype of PBMC was determined by antibody staining with NKp46-APC, CD56-PE, and CD3-PerCP-Cy5.5 [all antibodies purchased from BD Biosciences (Harlingen, San Jose, CA)]. NK cells were defined as NKp46+CD3− and CD56+CD3−. Cells were stained according to protocol provided by manufacturer. After staining, cells were washed twice with phosphate buffered saline (PBS), resuspended in 50–100 μl PBS and kept on ice before analyzing.

Preparation of Cytotoxicity Reactions

PBMC effector cells were collected, centrifuged, and resuspended in RPMI 1640 medium to cell density of 1 × 106 cells/ml and held on ice before use.

Target tumor cells were collected, centrifuged, resuspended in 1 ml RPMI 1640 medium, and stained with the CellTrace™ CFSE Cell Proliferation Kit (Molecular Probes, Leiden, Holland). CFSE solution was added to final concentration of 1 μM. Cells were incubated at 37°C for 15 min, centrifuged, and washed twice with RPMI 1640 medium. Before use, cells were resuspended in RPMI 1640 medium to cell density of 1 × 106 cells/ml and placed on ice.

Effector cells were mixed with constant number of CFSE labeled target cells (10,000) at different effector/target (E:T) ratios (varied from 12:1 to 0.5:1). Cells were seeded in 96-well microtiter plate with V-shaped bottom (Deltalab S.L., Barcelona, Spain) in RPMI 1640 of total volume of 100–120 μl for 4 h in a 5% CO2 atmosphere. Before incubations, cells were suspended and centrifuged at 500g for 2 min. After the end of incubation, 7-AAD (Molecular Probes, Leiden, Holland) solution was added to all cytotox reactions to final concentration of 3 μg/ml, mixed and left for 2 min to stain. Then, cells were centrifuged at 500g for 5 min, resuspended in 100–150 μl PBS and placed on ice.

In addition to cytotox reactions following control samples were prepared: unstained effector cells; target cells stained only with CFSE; target cells stained both with CFSE and 7-AAD; CFSE-stained target cells, permeabilized with 0.01% solution of Triton X-100™ (Sigma) and stained with 7-AAD.

Flow Cytometry Analysis

Samples were analyzed on Becton Dickinson LSRII flow cytometer using BD FACSDiva™ software biexponential display. Biexponential display provides good visualization of all cell subsets, including double negative cells (effector cells) that are usually “piled up” on the axis with logarithmic display. Better visualization of populations enables more exact compensation between 7-AAD and CFSE. In properly compensated sample, the location of the median 7AAD fluorescence in CFSE-population of effector cells matches the location of the median 7-AAD fluorescence in the CFSE+ target population.

First, live cell gate R1 (cells) was defined on side scatter (SSC) vs. forward scatter (FSC) plot to distinguish cells from cell fragments and debris (Fig. 1A). For defining the target cells gate (R2) on 7-AAD vs. CFSE plot, control samples of unstained effector cells and target cells stained both with 7-AAD and CFSE were analyzed (Fig. 1B). To define the permeabilized target cell gate (R3), TritonX-100-lysed target cells' control samples were used (Fig. 1C). Gates R2 and R3 were drawn rather exactly around populations to exclude cell fragments from the analyzed area. While analyzing 7-AAD-stained target cells, it was taken into consideration that the population of permeabilized cells may consist of two subpopulations on 7-AAD axis. Then, effector cells population (R4) was defined on 7-AAD vs. CFSE plot (Fig. 1D). 7-AAD positive dead effector cells were left out from R4 gate. To assure the exact results of analysis, the data of 10,000–20,000 cells from each sample were recorded.

Figure 1.

Defining cell populations on flow cytometer. A: R1 defines on SSC vs. FSC dot plot the cell population being analyzed. Dead cells (high SSC and low FSC values), but not cell debris in the lower left corner are included. B: R2 defines target cell population on 7-AAD vs. CFSE dot plot. C: R3 defines permeabilized target cell population on 7-AAD vs. CFSE dot plot. D: R4 defines effector cell population on 7-AAD vs. CFSE dot plot. Target cells permeabilized spontaneously (B), by Triton-X100 (C), and by NK cells (D) are comparable. The cells in C lie slightly to the left, compared to others, however, they all fall within the same gate. A representative set of more than 50 is shown. E: NK cells mixed with target cells. R3 shows experimental lysis (Lexp). F: Target cells without NK cells. R3 shows spontaneous lysis (Lspont). G: Target cells lysed with Triton-X100. R3 shows maximal lysis (Lmax). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

The hierarchy of the regions was defined as shown in Figures 1E–1G.

Data Analysis

Efficiency of NK cell directed cytolysis of target cells was calculated by comparing the lysis of target cell, coincubated with NK cells to the lysis of control sample target cells, incubated without effectors cells at the same conditions.

Spontaneous lysis of target cells without the presence of effector cells (Lspont) was the percentage of permeabilized cells in R3 gate in control sample stained both with CFSE and 7AAD (Fig. 1E, 4.1%). Maximum lysis of target cells (Lmax) was the percentage of permeabilized cells in R3 gate in control sample of CFSE-stained cells lysed with Triton X-100™ (Fig. 1F, 98%). NK-mediated target cell lysis (Lexp) was the percentage of permeabilized target cells in gate R3 in experimental samples (Fig. 1G, 74.7%).

On the basis of these data specific cytotoxicity (C) and actual E/T ratio (N) were calculated. C-value indicates percentage of target cells, lysed by NK cells and is calculated as follows:

equation image

N is the ratio of NK cells to target cells. Percentage of effector cells in cytotoxocity reaction was taken from gate R4 and percentage of target cells—from gate R2 (Fig. 1G). Please note that E:T ratio 2:1 is the intended ratio; the actual ratio is 1.573.Percentage of NK cells among effector cells (NK%) was obtained from immunophenotyping analysis. Thus, N = [(E:T ratio) × NK%]/100.

equation image

Data analysis were performed with GraphPad Prism4.

Results and Discussion

NK Cell Cytotoxic Function Follows Michaelis–Menten Law of Mass Action

We tested the sensitivity of three different NK target cell lines. 721.221 and K562 cell lines are considered sensitive to NK cell lysis, whereas Namalwa is considered more resistant. Relationship between specific cytotoxicity (C) and actual E/T ratio (N) was fitted to different nonlinear regression equations. The analysis shows that the best-fit model for all cell lines' data was one-site binding (hyperbolic) model (R2 = 0.97–0.99) (Fig. 2) Thus, NK cytotoxicty dependence on the number of NK cells per target cell is described by receptor–ligand model with one attachment position, that follows the law of mass action (Michaelis–Menten law). NK cells have a number of activating and inhibitory receptors, which all participate in the “decision” to release the cytotoxic granules. The single ligand–receptor model of NK cell target killing most likely represents the “all-or-none” decision of cyotoxic granule exocytosis.

Figure 2.

Target cell lysis by NK cells represented by plotting C (specific cytotoxicity) vs. N (NK:Target cell ratio). 721.221, K562, and Namalwa cells display twofold differences in their sensitivity to lysis, but all conform to the same hyperbolic model of single ligand–receptor interaction. Six representative datasets of more than 20 are shown. A, B, and C represent one donor (J63, 81% NK cells), D, E, and F represent another donor (D46, 89% NK cells).

Several groups have previously analyzed NK cell cytotoxicity in terms of mathematical models. Savary has analyzed rhesus monkey NK cells and suggested that the cytolysis displays Michaelis–Menten kinetics in a 3 h CRA. However, the error range of their method is rather large, as they estimate the Vmax to be 1–2 × 104 (11) Also, they do not specify a model. Cao (12) has analyzed NK activity using a four paramater logistic regression model, which has been proposed previously by Langhans et al. (13) This model fits reasonably well with most experimental data sets [although, we have obtained a data set (not shown) which does not fit this model, but fits the single ligand binding hyperbolic model]. However, the four paramater logistic regression model frequently predicts a plateau of specific cytotoxicity at exceedingly low E:T ratios, which has no biological basis. It is understandable that a model with more independent parameters is easier to make fit a data set. We have also compared one vs. two-ligand binding models, using GraphPad's built-in copmarison module, and the single ligand model fits the data better (Table 1).

Table 1. Comparison of models which may have generated the data according to Akaike's information criteria. Dataset D on Figure 2 was used
ModelOne-site binding (hyperbola)Sigmoidal dose-responseTwo- site binding (hyperbola)
Probability it is correct88.14%11.86%Did not converge
Ratio of probabilities7.43 
Preferred modelOne-site binding (hyperbola)
Difference in AICc−4.01 

Typically, cytotoxicity value was under 5% at N = 0.1 and became saturated between N = 5 and N = 10 (data not shown). On the basis of regression analysis, the N value for half maximum lysis (NC halfMax) and maximum lysis (Cmax) were calculated. NC halfMax shows at which E:T ratio 50% of maximum lysis is reached. This allowed us to compare the sensitivity of these cell lines numerically—NC halfMax of 721.221 cells was 0.277, whereas NC halfMax of K562 was 0.555 and NC halfMax of Namalwa was 1.158 revealing twofold differences in activity towards the different cell lines for this particular donor. Also, the Cmax is a good descriptor of NK cell potency when describing cytotoxicity against more resistant cell lines. In Figure 2, 721.221 Cmax is 96, K562 Cmax is 97, whereas Namalwa Cmax is 86. Two donors presented in Figure 2 display similar rank order of cytotoxicity toward the tested cell lines, whereas the donor 2 is more active towards all three. We find that 10,000 target cells per data point is an optimal number for analysis. To make a graph with 10 data points, one would need 100,000 target cells.

Quantification of Small Differences in Cytotoxicity

NK cells are inhibited by a different degree by various HLA alleles. Therefore, we analyzed the killing of three different transfected cell lines expressing different HLA-C alleles, all derived from 721.221 cells, which are HLA-negative. Analysis of cytolysis of these four cell lines showed that their lysis is described by the same model, but their sensitivity to NK mediated killing is different (Fig. 3).

Figure 3.

Nonlinear regression analysis (one-site binding model) of relationship between specific cytotoxicity (C) and actual E:T ratio (N) for four LCL cell lines [LCL 721.221-wt (A), LCL 721.221-0401 (B), LCL 721.221-1202 (C), and LCL 721.221-1403 (D)]. Bold line indicates the best-fit nonlinear regression line (that best predicts C values from N). Dashed curves demarcate the 95% confidence interval of the regression line. R2 quantifies goodness-of-fit of regression. One set of data from three is shown (D46, 89% NK cells).

The specific cytotoxicity was the highest against 721.221-wt, which does not express HLA antigens and, therefore, can not inhibit NK cells. 721.221 transfected with different HLA-C alleles express surface HLA antigens, which inhibit NK cell mediated lysis to a certain degree.

To estimate, whether the other lymphocytes present in the mixture might influence the outcome, we routinely analyzed the cells during the culture period. At early time points, the NK% is low and increases toward the end of the culture. This is donor specific, as is the intrinsic NK cell activity. However, when the cytotoxicity is analyzed at early timepoints and compared to data obtained later, the data fit on the same C/N plots.

The data presented in Figures 1–3 are done using cells from two donors. For purposes of ease and clarity, we chose donors, whose NK cells expand up to 80–90% purity. Different batches of NK cells (i.e., cells from different donors) have different intrinsic activity. We have tested tens of donors with our method, with similar results. The cytolytic activity of unstimulated, fresh NK cells in PBMC mixture is very low, achieving only a few percent cytolysis of K562 targets (data not shown). Two days activation, however, stimulates the cells to their full potential. When cytotoxicity is analyzed on different days, when NK% is different, the results all fit on the same graph (Fig. 4). The data fit a single ligand binding hyperbola model, the two-ligand model does not converge. The N c HalfMax is 0.469. It has been suggested that K562 cells may express CD32 antibody receptor and trigger T-cell cytotoxicity in the presence of OKT-3 antibody. In such case, cell preparations with low NK% should appear having higher cytotoxicity per NK cell. On the contrary, we observe good correspondence of specific cytotoxicity to NK% measured in the presence of different amounts of T-cells at different time points (Fig. 4).

Figure 4.

Cytotoxicity analyzed at three timepoints at different NK% fits the same curve. [Color figure can be viewed in the online issue which is available at wileyonlinelibrary.com]

Cytometric CFSE/7-AAD cytotoxicity assay is simple, reliable, and more sensitive than CRA, particularly at low E/T ratios. Particular attention must be given to discrimination between cells and the inevitable debris to correctly identify the effector target cell ratio. Therefore, double staining of target cells is the preferred approach, as it allows the elimination of DNA-free (7-AAD negative) target cell fragments from analysis. At very high levels of cell lysis, however, loss of target cells may pose a problem.

We have been able to refine the precision of the analysis to the point that allows testing of mathematical models of cell–cell interactions. The method allows precise quantification and comparison of NK cell activity towards closely related target cells expressing different cell surface receptors, as well as quantification of different cell surface molecules' ability to regulate NK cell cytotoxicity.