## How is the frequency estimated

The best way to understand LDA is to consider a specific example (Figure 1). Let us imagine that we have a population of lymphocytes from the blood of a potential bone marrow donor. We want to know the frequency of the precursors of cytotoxic T cells (pCTLs) that may react to the recipient. A simple in vitro approach would be to dispense these lymphocytes into many aliquots in microculture trays, each microculture tray getting a different input number. If one adds the best medium and all essential growth factors together with the recipient's irradiated stimulator cells, then any precursor lymphocytes should be activated to grow, and produce clones. Within the clones, the majority of daughter cells eventually differentiate into cytotoxic T cells lethal to the recipient's target cells. A simple chromium release assay can indicate which wells have made cytotoxic T cells and which have not. If the culture conditions have been good enough, then single precursors should have produced sufficient clone progeny to guarantee that all wells containing CTI.s would be measurable in the readout.

Limiting dilution analysis relies upon the estimation of negative wells, as clearly these must have had no precursors. A positive well could have had one or more precursors and so is not so useful in the statistical analysis.

The proportion of cultures that fails to respond (i.e. has no pCTL within the lymphocyte aliquot) at any given lymphocyte input can be defined by the expression:

where F„ is the number of negative cultures per total number of cultures, e the base of the natural logarithm, and u the average number of precursors (pCTLs) per culture (well). The above expression is the zero term of the Poisson distribution. The full Poisson distribution formula and its derivation is considered in detail in limiting Dilution Analysis o/ the Immune System by I.efkovits and Waldmann.

If one modifies the above formula, one can derive the following expression:

(this represents the conversion of formula ¡1; to its logarithmic form). Through this conversion, the tor-

Figure 1 A single-hit curve. Four trays, each of 60 cultures, were inoculated with graded numbers of human lymphocytes (1250, 2750, 6250, 8750 cells per culture). Some of the cells in the inoculum are antigen specific, while the majority are irrelevant third-party cells. The dots in the upper row of trays indicate just the distribution of the antigen-specific cells. Culture conditions were chosen that allowed the cells of interest (dots) to have an optimum chance of proliferating and generating clones of cytotoxic T cells (CTLs). If the original parental T cells are called 'precursor' CTLs or pCTLs, then in the lower row of trays the dots represent the clones arising from the pCTLs. The CTL activity of each well is measured in a chromium release assay on the desired target cells. The cultures which scored positive are shown as 'filled' wells. In the first tray of 60 cultures, 24 responded. Therefore, F0 is 36/60 = 0.6. In the next three sets, we see that as cell input per culture well increases, the number and the proportion of negative cultures F0 decreases. The results are plotted on a semilog plot and they fit a straight line. The vertical bars represent the 95% confidence limits. These represent the limits within which F0 would be expected to fall 19/20 times. (Reproduced with permission from Lefkovits and Wald-mann (1984).)

mula is reorganized around the value that interests us - u, the average number of precursor cells. In addition, it can be represented graphically, providing a straight line in a semilogarithmic plot. In other words, the negative logarithm of the fraction of nonresponding (negative) cultures is linearly proportional to the mean number of precursors per culture. The resulting straight line has two uses. First, it allows the estimation of frequency by a var iety of methods. Second, it acts as a sort of internal control to check that the system is measuring the activity of a single limiting cell type. A straight line means a single-hit curve, which means that only a single cell type is limiting (the pCTL). Any deviations from linearity determined by use of appropriate statistical methods would invalidate that starting assumption, confound the estimate of frequency, and preclude any further dependent analyses.

In the semilog plot in Figure 1, we could apply statistical methods to test the goodness of fit of the data points with the single-hit Poisson model by calculating x2 statistics and confidence intervals for the slope. If we assume that the line drawn in the figure is the best-fit line, then we can make a reasonable estimate of frequency from the plot by interpolating at 37% of negative cultures. This is best understood by substituting u = 1 in the zero term of the Poisson formula (1). We then derive

There are more sophisticated ways of estimating frequency, and the reader is referred to Taswell. From the example shown here, interpolation at 0.37 reveals a frequency of pCTL of 1/2500 or 4 X 10

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