A Venn Diagram Model That Allows Triple-Label
Immunophenotypic Analysis of Cells Based
upon Double-Label Measurements
GARY S. WOOD, M.D.
Using Venn diagrams derived from set theory, a mathematic
model is presented that allows triple-label immunophenotypic
analysis of cells based upon double-label measurements. Practical
applications of this method are illustrated and discussed. Using
this model, triple-label data concerning Leu-8 and Leu-9 (CD7)
antigen co-expression by Leu-4+ (CD3+) leukocytes were derived
from double-label measurements obtained with a single laser
flow cytometer. Mean values were as follows: Leu-4+8+9+ (62%),
Leu-4+8"9+ (36%), Leu-4+8+9" (0%), and Leu-4+8"9~ (0%).
Examples are given of additional models based upon similar
principles. (Key words: Set theory; Venn diagram; FACS; Immunophenotype) Am J Clin Pathol 1989;92:73-77
Departments of Pathology and Dermatology, Stanford
University, Stanford, and Department of Dermatology,
Veterans Administration Medical Center, Palo Alto, California
described for calculating triple-label immunostaining results from double-label measurements obtained with a
single-laser flow cytometer. This model is then used to
define human leukocyte subsets and compared with direct
triple-label measurements. Finally, examples are given of
additional, more complex models that can be generated
with the use of the same principles of set theory.
CYTOFLUOROMETRIC ANALYSIS has become a
widely used method of quantifying cellular subsets for
research and clinical use.4'5,7'8 A single-laser flow
cytometer4'8 is capable of single- and double-label (oneand two-color) immunostaining measurements; however,
the more expensive and less widely available dual-laser
flow cytometer5,7'8 is required for triple- or quadruplelabel (three- and four-color) measurements. Not infrequently, it is these latter types of measurements that are
needed for immunologic research (e.g., direct measurement of the percentages of CD3 + cells expressing various
combinations of CD8 and CD4 antigens requires triplelabel analysis). Without access to a dual-laser flow cytometer and the appropriate combinations of antibody reagents necessary for triple-label immunostaining, one alternative method of CD3/4/8 analysis would be to first
purify the CD3 + cells and then perform less-complicated
double-label staining for CD8 and CD4 with the use of a
single-laser flow cytometer. The disadvantages of such an
alternative approach include added time, expense, and
difficulty as well as inaccuracy related to the potential
impurity of the CD3 + population.
In this report, a method based upon set theory'-6 is
Materials and Methods
Triple Label Versus Double-Label Analysis
Fresh human peripheral blood was collected and shaken
in bottles containing glass beads to remove fibrin and
platelets. The plasma was removed after centrifugation at
2,500 rpm for 15 minutes. To lyse erythrocytes,3 the cells
were resuspended and incubated for 15 minutes in ACK
buffer (8.3 g ammonium chloride, 1.0 g potassium bicarbonate, 0.02 g disodium edetate [EDTA] per liter of water,
pH 7.4). After centrifugation, the cells were reincubated
with ACK buffer, centrifuged, and suspended in phosphate-buffered saline with 3% (v/v) fetal calf serum and
0.01% (w/v) sodium azide. This buffer was used for all
subsequent dilutions and washes; 5 X 105 cell aliquots of
the resulting fresh human peripheral blood leukocytes
were washed, incubated on ice for 20 minutes with various
combinations of monoclonal antibodies according to the
protocol in Table 1, and washed three times. Where appropriate, some aliquots were incubated for 15 minutes
with avidin-Texas Red (avidin-TR) and washed three
more times. The cells were passed through a nylon mesh
during the last wash and were resuspended in a final volume of 100 nL. All incubations were performed in the
dark. All Leu-series reagents were used at a concentration
of 10 /*L per cell aliquot. Other reagents were used at a
concentration of 10 ng per cell aliquot. All reagents containing fluorescein isothiocyanate (FITC) or TR, but not
phycoerythrin (PE), were ultracentrifuged in a Beckman
Received June 7, 1988; received revised manuscript and accepted for
publication November 21, 1988.
Supported by a grant from the Elsa U. Pardee Foundation and by
Merit Review funding from the Veterans Administration.
Address correspondence to Dr. Wood: Dermatology Service (111D),
Veterans Administration Medical Center, 3801 Miranda Avenue, Palo
Alto, California 94304. Reprints not available.
73
74
WOOD
Table 1. Double- and Triple-Label Flow Cytometric
Immunostaining Protocol*
PEf
TRf
Anti-Leu-4 (CD3)t
Anti-Leu-4§
Anti-Leu-4§
Anti-Leu-4§
Anti-Leu-4H
—U
—H
—H
IgG| controlH
—11
—H
Anti-Leu-3a (CD4)
FlTCf
Anti-Leu-8
Anti-Leu-3a
Anti-Leu-8
Anti-Leu-3a,
Anti-Leu-3a
A.J.C.P.-July 1989
+
Table 2. Derivation of Subsets of A Cells Denned by
Reactivity with B and C: Double-Label Measurements
Triple-Label
Subset
Double-Label Subset
Equivalent
Venn Diagram
Equivalent
A+BC"
A++B+C-+
A+ BC
A B+C+
A+BC"
+
A+C" - A+BC"
A B" +- A+BC"+
+ +
ADBnC* = Area If
A 0 B n C = Area 2
A n B n C = Area 4
A n B n C = Area 5
A B + A BC~ - A C
* A, B. and C correspond to sets A, B, and C in Figure I. Any set symbol with a bar over
(e.g.. B) represents all elements not belonging to it (i.e., B represents all elements that are no
members of set B). The elements that two sets (e.g.. A and B) share in common are designated
as " A D B" (read as the intersection of sets A and B). Therefore, the elements that belong to A
IgG2 control
but not B or C are designated as "A n B O C". This would be the cells expressing the A*B~C~
immunophenotype.
t Area numbers correspond to those given in Figure I.
* All antibody reagents were obtained from Bccton Dickinson. Mountain View, California.
Avidin-TR was obtained from Molecular Probes Inc.. Eugene. Oregon.
t Fluorochromcs abbreviated as follows: PE (phycoerythrin). TR (Texas red), FITC (fluorescein
isothiocyanate).
t Three-color staining combination for direct measurement of triple-label subsets with dual-of Stanford Desk System software. Forward and obtuse
laserflowcytometer.
scatter distributions were gated for lymphocytes and
§ Two-color staining combinations needed to calculate triple-label subsets from results measured
with a single-laserflowcytometer.
checked against distributions obtained from aliquots of
% Controls.
mononuclear leukocytes purified by Ficoll-Hypaque®
centrifugation.
Airfuge® at 26,000 psi for 10 minutes to remove particulate debris. The immunostained samples were then anQuantitation of Leu-4/8/9 Leukocyte Subsets
alyzed with an FACS II® (Becton-Dickinson) equipped
with dual lasers and modified for six-parameter applicaFive milliliters of peripheral blood were obtained from
tions at the Stanford University Medical Center Shared
each
of five healthy adult donors. Erythrocytes were lysed,3
FACS Facility.7 Data were analyzed with a Vax 11/780®
and leukocytes were subjected to simultaneous two-color
computer (Digital Equipment Corporation) with the use
cytofluorometric analysis as described previously4 using
an Ortho Cytofluorograf 50-H® single-laser flow cytometer. Forward and right-angle light scatter settings were
adjusted to gate the lymphoid subpopulation. Two-color
staining for Leu-4/Leu-8 and Leu-4/Leu-9 was performed
with the use of direct antibody conjugates of PE or FITC.
The anti-Leu-4 PE conjugates were obtained from BectonDickinson, whereas the anti-Leu-8 and anti-Leu-9 FITC
conjugates were prepared at the Stanford University Blood
Avidin
Anti-Leu-3a
Table 3. Fluorocytometric Analysis of Peripheral
Blood Leukocytes
Percentages cf Cells
Subset
Uu-4++3~8"
Leu-4 3+8"+
Leu-4++3"8
Leu-4 3+8+
FIG. 1. Venn diagram of possible cell subsets defined by three antigens.
The immunophenotypic subsets+ that correspond
to the numbered areas
in the+figure
are as follows:
1) A BC";
2) A+B+C~; 3) A"B+C"; 4) A+B"C+;
+ +
+ +
+
5) A B C ; 6) A'B C ; 7) A"B-C .
Leu-4+2"8Leu-4++2+8"+
Leu-4+2~8
Leu-4 2+8+
Direct
Triple-Label
Measurement
Calculated
Double-Label
Equivalent
12
5
20
32
9
7
42
9
14
3
19
29
9
8
48
1
VENN DIAGRAM MODEL
Vol. 92 • No. 1
+
Table 4. Discrepancies in Leu-4 Subset Quantitation
Are Related to the Type of AntibodyFluorochrome Reagent Used
Table 6. Triple-label Subsets of Leu-4 Peripheral
Blood Leukocytes Derived from Double-label
Measurements with the Use of a Singlelaser Flow Cytometer
Percentages of Cells
Leu-4 Subset
Triple Label
Measurement
Double Label
Measurement
Leu-2++
Leu-3+
Leu-8
16 (TR)
37 (TR)
51 (RTC)
9 (FITC)
32 (FITC)
49 (FITC)
+
Calculated Triple-Label+ Subsets
(percentage of Leu-4 Cells)
•4+8+9~
Double-label measurements were calculated as follows from triple-labelflowcytometric data:
Leu-4*2* = Leu-4*2*8* + Leu-4*2*8"; Lcu-4*3* = Leu-4*3*8* + Leu-4*3*8_; Leu-4*8* = Leu4*2*8* + Lcu-4*2"8* or Leu-4*8* = Leu-4*3*8* + Leu-4*3~8* (results were similar).
Double-label measurements were taken directly from double-labelflowcytometric data.
Center with the use of standard methods.2 Results from
50,000 gated events were displayed with the use of linear
amplification. Background control ranges were determined with direct PE and FITC conjugates of irrelevant,
isotype-matched monoclonal antibodies.
Results
Mathematic Model
With the principles of set theory, it is possible to construct a Venn diagram model for deriving triple-label immunostaining results from double-label measurements.
The possible combinations of reactivity with three hypothetical antibodies (A, B, and C) are illustrated in Figure
1. If subsets of A + cells are to be defined by reactivity with
Table 5. Double-label Flow Cytometric
Immunostaining Protocol for Defining Leu-4/8/9
Subsets of Peripheral Blood Leukocytes*
FITC
PE
Anti-Leu-8f
Anti-Leu-4f
Anti-Leu-4f
—t
—t
IgG2 control:):
lgG2 control:):
Anti-Leu-4t
-t
Anti-Leu-8:):
-t
-t
Anti-Leu-4 (CD3)
Anti-Leu-9 (CD7)
Anti-Leu-8, 9
IgG, control
—
IgG| control
—
Anti-Leu-4
—
Anti-Leu-8
Anti-Leu-9
75
+
Leu-4+8~9+
Leu-4+8+9+
49
47
25
31
30
36
49
53
74
69
70
62
1
0
1
0
0
0
B and C, then there will be only four possible subsets:
A + B - C \ A + B + C", A + B"C + , and A + B + C + .
The A+B"C~ subset can be measured directly in a double-label FACS system by immunostaining with A labeled
with one fluorochrome and B and C both labeled with a
second distinct fluorochrome (e.g., A-PE and B-FITC/CFITC).
The remaining subsets (A+B+C~, A + B~C + , and
+ + +
A B C ) can be derived from double-label flow cytometric data according to the equations given in Table 2.
Practical Application of the Model
In order to demonstrate the practical applications of
this mathematic model, human peripheral blood leukocytes were immunostained with anti-Leu-4-PE/anti-Leu3a-biotin:avidin-TR/anti-Leu-8-FITC according to the
protocol outlined in Table 1. They were then analyzed
with a dual-laser flow cytometer, as detailed in the "Materials and Methods." The results are summarized in the
upper portion of Table 3. There was good correlation between the direct triple-label measurements and the cal-
Table 7. Derivation of Subsets of A + Cells Defined by
Reactivity with B and C: Single-Label Measurements
Triple-Label
Subset
Single-Label Subset
Equivalent
A++B~CA B"C"
A++B+4C+
A B C"
+
+
(ABC)
- (BC)
*
+
+
+
(AB)
B
A
BX"+ +
+
+
+
A+ - (AC)
+
C
- A B"C
A+ - A++B"C-+ -+ +
A BX - A B C
Venn Diagram
Equivalent
A
A
A
A
n B n C = area 11
n B n C = area 4
D B n C = area 5
n B l~l C = area 2
* All reagents were obtained from Becton-Dickinson, Mountain View. California, except anti-* (ABC)* denotes the percentage of positive cells resulting when a single aliquot of cells is
stained with all three antibodies+(A. B, and C) each conjugated to the same type of lluorochronie,
Leu-8 (PI:) and anti-Leu-9 (PE). which were produced at the Stanford University Blood Center.
e.g.. FITC. Therefore. (ABC) corresonds to the "union" of sets A, B, and C that is areas 1-7. In
t These are the essential staining combinations needed to calculate the results given in Tacontrast A*B*C+ corresponds to the "intersection" of sets A, B, and C, which is area 5.
ble 6.
t Controls.
t Area numbers correspond to those given in Figure 1.
76
WOOD
A.J.C.P.- July 1989
+
Table 8. Derivation of Subsets of A Cells Denned by Reactivity with B, C, and D: Single-Label Measurements
Single-Label Subset Equivalent
Quadruple-Label Subset
A++B~CTDA B+C"D+
A++B~C
D+ +
A +B C D"
+
A BXD
+ +
A++B"C
D
A+B++C"D++
A B CD
(ABCD)+ - (BCD)+ *
(ACDf - (CD)+ - A+B"CrD+
+
(ABD)
- (BD) +- A+B-CTD- + +
(AD)+ +- D+ - A
B"C-D- A B C-D" - A+BX+D+
+
(ABC)
(BC)
A
B-C"D+
(AB)++ - B++ - A+B"C-D- - A++ BCD ++ - A++B~C
D+
(AC) - C - A*B-C"D- - A B"C D - A B C"D-
A + - (AC)+ + C + - A + B + C + D" A + B"C + D + - A + B"C +
* See Table 7 for an explanation of (ABCD)* versus A+B*C+D+.
culated double-label equivalents. All corresponding values
were within three percentage points of each other.
In order to illustrate the importance of reagent selection
to the accuracy of the results, an additional set of flow
cytometric analyses was performed, substituting anti-Leu2a-biotin:avidin-TR and anti-Leu-2a-FITC for the corresponding anti-Leu-3a reagents according to the protocol
in Table 1. The results are shown in the lower portion of
Table 3. In contrast to the good correlations observed for
Leu-4/Leu-3/Leu-8 subsets, there were larger discrepancies within the Leu-4 + 2 + 8 + and Leu-4+2~8+ subsets where
values differed by 8% and 6%, respectively.
As shown in Table 4, these discrepancies were probably
related to the different reagents used to detect the Leu-2 +
subset in the triple- versus double-label measurements. In
the triple-label measurements, Leu-2 + cells were detected
with TR using a two-step anti-Leu-2a-biotin:avidin-TR
immunostain. In the double-label measurements, Leu-2+
cells were detected with the use of a one-step anti-Leu2a-FITC immunostain. The potential signal amplification
inherent in a two-step staining system, together with the
enhanced fluorescent brightness of TR relative to FITC,
would tend to result in detection of a greater proportion
of Leu-2 + cells with the TR label. A similar but less-extensive discrepancy is also noted for the Leu-3 + subset.
This suggests that additional factors may have also contributed to the diminished ability of the anti-Leu-2a-FITC
reagent to detect Leu-2 + cells. In contrast, the Leu-8 + subset was always detected with a one-step anti-Leu-8-FITC
immunostain. Consequently, the results of triple- and
double-label measurements of the Leu-8 + subset were very
similar.
Use of the Model to Quantitate Leu-4/8/9
Subsets
Venn Diagram Equivalent
A n B n C n D = area If
A n B n C f 1 D = area 3
A n B n C n D = area 5
A f l B n c n 5 = area 4
A n B n C (~1 D = area 14
A n B n C n D = area 9
A n B D C n D = area 7
A n B n C n D = area 8
t Area numbers correspond to those given in Figure 2.
Leu-4 + subset were calculated as listed in Table 6. The
results indicate that very few cells belong to the Leu4 + 8"9" or Leu-4 + 8 + 9" subsets, whereas in most cases there
is a majority Leu-4 + 8 + 9 + population and a minority Leu4 + 8~9 + population.
Discussion
The current report describes a Venn diagramatic model
that can be used to derive triple-label immunostaining
results from double-label data. This technique should be
useful to those who have access to only single-laser flow
cytometric analysis because a dual-laser flow cytometer
Leukocyte
Peripheral blood was obtained from healthy donors and
stained according to the protocol outlined in Table 5 and
analyzed with the use of a single-laser flow cytometer.
Using the equations given in Table 2, the values for each
FIG. 2. Venn diagram of possible cell subsets denned by four antigens.
The immunophenotypic subsets that correspond to the numbered
areas
+ +
in the+figure
are
as
follows:
1)
A^CTD";
2)
AB+CTD-;
3)
A
B
C"D-;
+
+
+
4)+ A+ B++C++D~; 5)+ A+BC
D-; 6) A"BX
D-; 7) A+B++C"D
; 8)
+ +
+
+
+ +
A B C+ D+ ; 9) A BC D+ ; 10) +A"B CrD
; 11) A"B C D ; 12)
ABC D ; 13)A-BC-D ; 14) A B"CrD+; 15)A"B+C+D-.
Vol. 92 • No. I
VENN DIAGRAM MODEL
is required for direct triple-label measurements. The cost
of cytofluorometric analysis should also be reduced by
omitting the need for a dual-laser system.
The discrepancies in Leu-2+ (CD8 + ) subset data presented in Tables 3 and 4 highlight another potential advantage of this technique. Some fluorochrome-conjugated
reagents are significantly brighter than others. Therefore,
using two bright fluorochrome conjugates to measure
double-label data and then calculating the triple-label results may generate more accurate values than using two
optimal and one suboptimal conjugate to directly measure
triple-label results.
The principles of set theory used to construct the model
detailed in Table 2 can also be applied to create other
models applicable to cytofluorometric analysis. In Table
7, a model is given for the calculation of triple-label results
based upon single-label measurements requiring only one
type of flurorchrome. Similar methods can be used to
create more complex models such as are shown in Table
8, where single-label measurements are used to calculate
quadruple-label results based upon the subsets illustrated
in Figure 2. The main practical limitation of such models
is that the large number of separate terms in each subset's
equation creates the potential for any small inaccuracies
within individual cytofluorometric measurements to be
magnified into larger errors when the net value is calculated for that particular subset (see "Appendix" for details). The double-label model outlined in Table 2 minimizes this source of error by having relatively few terms
in its equations.
77
rescence analysis of mouse B lymphocyte subpopulations. Cytometry 1984;5:159.
8. Parks DR, Lanier LL, Herzenberg LA. Flow cytometry and fluorescence activated cell sorting (FACS). In: Weir DM, Herzenberg
LA, Blackwell C, Herzenberg LA, eds. Handbook of experimental
immunology, vol 1. 4th ed. Oxford: Blackwell Scientific Publications, 1986:29.1-29.21.
APPENDIX
The following statistical analysis, applicable to the Venn diagram models discussed above, was kindly provided by Dr. Bradley Efron, Professor of Biostatistics at Stanford University.
[ 1 ] Suppose you want to calculate some quantity like
a + b — c,
where perhaps
a = true Leu-4+ proportion
b = true Leu-8+ proportion
c = true Leu-4+ or Leu-8+ proportion
(This would give the true Leu-4+8+ proportion.)
[2] Suppose also that you can't directly observe a, b, c, but
only estimates
A = a + e(A)
B = b + e(B)
C = c + e(C),
where e(A), e(B), and e(C) are independent measurement errors,
all with mean 0 and the same standard deviation, say
standard deviation = sigma.
Acknowledgments. The author thanks Dave Parks, Tim Knaak, Dennis [3] Then the obvious estimate A + B - C has standard deSasaki, and Eva Pfendt for expert advice and technical assistance and
viation sqrt(3) X sigma. In general, if you use a formula with
Mary Lou McCourt for expert secretarial assistance.
"n" terms, instead of three terms as above, the standard deviation
will be
References
sqrt(n) X sigma.
1. Bruckheimer M, Gowar NW, Scraton RE. Mathematics for technology: a new approach. New York: American Elsevier Publishing,
1968:1-16.
2. Goding JW. Conjugation of antibodies withfluorochromes:modifications to the standard methods. J Immunol Methods 1976; 13:
215-226.
3. Hoffman RA, Kung PC, Hansen WP, Goldstein G. Simple and rapid
measurement of human T lymphocytes and their subclasses in
peripheral blood. Proc Natl Acad Sci 1980;77:4914-4917.
4. Lifson JD, Finch SL, Sasaki DT, Engleman EG. Variables affecting
T-lymphocyte subsets in a volunteer donor population. Clin Immunol Immunopathol 1985;36:151-160.
5. Loken MR, Lanier LL. Three-color immunofluoroescence analysis
of Leu antigens on human peripheral blood using two lasers on
a fluorescence-activated cell sorter. Cytometry 1984;5:151.
6. Nahikian HM. A modern algebra for biologists. Chicago: University
of Chicago Press, 1964:1-50.
7. Parks DR, Hardy RR, Herzenberg LA. Three-color immunofluo-
[4] If the measurements are independently replicated "k"
times on the same donor and the results are then averaged, the
standard deviation becomes
sqrt(n/k) X sigma.
In general, one is penalized in proportion to the number of
terms (n) in the formula and rewarded in proportion to the
number of replications (k).
Comment Regarding Statistical Analysis
Because sigma equals approximately 0.01 in the fluorocytometric analyses presented in this study, values obtained with the
use of the double-label model outlined in Table 2 have a standard
deviation of approximately 0.017 (sqrt[3] X 0.01).