Limiting dilution analysis: from frequencies to cellular interactions.

Abstract

Molecular biology techniques have provided the means to pursue the precise definition of molecular components in a single cell type and its clonal progeny. However, to study the workings of the individual components of the immune system as well as the system itself, additional tools and supplementary approaches are required.

Full text

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imiting dilution analysis (LDA)1
has gained widespread accept-
ance as a tool for quantifying
cells that possess observable
functional activities. Thoroughly planned
titration experiments can produce straight-
forward and interpretable single-hit kinetics,
whereas analyses of unfractionated cell
populations over a broader dilution range
result in data that deviate from linearity and
do not adhere to all-or-none functionality
(e.g. virgin and memory CD4?T cells2and
others3–10). However, by studying the factors
that cause the deviation from linearity, the
interactions between different cell types in
the population can be identified and charac-
terized. As a corollary, it follows that, along
with quantification of desired cells, LDA al-
lows an analysis of the regulatory processes
that underlie an observed activity.
Complex interactions and their
zigzag approximation
Because the immune system is a complex
network of cellular and humoral interac-
tions, it is not unreasonable to expect that
any unfractionated cell sample that is taken
from a functional organism and then dis-
persed into limiting dilution cultures will
yield non-linear titration curves. One of the
most frequent nonlinear phenomena that oc-
curs during LD titration of this type is typi-
fied by a zigzag curve (Fig. 1). Zigzag rela-
tionships have previously been noted in a
variety of experimental systems including
tests of T-helper-cell function in humoral
responses5,6, mixed lymphocyte responses
(MLR)3,9and generation of cytotoxic activ-
ity4,7. Investigators have argued that zigzag
curves reflect a superimposition of three dis-
tinct types of limiting precursor cells (LPC)
with alternative functions: effectors (LPC1),
followed by suppressors (sLPC) and then by
an additional class of suppressor-resistant
effectors or contrasuppressor cells (LPC2)5,6,11.
Notwithstanding, the existence of three pre-
cursor types and the validity of the equation
Z ? LPC1 ? sLPC ? LPC2 have never been
confirmed experimentally. Furthermore, no
satisfactory hypothesis has been formulated
to create a solid basis for validating this
phenomenon. A noteworthy exception,
however, is an attempt by Eichman et al. to
explain LDA data as evidence of a regu-
latory network existing between T-cell
populations12.
Recently, as a result of reinvestigation of
the zigzag phenomenon, a new model has
been proposed8,10. In short, zigzag-shaped
LDA plots were obtained from tests of the
following different T-helper-cell functions:
(1) proliferative responses against syngeneic
or allogeneic feeder cells; (2) interleukin 2
(IL-2) production in the presence of allo-
geneic cells; and (3) helper function for B
cells. To reveal the true titration curve shape,
the number of experimental points was in-
creased. The resulting LDA plots were in-
consistent with the linear approximation
model described above (see Fig. 1a). To ap-
proximate the third region of the zigzag
curve with a straight line that passed
through the origin was especially problem-
atic. The positions of the experimental points
between the second and third regions, the
kink, usually fell close to or laid on the x-axis
such that an approximating line for the third
part of the curve needed to be non-linear
(see LPC2 curve in Fig. 1b).
LPC types
Based on these results, the whole zigzag
curve might represent a competition (poss-
ibly for growth factors) between only two
distinct cell subpopulations (Fig. 1b). The
second subpopulation (LPC2) has n-hit ki-
netics (where only n or more cells of this
type are able to transform the culture into a
positive one), and is capable of suppressing
the proliferation of LPC1 with m-hit kinetics
(where m equals the number of LPC2 cells
able to suppress LPC1 activity). From this
definition, the proposed model predicts that
the zigzag phenomenon will occur when
n?m. When the conventional model that was
based on linear approximation of the curve
portions was compared with the newer non-
linear two-cell model, it became evident that
the latter was far superior in predicting and
conforming with observed results2,10.
We have also confirmed the validity of
the equationZ ? LPC1 ? LPC2 (from Fig. 1b)
experimentally in the following manner: (1)
the zigzag was split into its elementary con-
stituents – LPC1 and LPC2 – and each of the
individual components displayed the pre-
dicted kinetics (i.e. LPC1 with single hit and
LPC2 with multi-hits); (2) a mixture of two T-
cell populations with single-hit and n-hit ki-
netics produced a zigzag LD curve; (3) when
the single-hit kinetics of LPC2 were induced
by prolonged cultivation or by supplying nec-
essary growth factors, the zigzag curve degen-
erated into a straight line (this is the case as the
obligatory condition for the zigzag curve ap-
pearance is the multi-hittedness of LPC2); and
(4) blockade of the lymphokine that is pro-
duced by LPC2 (which is capable of inhibit-
ing LPC1) using anti-lymphokine antibodies
transformed the zigzag into a straight line2,10.
Two cell types providing three
functions
The two-cell-interaction model uses four pa-
rameters to define the zigzag shape: (1, 2) f1
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Limiting dilution analysis: from frequencies to cellular
interactions
Igor Dozmorov, Michael D. Eisenbraun and Ivan Lefkovits
Molecular biology techniques have
provided the means to pursue the
precise definition of molecular
components in a single cell type and
its clonal progeny. However, to
study the workings of the individual
components of the immune system
as well as the system itself,
additional tools and supplementary
approaches are required.
L

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and f2, the frequencies of LPC1 and LPC2,
respectively; (3) n, the minimal number of
cells that promote LPC2 effector function in
one culture; and (4) m, the minimal number
of LPC2 cells that promote an inhibitory
effect on LPC1 activity in a co-habitated cul-
ture. The model not only gives a better fit to
the experimental data8,10, but also provides
insight into the underlying mechanism of
cell interactions, because changes in LDA
curve shapes can be monitored as individual
parameters that affect either population are
altered.
To illustrate this point, four theoretical
sets of curves showing the effects of varying
the four parameters were created (Fig. 2a–d).
Under experimental conditions, real analogs
of these curves were produced, thus con-
firming the practical use of this model
(Fig. 2e–g). It should be possible to charac-
terize the regulatory interactions at the
clonal level quantitatively from the changes
that have been introduced through the par-
ticipating subpopulations (and through their
kinetic parameters). To overcome the gap
between the approach of ‘theoreticians’ and
‘practitioners’, a set of simplified algo-
rithms8, which provide more than adequate
parameter estimations without the need for
any profound mathematical armament, have
been developed.
Universality of the competition of
alternative activities?
It is astonishing that the zigzag curve shape
is obtained from population titrations using
a variety of different cell sources and exper-
imental systems. Is it possible that there is
something in all cell populations that func-
tions to ‘keep a balance’? Although we do
not go as far as to suggest that the immune
system functions entirely in a ‘ying–yang’
fashion, the need for homeostatic controls
certainly exists in complex biological sys-
tems; several immunological examples are
provided below.
Th1 and Th2 cell populations
The adaptive immune system is controlled
by two subsets of T helper (Th) cells, termed
Th1 and Th2, which can be distinguished
from each other by mutually exclusive
lymphokine production profiles. Strong
Th1-like responses (e.g. delayed-type hyper-
sensitivity) are often mutually exclusive of
strong Th2-like responses (e.g. strong anti-
body and allergic responses)14. Several lines
of indirect evidence have been obtained
from testing the responses of Th cells to al-
loantigen under LD conditions, indicating
that the competition between these two cell
types might be responsible for creating a
zigzag-type dependence15.
Virgin and memory T cells
The competition between virgin and mem-
ory effector B cells was described some time
ago for humoral immune responses16, and
was extended to T cells once the Th-cell re-
sponses to mitogen activation under LD con-
ditions were investigated2,10. Analyses of the
corresponding zigzag curves agree very well
with the two-cell model. Separated CD4?
virgin and memory cells represent different
kinetics of the response: single-hit for virgin
cells and multi-hit for memory ones. The in-
hibitory influence of memory T cells on
virgin-T-cell activity can be modeled by
adding exogenous interleukin 10 (IL-10) and
this, in turn, can be blocked by adding anti-
IL-10 antibodies. The multi-hittedness of
the response of purified memory T cells re-
sults from an IL-2 deficiency and can be
converted into single-hittedness through ad-
dition of exogenous IL-2. The conversion of
multi-hit into single-hit kinetics is – as pre-
dicted by our theory – accompanied by the
degeneration of the zigzag into a straight
line2.
T cells in young and old mice
Dose-response curves generated for CD4?T
cells from old – but not from young – ani-
mals have a zigzag-shaped relationship.
Using cell separation methods, treatment
with blocking antibody, or exogenously sup-
plied lymphokines, the zigzag phenomenon
in T-cell responses from old mice is due to a
regulatory interaction between virgin and
memory CD4?T cells2. This effect is not seen
when T cells from young animals are exam-
ined, as the frequency of memory cells that
are derived from young mice is insufficient
to induce significant nonlinearity in the
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Fig. 1. Zigzag limiting dilution analysis (LDA)
plot. Analysis of the contributing elements
using the partial linear approximation method
(a) and two-cell model (b) of data from
experiments with concanavalin A (ConA)-
induced proliferation of mouse CD4?T cells.
(a) The resulting zigzag dependence is presented
as a superimposition of three distinct cell types
all possessing single-hit kinetics: (1) limiting
precursor cells with effector functions (LPC1),
which are present at the highest frequency;
(2) suppressor cells (sLPC), which inhibit LPC1
activity in shared cultures; and (3) suppressor-
resistant effectors or contrasuppressor cells
(LPC2), which are insensitive to or capable of
canceling the inhibitory influence of sLPC. If
their activities are considered to possess single-
hit kinetics, the frequency of all three precursor
cell types can be estimated by traditional
methods using approximating straight lines that
pass through the coordinate origin. In the case
of sLPC, an analogous linear dependence can be
obtained if the LDA plot is transformed to
reflect the dependence of the fraction of positive
cultures (instead of negative cultures) with respect
to responder cell input. (b) The resulting zigzag
dependence is presented as a superimposition of
only two distinctive cell types with different
activities and kinetics: (1) LPC1 with single-hit
kinetics and high individual activity; and (2)
LPC2 exhibiting multi-hit kinetics and an
ability to inhibit LPC1 activity in shared
cultures.
0.4
0.6
0.8
1.0
0.2
LPC1
LPC2
sLPC
10 20
Cells/well
30 40 500
0.4
0.6
0.8
1.0
0.2
LPC1
LPC2
1020
Cells/well
Z = LPC1 + sLPC + LPC2
Z = LPC1 + LPC2
30 40 500
Fraction of non-responding cultures
(a)
(b)
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dose-response10,17. The accumulation of
memory cells upon ageing18shifts the pro-
portions of virgin and memory cells into the
range where the memory cells cause a non-
linearity in the dose-response. Moreover,
this zigzag phenomenon can be generated
for T cells that are derived from young
animals by increasing the proportion of T
memory cells experimentally.
Unmasking hidden activities
At a high cell input (e.g. bulk cultures), com-
petition among cell populations might mask
the expression of some cellular phenotypes.
However, LDA systems can reveal that cer-
tain activities are still present (see examples
of hidden and unmasked autoreactivities in
Refs 9, 19). It is plausible that such an un-
masking (of inhibitory activity) in vitro has
its pendants in certain pathologies in vivo. If
this turns out to be the case, the two-cell
model could provide the basis to estimate
the level of pathology, its development and
efficacy of treatments.
Concluding remarks
Neither molecular methods nor cellular as-
says alone offer a full description of the fun-
damental mechanisms of immunological
phenomena. In fact, both approaches are
needed, as analysis of immunological re-
sponses requires a combination of molecular
and cellular techniques to fully understand
the participation of a single cell in the con-
text of a much larger and intricate system.
Jerne noted 30 years ago that biological
science progresses by alternating the effort
in studying function and structure20. The
science of today reveals that the clear dis-
tinction between these two worlds has
already begun to disappear. For example,
LDA is a cellular assay, polymerase chain reac-
tion (PCR) is a molecular one and LD-PCR
(Refs 21, 22) is the result of the convergence
of these two distinct approaches.
The behavior of cells and their complex
interactions are difficult to understand and
to monitor in situ because only a few of the
many parameters can be controlled. By con-
trast, systems that are controlled strictly,
such as work with purified clones or trans-
formed cells, are highly artificial and raise
V o l . 2 1 N o . 11 7
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Fig. 2. Theoretical and experimental sets of limiting dilution analysis (LDA) plots. Theoretical plots
(a–d) were generated using the ‘two-cell’ model by varying one of the four model parameters and keep-
ing the other three fixed. Lower panels (e–h) present experimental data that are analogous to the pre-
dicted outcomes and are arranged to correspond with the upper set of panels. (e) Allogeneic mixed
lymphocyte reaction (MLR) of non-adherent BALB/c splenocytes against adherent B6 spleen cells,
treated (red) or not treated (blue) with all-trans retinoic acid before initiation of the LDA culture.
Such treatment of antigen-presenting cells (APCs) results in alteration of their membrane structures
and an increase of their stimulatory potency for autoreactive T lymphocytes (see discussion in Ref. 13).
(f) Concanavalin A (ConA)-induced proliferation of naïve and memory CD4?T cells at ratios of 9:1
(blue) or 7:3 (red). (g) ConA-induced proliferation of a 1:1 mixture of naïve and memory CD4?T cells
in the presence (red) or absence (blue) of anti-IL-10 antibodies. (h) Autologous MLR of BALB/c non-
adherent spleen cells versus mitomycin C-treated BALB/c adherent spleen cells at concentrations of
104stimulators (blue) or 2 ? 104stimulators (red) per well. Data taken, with permission, from Ref. 8.
0.7
0.6
0.5
0.4
1.0
0.9
0.8
(e)
1.0
0.8
0.6
0.4
0.2
(g)
0.7
0.6
0.5
0.4
1.0
0.9
0.8
(f)
m
0010 20 3040
0
1.0
0.9
0.8
0.7
0.6
0.5
(h)
0200 400 600 800 1000 1020 30 40
200 400600
n
f2
f1
(a)
Cells/well
0
1.0
246
(b)
Cells/well
0246
0.1
(c)(d)
1.0
0.1
Fraction of non-responding cultures
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some doubts about the applicability of the
observations to real processes in vivo. LDA
does not offer a cheap solution, but it is
one of the few methods in which a large
number of parameters can be controlled
simultaneously, while also allowing a large
number of variables to be monitored.
There are many relevant LDA examples
that could not be dealt with here because of
space constraints. These are given in the
new edition of the LDA monograph (see
Ref. 1) in which a software package for LDA
calculations and LDA simulation is included.
The Basel Institute for Immunology was founded
and is supported by F. Hoffman-La Roche & Co. Ltd,
Basel, Switzerland.
Igor Dozmorov and Michael Eisenbraun are
at the Dept of Pathology, University of Michigan,
Ann Arbor, MI 48109, USA; Ivan Lefkovits
(lefkovits@bii.ch) is at the Basel Institute for
Immunology, CH-4005 Basel, Switzerland.
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