Mathematical Modeling Reveals the
Biological Program Regulating
Lymphopenia-Induced Proliferation
This information is current as
of June 17, 2017.
Andrew Yates, Manoj Saini, Anne Mathiot and Benedict
Seddon
J Immunol 2008; 180:1414-1422; ;
doi: 10.4049/jimmunol.180.3.1414
http://www.jimmunol.org/content/180/3/1414
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References
The Journal of Immunology
Mathematical Modeling Reveals the Biological Program
Regulating Lymphopenia-Induced Proliferation1
Andrew Yates,† Manoj Saini,* Anne Mathiot,* and Benedict Seddon2*
T
he peripheral T cell compartment is maintained in a remarkably stable state of homeostasis despite its heterogenous composition and the constant influx of cells into the
different pools (1). This stability is achieved by close regulation of
cell survival and proliferation of the constituent subsets. For naive
CD4 and CD8 T cells, these homeostatic survival and proliferative
mechanisms are regulated by cytokines and TCR signals. In contrast, memory cell homeostasis appears to depend only on cytokines including IL-7 and IL-15. During Ag responses, specific T
cells are released from normal homeostatic control and proliferation and survival is governed by the presence of specific Ag. Upon
resolution of a response and clearance of Ag, specific T cells return
to normal homeostatic control.
Recognition of self-peptide MHC complexes by naive T cells
delivers signals that are important for T cell survival (2– 8), and in
conditions of lymphopenia, these same signals can induce T cells
to undergo so-called homeostatic or lymphopenia-induced proliferation (LIP)3 (4, 9 –11). Although induction of proliferation in
lymphopenia is critically dependent on TCR signals, and therefore
could rely on similar proliferative mechanisms that govern responses to Ag, the precise relation between these two responses
remains unclear. Studies using different CD8⫹ TCR transgenic
*Division of Immune Cell Biology, National Institute for Medical Research, London,
United Kingdom; and †Department of Biology, Emory University, Atlanta, GA 30322
Received for publication July 26, 2007. Accepted for publication November 20, 2007.
The costs of publication of this article were defrayed in part by the payment of page
charges. This article must therefore be hereby marked advertisement in accordance
with 18 U.S.C. Section 1734 solely to indicate this fact.
1
This work was supported by the Medical Research Council (U.K.) and the National
Institutes of Health.
2
Address correspondence and reprint requests to Dr. B. Seddon, Division of Immune
Cell Biology, National Institute for Medical Research, The Ridgeway, London NW7
1AA, U.K. E-mail address: [email protected]
3
Abbreviations used in this paper: LIP, lymphopenia-induced proliferation; flu, influenza; SM, Smith-Martin; GH, Gett-Hodgkin; WT, wild type.
Copyright © 2008 by The American Association of Immunologists, Inc. 0022-1767/08/$2.00
www.jimmunol.org
models have revealed a complex picture in which a broad range of
proliferative responses to lymphopenia can be observed. Some
TCR transgenic T cells proliferate slowly (12) or not at all (6),
whereas others undergo extensive cell division accompanied by
development of effector function (13, 14). LIP may also have a
strong stochastic (random) component because even within an apparently homogeneous monoclonal population, some cells divide
extensively, whereas in the same environment and time frame others will not divide at all. In the instances where lymphopenia induces the most rapid division, the similarity with true Ag-driven
proliferation is quite considerable both in terms of dependency on
CD4 T cell help (15) and at a genetic level where they share similar
profiles of gene expression (16). However, in other regards,
such as the lower rate of division and its dependence on IL-7
(17), lymphopenia-induced proliferation is distinct from that
induced by Ag.
Despite the extensive molecular and cellular characterization, it
remains unclear whether proliferation induced by lymphopenia
follows a pattern similar to that induced by Ag, differing from it
only quantitatively, or whether induction and/or maintenance of
cell division in these two types of proliferation are regulated in
fundamentally distinct ways. To gain a better understanding of
how LIP is regulated, we took a different approach, constructing
mathematical models to describe the proliferative process quantitatively and thus provide new insights into the control required to
achieve the complex responses observed. Previous studies have
successfully used mathematical modeling to better understand T
cell proliferation to Ag (18 –21), in which cells divide in a “programmed” manner, largely independent of the requirement for
continued antigenic stimulation (22, 23). This program can be represented mathematically as a rapid and deterministic or autopilot
series of divisions that follow a relatively slow and stochastic
(probabilistic) entry into the first division, the mean time to which
can be affected by costimulation and cytokines. This model can
accurately describe the processes involved in T cell activation and
provide quantitative insights into how costimulation shapes the
proliferative response.
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Recognition of peptide-MHC by the TCR induces T lymphocytes to undergo cell division. Although recognition of foreign peptide
induces a program of cellular division and differentiation by responding T cells, stimulation by self-peptide MHC complexes in
lymphopenic conditions induces a slower burst of divisions that may or may not be accompanied by effector differentiation.
Although both responses are triggered by signals from the TCR, it is not known whether they represent distinct programs of cell
cycle control. In this study, we use a mathematical modeling approach to analyze the proliferative response of TCR transgenic F5
T cells to lymphopenia. We tested two fundamentally different models of cell division: one in which T cells are triggered into an
“autopilot” deterministic burst of divisions, a model successfully used elsewhere to describe T cell responses to cognate Ag, and
a second contrasting model in which cells undergo independent single stochastic divisions. Whereas the autopilot model provided
a very poor description of the F5 T cell responses to lymphopenia, the model of single stochastic divisions fitted the experimental
data remarkably closely. Furthermore, this model proved robust because specific predictions of cellular behavior made by this
model concerning the onset, rate, and nature of division were successfully validated experimentally. Our results suggest cell
division induced by lymphopenia involves a process of single stochastic divisions, which is best suited to a homeostatic rather than
differentiation role. The Journal of Immunology, 2008, 180: 1414 –1422.
The Journal of Immunology
1415
In the present study, we show that F5 TCR transgenic T cells
undergo a distinct and defined program of cell division in response
to lymphopenia. Proliferation is slower than compared with Aginduced proliferation and cells fail to up-regulate activation markers or develop effector function, a response profile found to be in
common with a large proportion of the CD8 T cell repertoire in
polyclonal mice. Significantly, we show that homeostatic proliferation of F5 T cells is best described by a remarkably simple
model of infrequent, stochastic single divisions and does not fit a
model of an autopilot burst of proliferation. Our data suggest that
Ag- and lymphopenia-induced proliferation may represent fundamentally distinct programs of cell division.
Materials and Methods
Mice
pi ⫽ 0共t兲 ⫽ 1 ⫺
Lymphocytes were teased from lymph nodes and spleen of donor mice and
single cell suspensions were prepared. Cells were labeled with 2 ␮M CFSE
(Molecular Probes) in Dulbecco’s PBS (Invitrogen Life Technologies) for
10 min at 37°C and washed twice. Cells were transferred into various
recipient mice via tail vein injections. Mice further challenged with influenza virus (A/NT/60-68) were injected i.v. with 107 hemagglutinating units
of virus.
Cellular expansion of CFSE-labeled cells was calculated by determining
the frequency (Fi) of recovered donor cells that had undergone i divisions,
冱Fi ⫽ 1. Adjusted frequencies, or precursor frequencies, fi were calculated
by dividing Fi by 2i to remove the effects of expansion (冱fi ⱕ 1). The fold
expansion was then given by the quantity 1/冱fi. Mean divisions were calculated as 冱i fi/冱fi.
In vivo killing assay
Splenocytes from Ly5.1 C57BL/6J were incubated with either 10⫺6 M,
10⫺7 M, 10⫺8 M, 10⫺9 M, or no Ag as control for 2 h at 37°C, then labeled
with 4 ␮M, 1 ␮M, 250 nM, 62 nM, and 16 nM CFSE, respectively, for 10
min at 37°C and washed twice. Four ⫻ 106 of each population of cells were
mixed and injected into Rag1⫺/⫺ recipients. Rag1⫺/⫺ and F5 Rag1⫺/⫺
recipients received targets alone as controls. Twenty-four hours later, recipients were killed and splenocytes isolated. Target cells were stained with
Ly5.1 and analyzed by FACS. Percent killing was determined by comparison of target populations remaining in experimental hosts with control
hosts that received targets alone and normalized to the unpulsed control
population.
Mathematical modeling of homeostatic proliferation
Gett-Hodgkin (GH) (programmed divisions) model. In this model, used
successfully to describe programmed T cell responses to Ag stimulation
(18), the first division for each cell takes a time t drawn from a distribution
with probability density ␾(t), and subsequent divisions for each cell are
deterministic, each division taking a time ⌬. We assumed no cell death
over the course of the experiments. We attempted to fit the data using three
left-bounded unimodal distributions ␾(t)-gamma, lognormal, and Weibull.
␾共s兲 ds, pi ⬎ 0共t兲 ⫽
冕
t ⫺ T ⫺ 共i ⫺ 1兲⌬
␾共s兲 ds.
t ⫺ T ⫺ i⌬
Stochastic single divisions model. To describe cells undergoing single
divisions randomly, we used a version of the Smith-Martin (SM) model of
the cell cycle (24) in which cells spend exponentially distributed times in
a resting “A phase” with rate constant ␭ (such that the average waiting time
for a cell in A phase is 1/␭). In this model, all cells have an equal probability of committing to division in any time interval. After committing to
division, cells transit deterministically through a “B phase” of duration ⌬
before undergoing mitosis and returning to A phase. Let the expected numbers of cells in generation i (that is, cells that have divided i times since
transfer) in the A and B phases at time t after transfer be xAi (t) and xBi (t),
respectively. Assuming N cells survive the transfer and are capable of
dividing in the recipient, these quantities can be calculated using the following system of equations:
d A
x 共t兲 ⫽ ⫺␭xA0 共t兲,
dt 0
d A
x 共t兲 ⫽ 2xB0 共t, ⌬兲 ⫺ ␭xA1 共t兲
dt 1
6
Labeling and adoptive transfer of T cells
t⫺T
0
Flow cytometry
Flow cytometry was conducted using 2–5 ⫻ 10 lymph node or spleen
cells. Cell concentrations were determined using a Scharf Instruments Casy
Counter. Cells were incubated with saturating concentrations of Abs in 100
␮l of PBS-BSA (0.1%)-azide (1 mM) for 1 h at 4°C followed by three
washes in PBS-BSA-azide. Monoclonal Ab used in this study were as
follows: allophycocyanin-TCR (H57–597) (BD Pharmingen), allophycocyanin-CD44 (Leinco Technologies), bio-CD44 (BD Pharmingen), and
PerCP- and PE-CD8 (BD Pharmingen). Biotinylated mAb staining was
detected using PerCP steptavidin (BD Pharmingen) and PE-TR steptavidin
(Caltag Laboratories). Four- and five-color cytometric staining was analyzed on a FACSCalibur and LSR Instruments (BD Biosciences), respectively, and data analysis was performed using FlowJo V8.1 software
(TreeStar).
冕
⯗
d A
x 共t兲 ⫽ 2xnB ⫺ 1共t, ⌬兲 ⫺ ␭xAn 共t兲,
dt n
冉
冊
⭸ B
⭸
⫹
x 共s, t兲 ⫽ 0
⭸t ⭸s i
for all i ⫽ 0. . .n,
where xBi (s, t) is the population density of cells at time t that have already
divided i times and have been in B phase for a time s, 0 ⱕ s ⱕ ⌬. The
solution is subject to the boundary conditions xBi (t, 0) ⫽ ␭xAi (t), i ⫽ 0. . . n,
xA0 (t ⫽ T) ⫽ N, xAi (T) ⫽ 0, i ⫽ 1. . .n, and xBi (s, 0), i ⫹ 1. . .n, for all s. The
subscripts i ⫽ 0. . . n refer to division or generation number, n is the maximum number of divisions that can be resolved (typically 8 –9 with CFSE
labeling), and we assumed no division was possible during an initial lag
period of duration T after transfer. With CFSE information alone only
the sum of the populations in the A and B phases are observable; that
is, the number of cells in CFSE peak corresponding to cells divided i
times is
Yi共t兲 ⫽ xAi 共t兲 ⫹
冕
⌬
xBi 共t, s兲 ds.
0
These quantities are proportional to the initial undivided cell number, N, or
the number of transferred cells that survive the transfer and participate in
the homeostatic expansion. This number is unknown and may vary from
animal to animal. The assumption of no cell death allowed us to remove
this dependence on N and work instead with the precursor frequencies of
cells in each generation predicted at time t,
pi共t兲 ⫽
2⫺iYi共t兲
冘
n
.
2⫺jYj共t兲
j⫽0
We also fitted a modified SM model with an additional parameter ␮ that
allowed a progressive reduction in the division rate with time to mimic
competition for homeostatic resources as the population expands. We assumed an exponential decline, such that ␭(t) ⫽ ␭0 exp(⫺␮t).
Estimation of parameters. Best-fit values of the parameters (that is, for
the GH model, the parameters describing ␾(t) together with the lag time T
and the division time ⌬; for the simple SM model, ␭, ⌬, and T; and for the
modified SM model; ␭0, ␮, ⌬, and T) were found using a custom-written
Mathematica script. The model equations were solved numerically to generate the predicted precursor frequencies pi(t); independent components of
the frequency data pi(t) and their observed counterparts fi(t) were arcsinsquare root transformed to normalize their distributions across animals at
each division at each time point; the weighted sum of squared residuals
between the pi(t) and the fi(t) was then minimized with respect to the
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C57Bl6/J wild-type (WT), Ly5.1 C57Bl/6J, RAG-1-deficient (Rag1⫺/⫺)
mice, Il7⫺/⫺ (The Jackson Laboratory), F5 Rag1⫺/⫺ mice, and breeding
combinations thereof were bred in a conventional colony free of pathogens
at the National Institute for Medical Research (London, U.K.). All lines
used were of H-2b haplotype. Animal experiments were done according to
institutional guidelines and Home Office regulations.
Whatever the specific form of ␾(t), if cells are exposed to a stimulus at a time
T after transfer, this model predicts that the precursor frequency of cells that
have divided i times (that is, the frequency of cells in generation i after adjustment for their expansion) at time t after transfer is (18):
1416
MODELING LYMPHOPENIA-INDUCED T CELL PROLIFERATION
parameters using a Nelder-Mead optimization algorithm. By fitting the
model to 500 datasets generated by resampling with replacement, we obtained empirical (bootstrap) distributions of the parameter estimates and
95% confidence intervals were obtained using the percentile method (25).
The Mathematica script is available on request from the authors.
Comparing models. The improvement in fit obtained by adding a parameter to the SM model was assessed with an F-test. If SSR denotes the sum
of the squares of the weighted residuals, each term in which is assumed be
the square of a normally distributed variable with unit variance, then the
quantity (SSR1 ⫺ SSR2)/(SSR2/(m ⫺ q)) is F-distributed on (1, m ⫺ q)
degrees of freedom, where SSR1 is the original model, SSR2 is the extended model with extra parameter ␮ (SSR2 ⬍ SSR1), m is the number of
independent frequency measurements used in the fits (m ⫽ 186), and q ⫽
4 is the number of parameters in the larger model. Measures of relative
support for the GH and SM models were obtained with the Akaike information
criterion (AIC), using the definition AIC ⫽ m log (SSR/m) ⫹ 2(K ⫹ 1) where
K is the number of parameters in the model (26).
Results
Lymphopenia and Ag induce distinct programs of cell division
and differentiation in F5 T cells
input indicates the proportion of the precursor population that gave rise
to the corresponding CD44high and CD44low populations on the dot plots
as determined by current frequency and CFSE labeling.
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FIGURE 1. F5 T cells maintain their naive state following lymphopenia induce proliferation. CD8⫹ T cells from F5 Rag1⫺/⫺ mice
were labeled with CFSE and transferred i.v. to Rag1⫺/⫺ recipients (2 ⫻
106/mouse) with or without influenza virus. At days 3, 7, and 14,
splenocytes were recovered from recipient mice and analyzed by FACS
for expression of CD8, TCR, CD44, and CFSE labeling. A, Histograms
show CFSE labeling of CD8⫹ TCRhigh cells 3 and 7 days post transfer,
either in the absence (top) or presence (bottom) of challenge with influenza virus. B, Dot plots show CD44 expression vs CFSE labeling of
F5 T cells either 3 days after transfer and influenza challenge, or 14
days post transfer in the absence of influenza challenge. C, 1.5 ⫻ 107
F5 T cells were transferred to Rag1⫺/⫺ hosts or 2 ⫻ 106 F5 T cells were
transferred to Rag1⫺/⫺ hosts and challenged with flu. Eleven days later,
killing capacity of F5 T cells in these mice was determined by injecting
hosts with a mixture of target splenocyte populations pulsed with varying concentrations of NP68 peptide. Dot plots show the frequency of F5
T cells in the lymph nodes of recipient mice. The graph shows the
percentage killing of targets by F5 T cells undergo homeostatic (E) or
Ag-induced (F) proliferation. D, CD8 T cells were purified from WT
mice, labeled with CFSE and transferred to Rag1⫺/⫺ hosts. At days 6
and 13, lymph node cells from recipients were stained for CD8, CD44,
TCR and analyzed by FACS. Dot plots show CFSE vs CD44 expression
by CD8⫹TCRhigh cells. Percentages indicate frequency of CD44high and
CD44low cells. Histograms show CFSE labeling of CD44high and
CD44low populations from the dot plots. Numbers on CFSE histograms
indicate mean division within the precursor population. Percentage of
A mouse model extensively used in our laboratory for studying the
processes that govern homeostatic survival and proliferative responses is the F5 TCR transgenic mouse strain that expresses a
class I restricted TCR specific for a peptide of influenza nucleoprotein (27). F5 T cells make relatively weak proliferative responses under conditions of T cell deficiency (12) as compared
with other TCR transgenic strains (14). However, to directly compare the proliferative response of F5 T cells in lymphopenia with
that induced by Ag, we transferred F5 T cells labeled with the cell
dye CFSE into Rag1⫺/⫺ T cell-deficient hosts either in the presence or absence of influenza (Flu) virus. At various days post
transfer, cells were recovered from host mice and analyzed by
FACS. In the absence of Flu virus, T cells underwent a sustained
but slow process of cell division. In hosts additionally challenged
with flu virus, F5 T cells underwent a rapid and robust proliferation
with most cells losing CFSE dye by day 7 (Fig. 1A). In fact, in
lymphopenic hosts it took approximately 2 wk for F5 T cells to
proliferate to a similar degree as observed in only 3 days in the
presence of Ag (Fig. 1B). Furthermore, F5 T cells failed to upregulate activation markers such as CD44 compared with the same
cells challenged with Flu virus (Fig. 1B) and failed to elicit any
effector function when challenged with Ag pulse targets in vivo
(Fig. 1C).
Because of the wide variation in lymphopenic proliferative responses reported in different TCR transgenic systems, we more
closely examined homeostatic proliferation of WT CD8⫹ T cells to
collectively quantify the variety of responses made within a polyclonal repertoire. Purified CD8⫹ T cells from WT mice were labeled with CFSE and transferred into Rag1⫺/⫺ hosts. At 6 and 13
days post transfer, cells were recovered and the lymphopenia-induced proliferation assessed by FACS. By day 13, it appeared the
majority of cells recovered were CFSE negative and CD44high
(Fig. 1D) as described previously (9, 13, 28). However, a significant proportion of T cells were still labeled with varying levels of
CFSE and remained CD44low. Examination of responses at day 6
allowed more detailed quantitative analysis because all cells were
still CFSE positive. More than 40% of T cells had up-regulated
CD44 at this stage and had undergone more divisions than the
CD44low population. However, using the CFSE profile to calculate
the precursor frequency (see Materials and Methods) revealed that
The Journal of Immunology
1417
the CD44high population arose from only 17% of the precursor
population. The remaining 83% of the precursor pool remained
CD44low and divided more slowly or not at all. In fact, 37% of
the injected T cells were still undivided by day 6. Therefore, it
appears that the abundance of the CD44high population at day 13
reflects the extensive proliferation of a small proportion of the
starting population rather than a majority response by the entire
initial cohort. These data suggest that the ability of T cells to
undergo extensive proliferation and phenotypic conversion is a
feature of a relatively small proportion of the normal WT repertoire. Rather, the majority of CD8 T cells divided slowly, or
not at all, without up-regulating activation markers. Therefore,
the slow division rate of F5 TCR transgenic T cells in Rag1⫺/⫺
hosts closely resembles the slow dividing population of WT
CD8 T cells, further highlighting the suitability of F5 T cells for
the present analysis.
Quantifying the F5 T cell response to lymphopenia
F5 T cells proliferate in response to lymphopenia in a manner quite
distinct from that induced by cognate Ag (Fig. 1A). Previous studies have demonstrated that T cell stimulation with cognate Ag
results in the induction of a predetermined program of cell division
with distinct characteristics, such as time to first division and subsequent interdivisional durations. To gain an insight into the cell
cycle regulation controlling homeostatic cell division and how it
compares to that described following T cell activation, we aimed
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FIGURE 2. Kinetics of F5 T cells homeostatic proliferation. CD8⫹ T cells from F5 Rag1⫺/⫺ mice were labeled with CFSE and transferred to Rag1⫺/⫺
hosts (106/mouse). At different time points, lymph nodes were recovered from recipient mice and analyzed by FACS for expression of CD8, TCR, and CFSE
labeling. A, Histograms show CFSE labeling of CD8⫹TCRhigh cell at the days indicated. B, The line graphs show the mean divisions by CD8⫹TCRhigh cells
in lymph node at different days post transfer (left) and the population expansion predicted by the profile of cell divisions revealed by the CFSE (right). C,
In a similar experiment described in A, cell recoveries and CFSE labeling of F5 T cells transferred to Rag1⫺/⫺ hosts (n ⫽ 5) was determined at different
times post transfer. The graph shows total cell recoveries (F) and cell recoveries of the precursor population that excludes the expansive effects of cell
division (E). D, CD8⫹ T cells from F5 Rag1⫺/⫺ mice were labeled with CFSE and transferred to Rag1⫺/⫺ hosts at a range of doses. The graph shows the
mean divisions of CFSE-labeled F5 T cells at day 14 after transfer of the cell doses indicated. E, CFSE-labeled F5 T cells (2 ⫻ 106/mouse) were transferred
to either Il7⫹/⫹ Rag1⫺/⫺ or Il7⫹/⫺Rag1⫺/⫺ recipients. Two, 7, and 13 days later, lymph node cells were harvested from groups of mice and analyzed by
FACS for expression of CD8, TCR, and CFSE labeling. The graph shows mean division of F5 T cells with time in Il7⫹/⫹ Rag1⫺/⫺ (filled squares) or
Il7⫹/⫺Rag1⫺/⫺ (half filled squares) hosts. Data are representative of at least three experiments.
1418
MODELING LYMPHOPENIA-INDUCED T CELL PROLIFERATION
Modeling the homeostatic proliferation of F5 T cells
To test different hypotheses for the regulation of LIP, we fitted a
series of mathematical models of cell division to these time series
data. For each model, we searched for values of the model parameters that gave predicted patterns of cell division over time that
best fitted our experimental time course data. In all cases we assumed that cell death among the cohort of cells that survived the
transfer was negligible. This seemed reasonable on biological
grounds as the lymphopenic host is likely to present an abundance
of T cell survival signals or resources, but this assumption was also
validated by the observation that the precursor number calculated
from cells recovered from spleen and lymph nodes was approximately constant during the first 14 days after transfer (Fig. 2C).
We attempted to describe the data with two classes of stochastic
model. The formulation of the models and the fitting procedure are
described in detail in Materials and Methods. The first class of
model assumed that the mode of proliferation observed was analogous to the programmed proliferation of Ag-stimulated T cells
observed in vitro, a model first studied quantitatively by Gett and
Hodgkin (18) and extended in further studies (19, 21). We refer to
it here as the GH model. To apply this model in the context of
lymphopenia-induced proliferation in vivo we assume cells receive
a proliferative stimulus from their environment soon after transfer
to the lymphopenic host. For each cell, the time between receipt of
FIGURE 3. Models of lymphopenia-induced proliferation of T cells. A
model of asynchronous “programmed” T cell proliferation based on the
GH model (18) (upper panel). Cells transferred to the lymphopenic host
take a time to their first mitosis drawn from a probability distribution
(shown schematically) and subsequently divide deterministically. Our
model includes the additional assumption that division is not possible during a time interval T immediately following transfer. Two cells are illustrated here: cell 1 undergoes its first division at time t1, cell 2 at a later time
t2. Both continue to divide with each subsequent division taking a fixed
time ⌬. An alternative model of homeostatic proliferation, based on the
stochastic SM model (24) (lower panel). T cells in a lymphopenic environment receive signals that induce cell division. However, not all cells
divide at once. As in the model above, entry into division (here denoted B
phase) is a process of chance, and in this model within a given time interval
all cells have an equal probability of receiving the necessary signals to
stimulate division. However, cells that receive the necessary stimulus only
divide once (taking a time ⌬) before returning to the resting state (A phase).
In this figure, we illustrate a single cell dividing, but recruitment into B
phase is continuous and at any given moment, many cells may progress
through a single division. The process is memoryless in the sense that cells
that have divided have the same chance as other resting T cells of receiving
further division signals.
this stimulus and first mitosis is assumed to be a random variable
drawn from a left-bounded unimodal distribution. We used lognormal, gamma, and Weibull distributions as candidates. Cells that
have divided once then embark on a program of divisions each of
duration ⌬ before returning to a quiescent state (Fig. 3A).
The free parameters in this model were the mean and variance
specifying the probability distribution of times to first division, the
division time ⌬, and a delay before any cell could divide at all (T).
The Weibull distribution provided the best fit of the three candidates (⌬AIC values; log-normal-Weibull ⫽ 75; gamma-Weibull ⫽
35). The best fit parameters were as follows: distribution shape
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to use models to describe the cell division processes quantitatively.
To do this, we required a precise experimental description of lymphopenia-induced proliferation by F5 T cells to underpin any modeling. Therefore, we first attempted to measure the F5 T cell response to lymphopenia in detail.
F5 T cells were labeled with CFSE and transferred into T celldeficient Rag1⫺/⫺ hosts. At various time points, cohorts of mice
were taken and the proliferative and expansive behavior of the F5
T cells determined by cellular enumeration in lymph nodes and
spleen and analysis of CFSE labeling by FACS. Over the duration
of the experiments, F5 T cells were seen to undergo steady progressive divisions (Fig. 2A). Calculating the mean number of divisions undergone revealed a brief lag followed by a near linear
increase in mean division with time corresponding to roughly exponential growth (Fig. 2B), confirmed by measurement of absolute
cell numbers (Fig. 2C). The F5 T cell precursor number, calculated
by removing the effects of expansion predicted by the cells’ CFSE
profile from the cell recovery, did not decline during the course of
the experiment, suggesting that there was negligible cell death of
F5 T cells in the lymphopenic hosts (Fig. 2C).
We next assessed the sensitivity of the homeostatic proliferative
response to host environment. First, we compared the proliferative
response when different numbers of F5 T cells were transferred to
Rag1⫺/⫺ recipients. Hosts were injected with cell doses between
5 ⫻ 105 and 1.5 ⫻ 107 CFSE-labeled F5 T cells and cell division
assessed 14 days post transfer. F5 T cell proliferation as measured
by mean division number was inversely proportional to inoculum
size and the response was indeed exquisitely sensitive to initial cell
dose, with reproducible differences observed between as little as
2-fold variations in initial cell dose (Fig. 2D). Since IL-7 is critically required for homeostatic cell division, we asked whether
varying IL-7 availability would have a quantitative effect on F5 T
cell proliferation. We tested this by transferring CFSE-labeled F5
T cells into control Rag1⫺/⫺ hosts or into Rag1⫺/⫺ Il7⫹/⫺ hosts
that have only one functional Il7 allele. F5 T cells in Il7⫹/⫺ hosts
proliferated more slowly than in controls as measured by mean
division number versus time (Fig. 2E), further illustrating that homeostatic proliferation was very sensitive to the availability of
resources in the host.
The Journal of Immunology
1419
parameter ⫽ 2.26, scale parameter ⫽ 8.27 (mean time to first
mitosis ⫽ 7.33 days, SD ⫽ 3.43 days), division time ⌬ ⫽ 2.88
days, lag time T ⫽ 0. On comparing the time course of cell division predicted by this GH model to the response observed experimentally, the model appeared to provide a very poor description
of the data (Fig. 4A), both in terms of the kinetics of mean division
for the whole population and the predicted profiles of cell divisions
at the various time points.
The second class of model assumed that cells circulate and compete for homeostatic division signals such that a given cell has a
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FIGURE 4. Models of lymphopenia-induced proliferation. The upper
graphs show the mean and variance of
division number in each animal (solid
triangles); these quantities averaged
over independent experiments at each
time point (gray lines); and the bestfit model predictions of the mean and
variance of division number (black
lines). Histograms indicating the proportion of F5 T cells that have undergone different numbers of divisions at
the days indicated. Gray bars show
the observed experimental profiles,
each bar corresponding to one CFSE
peak and with height equal to the proportion of cells observed in each division (averaged over several independent experiments and not adjusted
for expansion). Black bars indicate
the corresponding model predictions
using the best-fit parameters. In A, we
show the best fit from a programmed
divisions model, a variant of the GH
model in which times to first division
are Weibull-distributed and subsequent divisions are deterministic,
each with duration ⌬. B and C, Variants of the SM model. In B, we assumed constant mean rate of entry
into division (␭). An extended SM
model (C) assumed an exponential
decline in ␭ with time as the cells divide to fill the host, and provided a
significantly better fit.
constant probability of entering the cell cycle in any short time
interval and dividing once. We used a version of the canonical SM
model of the cell cycle (24), illustrated schematically in Fig. 3B, in
which cells spend exponentially distributed times in a quiescent A
phase (corresponding approximately to the G0/G1 states of the cell
cycle) and then progress through a B phase (S, G2, and M) before
dividing and returning to the A phase. The B phase was taken to
be of fixed duration. The rate of transition from A to B phase, i.e.,
the rate at which T cells are triggered into cell division, is represented by the term ␭, and duration of the B phase by ⌬. The rate
1420
MODELING LYMPHOPENIA-INDUCED T CELL PROLIFERATION
Testing the biological predictions of the mathematical model in
vivo
Although the SM-based model of stochastic cell divisions appears
to provide the best description of the observed response by F5 T
cells to lymphopenia, this observation neither proves the model is
correct nor indeed excludes the possibility that other models might
equally well or better describe the data. The model did, however,
make several specific predictions about the nature of the F5 T cell
response to lymphopenia. Therefore, we sought to further validate
the model by directly testing its predictions experimentally.
A key biological prediction of the model was that TCR stimulation by self-peptide MHC should induce independent, single stochastic cell divisions. That being the case, ongoing homeostatic
cell division should rely on constant signals from the lymphopenic
environment to maintain continued cell division. Reversal of lymphopenia should, therefore, bring about a cessation of cell division,
which would not be the case if the divisions were programmed or
FIGURE 5. Validation of predictions made by the mathematical model.
A, F5 T cells were labeled with CFSE and injected (106/mouse) into
Rag1⫺/⫺ hosts. At day 3, one group of mice was injected with 108 WT T
cells. Mice were bled at days 3, 5, 6, and 11 and expression of CD8 and
CFSE labeling analyzed by FACS. The graph shows the mean divisions of
F5 T cells in the two different groups at the days indicated. B, Rag1 ⫺/⫺
hosts were injected with 106 F5 T cells. Two weeks later, CFSE-labeled F5
T cells (106/mouse) were injected into the pretreated hosts or empty
Rag1⫺/⫺ mice as control. Groups of these mice were taken at different
times and the mean division of CFSE-labeled F5 T cells determined. The
graph shows the mean divisions with time of CFSE-labeled F5 T cells in
empty Rag1⫺/⫺ controls (F) and in Rag1⫺/⫺ mice pretreated with unlabeled F5 T cells (ˆ). Data are representative of three independent experiments. C, F5 T cells were injected (2 ⫻ 106/mouse) into either Rag1⫺/⫺ or
C57BL/6 Ly5.1 WT hosts. At days 1, 2, and 3 post transfer, cells were
recovered and labeled for CD8, TCR, Ly5.2, and CD5. Histograms are of
FSc (top row) and CD5 (bottom row) for CD8⫹ Ly5.2⫹ TCRhigh gated F5
T cells from either Rag1⫺/⫺ hosts (solid lines), WT hosts (broken line) or
from F5 control mice (gray filled). Data are representative of two independent experiments with 4 mice per group.
on autopilot. To test this directly, we injected groups of Rag1⫺/⫺
mice with CFSE-labeled F5 T cells. Three days later, when cell
division of F5 T cells had already started and was at its highest
rate, we injected cohorts of mice with very high doses of WT T
cells (108/mouse) to quickly reverse the lymphopenia. Strikingly,
such treatment abruptly halted the observed cell division of F5 T
cells in these hosts, whereas the same cells in untreated Rag1⫺/⫺
hosts continued to divide (Fig. 5A). Although F5 T cells had divided when examined 2 days later in treated hosts, they had undergone fewer divisions than F5 T cells from untreated control
hosts and there was no further cell division detected in the treated
hosts up to 17 days later. Only 27% of cells in treated hosts underwent any cell division, compared with 83% in controls. Were a
deterministic program of division at work, then cells triggered into
division before treatment would be expected to complete their program of divisions. Rather, in treated mice, the vast majority of the
dividing population (80%) underwent only a single division.
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␭ is defined such that the probability of any given cell entering B
phase in a short time interval ␦t is simply ␭␦t. We also estimated a lag
time T after transfer during which no entry into B phase is possible.
The assumption of a constant mean rate of entry into division (␭)
after the lag time T gave a reasonable fit to the data (Fig. 4B). This
model predicts a linear increase in the mean and variance of division
number with time after a brief transient. The best-fit parameter estimates with 95% confidence intervals were ␭ ⫽ 0.210 day⫺1 (0.157,
0.266), ⌬ ⫽ 6.65 h (3.29, 18.8), and T ⫽ 2.03 days (0.172, 2.72).
Therefore we estimate the mean duration of the A phase, ␭⫺1, to be
around 4.8 days, and the average time between divisions for each cell
(␭⫺1 ⫹ ⌬) as just over 5 days. The trend in the mean division number
averaged over all animals at each time point (Fig. 4) suggested a
progressive slowing of the division rate with time. To investigate this
possibility we extended the model to allow a time-dependent change
in division rate, defined by an additional parameter ␮ such that ␭ ⫽
␭0 exp(⫺␮t). This improved the fit significantly (partial F-test, p ⬍
10⫺4; ⌬AIC ⫽ 14; Fig. 4C). Best-fit parameter estimates were ␭0 ⫽
0.455 day⫺1 (0.228, 3.58), ␮ ⫽ 0.115 day⫺1 (0.020, 0.683), ⌬ ⫽
6.19 h (2.26, 15.1), and T ⫽ 2.44 days (1.99, 2.90). The slowing of the
division rate means that at day 3 the mean interdivision time was 3.4
days, increasing to 11.3 days at day 14.
Both the simple and modified SM model provided a significantly better fit to the data than the best fitting “programmed divisions” GH model (⌬AIC values; GH ⫺ basic SM model ⫽ 214;
GH ⫺ modified SM model ⫽ 227). Despite the poor fit of the GH
model, however, we clearly cannot rule out this model with other
distributions for the times to first division. In particular the assumption that all cells receive their first division stimulus at the
same time, as is likely in in vitro Ag-driven proliferation assays,
may be questioned in vivo. Nevertheless we note that the gradient
of the curve of mean division number with time is the inverse of
the average time between divisions, which is roughly 5 days for the
linear fit predicted by the simple SM model (the quantity ␭⫺1 ⫹ ⌬)
and between 3 and 11 days for the extended SM model with slowing division. Irrespective of the precise form of the distribution of
times to first division, the GH model predicts that a plot of mean
division number versus time tends to a straight line with slope 1/⌬
after all cells have entered the proliferative program and are dividing asynchronously but uniformly with constant division time
⌬. Crucially, the slope we observe indicates an interdivision ⬎10fold slower than that observed in Ag-driven T cell responses (18)
and such slow proliferation is unprecedented in programmed responses. In summary, the SM model is a good description of the
data, with even the more parsimonious three-parameter version
describing proliferation better than the GH autopilot model.
The Journal of Immunology
Discussion
Cell division induced in T cells by cognate Ag or by lymphopenia
both depend on TCR signaling but result in distinct patterns of
proliferation and, for many cells, differences in development of
effector function. Previous studies have successfully used mathematical modeling to describe T responses to cognate Ag and
greatly facilitated our understanding of the proliferative control
involved. Here, we used a mathematical modeling approach to
better understand the program of cell division that is induced in
lymphopenia. In contrast to the autopilot program of deterministic
cell divisions induced following stimulation by cognate Ag, we found
that a model in which T cells underwent independent single stochastic
divisions better described the F5 T cell response to lymphopenia.
Variants of the SM stochastic single divisions model have been
used by other authors to describe turnover of both T (31, 32) and B
(33) cells and implicitly held it to be a good description of their data
but did not formally compare alternative proliferation mechanisms.
Our successful application of the SM-based model of single stochastic
divisions (Fig. 3), however, still does not prove it is correct or exclude
other possible models. What did provide strong support for the SM
model was the predictions it made that were successfully validated
experimentally and that some of these predictions were incompatible
with an autopilot model. Reversing the lymphopenia at the peak of the
proliferative response had a profound inhibitory affect on the response, consistent with the SM model in which cells underwent in-
dependent single stochastic divisions and that additional divisions required further stimulus from the lymphopenia. Were cells stimulated
into a deterministic program of divisions, they would have already
received their proliferative stimulus and would have continued upon
the deterministic program of division that could not be reversed by
later removal of the inducing stimulus, i.e., by reversing lymphopenia.
In contrast, this is certainly the case for activation of CD8⫹ T cells by
cognate Ag that requires as little as 4 h stimulation to evoke a full
proliferative response (34).
Another prediction made by the SM model that we tested was
that the division rate slowed with time. Both models were compatible with a strictly linear increase in mean division that was
close to what was observed experimentally. However, the fit of the
SM model was improved by introducing a time dependent reduction in rate of entry of cells into division. Closer experimental
scrutiny revealed that lymphopenia-induced proliferation of F5 T
cells did indeed slow with time (Fig. 5). The magnitude of the F5
T cell response to lymphopenia was exquisitely sensitive to the
lymphopenic environment, such that altered levels of IL-7 or even
increasing T cell competition by varying T cell numbers was sufficient to have appreciable impact on proliferation. Therefore, it
seems that the strength of the environmental signals that induce T cell
proliferation in lymphopenia are subject to dynamic change as a consequence of T cell expansion, resulting in increasing cellular competition for key resources such as IL-7 and self-peptide MHC. Requiring
T cells to re-audition after single divisions for further lymphopeniainduced divisions, as described in the model, also explains how the
proliferation can be so finely tuned to relatively subtle changes in the
competitive environment. The SM model also predicted that F5 T
cells required an initial 24 – 48 h period of exposure to lymphopenic
environment before undergoing LIP responses, a prediction also validated experimentally by the kinetics of phenotypic changes observed
in F5 T cells. Such a delay may allow for transient, localized fluctuations in microenvironment that might otherwise stimulate LIP but
not in fact necessitate a homeostatic response.
Although the GH model describes Ag-induced proliferation
well, our study required an entirely distinct model of proliferation
to describe lymphopenia-induced proliferation and yet both responses are triggered through TCR stimulation. This raises the
question of how it is possible for signals from the same receptor to
elicit such distinct programs of cell division? One factor might by
that the TCR signals required for induction of lymphopenia-induced proliferation depend on self-peptide MHC recognition
which are likely to be low avidity interactions compared with cognate Ag. However, even low affinity TCR ligands can induce autopilot proliferation, just requiring a longer duration of stimulus
(34). Another possibility may be the differential requirements for
IL-7 and IL-2 for lymphopenia vs Ag-induced proliferation. T cell
responses to cognate Ag have no apparent requirement for IL-7,
either in vivo (17) or in vitro (B. Seddon, unpublished data). Indeed, IL-7R expression is largely lost following activation (17,
35). A key step in the Ag-induced program of proliferation is the
induction of IL-2 and its receptor, following TCR stimulation. This
autocrine axis plays an important role in promoting cell division
and survival following T cell activation. In contrast, lymphopeniainduced proliferation is highly dependent on IL-7 (17, 36, 37).
Like IL-2, IL-7 is also an important T cell survival factor (17, 38)
and has also been described to affect cell cycle by its effects on
expression of negative regulators such as p27kip (39) and FOXO
proteins (40). Both IL-2 and IL-7 are members of the common
␥-chain (␥c) family of cytokines, and consequently, use similar signaling pathways to mediate their biological properties, such as Stat5
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Although the simple SM model described the data well (Fig.
4B), introducing a time-dependent reduction in division rate resulted in a significantly better fit (Fig. 4C). The slowing in division
rate indicated by this analysis was also supported by our observation that F5 T cell homeostatic division is very sensitive to cell
numbers in the host environment (Fig. 2D). It seems reasonable
that the rate of cell division should slow as the T cells expand in
number, either due to the effects of competition or adaptation to the
host. This slowing was not always evident in longer time courses
because at later time points many cells had become CFSE negative, and mean cell divisions could not be accurately determined.
To overcome this limitation and examine the cell kinetics of expansion over longer time periods, we measured the division rate by
injecting hosts first with unlabeled F5 T cells and then transferring
CFSE-labeled F5 sensor cells 2 wk later and measuring their cell
division over time compared with the same cells in untreated
Rag1⫺/⫺ hosts. As the data show, F5 T cells injected into the
pretreated hosts proliferated at a significantly reduced rate compared with cells transferred to empty hosts (Fig. 5B). The division
rate also further reduced with time in both hosts.
Finally, another prediction of the model was that there is a lag
phase of ⬃2 days before cells started to respond to lymphopenia. To
investigate whether there was evidence of such a delay in responsiveness of F5 T cells to the lymphopenic environment, we assessed cell
size and CD5 expression by F5 T cells following transfer to lymphopenic hosts. Previous studies have shown that environmental signals in lymphopenia have trophic affects on T cells (29), resulting in
increases in cell size, while the level of CD5 expression by peripheral
T cells has been shown to be positively regulated by contact with
self-peptide-MHC complexes (5, 30). Upon transfer to Rag1⫺/⫺
hosts, F5 T cells did indeed increase their cell size and up-regulated
CD5 expression compared with the same cells transferred to C57Bl6
controls, indicative of enhanced contact with self-peptide-MHC that is
required to induce LIP (Fig. 5C). However, these changes in F5 T cell
phenotype were not evident until 2 days post transfer and not maximal
until day 3. By 24 h post transfer, the phenotype of F5 T cells in
lymphopenic Rag1⫺/⫺ hosts was almost indistinguishable from the
same cells in replete C57BL/6 controls.
1421
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MODELING LYMPHOPENIA-INDUCED T CELL PROLIFERATION
Acknowledgment
We thank Roslyn Kemp for critical reading of the manuscript.
Disclosures
The authors have no financial conflict of interest.
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(41) and PI3K (42). Therefore, IL-2 and IL-7 may play parallel biochemical roles in supporting survival and proliferation of T cells during Ag and lymphopenia-induced proliferation, respectively, even if
their administration is different. Also, it is interesting to note that activated T cells do not express IL-7R, while cells undergoing lymphopenia-induced proliferation do not express IL-2R.
Crucial differences in the biology of IL-2 and IL-7 may ultimately account for the fundamentally different programs of cell
division they support. Following T cell activation, induction of
IL-2 and IL-2R forms part of an autocrine loop that plays a key
role in the burst of proliferation T cells undergo following antigenic stimulation, and it is therefore possible that it is the maintenance of just such a loop that underlies the deterministic program
of divisions that is characteristic of Ag-induced responses. In contrast, enhanced self-peptide MHC recognition and increased bioavailability of IL-7 in lymphopenia are the key events that trigger
some T cells into cell division. However, since IL-7 is made by
stromal cells within the lymphoid compartment (43) and not by T
cells and all T cells express IL-7R, dividing T cells are therefore in
equal competition with nondividing T cells for continued stimulation
from IL-7 to support continued division. Since T cells cannot make
IL-7, there is no autocrine loop that might permit a deterministic burst
of cell division. Demanding that T cells repeatedly audition for further
lymphopenia-induced cell division permits the homeostatic responses
to be highly reactive to the current size of the T cell compartment. In
conclusion, the SM model of cell division not only provides a remarkably accurate description of lymphopenia-induced proliferation
by F5 T cells but also has many of the features required for dynamic
homeostatic regulation of T cell numbers.