Wrote the paper: SMC BHD MLM TBK. Conceived the study: MLM TBK. Developed mathematical components: SMC TBK. Interpreted results: SMC BHD MLM TBK. Developed empirical components: BHD MLM.
The authors have declared that no competing interests exist.
T cell populations are regulated both by signals specific to the Tcell receptor
(TCR) and by signals and resources, such as cytokines and space, that act
independently of TCR specificity. Although it has been demonstrated that
disruption of either of these pathways has a profound effect on Tcell
development, we do not yet have an understanding of the dynamical interactions
of these pathways in their joint shaping of the T cell repertoire. Complete
DiGeorge Anomaly is a developmental abnormality that results in the failure of
the thymus to develop, absence of T cells, and profound immune deficiency. After
receiving thymic tissue grafts, patients suffering from DiGeorge anomaly develop
T cells derived from their own precursors but matured in the donor tissue. We
followed three DiGeorge patients after thymus transplantation to utilize the
remarkable opportunity these subjects provide to elucidate human Tcell
developmental regulation. Our goal is the determination of the respective roles
of TCRspecific vs. TCRnonspecific regulatory signals in the growth of these
emerging Tcell populations. During the course of the study, we measured
peripheral blood Tcell concentrations, TCR
Protective adaptive immunity depends crucially on the enormous diversity of the
Tcell receptor repertoire, the antigen receptors expressed collectively on
Tcell populations. T cells develop from Tcell precursors that originate in the
bone marrow and migrate to the thymus, where their T cell receptors are
constructed stochastically, and tested for autoreactivity against a host of self
antigens. Complete DiGeorge anomaly is a rare congenital disease in which the
thymus fails to develop, blocking all T cell development and causing profound
immunodeficiency. Thymus transplantation, performed in the first two postnatal
years, allows the patient's own T cell precursors to develop in the
engrafted thymus tissue into normal, functioning T cells. In addition to saving
patients' lives, this procedure provides an extraordinary opportunity
to study the
An essential characteristic of T lymphocytes is their ability, as a population, to
recognize an enormous number of peptide antigens. This capability is essential to
the function of the adaptive immune system and is attributable to the diversity of
the Tcell receptors (TCR) they express. This diversity in turn comes about through
the stochastic assembly of TCR from genomic “libraries” of gene
segments for the TCR alpha and beta chains, the two polypeptides that together make
up the most common form of the TCR
The peripheral Tcell population is maintained at a constant size in spite of
substantial, continual turnover, ongoing thymic production, and clonal expansion in
response to immunological challenges
Diverse lines of investigation imply that signals delivered, and resources provided,
through both TCRspecific and TCRnonspecific channels are essential for the
establishment and maintenance of the size and diversity of T cell populations
Our intent in this paper is to elucidate the mechanisms that lead to expansion and diversification of the TCR repertoire in a recovering Tcelldeficient host: in our case, a complete DiGeorge subject that has received an unrelated, unmatched thymus transplant. After transplantation, host Tcell precursors migrate from the bone marrow to the donor thymus where they develop via thymopoiesis into host T cells that then emigrate into the peripheral blood. Here, T cells have the capacity to undergo spontaneous clonal expansion, thus leading to restoration of the peripheral Tcell pool. The expansion continues until a steady state is reached. Throughout this process, selective pressure for survival emerges among and within the clones through competition for stimulatory signals.
The steady state Tcell population size arises in the balance among several
phenomena, including the rapid and extensive expansion of rare clones through
activationinduced peripheral division and their subsequent contraction (which
constitute, in part, the Tcell immune response) and the relatively slow
turnover of a diverse pool of naive cells through continuous thymic emigration
and cell death. The specific memory T cells that arise as the result of clonal
expansion appear to be regulated largely independently of the naive cells
Both the size and diversity of peripheral T cell populations are controlled
through competition for limiting resources. (In the interest of simplicity, we
will use the term “resources” familiar from ecological
population studies to refer to all of the factors that mediate growth
regulation. We do not intend by this to exclude factors that are more accurately
referred to as “signals”)
Our current understanding of lymphopeniainduced proliferation is due to studies
in mice demonstrating that T cells divide rapidly after transfer into T
celldeficient (usually due to RAG or CD3
The importance of TCR specific signals has been studied at length, showing that
competition within T cell clones is important in maintaining TCR repertoire
diversity. It has been shown, for example, that in a Tcelldeficient host, a T
cell must interact with antigenpresenting cells bearing the MHC allele
responsible for that cell's thymic selection in order to proliferate
TCRnonspecific signals include cytokines such as the cytokine interleukin7
(IL7), which is necessary for the survival of nave T cells
To assess the contributions of thymic emigration rate on the steadystate Tcell
population size, Berzins et al
Complete DiGeorge Anomaly (cDGA) is a congenital condition, the hallmark of which
is profound immunodeficiency arising from abnormal development of the third and
fourth pharyngeal pouches resulting in thymic aplasia (complete absence of the
thymus). This developmental irregularity may also cause various other anatomical
abnormalities including heart defects, hypoparathyroidism, and craniofacial
malformations
Thymus transplantation is emerging as a valuable treatment for athymia generally. There is therefore substantial medical interest in elucidating the immunological recovery of thymus transplant patients quite apart from the basic immunology these patients may reveal.
To illustrate the connection between these alternative homeostatic mechanisms and
the observations one might make on the DiGeorge subjects, consider the following
two scenarios as a thought experiment. First, suppose that TCRspecific
resources such as spMHC are not limiting, either because they are produced in
great excess or because they are rendered unnecessary. Under these conditions,
homeostasis will be due exclusively to competition for TCRnonspecific
resources. The first T cells leaving the thymus will expand rapidly, consuming
the TCRnonspecific resources required for growth of their own growth and for
the growth of all other clones. Subsequent T cells leaving the thymus will
encounter a more impoverished environment and will grow more slowly, leading to
early dominance of one or a small number of early clones and therefore a limited
TCR repertoire (
Clone sizes over time as computed under the model of Eqs.(1,4)
(A–C) and the corresponding T cell concentrations (D) and TCR
repertoire diversity (E) as a function of time past transplantation.
computed under the model of Eqs.(1,4) as a function of time past
transplantation. Parameter
Reality will lie somewhere between these two limiting cases: TCR repertoire diversity will be shaped by both TCRspecific and TCRnonspecific resources. Our goal is to explore, quantitatively, how these two sets of signals interact dynamically to shape the mature functional T cell population.
The quantitative contributions of different signals in the regulation of Tcell number and the diversity of the TCR repertoire are very difficult to determine experimentally in the context of an intact system. To examine the interplay among the mechanisms responsible for the heterogeneity of the Tcell repertoire, we have developed a mathematical model for the effective interactions among Tcell clones. We model the temporal evolution of Tcell clones and their dynamics under the combined effect of TCRspecific and TCRnonspecific signals. In particular, we consider competition within clones for spMHC complexes presented by antigenpresenting cells as well as competition among cells in all clones for cytokines, space and other TCRnonspecific resources.
Such mathematical models have been used extensively to study the dynamical
interactions between different branches of the immune system
We use a populationdynamic model for the analysis of patient data. The state of the
model at any time is given by the variables
The maximum growth rate for model cells is denoted
Eq.(1) provides the basis for the numerical data fitting we perform, but analysis of
the model itself is facilitated by replacing the the Poisson process in Eq.(1) by
its timedependent mean value, ie, neglecting the fluctuations due to the
discreteness of the emigration process. This approximation is valid near steady
state, where each of the clones is substantially filled. In this approximation, the
dynamic equations for, eg, the clone proportions
Using this same approximation, the dynamics of the total Tcell population size is
given by
At the single stable steady state, which is unique and stable, all clones have the
same size, independent of
When
Because the thymic graft is likely to become functional only gradually over time, we
model the thymic emigration rate as a Hill function,
This parameterization is atypical, but proves to be practical for performing the
fitting, as will be described in the
Sample paths under the model are generated using software written in C# under Visual
Studio 2008, and the Troschuetz pseudorandom number generator classes (
We then draw a Poisson random variable, call it
This study was conducted according to the principles expressed in the Declaration of Helsinki. All subject samples were obtained under protocols approved by the Duke University Medical Center Institutional Review Board (IRB).
Between 13 and 17 blood samples per subject were collected over a period of 2–3 years following thymus transplantation in three infants with complete DiGeorge anomaly. T cell receptor diversity in CD4 T cells was assessed by obtaining frequencies of TCRBV family usage by flow cytometry, and of complementarity determining region 3 (CDR3) length distributions within TCRBV families by spectratyping and using these frequencies to estimate the diversity as described below.
Standard flow cytometry on whole blood samples was performed as previously
described
Spectratype analysis provided information about CDR3 length diversity within
each functional TCRBV family. Briefly, CD4 T cells were isolated from the
peripheral blood of subjects and controls. RNA was prepared and used for
complementary DNA (cDNA) synthesis. The cDNA was used as a template for 23
TCRBVspecific primer pairs covering 21 TCRBV families to amplify the
complete CDR3 region by PCR
Raw CD4^{+} spectratype data (upper panel) for subject 1 on days 70 (left) and 183 (right) posttransplantation. CD4^{+} TCRBV usage frequency for average over 10 healthy controls (solid bars), and subject 1 (striped bars) on the same two days. The raw spectratype profiles are not represented on a consistent scale. Assays that had no peaks above 500 fluorescent units are routinely excluded from subsequent analysis. These are marked with an asterisk.
The KullbackLeibler divergence (
Furthermore,
The
The inverse of the
The sample
Subject  days posttransplantation  # cells/RT reaction  CD4^{+} T cell events 
1  70  63,636  3,766 
88  200,000  1,130  
117  38,889  4,437  
145  125,000  6,396  
183  58,696  8,106  
398  441,176  17,261  
2  175  109,091  20,302 
209  71,429  26,329  
286  67,273  36,712  
3  102  126,667  9,090 
130  230,000  15,058  
166  51,020  20,669  
372  892,857  43,371 
“# cells/RT reaction” is the number of cells used in RNA extraction for spectratyping. The numbers were estimated from the starting number of CD3^{+} cells and RNA yield. All spectratyping used 150 ng of RNA regardless of the cellular sample size. “CD4^{+} T cell events” refers to the total number of flow cytometric events in the CD3^{+}CD4^{+} gate used to determine the proportion of cells in each of 22 TCRBV families as described in the text.
The model is based on absolute Tcell numbers, but the data are blood T cell
concentrations. We must use a conversion factor
Assuming that a 10 kg subject has 600 ml of blood, and that there are 45 times
more lymphocytes in the tissues than in the blood, based on adult data
Furthermore, some subjects have a relatively small number of anomalous T cells at
the time of transplantation. We denote the concentration of such cells
We estimate the remaining parameters
We estimate parameters by using the likelihood function given by
The parameters are all manifestly positive, and
The maximumlikelihood parameters estimates are shown in
meaning  units  subject  
1  2  3  

thymic emig. amplitude  10^{−4} cells/day 
6.5, (6.2,6.9)  9.5, (8.4,10)  21.5, (16.9,21.8) 

max. growth rate  day^{−1}  2.67, (2.58,2.72)  0.52, (0.41,0.72)  0.45, (0.39,0.98) 

carrying capacity ratio  10^{−4}  6.2, (5.2,7.7)  27, (25,58)  1.4, (1.4,3.4) 

carrying capacity  10^{10}  2.6, (2.4,2.7)  3.7, (3.3,3.9)  2.9, (2.4,3.3) 

emigration rate exponent  1  1.9, (1.8,2)  1.9, (1.9,2.1)  1.9, (1.6,2.1) 

baseline T cell conc  cells/µl  59, (53,60)  10, (10,11)  26, (21,35) 

1  0.07, (0.069,0.081)  0.077, (0.074,0.1)  0.12, (0.11,0.23)  

error variance, T cell conc  (cells/µl)^{2}  13, (13,16)  1.8, (1.7,2.4)  16, (16,34) 

error variance, 
10^{2}  10, (10,16)  0.38, (0.07,0.58)  19, (19,56) 
Maximumlikelihood estimates and upper (UL) and lower (LL) limits on the 95% credible intervals for model parameters.
The estimate of greatest interest for us is
The parameter governing total thymic emigration,
We can examine the estimated values of the the
Maximumlikelihood fits of the populationdynamic model given by
Eqs.(1,4) to patient data. The curves are the trajectories of
In order to gain a deeper understanding of the role of
A reasonable concern is that thymic emigration rate and
Maximumlikelihood fit of data from subject 1 to the populationdynamic
model given by Eqs.(1,4). The black curve represents the maximum
likelihood solution; the orange and blue curves represent the maximum
likelihood solutions subject to a constraint on
Model 










10^{−4} cells/day 
day^{−1}  10^{−4}  10^{10} cells  cells/µl  (cells/µl)^{2}  

10  2.6  65  2.6  2.3  51  0.073  0.15  1.27  8.7 

5.9  2.5  0.65  1.4  1.9  46  0.069  0.32  0.12  7.7 
A focused analysis of the timedependent sensitivity of the model trajectories to
parameter variation may provide further insight into the mechanisms of
regulation. We examined the sensitivity of our model to changes in
Define the sensitivity functions
The sensitivity to
We know that mean sensitivity to
Relative sensitivity trajectories for the model fit to data from subject
1. The left panel shows the relative sensitivity to changes in
Increases in the thymic emigration rate
The model does not account for the microenvironmental perturbations invariably
encountered by people, including our study subjects. Infection produces a
transient change in the steady state of the T cell repertoire, typified by a
decrease in the diversity. Once the infection has been resolved, the Tcell
population returns to its steady state, and the diversity relaxes back to its
original value. The rate at which the return to steady state values occurs
depends on
We start with a system at steady state, and suddenly increase the size of a
single clone by a factor of 10^{4}, allowing it to consume the nonTCR
specific resources its new size requires. The diversity decreases as a result.
Over a few days, the other clones adjust to the new steady state. After ten days
we remove the artificial support for the enlarged clone and allow the system to
return to steady state (
The system was initialized at steady state, and was subjected to the
sudden enlargement of a single randomly chosen clone by 10,000fold.
This clone was held artificially high for 10 days, after which the
system was allowed to relax back to steadystate. The artificially
enlarged clone consumes TCRnonspecific resources at a rate appropriate
to its size. The TCR repertoire diversity is shown in the bottom panel,
the T cell concentration is shown in the panel above.
The functional integration of the manifold processes involved in the development and maintenance of the mature T cell repertoire has not yet been fully elucidated. These processes include thymic export, competition for TCR ligands, and competition for nonspecific stimulatory factors. Here, we study T cell homeostasis in complete DiGeorge Anomaly patients during the establishment of T cells following thymic transplantation.
To quantify the balance between TCRspecific and TCRnonspecific factors that act to
limit Tcell population growth, we developed a mathematical model that accounts for
intra and interclonal competition. The key parameter in this model for the present
purpose is
At steadystate, TCR diversity is maintained through competition for self peptide MHC
Our models fit the data adequately only when the thymic emigration is an accelerating
function of time through the initial posttransplantation phase, as would be
expected from the biology of thymus transplantation
The current study was not able to treat naive and memory T cells separately, but
doing so is clearly of great interest given the indications that these pools are
regulated independently of each other
Our aim has been to study T cell homeostasis in a more natural human system than has
otherwise been available thus far. It remains to be shown just how natural the
posttransplantation environment is in this regard, and therefore how broadly
applicable our findings are. We are encouraged, however, by the fact that complete
DiGeorge patients receiving thymus transplants recover adaptive immune function. It
is important to note that total Tcell counts in these recovered patients remains
below normal, typically at about the 10th percentile for children of the same age
Our study enabled us to discriminate between two qualitatively different forms of population regulation: TCRspecific regulation and TCRnonspecific regulation, and to elucidate their roles in jointly maintaining the dynamic homeostasis of the T cell population. The parameters estimates obtained by analyzing data from three DiGeorge Anomaly patients suggest that Tcell population size is maintained by TCRnonspecific mechanisms, and TCR repertoire diversity is maintained by TCRspecific mechanisms.
Supporting Information
(0.06 MB PDF)
We acknowledge the help of John Tomfohr, the technical assistance of Marilyn Alexieff, Jie Li, ChiaSan Hsieh, Jennifer Lonon and Julie E. Smith, the clinical research assistance of Stephanie Gupton and Alice Jackson, and the regulatory affairs assistance of Elizabeth McCarthy and Michele Cox are appreciated as is the clinical care by the faculty and fellows of the Duke Pediatric Allergy and Immunology Division. We acknowledge the collaboration of surgeons James Jaggers, Andrew Lodge, Henry Rice, Michael Skinner, and Jeffrey Hoehner. We appreciate the assistance of Drs. Michael Cook and Scott Langdon in the Duke Comprehensive Cancer Center flow cytometry and sequencing facilities.