Comput. Math. Applic. Vol. 14, No. 9-12, pp. 657-697, 1987 0097-4943/87 $3.00+0.00 Printed in Great Britain Pergamon Journals Ltd

REVIEW ARTICLE DIFFERENTIATION, REPERTOIRE DEVELOPMENT AND MIGRATION: THE NEED FOR MATHEMATICAL MODELS

J. R. LUMB Department of Biology, Atlanta University, Atlanta, GA 30314, U.S.A.

INTRODUCTION The immune system is a tremendously complicated and important system, whose central players, the , are able to recognize specific foreign invaders (antigens) with their specific cell surface receptors (immunoglobulins on B-lymphocytes and receptors on T-lymphocytes). A number of different subpopulations of lymphocytes assume the various functions of the immune response: / production of specific immunoglobulin, helper/inducer T cells, suppressor T cells and specific cytotoxic T cells. Each of these lymphocyte subpopulations is composed of a large number of clones which each respond to specific antigens via their cell surface receptors. The study of lymphocyte kinetics involves cellular proliferation and differentiation under two different conditions: (1) the initial generation of the repertoire of lymphocytes, each of which displays a specific cell surface receptor, occurs in the absence of triggering by antigen; and (2) the specific clonal lymphocytic proliferation and differentiation occurs in response to antigen recog- nition by these receptors. Both processes involve considerations of cell cycle kinetics and population dynamics, and neither is completely understood. One approach to the study of complex systems, along with the experimental approach, is to construct mathematical models of the processes. Most of the mathematical models of the immune system have focused on the second phase, the antigen-induced activities (for reviews, see Refs [1-6]). This work will focus on the first phase, the development of the lymphocyte repertoire. Mathe- matical treatment of the development of the lymphocytes is limited to early models of thymus cellular kinetics constructed before the lymphocyte subpopulations and precursors were described. A thorough search of the literature showed no work in this field for the last 10 years. The current level of understanding of lymphocyte differentiation makes this an untapped fertile field for theoretical research. It is the purpose of this work to encourage such theoretical research by

(1) reviewing mathematical treatment of lymphocyte differentiation or analogous systems; (2) providing a coherent review of lymphocyte differentiation as it is now under- stood; (3) raising the crucial questions which remain to be solved in this field; and (4) suggesting mathematical treatment which may be helpful in resolution of those crucial questions. First, an overview of the immune response will be provided as background. Then, using selected models of the antigen-induced phase of the immune response as examples, theoretical methods will be reviewed. The current understanding of the differentiation sequence of each subpopulation of lymphocytes will be described in detail, along with an examination of relevant models. Similarly, lymphocyte migration, including the only existing theoretical treatment, will be discussed. Finally, the conclusions will include a summary of the crucial questions identified and suggestions of possible theoretical treatments. In order to construct a mathematical model of lymphocyte differentiation, certain pieces of 657 658 J.R. LUMB information are required. The basic questions are: (1) What are the stages of lymphocyte differentiation as defined by markers of specific cell types? (2) What is the sequence of differentiation in each lymphocyte subpopulation? (3) How many cells of each type are in the lymphoid tissues at different times during ontogeny? (4) How are these cells distributed between dividing cells and resting cells? (5) What is the generation time for the dividing cells? (6) What is the product of the division, i.e. is the cell a self-renewing population or does it divide into cells which are further differentiated? (7) How long does a particular cell remain in each population? (8) How many cells die at each stage of differentiation? (9) How is the repertoire of cells with specific receptors generated and controlled? (10) Which cells migrate and to which tissues? (11) What is the rate of migration of each cell type into and out of each tissue? The process of model construction makes quite apparent which pieces of information are lacking and organizes those which do exist. The reader will find that the information available about lymphocyte differentiation varies from cell populations which are only hypothesized to relatively well-described populations with known cellular kinetic parameters.

L ymphocytes Mature lymphocytes are divided into two major populations, B-lymphocytes (B cells) which display specific antibody structures, and T-lymphocytes (T cells) which also display specific antigen receptors called T cell receptors. Both populations have several subpopulations which will be described in detail in the sections B Cell Differentiation and Intrathymic T cell Differentiation. B cells mature in a dedicated organ in birds, called the Bursa of Fabricius, which is the origin of the name B cell. However, in mammals the differentiation of B cells occurs in the bone marrow which also provides other hemopoietic precursors including the cells that migrate to the thymus to become T cells. Upon an encounter with an antigen which will bind to the specific receptor, the mature, non-dividing lymphocyte is activated to become a dividing cell, proliferates through several divisions (5-10) and differentiates into a mature cell. Different subpopulations of T cells either perform a control function, inducing or suppressing specific effector lymphocytes (B cells or T cells), or differentiate into specific cytotoxic effector cells. B cells differentiate into plasma cells which secrete large amounts of the particular antibody molecule which has been displayed.

Early studies on lymphocyte kinetics Detailed cellular kinetic studies were done on lymphocytes over 20 years ago, before the discovery of the different subpopulations. Everett et al. [7, 8] demonstrated both short-lived lymphocytes with half-lives of less than 2 weeks and long-lived lymphocytes with half-lives of at least several months. The lymph had a lower percentage of short-lived lymphocytes (10%) than the (35%). The blood contained lymphocytes from a source that had a direct entry (i.e. without going through the lymph first) and that this source was highly proliferative. They observed that this could not have been the thymus because the cells leaving the thymus were weakly labeled with tritiated thymidine while the cells in the blood were strongly labeled. This indicates that the cells coming from the highly proliferative source (probably the bone marrow) did not divide as many times, diluting out the strong label. The half-time turnover of cells in the bone marrow was 24 h.

Immune response Extensive research has been applied to study the immune system's response to a foreign antigen. As a result, this area is relatively well-understood. The brief synopsis which follows is necessarily a vast simplification. The reader is referred to recent textbooks for more detailed descriptions [9-11]. Lymphocytedifferentiation: the need for mathematical models 659

There are two arms of the immune response, distinguished by the nature of the final effector. Humoral immunity can be transferred from an immune animal to another animal by the injection of serum which includes antibody molecules, but is free of cells. Cell-mediated immunity, on the other hand, requires the transfer of lymphocytes from an immunized animal. The immune response begins with the introduction of an antigen which is taken up by an antigen-presenting cell (either a macrophage or a B cell) and degraded with the fragments presented to the helper T cell in the context of a self-marker (cell surface class II major histocompatibility antigen called Ia). If the specific receptors on the surface of the helper T cell recognize both the antigenic fragment and the self Ia, the helper T cell is activated and proliferates. The mature B cells that recognize the antigen through their surface receptors interact with the helper T cells, and the antigen-presenting cells, and pass through the same activation and proliferation stages prior to differentiation. The resulting amplification produces a large number of B cells responding specifically to the current antigenic invasion. During the amplification somatic mutation of the genes coding for immunoglobulin allow for the generation of additional diversity in the antigen receptor. After activation and proliferation, B cells differentiate into plasma cells which secrete large amounts of the same antibody which was displayed on the surface of the progenitor B cell. Plasma cells are non-dividing and short-lived, having the single purpose of producing antibody specific for the invading antigen. Their demise at the end of a cycle of immune response is one means of control of the humoral immune response. The T inducer cells allow the production of the cytotoxic T cells responsible for the destruction of cells infected with an intracellular parasite, such as a virus, or displaying some other abnormality, such as malignant cells. This is a very important component of the immunity against intracellular parasites such as viruses because viruses are unaffected by antibody once they have infected a cell. The only way to control the infection at that point is to rid the host of the infected ceils. For this response it is necessary that a specific antigen, foreign to the host, be on the surface of a cell which also carries the self class I major histocompatibility antigens. A proportion of the responding lymphocytes, both B and T cells, differentiate into memory cells rather than effector cells. These memory cells are part of the long-lived lymphocyte population and are mature, resting B- and T-lymphocytes that are ready to respond to a second invasion of the same antigen. The response is controlled by the induction of T suppressor cells. This is thought to be the mechanism of the self-tolerance which prevents the immune system from responding to its own antigens and self-destructing. Numerous soluble factors have been isolated that mediate the activation, proliferation and differentiation of lymphocytes. Interleukin-1 (IL-1), produced by the macrophage, facilitates the activation of T inducer/helper cells which have recognized the specific antigen-self Ia complex. These activated T inducer/helper cells then make receptors for interleukin-2 (IL-2) and sub- sequently produce IL-2 molecules which are required for proliferation of T cells and which also stimulate the proliferation of B cells. Other factors involved in specific reactions include ~-interferon (which acts as a B cell differentiation factor), another B cell growth factor, helper factors and suppressor factors.

MATHEMATICAL MODELING METHODS Most of the mathematical treatments of lymphocyte kinetics are in the context of the antigen-induced phase of the immune response. This work has been reviewed extensively and will not be duplicated here [1-6]. Instead, this section will concentrate on selected works to demonstrate the different approaches which have been taken. Cellular kinetic models Most models of cellular kinetics can be applied to the study of lymphocyte kinetics. While it is beyond the scope of this review to consider all available models, a comparison of the types of mathematical models of cell growth is instructive [12]. Partial differential equations were used to consider two variables: time and age of each cell [13], On(a, Q/at + ~n(a, t)/aa = O, t ~ [0, T], 660 J.R. LUMB

where n(a, t) is the cell density per age unit, t is time and a is cell age which ranges from a = 0 just after mitosis to a = T just prior to the next mitosis. The boundary condition, n(0, t), equals 2n(T, t) because at a = T, binary fission occurs. Further refinement of this model added a coefficient for the loss of cells, l(a, t) = li(a) + ld(a, t), where li is the probability of loss of cells from a population due to mitosis and ld is the probability of loss of cells due to death [14]. Thus, the partial differential equation becomes On(a, t)/Ot + On(a, t)/Oa = -l(a, t)n(a, t). Since the age of a cell is assumed to range from 0 to ~, the boundary condition

n(O, t) = .If 2n(a, t)li(a) da was used. Another variation of this model changed the age variable to a maturity variable (m) which considered size or DNA content of the cell and varied from m = 0 just after mitosis to n = 1 at full maturity just prior to mitosis. The maturity variable had a velocity function, v(m, t), which was used to explain the distribution of generation times in a cell population: On(m, t)/Ot + O[v(m, t)n(m, t)]/Om = -ld(m, t)n(m, t), with the boundary condition v (0, t)n (0, t) = 2v (1, t)n (1, t). When the additional variables are set to zero and the additional functions assumed to be constant, these refined models are equivalent to the first one described. Models involving difference equations are especially conducive to the use of the digital computer. Cells are assumed to be contained within N compartments, each corresponding to a cell age between the prior mitosis and the next mitosis [15, 16]. If the generation time of the cell population is To and the initial time is to, at t = to + To~N, the cells will have advanced one compartment. If k is taken as the number of time intervals, then at the time interval k + 1 the cells will have advanced one component. The transition from x(k) to x(k + 1) is accomplished using the unit time shift operator, S. To account for the probability that some cells will age faster or slower than others, a unit dispersion operator, D, is constructed which is a matrix subtracting the fraction of cells that would remain in the same compartment instead of advancing and those that would advance two compartments rather than one. The difference equation relating the state vector at the (k + l)th time interval with that at the kth time is x(k + 1) = P(k + 1, k)x(k), where P(k + 1, k) = DS. This model has been generalized to separate the cell population into the proliferating pool and the non-proliferating pool. Differential equations, derived from population studies, have been used to study cellular kinetics. The general form of these equations is N(t) = g(N(t)), N(O) = No, where N(t) refers to the number of cells at time t and g(N) is the growth function, which allows the cell population total to approach some critical size. An example of such a function is that proposed by Simpson-Herren and Griswold [17]: N(t ) = -- aN(t) In [N(t)/Nm ], where a is a constant and Nm is the maximum number of cells expected. Sundareshan and Fundakowski [12] use the multicompartmental model of Takahashi [18, 19] to determine the conditions under which the models using partial differential equations, those using difference equations and those using ordinary differential equations are equivalent.

X- Y-Z models Jilek and Prikrylova [20] examine the models which use the X-Y-Z scheme first described by Sercarz and Coons [21] and Sterzl [22]: X represents the circulating cells with the potential to Lymphocyte differentiation: the need for mathematical models 661

respond to a particular antigen, Y cells are the progeny of X cells activated by the antigen to proliferate and Z cells are the differentiated progeny of Y cells that are now able to produce antibody for a limited time. The authors provide a very useful comparison of X-Y-Z models showing the varying terminology used for X, Y and Z cells. They concluded that the X-Y-Z scheme works well in representing the immune response and probably will continue to be used. An example of such a model is ABIGAIL (Table 1) which was constructed by Bell [23-25]. In ABIGAIL, considerations of chemical equilibrium are used to simulate the interactions of antigen and antibody, whether on the surface of X cells or as free molecules in the circulation. The model considers a number (J) of cellular populations having receptors with a range of affinity constants for binding with a particular antigen. Binding constants (kj) were used to determine the fraction of group j receptor sites bound to antigen (rj): rj = kjf /(1 -1- kjf ),

where L is the number of unbound antigen sites. A threshold fraction (frnin) of occupied receptor sites on the surface of X cells triggered the transformation of that cell into a Y cell that divides through a given number of divisions before differentiating into a Z Cell which produces antibody. The proportion of cells stimulated is determined by Ry = r/fmin, Fj = Ry/(1 q- Rj), tty= (Ry- 1)/(Ry-t- 1). The parameters Fy and/-/y determine the stimulation of target and proliferating cells, respectively, in the cellular kinetic equations shown in Table 1. As more antibody is produced, the antigen is bound by the free antibody and is not available for binding to the membrane-bound antibody on X cells. Thus, the clearance of antigen and cessation of the immune response is simulated. In a report which combines considerations of the development of lymphocytes with the effect of the immune response, Lumb and Morrison [26] used THYMS [27, 28], which simulates the differentiation of mature lymphocytes, as the source of immunocompetent cells (X cells) in the ABIGAIL model. The THYMS equations, shown in Table 2, will be derived in later sections (Theoretical Approaches to the Progenitor Cell Problem and Theoretical Approaches to the Thymus Problem). Although THYMS was designed as a model of the production of T cells by the thymus, it was used in this interfaced model (THYMS/ABIGAIL) as an approximation of the production of B cells in a thymus-independent immune response. Since THYMS was capable of simulating the recovery of the immune system from irradiation, it was possible to test the effects of different levels of irradiation upon the immune response in the model and compare the data with that obtained in experimental systems. The results were similar, verifying this model. Kaufman et al. [29] use formal logic in a fresh approach to the derivation of a system of differential equations. This model gives values of 0 or 1 to the elements of the system (virgin B cells, mature B cells, T helper cells, T suppressor cells, antibody and antigen), depending on whether or not the concentration is greater than a threshold value. The rate of production of each variable is also given a logical value of 0 or 1, depending upon whether or not it is great enough to be considered. The differential equations derived to assign the logical value are similar to those described in ABIGAIL except that this model also includes the inhibition of responses caused by T suppressor cells.The advantage of this approach is that the state of the system can be considered separate from the quantitative results of the differential equations.

Predator-prey models The Volterra-Lotka equations [30-32] developed for the mathematical treatment of the predator-prey relationship have been applied to the immune response where the antibody molecule [33] or the B cell/plasma cell [34] is considered to be the predator and the antigen to be the prey. The similarity to the predator-prey situation lies in the fact that both the predator and prey are self-replicating or are produced by self-replicating entities. When an interaction occurs, the prey (antigen) is eliminated from the system. The equations from Dibrov et al. [34] for antibody, A (t), antigen G(t) and target cell X(t) concentrations at time t are dA (t)/dt = aX(t - T)g (t - T)S(t - T) - rG(t)A (t) - eA (t), dG(t)/dt = kG(t ) - qG(t)A (t)

C.A.M.W.A. 1419-12--B 662 J.R. LUMB

Table 1. Equations used in ABIGAIL (shown as modified for the interface with THYMS) J ffi Number of classes of cells with different affinity constants for binding to antigen N~.j(t) = Concentration of type i cells in group j, where i = 1, 2, 3, 4 denote respectively target, proliferating, plasma and memory cells, j = 1, 2 ..... J Fp = Fraction of proliferating cells that become plasma cells Fj = Parameter reflecting how close a target cell is to being sufficiently bound to antigen to become a proliferating cell Hj = Parameter reflecting how close a proliferating cell is to being sufficiently bound to antigen to divide T I = Mean time for the target cell to become a proliferating cell T 2 = Mean time for division of a proliferating cell T~, T~, T 3, T4 = Mean time (respectively) for death of target, proliferating, plasma and memory cells in the absence of antigen T2(t) = Function which limits proliferating cells to a specified maximum Elimination of antigen--antibody complexes Abj(t) = Concentration of antibodies in group j Ag(t) = Concentration of antigen molecules T 5, T~, T6 = Mean time (respectively) for removal of antigen combined with antibody, free antibody and free antigen r/= Fraction of antibody sites bound to antigen c2, c 3 = Number of antibody molecules produced per unit time (respectively) by proliferating cells and plasma cells n = Number of sites per antigen molecule for binding to antigen Concentration of antibody dabj dt = c2N2"j (Source of antibody produced by proliferating cells?)

+c3N3.j (Source of antibody produced by plasma cells)

2rj 2l~.,rjAby(1-;)+l nr 5 Ag Ab, (Elimination of antibody bound to sufficient antigen) --AbJr; (Elimination of all antibody) Concentration of antigen

dAg = S(t) (Imposed source of antigen) dt - 2~rjAbl/nT5

-2 ~ rjAbj/nT5 (Elimination of antigen bound to antibody) j=l -- Ag /T6 (Elimination of all antigen) Concentration of target cells dNl,~(t) dt -FjNLJ (Loss of target cells to proliferating population) +(1 - Fp)(1 - Hj)N2,j/T2 (Addition of memory cells to target cell population) -- NI,j/T'I (Target cell death) + Lym (t) (Target cell input from THYMS) Concentration of proliferating cells dN2,j(t) dt F/NI"j/TL (Source of proliferating cells due to transformation of target cells) + HjN2,i/T2(t) (Multiplication of proliferating cells) -- N2, j~ T~ (Proliferating cell death) Concentration of plasma cells dN3,j(t) dt Fp(l - Hy)N2,j/T2(t ) (Source of plasma cells due to differentiation of proliferating cells) -- N3.y/T3 (Plasma cell death) Concentration of memory cells

dN4'/(t) (I - Fp)(l - Hj)N2,j/T2(t ) (Source of memory cells due to differentiation of proliferating cells) dt --N,,j/T4 (Memory cell death) and

dX(t)/dt = J - X(t)/d -pX(t)G (t) + mX(t - M)G (t - M)S(t - M), where a, r, e, k, q, p and m are arbitrary coefficients; T is the time delay between the stimulation of X cells and the production of antibody by Z cells; M is the time delay between the stimulation of X cells and the appearance of memory cells in the target cell pool; S(t) is a step function which is equal to 0 if t < 0 and equal to 1 if t >/= 0; J is the rate of appearance of target cells from precursor populations; and d is the average lifespan of target cells. This model and that of Bell [24] both result in oscillating systems, which has been also observed in natural immune responses [351. Lymphocyte differentiation: the need for mathematical models 663

Table 2. Equations used in THYMS (shown as modified for the interface with ABIGAIL) Bone marrow compartment Li dL, ~ 2(i -~)G,L, t = Time dt k L~ = Stem cell population Sm= Maximum number of stem cells allowed G I = Reciprocal of generation time of stem cells

L 2 = Number of dS~ -- = - DG~ Sm D ~ Rate of decrease of S m (degeneration parameter) dt Thymus compartment

dLj+, = 2GjL)--Gj+,Lj+, Lj = Number of cells in the jth generation dt Gj = Reciprocal of generation time of the jth generation where 2j = n - 1 dL~=2Gn iL. i--0.017Ln L, = Number of cells in the last generation in the thymus dt G,_ ~ = Reciprocal of generation time of the (n - l)th generation Blood vessel compartment dE d~ = 2GeE E = Total number of endothelial cells G e = Reciprocal of generation time of endothelial cells 1 NPT k -- ~-50 NP = Nutrition parameter G~ Tk--Tk_1 T~ = Total number of lymphocytes in the kth iteration of the model Irradiation compartment Calculate fraction of cells remaining (F) F/= 0.37 exp(D/D37) D = Dose of irradiation D37 = Dose of irradiation ofjth type of cell at F/= 0.37 Decrease number of cells in population ofjth type of cell by F/ /) = F:L/ Decrease cell division following irradiation

/120- T'~ G,=Go~-) - F/

Optimization and control theory Optimal control theory has been used to advantage in the design of models of the immune response [36-38]. The assumption that the immune system has evolved by natural selection suggests that optimal conditions would be selected for. However, evolutionary changes are only miniscule changes in any one time interval. Therefore, one would expect that the system would be optimized within a limited range of possibilities which are bounded by lethal conditions. Thus, mathematical treatments which determine the general optimal condition may not be applicable to the natural situation [39]. Nevertheless, optimal control theory seems applicable within the boundaries of specific aspects of the immune response. For example, one can ask the question: What is the optimal number of divisions of lymphocytes during the proliferative stage of an immune response. If too many divisions occur, the response will be high because of all the effector cells eventually produced, but it will be too late to be of any use in the immune response. If too few divisions occur, the response will be timely, but too low. Perelson et al. [36] designed a model to address this question: dA/dt = k(L + nP), dL /dt = bu( t )L - m[1 - u( t )]L -dL, dP /dt = m[1 - u( t )]L - eP, where A, L and P are the concentrations of antibody, lymphocytes and plasma cells; k is the rate of production of antibody by proliferating cells and n is the incremental increase of this rate in plasma cells; b is birth rate of the fraction u(t) which continues in the lymphocyte population and 664 J.R. LUMB m is the maturation rate of the 1 - u(t) fraction of lymphocytes which mature into plasma cells; and d and e are the death rates for lymphocytes and plasma cells, respectively. The objective was to select the function u (t) which minimizes the time required to reach the concentration of antibody which will neutralize the given amount of antigen. In control theory this is called the cost function. They applied the maximum principle of Pontryagin which involves the definition of a Hamiltonian (H), a scalar function (for detailed descriptions the reader is referred to texts by Athans and Falb [40], Leitmann [41] and Clark [42]):

H = Vofo + Vt[k(L +nP)] + V2[bu - m(1 - u) - d]L + V3[m(1 - u)L - eP], where 110 = const/> 0 and V,, 1/2 and 1/3 satisfy

dV,/dt = -H/A =0

dV2/dt = - H/L = --kVl -- [bu - m(1 - u) - d]V2 - m(1 - u)V3 and

dV3/dt = -K/P = -nkV~ + eV3, with the final conditions Vj (T) = 1, V2(T) = V3(T) = 0. They found the control to be "bang-bang" which means that it is an on-off switch. Collecting all the terms in H which involve u,

H = [(b + re)V2 - mV3]Lu + all other terms, provides a switching function, f(t) = (b + m)V2(t) - mV3(t), which maximizes H if u(t) = 1 when f(t) > 0 and u(t) = 0 whenf(t) < 0. In this way the model will predict the time when the u(t) should change from 1 to 0, i.e. the time when the natural system will change from the proliferating lymphocyte stage to the stage where the lymphocytes differentiate into plasma cells. Similar methods have been applied to models of the immune system including memory cells and optimizing the IgM-IgG switch [43, 44]. Another optimization method is the iterative solution to the problem of the traveling salesman who needs to visit N cities in a particular area and wants to minimize the total distance traveled. Brady [45] used this problem to learn some optimization strategies from evolution. There are N!/2 possible solutions. In the first strategy, a trial solution was selected at random and changes were made in this solution only if the total distance traveled was reduced. To consider the effect of competition, the simulation then tested two trial solutions against each other, duplicated the better of the two, and discarded the remaining one. This strategy did not work as well as the one without competition because it reduced diversity. To increase diversity, a strategy simulating mating and genetic recombination was devised. Sections of each of the two competing solutions were compared and the shortest was duplicated for both solutions. In this way, the competition was at the gene level, rather than at the level of the individual organism. The gene level competition produced the most efficient results of the three strategies used. Although this method requires a tremendous amount of computer time, because of the faster supercomputers available today, this is not a severe limitation.

Network theory Jerne [46, 47] described a theory of immune system control based upon the observation that specific antibodies could act as antigen in the syngeneic host. The idiotypic determinants (antigenic determinants located in the highly variable regions of the antibody structure) are not subject to the normal self-recognition and anti-idiotype antibodies are produced. Jerne suggested that this allowed a mechanism for control of the immune response. The sequence goes like this: a foreign antigen(+) induces the production of specific antibody with an idiotype(-); when the concen- tration of that idiotype(-) gets large enough, it acts as an antigen and induces anti-idiotypic(+) antibody which is suppressive of the production of the original antibody(-); then the concen- tration of the anti-idiotype(+) antibody with its own specific idiotype gets high enough to induce a second level anti-anti-idiotype(-) antibody which can go on to suppress the anti-idiotype Lymphocytedifferentiation: the need for mathematical models 665 antibody. The theory is called the "network theory" because of the network of interacting complexes of idiotype-anti-idiotype-anti-anti-idiotype. The model by Richter [48] uses the network theory to investigate the experimentally-induced tolerance with very small, continuous doses of antigen (low zone tolerance) and with very high doses of antigen (high zone tolerance). In this model they assume that all receptors bind with the same binding constant, K, and consider the birth and death of I cells with the original idiotype, which occur at rates b and d. The following interactions of the receptors on lymphocytes (I) with antigen (I-) or anti-idiotype antibody (I+): (1) activation of I with I- (2) inhibition of (1) with I ÷ (3) destruction of I by I ÷ (4) protection against (3) by I- affect the I cells. These are described by the differential equation:

dI/dt = I[bB(I +, I-) - dD(I +, I-)],

where B is the fraction of lymphocytes (I) currently being activated by I- and D is the fraction of I currently being destroyed by I +. Using chemical equilibrium, the concentration of receptors on the I cells that are bound with antigen (RI-), with anti-idiotype antibody (RI +) or with both (I-RI +) are: RI- --- K[i -][R + ], RI + = K[i +] [R+ ] and I-RI + = qK[i-][i+][R+ ], respectively, where i, i- and i + are the free antibody, antigen and anti-idiotype antibody concentrations, R+ is the unbound receptor and q is a parameter decreasing the binding of two antibodies to one receptor because of steric hindrance. The fractions B and D are given by

B=I- k yk]/ (l+y;)M and

D = 1 -- z k (1 +Zl)M, 0 k ]/ where y, = [I-](l + qK[I+])/(1 + K[I+])

and

z, = [I+](1 + qK[I-])/(1 + K[I-]), and u, M and v are parameters. They demonstrate that there is a sharp threshold for activation at u/M and for suppression at v/M. This model was applied to conditions under which tolerance was induced, i.e. with either very low doses of antigen or very high doses of antigen. They demonstrated that network theory could explain these phenomena. There is also experimental evidence that idiotypes will induce the production of suppressor cells and contrasuppressor cells [49, 50]. This suggests that not only antibody, but also lymphocytes may participate in a network of control of the immune response. One model assumes that clusters of lymphocytes form during the idiotype-induced suppression reactions [51]. Lymphocytes within these clusters were suppressed. Free lymphocytes were allowed to develop into effector cells. All 666 J.R. LUMB reactions were also reversible and a cluster (chain) was also allowed to decay. A series of differential equations were developed for the single cells, x, and chains of length n cells varying from 2 to N:

N-I N-I dx/dt =-kx2-kx ~ Cn+2r ~ C n n=2 n=2

dCn/dt = -kxCn + kxCn_ I - 2rCn + 2rC~+ 1

dCN/dt = kxCN_ i -- 2rCN, where k is the rate of addition of one cell to a chain, k/2 is the rate of association of two single cells, r is the rate of decay of the association between two cells and 2r is the rate of decay of a chain by removal of one cell. The simulation was run until it reached equilibrium, so the system satisfied the condition

dN/dt=d(x+ ~ with the solution C~ = x(kx/2r) ~-I. This model was applied to the interpretation of limiting dilution experiments in which T cells are exposed to polyclonal activators, growth factors and maturation factors, and then assayed for resulting effector cells. These experiments provide information as to the number of precursor cells which were available for development into the effector cells measured and the size of clones developing under these conditions. These data allowed the construction of a model of the lymphocyte control network. Another model of network regulation of suppression determines what configuration of suppressor cells, effector cells and contrasuppressor cells is most advantageous [52, 53].

Stochastic approach A refinement of previous stochastic models [54-59] focuses upon the transitions from X cells to Y cells and from Y cells to Z cells. The transition functions determine the probability of transition using the recent information about the role of macrophages, IL-1, IL-2 and T helper cells in the antibody response [60]. The standard form of the transition function is the probability P(Q) = Or~(1 + Q~), where Q = A/(qB); A and B are amounts of the appropriate substance or cells, q is the value for P(Q) = 1/2 and v is a constant which determines the curvature of the function. The transitions which are considered include the transition of mature B cells to proliferating B cells, the transition from proliferating B cells to plasma cells, the transition from target T helper cells to proliferating T helper cells with concomitant production of IL-2, and the probability that macrophages will produce IL-1. Look et al. [60] used discrete-event simulation with the General Purpose Simulation System for the IBM 360 to construct a model of the cytotoxic response to malignant lymphoma cells. The model includes the circulation of the lymphoma cells to different tissues, the proliferation of the lymphoma cells, the activation and proliferation of cytotoxic effector cells and the killing of lymphma cells with the cytotoxic cells. Beyond the lag between the antigenic stimulation of sensitized lymphocytes and their proliferation, dN/dt = kloNe nt,

where N is the number of tumor cells, I0 is the intial number of cytotoxic lymphocytes, k is the rate of tumor cell killing and n is the proliferation rate for the cytotoxic lymphocytes. The stochastic aspect of this model is the distribution function which characterizes target cell kill F(t') = 1 - N(t')/No, where t' is the time since the tumor cell entered a specific organ, t' = t - to where to is time between injection of tumor cells and their entry into a particular organ. The authors describe experimental Lymphocytedifferentiation: the need for mathematical models 667 verification using ~Cr-labeled tumor cells injected into an animal and detected in the different organs as a function of time. The mechanism of target cell lysis by cytotoxic T cells has been the subject of detailed mathematical treatment [61]. Both a deterministic and a stochastic model were developed and gave similar results. The stochastic model was an application of queueing theory, where arrival time and service time show a two-stage model. They advocate a scheme which involves the delivering a single, lethal hit upon arrival after which the T cell facilitates the disintegration of the target cell, the service. A probability distribution for the target cell disintegration was determined [62] and the model tested by comparison with experimental results [63]. Another stochastic method is the Monte Carlo method, where the model is constructed in terms of probability functions. An iterative computer simulation chooses random numbers which are tested against the probability functions to make decisions. For example, a stochastic model of stem cell proliferation was constructed using birth and death probabilities for individual cells in analogy with population models [64]. Variables were evaluated as a function of time by a Monte Carlo method in which random numbers are used with the given probability to decide whether an event occurred.

Summary of mathematical modeling methods Most of the methods described above involve the use of differential equations in one form or another. The equations for the growth of cells are in the form dN/dt = rN, where N is the number of cells and r is the rate at which they are growing. For a population with incoming and outgoing cells the form of the equations becomes

dN/dt = (r + b - m - d)N, where b is the birth rate by which cells enter this population, m is the maturation rate by which cells leave this population for a more mature population and d is the death rate. The derivation for a particular model depends on the cell types, the assumptions and the level of detail used. For example, the detailed cellular kinetic models considered only one cell type and modeled the stages of the cell cycle. In the immunological models, where several populations of cells were used (including target cells, proliferating cells, effector cells and controlling cells), the details of the cell cycle were usually neglected. An exception is Prikrylova's model [59] that describes transitions of cells through the cell cycle as controlled by growth factors, T helper cells and T suppressor cells. With increased understanding of the molecular events in an immune response, there will be more information on the effects of mediators at different stages of the cell cycle. Thus, approaches to the description of the cell cycle will become more important to theoretical immunologists. Models can be deterministic and the equations exactly describe the biological phemomena, such as Bell's ABIGAIL [23-25], or the equations may be stochastic and describe the probability of an event. In the latter case, a Monte Carlo method may be used to describe events for which the mechanistic information is unavailable. Some of the differential equations can be solved analytically, but most are solved through computer simulation using numerical methods such as Euler's or Runga-Kutta. The iterative potential of the digital computer makes it possible to use sophisticated random number generators for a Monte Carlo simulation or to try many solutions to the traveling salesman problem. The cellular kinetic models using difference equations are especially applicable to solution by digital computer. Among the most useful recent applications in are the use of economic and electrical engineering mathematical methods for optimization and description of control networks. Both of these methods are highly applicable to immunology using information currently available. For example, optimization was used to determine the point in the immune response when lymphoblasts stopped proliferating and differentiated into immunoglobulin-producing cells. The following sections will review the experimental data which is currently available on the differentiation of lymphocytes, raise the crucial questions and suggest theoretical approaches which will aid in the resolution of these questions. 668 J.R. LUMB

LYMPHOCYTE PROGENITOR CELLS It is generally assumed that all blood cells originally descend from the pluripotential stem cells which are capable of repopulating the blood and lymphoid tissues of an experimental animal which has been lethally irradiated or given equivalent chemical treatment.

The pluripotential stem cell The pluripotential stem cell (PSC) is that cell which is able to regenerate all the bemopoietic cell lineages--erythrocytic, myelocytic, granulocytic, lymphocytic and mega-karyocytic/platelet. In an adult, the PSCs are found in the bone marrow. When bone marrow cells were i.v. injected into a lethally irradiated animal, white nodules appeared in the spleen [65, 66]. It was assumed that these nodules were formed from the division of one cell, forming a colony. Since these colonies contained all types of bemopoietic cells, it was concluded that there are cells in the bone marrow which are pluripotential. Thus, the spleen colony assay became the accepted method for determination of PSCs. It is generally accepted that these cells have the morphology of atypical, immature lymphoid cells [67-73]. Cells with this description make up 1.6-2.0% of human bone marrow cells [74] but estimates from spleen colony assays of mouse bone marrow are much lower (0.1-0.7%) [75]. These data and that from other work suggest that the cells detected by the spleen colony assay may not represent all the pluripotential cells of the bone marrow [76-79]. PSCs are also found in the blood, spleen and brain [80, 81]. The understanding of the PSC is severely limited by the current methods. The spleen colony assay can only be done in experimental animals and there is no equivalent method to study human PSCs. Thus, it is difficult to obtain accurate estimates of the number of PSCs, but 2% of the bone marrow cells is the upper limit of that tissue's contribution. Several investigations have shown that in a normal, steady-state condition, most of these cells are non-cycling, but they are capable of proliferating when triggered by irradiation, steroids or toxic chemicals [82-84]. Lord [85] showed that the non-cycling PSCs are out of cycle in a GO state which is located at the end of the GI phase and, therefore, ready to proceed rapidly into DNA synthesis upon receiving the proper signals. Since dividing cells are especially vulnerable to destruction by various agents such as irradiation and toxic chemicals, there would be survival benefit in having a highly resistant, resting cell population which is capable of restoring all the blood cells. To provide a quick response to blood cell destruction, the logical point for the cells to be resting is at the end of the G1 phase of the cell cycle. Soluble mediators are involved in the regulation of the PSC proliferation. Factors have been isolated from the supernatants of normal bone marrow cells which inhibit the PSC proliferation [86], and from regenerating bone marrow cell supernatants which stimulate the PSC proliferation [87]. Thymus humoral factor will restore the spleen colony-forming cells to thymectomized mice [88]. It has also been observed that are needed for restoration of the full proliferation potential of separated bone marrow cells [75]. Furthermore, it appears that the PSC in mouse bone marrow expresses a low level of thyl antigen [89, 90]. The low thyl population represents 0.1% of the adult bone marrow cells and contains all the spleen colony-forming cells. The establishment of long-term cultures from these cells indicates that they may be self-renewing. This is in contrast to the in vivo kinetic data which indicates that the spleen colony-forming cells are non-dividing. In summary, the PSCs appear to be a non-cycling population of cells found primarily in the bone marrow which is capable of replenishing the blood cell precursors in case of their destruction. Current evidence indicates that these cells are not responsible for the renewal of blood cells under normal conditions. This conclusion raises several questions in regard to the generation of lymphocyte precursors. What cells are responsible for the renewal of lymphocytes? Is there one common precursor responsible for renewing lymphocyte populations? Are the self-renewing cells descended from the PSCs? Are the self-renewing cells committed to a particular differentiation pathway?

Theoretical approaches to the progenitor cell problem The first mathematical treatment of self-renewing stem cell populations assumed that stem cells were capable of an asymmetric division, one that produced another stem cell and a differentiated Lymphocyte differentiation: the need for mathematical models 669 cell [91, 92]. In order to accommodate the ability of stem cells to recover from irradiation or other insults, it was suggested that stem cells were also capable of symmetric division into two stem cells. The distribution between the two types of divisions was determined statistically. This approach was criticized because there was no experimental evidence of the ability of ceils to divide asymmetrically [93]. Therefore, Lajtha [93-95] designed a mathematical model in which the differentiation of a cell was a separate event from cell division. The stem cells were apportioned between available stem cells (S,) during a given time interval (n) and the total triggered stem cell stock (T,). Under a normal (steady-state) condition, as many cells will be differentiating (d~) as will be triggered to divide (t~): d~= t,=pS,, where p is the proportion differentiating per time interval. If q time intervals equal the total cell cycle time, the daughter ceils arising from division are described by x. = 2(t._ q). Therefore, the available stem cells at time n are calculated by S.=S._l-d._,-t._,+x.. This system was quite stable and responded to changes in p within two cells cycles. The system was also self-protective because the proportion of differentiating cells reached a maximum after which further increases in the p value did not increase the number of differentiated cells. In order to simulate irradiation, a proportion of the available stem cells were declared sterile cells (no longer capable of division) and extra triggered cells were added to compensate for the dying cells. After a single dose of irradiation which sterilized 90% of the stem cells, the system recovered within 10 cell cycles. Continuous irradiation was also considered. In this case, the number of stem cells available only returned to a fraction of the original level. The results of both of these irradiation simulations are similar to experimental data from irradiated animals. However, it was necessary to specify the number of extra triggered cells to attain these results.

L, POPULATION I BLOQD VESSEL BONE MARROw

THYMUS TL=LI+L2+'''+ Ln 1 I

Fig. 1. A schematic representation of the computer simulation THYMS. L, is the number of stem cells at time t; L~ is the number of cells in the ith thymus cell population at time t; E is the number of blood vessel cells; Sm is the maximum number of stem cells allowed; TE is the total number of blood vessel cells in the thymus and T L is the total number of lymphocytes in the thymus. This figure was reproduced with permission from Lumb [28]. 670 J.R. LUMB

A regenerating model of the thymus depended upon a stem cell compartment to provide the continuous supply of prothymocytes and to allow recovery from irradiation (see Fig. 1, Table 2 and Refs [26-28]). Each stem cell division was distributed between those producing one stem cell and one differentiated cell (S/Sm), and those producing two stem cells (1 - S/Sm), where S was the number of stem cells at time t and Sm was the maximum number of stem cells allowed, a parameter which was varied. The differential equations which describe the changes in stem cell population and differentiated lymphocyte precursors (L) are dS/dt = 211 - (S/Sm)]GS and dL /dt = ( S / Sm)GS, where G is the reciprocal of the generation time for the stem cell and L is the differentiated progeny of the stem cell. In order to simulate the recovery from irradiation, the number of cells in each population was multiplied by the appropriate fraction, depending upon their sensitivity to irradiation. The simulated irradiation occurred at a designated time and then the model proceeded without any changes. The degeneration of the thymus was simulated by decreasing the Sm value as a function of the number of population doublings which had taken place:

dSm/dt = - DGt Sin, where D is the rate of decrease (degeneration parameter) of the maximum number of stem cells; D was adjusted until the simulated degeneration was similar to the natural degeneration. The results of this simulation, calculated using a digital computer, compared favorably with the data for the natural development of the thymus and for recovery from low doses of irradiation. However, natural systems showed a slower recovery from high doses of irradiation than the model predicted. It was suggested that the differences resulted from secondary toxic effects of the irradiation such as the formation of peroxides and other toxic chemicals. A further refinement of the model took these effects into account by using the time since irradiation (t) and the fraction of cells surviving that dose of irradiation (F) to decrease the generation time of stem cells after irradiation:

F t = (70[(120 -/)120 -- F]/F, where 0 < t > 120 h, Gt is the generation time of stem cells at time t and Go is the unirradiated stem cell generation time. By this equation, the generation time of cells immediately following irradiation is increased to a maximum value and then decreases gradually to the original value by 120 h after irradiation. Because generation time was dependent upon the dose of irradiation via the value of F, the toxic effect of increased irradiation doses was simulated. With this refinement the model simulated the response to high doses of irradiation more closely [26]. One of the most puzzling questions in the study of lymphocyte differentiation is the nature of the lymphoid precursor cells. Since the lymphoid populations are restored in lethally irradiated animals by the administration of bone marrow cells with markers indicating descendance from donor cells, it has been concluded that cells in the bone marrow are capable of differentiating into lymphocyte lineage cells [96-98]. However, lymphoid precursor cells derived from PSCs have not been directly isolated because of the lack of appropriate markers. Therefore, studies of lymphocyte ontogeny invol'/e working backwards from the circulating, mature lymphocytes. This is an area where a theoretical treatment may be able to lead researchers to the identification of the lymphocyte precursor. Such a treatment would involve a series of differential equations describing the current understanding of the kinetics of known lymphocyte subpopulations. To predict the characteristics of the lymphocyte precursor, equations describing precursors according to different hypotheses would be added and the number, generation time and percentage in cycle of the input cells determined. For example, one hypothesis states that there is a single lymphocyte precursor derived from the PSC which generates all the subpopulations of both T- and B-lymphocytes. Other hypotheses include the suggestion that two or more precursors are derived directly from the PSC, or that the precursors are independent of the PSC. Lymphocytedifferentiation: the need for mathematicalmodels 671

B CELL DIFFERENTIATION lmmunoglobulin gene rearrangement The genetic events allowing the differentiation of B cells are relatively well-understood [99]. The selection of a particular immunoglobulin specificity occurs by the selection of one of a number of genes for the variable portion, which, through a series of DNA rearrangement events, is brought into the proximity of the genes for the constant portion of the immunoglobulin. The two polypeptide chains (light and heavy) of immunoglobulin are coded by separate genes, each of which undergoes separate rearrangement events. The pathway of differentiation of B cells is directed by these events. Three regions in the heavy chain gene cluster--the variable region (Vh), the diversity region (Dh) and the joining region (Jh)--code for the variable domain of heavy chains. In each of these regions, one of a number of genes is selected and the DNA between the two selected genes is looped out, bringing the two selected genes in proximity to each other. The Dh and Jh genes rearrange first (DJh), and then the Vh gene rearranges with the DJh, forming a VDJh segment [100, 101]. Since the region between the Jh genes and the genes for the constant portion of the heavy chain is an intervening sequence (IVS), these rearrangements result in an immunoglobulin gene (VDJh-IVS-Ch) ready for transcription into hnRNA, the primary transcript. The IVSs are spliced out of the hnRNA and the cell is then able to synthesize the mu chain, the most primitive of the heavy chains. Once this occurs, the mu chain can be detected in the cytoplasm of the cells (cmu), which is the working definition of a pre-B cell. The genes for the light chains (either kappa or lambda) are organized in a similar way except that they have no diversity regions. A successful rearrangement of the genes for one of the light chains (VJ1) then allows the synthesis of the light chain and, finally, the formation of a complete immunoglobulin molecule (IgM) (2 light, 2 heavy chains) to be displayed on the surface of the cell. The working definition of a B cell is that it has surface immunoglobulin (slg). Thus, at the genetic level there are four designations of cells in the B cell differentiation pathway: (1) no immu- noglobulin genes rearranged, (2) DJh rearrangement, (3) VDjh rearrangement and (4) both VDJh and VJI rearrangements.

Cell separation and cloning While the rearrangement of a cell's DNA allows a relatively certain assignment of the differentiation status of the cell, it requires a large number of identical cells from which to isolate DNA. There is no current means of isolating cells on the basis of DNA rearrangements. Other characteristics of the cells, such as cell size (density) and the presence of cell surface antigens and lectin receptors, are used for the separation of lymphocyte subpopulations. Two innovations in experimental technology have made it possible to obtain more homogeneous cell populations. First, the production of monoclonal antibodies has allowed the identification of many specific antigens which are used as markers of cells in different stages of lymphocyte differentiation. The monoclonal antibody can then be used to isolate specific subpopulations with the fluorescence-activated cell sorter. These two techniques have revolutionized experimental work in cellular immunology because it is now possible to obtain homogeneous reagents, both antibodies and cells. Con- sequently, in the last 10 years the accumulation of information on lymphocyte differentiation has been exponential. Once separated the cells can be tested for various functions such as the ability to restore a depleted cell population, the presence of cell surface markers, and cellular kinetic parameters. One indication of the normal sequence of cellular differentiation is the chronological order in which the cells reappear in an animal that has been depleted of certain populations. Alternatively, the immortality of malignant cells can be used to obtain a large number of cells with the same characteristics. There are three sources of relevant malignant lymphoid cell lines. A number of cell lines have been established from experimentally induced or naturally occurring and lymphomas. Such cell lines can be induced specifically by viral infection of the appropriate cells, especially Epstein-Barr virus in humans and Abelson virus in mice. Finally, hybridomas that are constructed by fusing normal cells of interest with an immortal cell line have the characteristics of both parental cell lines. Some normal cell populations such as activated T and B cells can be cloned, providing a large 672 J.R. LUMB

number of identical cells. This usually involves the use of growth factors such as IL-2, the T cell growth factor. The cloned cells generally retain their natural functions.

cmu + Cells The bone marrow is assumed to be the location where B cell differentiation takes place in adult mammals. However, in the developing embryo, the bone marrow does not contain lymphocytes until after birth. The liver seems to be the sources of B cells in the fetus. Thus, both the adult bone marrow and the fetal liver have been studied to determine the pathways leading to the mature B cell. In both tissues, pre-B cells have been demonstrated by the presence of cmu [102-107]. Large pre-B cells are dividing cells and give rise to the small pre-B cells [108, 109], which are non-dividing cells [108, 110-112]. The bulk of the evidence indicates that large pre-B cells divide once and are not a self-renewing population [107, 113].

B220 molecule (14.8 antigen) The cell surface molecule B220, which is recognized by the 14.8 monoclonal antibody, is found on pre-B cells and mature B cells [114]. Using a monoclonal antibody against B220 to separate pre-B cells, Coffman and Weissman [115] showed that heavy chain genes were rearranged in large pre-B cells, but they found no evidence of light chain gene rearrangements in large pre-B cells. However, in small pre-B cells they found some light chain genes rearranged.

Peanut agglutinin Another marker used in the study of pre-B cells is the receptor for the non-mitogenic lectin, peanut agglutinin (PNA). This marker was first found on some cells in the thymus and peripheral lymphoid tissues and thought to be involved in T cell differentiation, but it was also found on pre-B cells and some early B cells in adult bone marrow [116, 117] and fetal liver [118-120]. Osmond [121] showed that the PNA ÷ bone marrow cells were mostly pre-B cells or cells which had just begun to synthesize slg. Some cmu- cells were also agglutinated by PNA. There is some evidence that bone marrow precursor cells to the pre-B cells are also PNA +. Osmond et al. [122] sorted and cultured PNA ÷ cells from the adult bone marrow and found that functional B cells were produced for a longer time than would have been expected if they differentiated from the large pre-B cells, which divide only once. It has also been shown that spleen colony-forming cells are among the PNA + population [116]. Two cell populations were identified in mouse bone marrow which would give rise to B cells in the Whitlock-Witte culture system. This system involves the establishment of a layer of bone marrow stromal cells to which additional bone marrow cells are added [123]. Under these conditions the differentiation of B cells occurs to the exclusion of other cell types. Long-term B cell-producing cultures were established from a population of cells which were negative for the B220 antigen but expressed a low level of thyl antigen. This population contains the spleen colony- forming cells and is self-renewing, at least in this culture system. A population without B220 granulocyte marker, macrophage marker or thyl contained cells which grew for a limited time and differentiated into cells with the B220 antigen and surface IgM within 1 week [90]. The suggestion is that these B220- cells represent the immediate precursor of the large pre-B cell. In culture, 25-40% of these cells incorporated tritiated thymidine, indicating that they were dividing cells. Kinetic studies have identified a population of cells in the adult bone marrow that precedes the large pre-B cell [107]. These pre-B progenitor cells have a labeling index of 35% which is similar to the labeling index of the population identified in the Whitlock-Witte culture. The population in the culture had a limited lifespan while the kinetically-defined population did appear to be self-renewing. There is indirect evidence that pre-B progenitor cells are found in the fetal liver during development. Pre-B cells (cmu +) are first found in the fetal liver at 16 days gestation, while slg + cells are first seen at 19 days [103, 124]. Cells which are capable of producing pre-B cells in culture, without the benefit of migrating cells, are found in the fetal liver as early as 11.5 days gestation, at the same time as B220 ÷ cells and PNA ÷ cells appear [118-120]. Such cells were not found in either the blood or the yolk sac. Lymphocyte differentiation: the need for mathematical models 673

Depletion studies The order of reappearance from irradiation was used to determine the pathway of differentiation. The large pre-B cells appeared first, then the small pre-B cells and, finally, the slg ÷ cells [112]. Burrows et al. [104] used the treatment with antibody directed against IgM to deplete an animal of cells with slgM. They found that the bone marrow is depleted of B cells, but the pre-B cells with cmu remain, which will give rise to slg cells. Other animals were treated with cyclo- phosphamide which destroys both B and pre-B cells. Again, the same order of appearance was observed. It was 3-4 days before the large pre-B cells began to recover; the small pre-B cells appeared 1 day later; and, finally, the B cells, which began to recover in another 24 h, were fully recovered a total of 9 days after the treatment. In another experiment, the time required for complete turnover of the population, called the renewal time, was determined for B cell precursors [107]. For small cmu ÷ slg-lymphocytes the renewal time was 48h; for small cmu÷slg ÷ lymphocytes, 96 h; and the cmu-slg ÷ are long-lived and, therefore, do not renew in a measurable time.

Pre-B cell differentiation sequence A relatively straightforward picture of B cell differentiation emerges from the data discussed above (Figs 2 and 3). The earliest detectable B-lymphocyte precursor is found in the fetal liver at 11-12 days and is B220 ÷, 14.8 + and cmu- (Table 3). Pre-B cells with demonstrable cmu are detected by 16 days gestation, and B cells with slg appear by 19 days gestation in the fetal liver. In the adult bone marrow a self-renewing progenitor cell which is low thyl ÷ but negative for all other markers differentiates into a B220-, committed pre-B progenitor cell through one or several steps (Table 4). These pre-B progenitor cells differentiate into large B220 ÷ PNA ÷ pre-B cells with rearranged heavy chain genes. The large pre-B cells proceed through one more division and differentiate into small PNA+B220 ÷ cells with cmu. It is at this stage, the small pre-B cell, that the light chain gene rearrangement takes place. The resulting synthesis of the light chain allows the completion of the IgM. As soon as the IgM is displayed on the surface, the cell fulfils the definition of a B cell. p•YOlkSac

~ Bone Marrow Lymph Nodes i~ Sp,een 11 14 16 19 21 Birth Days of Gestation (mouse) Fig. 2. A schematic representation of fetal lymphocyte differentiation. The migration and differentiation of lymphocytes in the fetal mouse is shown as a function of gestation, where: + = presence of marker; - = absence of marker; PNA = receptor for peanut agglutinin; 14.8 = antigen reacting with the mono- clonal antibody named 14.8; cmu--mu polypeptide chain found in the cytoplasm; sIg = cell surface immunoglobulin; ~t, fl and ~ = polypeptide chains of the T cell antigen receptor; APase = alkaline phosphatase; TdT = terminal deoxynucleotidyl transferase. 674 J. R. LuMB

Spleen and Bone Marrow =. -.- Lymph Nodes 3-4 days =*= 1-2 days..=, ~1-3 days =. = Ionglived

Large Pre-B Cell ge B Cell L.I. =35% ~ LA.=31% Q heaw chain gene rearrangement Smell one cell divisior Pre-B Cell L.L =(

Sessile L.), = 0.8% ~,~ Marginal Zone light chain gene _ I-- B Cell rearrangement >- L.I. =0.2%

Recirculating Follicular ~ C- O'r I Y Germinal¢ Center .,>o- /,~ T Cell - dependent B Cell heavy chain switch )-~

Fig. 3. A schematic representation of B cell differentiation, where: + = presence of marker; 1o = low expression of marker; - = absence of marker; T = thyl antigen; P = receptor for peanut agglutinin; B = B220 antigen; C = cytoplasmic expression of immunoglobulin mu chain; M = cell surface IgM; D = cell surface IgD; L.I. = labeling index; and ? = unknown. The tissue in which the indicated events take place is shown at the top of the figure. The turnover time is the time required to completely replace the population with new entrants. The circled numbers indicate DNA rearrangement events: (1) the heavy chain gene rearrangement occurs between the pre-B progenitor cell and the large pre-B cell; (2) the light chain gene rearrangement occurs between the small pre-B cell and the small B cell; and (3) the heavy chain switch occurs between the recirculating follicular B cells and the germinal center memory B cells.

Two observations do not fit this emerging picture. First, only 70% of the small B cells in the bone marrow were the immediate division products of the pre-B cells. The other 30% were derived from a source other than direct differentiation from pre-B cells. The second observation is of large cmu÷slg + cells in the bone marrow which are neither descendants nor precursors of pre-B cells. Figure 3 shows this large B cell with questionable interactions with the subpopulations of mature B cells. The large B cell may represent an intermediate cell in a separate pathway of B cell differentiation that leads to the small B cells not directly descended from small pre-B cells. Alternatively, the first small B cell may transform into a large cmu+B cell, perhaps in the process of controlling the repertoire (see the Control of B Cell Repertoire section). However, the bone marrow is not exclusively a primary lymphoid organ. In fact, it has been shown that the bone marrow is responsible for a large proportion of serum antibody synthesis [125].

Table 3. Fetal liver development Incidence (%) 14.8 +a Day of gestation PNA + 14.8 + cmu + slg + cmu + slg ÷

12 -- 0.2 14 -- 1.2 16 19 3.5 0.1 0.2 0.3 <0.1 19 15 2.2 0.6 4.6 <0.1 B~ al4.8+ ceils wore separated and then tested for cmu or slg [124]. References: PNA + cells [117]; 14.8 + cells [120]. Lymphocyte differentiation: the need for mathematical models 675

Table 4. Distribution of bone marrow cells Cell type Incidence (%) Number (x 10 -5) Reference No. Bone marrow (total) 188 83 Small lymphocyte 25 46 122, 128 emu + 3-16 6-27 122, 128 slg + 7-11 11-19 122, 124, 128 slg +, cmu + 8.8 13 128 14.8 + 18 30 124 14.8 +, cmu + 8.7 14 124 PNA + 20 33 I 17 PNA +, emu +, small 11 20 121 PNA +, cmu +, large 5.7 9.4 122 TdT +, rat 1.6 128 human 1.1 183 PSC, morphology 1.8 74 human tbyl, low 0.1 0.19 90 CFUs 0.018 0.04 83

Thus, the large B cells may represent activated B cells and their descendants might be the 30% of the small B cells not directly descended from pre-B cells. The system is further complicated by the effects of antigen on the differentiation of pre-B cells in the bone marrow. The administration of antigen influences the pre-B cell differentiation in the bone marrow [126-128]. Since this activity is dependent upon spleen macrophages and not upon those of the bone marrow, there appears to be feedback control of B cell differentiation and production.

Control of B cell repertoire The question of control of the development of the B cell repertoire is an extremely important, yet unresolved issue in B cell differentiation. How are the potentially self-reactive clones removed? The earliest B cell which still has demonstrable cmu appears to be destroyed by interactions with the newly formed immunoglobulin receptors on its surface [129, 130]. The calculated rate of production of B cells based upon the kinetic data within the bone marrow is greater than the rate of appearance of cells in the peripheral tissues or the renewal rate of the circulating cells [111,130, 132]. The steady-state rate of migration of lymphocytes from the bone marrow into the peripheral tissues is estimated to be 4 x 106cells/day in an adult rat and 106cells/day in a mouse [7, 133, 134]. However, results vary as to the turnover of circulating B-lymphocytes [131,132, 135] so it is difficult to determine whether the export from the bone marrow is in excess of that required. This is an area where theoretical treatment could be very helpful in resolving an important question. Within the bone marrow the number of large pre-B cells is twice that required to produce the number of small pre-B cells [111]. If editing were to occur at this level, it could not be via the cell surface receptor interaction with antigen because these cells have no slg. A more likely explanation for this disparity is this author's suggestion that the division of large pre-B cells is an asymmetric one, resulting in one small pre-B cell and another unidentifiable cell.

B cell subpopulations An early subpopulation of B cells is characterized by its high level of PNA receptors [112] and its destruction by antigens or anti-immunoglobulin antibodies when B cells are exposed during this early stage of development [136-138]. These data are interpreted to mean that there is a stage in the differentiation orB cells, fight after the immunoglobulin is first displayed on the surface, when cells which would react with self-antigens are destroyed. With certain antigens, mainly polysaccharides, T helper cells are not required. This response is called the thymus-independent response and the antigens inducing it are of two types, called TI-1 and TI-2. TI-1 antigens induce a response in both adult and newborn spleen cells, while newborn spleen cells are unresponsive to the TI-2 antigens which induce a substantial response in adult spleen cells. Several markers associated with these differences allow subpopulations of B cells to be defined. Cells which respond only to TI-2 antigens appear earlier in ontogeny, localize in the marginal zone of the spleen, have only IgM on their surface and no IgD and are Lyb-3- and Lyb-5- 676 J.R. LUMB

[139]. The role of the marginal zone B cells in the response to polysaccharide antigens is supported by the association of these cells with the type of immunoglobulin subclass found in thymus- independent responses [140]. Although the marginal zone B cells are not recirculating cells [141-143], they are derived from recirculating cells. Recovery from depletion is considerably delayed and is not accelerated by the administration of fetal liver or bone marrow cells, which indicates that these cells are far removed from early B cell differentiation steps [144, 145]. The majority of the B cells in the lymph and blood are IgM+IgD ÷ recirculating B cells also found in the follicles of the spleen and lymph nodes [146, 147]. The recirculating, follicular B cells are not direct descendants of progenitors of the marginal zone B cells [147], but the two populations seem to be derived from separate lineages of B cells. Finally, the subpopulation of B cells found in the germinal centers are distinguishable by the expression of a high concentration of PNA receptors, a low level of slgM expression and lack of surface IgD [148-150]. These cells bind specific antigen and have undergone the heavy chain switch to other classes and subclasses of immunoglobulin. The class of immunoglobulin is defined by the type of heavy chain which is combined with either of the two light chains, lambda or kappa. Besides the IgM, which has a mu chain, the other classes and subclasses are IgGl, IgG2, IgG3, IgG4, IgA, IgD and IgE, which have gamma-1, gamma-2, gamma-3, gamma-4, alpha, delta and epsilon heavy chains respectively. The heavy chain gene cluster includes constant genes for all types of heavy chains and one set of V, D and J genes. The same VDJh rearrangement which has already occurred for the production of the mu chain continues to be used. To switch from mu to any of the other heavy chains another DNA rearrangement occurs between the switch region in front of the mu chain constant gene and the switch region in front of the other chain constant gene. The mechanism of rearrangement appears to be the same as that inducing the initial production of the mu chain (see the Immunogiobulin Gene Rearrangements section). This differentiation step in the B cell response requires the presence of specific T helper cells and generates the B cells found in germinal centers that are responsible for the transfer of memory [151]. In summary, the B cell subpopulations include:

(1) early B cells (IgM ÷, IgD-, PNA ÷, killed by anti-IgM); (2) marginal zone B cells (PNA-, IgM ÷, IgD-, Ly5-, sessile); (3) follicular B cells (PNA-, IgM ÷, IgD ÷, Ly5 ÷, recirculating); and (4) germinal center B cells (PNA ÷, IgM l°w, IgD- heavy chain switched, memory.

Theoretical approaches to the B cell problem A model of B cell development with an emphasis on the removal of self-reactive cells has been constructed [58]. This model adds the immature B cell (I cell) to the X-Y-Z scheme to simulate the control of the repertoire development. Thus, the new scheme is I-X-Y-Z. The equations for the concentrations of the three cell types considered (I cells, X cells and Z cells) are

dI(t)/dt = p - [m + Ka(t)]I(t), dX(t) = ml(t) - [d + a(t)]X(t) and dZ(t) = ka(t - tl )X(t - tl) - hZ(t), where I(t), X(t) and Z(t) are the concentrations of I, X and Z cells at time t; p is the rate of differentiation of I cells from precursor cells; m is the rate of maturation of I cells into X cells; d is the natural rate of death of X cells; h is the death rate for Z cells; a(t) is the rate of elimination of X cells by antigen; Ka(t) is the rate of elimination of I cells by antigen (K = const > 1, which indicates that the I cells are more sensitive to elimination by antigen than the X cells); and t~ is the time lag between the stimulation of an X cell and the generation of the antibody-producing Z cell. This model has been verified by testing against experimental results in chickens [4] and in mice [59]. Lymphocyte differentiation: the need for mathematical models 677

The unresolved questions with regard to B cell differentiation are: (1) What are the early phases of B cell differentiation between the thy t°w cell, the committed pre-B progenitor cell and the first identifiable cmu ÷ cells? While experimental data in this area is limited by the lack of relevant cellular markers, a theoretical treatment could be constructed to predict the kinetic parameters of intermediate stages from the information available. Such information would be helpful to researchers interested in filling these gaps in existing knowledge. (2) How do the large cmu+B cells and the other 30% of small B cells fit into the B cell differentiation pathway? A mathematical model of the bone marrow is needed which would include the major B cell differentiation pathway, events leading to antibody synthesis in the bone marrow and some consideration of feedback control by an ongoing immune response in the spleen. The model would consist of a series of differential equations combining the ideas of Bell [23-25], Lumb [26-28] and Prikrylova [59]. The three hypotheses suggested to account for the large cmu+B cells and the other 30% of small B cells could be tested with this model. (3) How is the B cell repertoire controlled to eliminate self-reactive B cells? Is there a threshold level of binding of self-antigens above which the system rejects? It appears that perhaps half of the B cells are eliminated in the bone marrow. Is that adequate to eliminate the potential self-reactive cells? Where and when does the elimination occur? Because the actual binding of self-antigens and foreign antigens is completely dependent upon the molecular structures of the antigen and the antibody, and because there are very sophisticated molecular structure programs available, it seems that the questions of how much binding is too much and whether the elimination of half the B cells is adequate might best be approached at the molecular level. Using estimates of the number of different antibodies derived from the numbers of variable genes, along with estimates of the number of possible antigenic structures (considering the specificity limits of self vs foreign) and extrapolations from known structures of antigen-antibody complexes, it should be possible to estimate the number of self-reactive B cells which could be produced. Either the traveling salesman iterative method or optimization methods might also be applied to determine the optimum position between the conditions of auto-immunity and of too little anti-foreign immunity.

PROTHYMOCYTES AND pre-T CELLS The study of precursors to T cells has developed in a different way from that of B cell precursors. While there was no functionally defined marker such as cmu, the thymus has provided a known organ devoted almost exclusively to the differentiation of T cells. The terms prothymocyte and pre-T cell have different experimental definitions. A prothymocyte is a cell which will migrate to the thymus and repopulate the lymphoid elements of that organ. A pre-T cell is one that will respond to treatment with differentiating agents, such as thymic hormones, with the production of T cell markers, such as thy 1 in mice or T3 in humans. In the adult, steady-state condition, the prothymocytes migrate to the thymus from the bone marrow, where they make up only a very small proportion of the lymphoid cells. Bone marrow cells migrating to the thymus have been determined [152]. They lack thyl and Lyt antigens, are separated from the cells of the B cell lineage, and constitute 0.1% of the bone marrow cells. During fetal development, the thymus is the first lymphoid organ to contain lymphocytic lineage cells, which migrate to the thymus from the yolk sac [153-156]. T cell antigen receptor The recently discovered T cell antigen receptor allows more precise definition of the steps of T cell differentiation [157-162]. As with immunoglobulin, the genes for the T cell receptor have the V-D-J-C organization and the differentiation in the thymus involves rearrangement to

C.A.M.W.A. 14/9-12~ 678 J.R. LUMB

expressible genes with the selection of particular V, D and J segments [163]. Three types of polypeptide chains have been identified: alpha, beta and gamma. In mature T cells, the T cell receptor is composed of one alpha and one beta chain. The gamma chain is thought to be involved in early T cell development [164, 165]. In keeping with the analogy of the T cell receptor gene rearrangements to that of the immunoglobulin genes, three distinct populations of thymocytes can be identified: (1) those with germline configuration for both the alpha and beta T cell receptor genes, called pro-T cells (for progenitor T cells); (2) those with germline alpha genes and rearranged/expressed beta genes, called pre-T cells; and (3) those with both alpha and beta chain genes rearranged/expressed, called T cells [166, 167]. Expression of the gamma chain gene appears first in the fetal thymus at 14 days and then decreases rapidly by the 17th day of gestation when the beta and alpha chain genes are expressed [168, 169].

Terminal deoxynucleotidyl transferase The first identified marker of pre-T cells was the enzyme terminal deoxynucleotide transferase (TdT) which was associated with lymphoid bone marrow cells and approx. 75% of thymus cells [170-172]. TdT ÷ cells first appear in the mouse fetal thymus at 17 days (4 days before birth) and at birth in the spleen and bone marrow [173]. Transient populations of TdT + cells are also seen in the liver, lung and peripheral blood in the first 10-14 days after birth [174]. The TdT + cells are present in the normal mouse spleen only up to 6 weeks of age. That these cells include pre-T cells is evidenced by two findings: half of the TdT+thyl - cells from normal mouse bone marrow have been shown to express thyl after incubation with thymic hormones [175-177]; and populations enriched for prothymocyte activity are also enriched for TdT + cells [178]. Thymic hormones which act early in T cell differentiation will induce TdT activity in spleen cells from athymic (nude) mice [179, 180]. TdT activity in normal thymocytes is decreased by a T cell differentiating factor isolated from blood, which acts later in T cell differentiation, and by retinoic acid, a general differentiation inducer [181]. There is also evidence that TdT is a marker of human bone marrow prothymocytes [182-184]. Thus, it appears that the TdT ÷ cell represents a stage of T cell differentiation intermediate between the PSC and the mature T cell [185, 186]. However, TdT is not limited to T cell differentiation and is also found in pre-B cells [176, 187]. In addition, it has been shown that a majority of the cells capable of repopulating the thymus (75%) are found in a population which had been depleted of TdT ÷ cells [190]. Thus, while some prothymocytes are TdT +, not all are. Peanut agglutinin The PNA receptor in a useful marker in T cell differentiation because it binds to most lymphocyte precursors and immature lymphocytes, but not to mature T cells [188]. Since agglutination is a convenient separation technique, most marker systems have been combined with PNA for a more detailed analysis. The earliest PNA + cells reported were in the yolk sac at l0 days gestation [189]. These cells had the morphology of an undifferentiated cell, but compared to cells with euchromatic nuclei. PNA ÷ cells appear in the fetal liver (11 days) and the fetal thymus (14 days) at the same time as the first functional lymphocyte precursors [119]. In the adult bone marrow, there is evidence that the PSC [120], the large pre-B progenitor cells, all pre-B cells, and early B cells [122] are all PNA ÷. In the thymus, PNA agglutination is used to separate the immature cells, assumed to be found in the cortex, from the mature, cortisone-resistant T cells, and can be used with other markers to identify subsets of thymocytes [191,192]. This will be discussed in more detail later in the section intrathymic T Cell Differentiation. The receptor for PN6 is a specific sugar moiety which is on the outer surface of the cells. It seems that this receptor appears early in the differentiation sequence leading to mature lymphocytes. Mature B cells actually lose the receptor while mature T cells merely cover it up with sialic acid: Therefore, treatment with an enzyme that hydrolyzes sialic acid (neuraminidase) transforms mature T cells from PNA- to PNA + [117]. Other markers of pre-T cells A placental isozyme of alkaline phosphatase is found on fetal thymocytes up to 17 days gestation [193-196]. Alkaline phosphatase is also found on a few bone marrow cells (< 1%), spleen cells Lymphocyte differentiation: the need for mathematical models 679

(marginal zone) and some activated B and T cells [197, 198]. There is also evidence by electron microscopy that a few subcapsular cells in the adult thymus are alkaline phosphatase ÷ [199]. The significance of this marker is not yet determined although the distribution among cell types suggests alkaline phosphatase may be useful in the isolation of early lymphocyte progenitor cells. FT-1 and FT-2 are recently discovered fetal thymus markers which are present in the developing thymus up to 17 days gestation [200, 201]. Unlike other markers described, these are exclusively localized in the fetal thymus and on cells. Using anti-mouse brain antibody, it was discovered that a neutral glycolipid (asialo GMI) was on fetal thymocytes but not on adult thymocytes or mature, circulating T cells [202, 203]. Growth factors IL-2 functions as the T cell growth factor. In order to respond to this growth factor, cells must carry specific IL-2 receptors on their surface. It was found that the most immature thymocytes (Ly-2-, L3T4-) and fetal thymocytes at 14 days gestation have IL-2 receptors but do not proliferate in response to IL-2 unless otherwise stimulated [204-206]. Lymphocyte precursors in the human bone marrow were identified by their response to IL-2 or to thymic hormones [207]. The IL-2 responsive cells already expressed both the IL-2 receptor and the T8 human cytotoxic cell marker prior to exposure to IL-2, while the thymic hormone-dependent population expressed the T4 human helper cell marker. The production of IL-2 is generally associated with helper T cells; however, in fetal organ culture, IL-2 production and the ability to develop into cytolytic cells were both found in both Lyt2 ÷ and Lyt2- populations [208]. This suggests that the Lyt2 marker may have a different meaning in fetal lymphocytes. This is also shown with the mature marker, thyl, which is found early in fetal development, prior to the development of any other mature T cell functions (Table 5). Role of fetal liver in thymus development The extent to which the fetal liver is involved in thymic differentiation has been in dispute. Initially, PNA was thought to be a thymus-specific marker so that the presence of PNA ÷ cells in the fetal liver of mice was taken as evidence of involvement in thymic development. However, since the PNA + cells in the fetal liver could be B cell precursors, that conclusion was less definite. In humans there is more direct evidence of fetal liver involvement in thymus development [209, 210]. Gene products of a gene cluster linked to the IgH-1 gene have been shown to be associated with development and are also found in the fetal liver [211]. Summary of fetal thymus development The cell distribution during development of the fetal thymus is shown in Table 5. There are three phases in the development of the thymus, as illustrated in Fig. 2. In phase 1, cells are produced in the mouse yolk sac and migrate to the thymus at 14 days gestation. In phase 2 (14-16 day fetal thymus), cells are identified by several markers (FT-1, FT-2, asialo GM1, alkaline phosphatase)

Table 5. Fetal thymus development Number of Incidence (%) Days of lymphoid gestation cells" TCR b APase ¢ GM1 +d TdT ÷c thyl +f Lyl +r Lyt2 +f 13 1 × 104 - ND 68 0 39 36 10 16 3 x 105 y + 17 84 80 62 17 6× l0 s ),,fl - 2.0 18 9 × 105 ~, B - 2.5 19 1 × 10 6 0~, fl -- 8 4.5 95 95 85 20 3 X 106 ~, p -- 8.0 Birth 5 × l0 6 ~, fl - 26 1 Week 5 x 107 ~, fl - 48 Adult 2 x 108 ct, p +/_s 5 65 95 95 83 'Lumb [26-28]. b'l" cell antigen receptor chain expressed/rearranged [164, 165, 168, 169]. CLumb [194, 195]; Floyd and Lumb [196]. dH~bu and Okumura [202]. CGregoire et aL [175]. fMathieson and Fowlkes [219]. SGotiinet [199], a few subeapsular APase + cells seen only by EM. 680 J.R. LUMB and by expression of the gamma T cell receptors gene. In phase 3 (17-20 day fetal thymus), cells are TdT +, and have alpha and beta T cell receptor genes rearranged/expressed, but FT-1, FT-2 and alkaline phosphatase have essentially disappeared. In phase 4 (birth-adult), the bone marrow has begun to function and provide thymic precursor cells.

Kinetics studies of bone marrow prothymocytes Studies of regeneration from irradiation using bone marrow cells have provided cellular kinetic information about prothymocytes. Bone marrow that is in the process of regenerating has fewer prothymocytes than normal bone marrow, suggesting that the prothymocytes, or the cells which produce them, are diverted to another function during regeneration [212]. Another remarkable finding was that the repopulation of the thymus which had been irradiated showed a gated phenomenon; i.e. up to a level of 450 rad of irradiation no donor bone marrow cells were found repopulating the thymus. However, with doses of />450 rad, the thymus was repopulated with numbers of donor cells that were proportional to dose [190]. Thus, the putative prothymocyte is contained in a population of host cells that survives irradiation of <450 rad. In summary, cells capable of repopulating the thymus are found in the bone marrow. Some of these cells (approx. 3 x 105 in mice) are TdT- and the rest constitute half the TdT + bone marrow cell population, approx. 105 cells.

Theoretical approaches to the pre-T cell problem The only theoretical treatment of the pre-T cell problem is THYMS (cf. Table 2, Fig. 1 and Refs [26-28]). This model has been described above (under Theoretical Approaches to the Progenitor Cell Problem). One of the results from this work is the calculation that 105 cells must enter the mouse thymus each day to maintain the observed steady-state conditions. Clearly, this area suffers from a lack of dependable cell markers. In the fetal thymus development, the markers FT-1, FT-2 and asialo GM 1 are useful, but are not described in the bone marrow. There are a few alkaline phosphatase ÷ cells in the bone marrow, but no studies have been done to determine if these cells are prothymocytes or prothymocyte progenitors. Since the TdT ÷ cells are divided evenly between those which are prothymocytes and those with B cell lineage markers, the immediate precursor of these cells may be a common one between the two lineages. In the fetal thymus, the immediate precursor of the TdT ÷ cell was alkaline phosphatase +. This marker should be exploited in the bone marrow. The unresolved questions concerning the pre-T cell/prothymocyte are: What is the nature of the prothymocyte? What is the sequence of differentiation leading up to the prothymocyte? Is there a common precursor cell with the B cell differentiation pathway? The prothymocyte should be included in the model of the bone marrow described above (under Theoretical Approaches to the B Cell Problem). An expanded model could be used to predict the characteristics of the progenitor cells and to test alternative hypotheses of prothymocyte differentiation. Because experimental data is lacking in this area, a theoretical approach can yield significant directions for laboratory research.

INTRATHYMIC T CELL DIFFERENTIATION

The basic function of the thymus is the differentiation of T cells. Cell surface markers define two major subpopulations of mature, circulating T cells with distinct functions [213]. Helper and inducer T cells have the L3T4 marker in mice and the T4 marker in humans. Cytotoxic and suppressor T cells have the Lyt-2 marker in mice and the T8 marker in humans. Most of the experimental work has been done in mice so that system of nomenclature will be used. The thymus is composed of an outer capsule, a cortex that contains 850 of the lymphocytes and a medulla that contains the other 15%. Generally, the cortical lymphocytes are less mature than those in the medulla. Treatment with hydrocortisone destroys 950 of the cells of the thymus, all of the cortical cells and two-thirds of the medullar cells, but not the cells which will initiate a graft-vs-host reaction when injected into a non-responding host. Blomgren and Andersson [214] studied the recovery from cortisone destruction of 950 of the cells in the thymus. The cortisone-sensitive cells were the short-lived cells, and therefore, the cortisone selected for the long-lived lymphocytes, which were shown to contain all the immunocompetent lymphocytes o! Lymphocyte differentiation: the need for mathematical models 681 the thymus capable of mounting a graft-vs-host reaction. Borum [215] demonstrated that small thymocytes arise from the mitotic division of large lymphocyte in the thymus. Using different marker systems, at least six subpopulations have been identified in the thymus [216--219]. There is no general agreement on a single explanation for all the results. The relevant questions are: (1) How many different thymic precursor cells migrate from the bone marrow and what is the identity of each? (2) How many different lineages mature in the thymus? (3) What are the identities of the cells at different stages in the pathway of differentiation for each lineage? (4) What are the identities of the thymus emigrant cells? thyl The marker which is associated with all mature T cells in mice is thyl. In fact, that is the working definition of a mature, circulating T cell. In the thymus, two subpopulations of cells can be distinguished by the amount of thyl and H-2 antigen they carry on their surface. Generally, the immature cortical cells have high thyl and low H-2, while medullary cells have low thyl and high H-2 [220, 221].

Ly antigens Ly antigens are markers in the mouse confined to lymphocyte lineages and are found on both B and T cells, while Lyt antigens are found only on T cells and Lyb antigens are found only on B cells. Specific antigens are designated by numbers. Lyl was originally thought to be a specific marker of T helper cells, but it is now known that Lyl is on some B cells [222, 223], and a more specific marker for T helper cells in mice, called L3T4, was developed [224]. Lyt2 and Lyt3 are generally found together on cytotoxic and suppressor T cells. Recent results delineating thymic subsets using Ly antigens have been reviewed [213, 219, 225]. In addition to the mature phenotypes, Lyt24 L3T4- and Lyt2- L3T4 4, both double negative and double positive cells exist in the thymus. The subcapsular population with low levels of Ly I (called dLyl for dull Lyl), which is also Ly2- L3T4-, has been shown to transiently repopulate the thymus after irradiation [225]. Six thymic subsets have been identified by localization of Ly 1, Lyt2, thy 1 and T200 (a cell surface molecule in the same family as B220) [226]. The two major populations were the cortical cells with high thyl, high T200, high Lyt2 and variable Lyl, and the medullary population with low thy 1, high Ly 1 and no Lyt2. A minor population of medullary cells had low thy 1, moderate T200, high Lyl and high Lyt2. Three subcapsular populations were found: (1) one that was negative for all markers tested; (2) one that was thyl 4, but negative for all others; and (3) one that was thyl 4, T2004, Lyl- and Lyt2-. Depending upon the relative sensitivities of the assays involved, these three populations could all be included in the dLyl population of Fowlkes et al. [227]. Another study asked the question, "Are there any mature, medullary cells in the outer cortex?" [228]. The outer cortex was labeled by dipping in dye and the stained cells analyzed. Neither cells capable of responding in functional assays nor cells of the complete medullary phenotype were found. Peanut agglutinin PNA has been used to separate the cortical cells (PNA 4) from the medullary cells (PNA-) of the thymus. At the time of birth 85% of the thymus cells are PNA ÷ [188]. The PNA 4 cells have the same phenotype as the majority cortisone-sensitive, cortical thymocyte (high thyl, low H-2) and are unable to participate in immune reactions [228]. The antigen Ly6.2 allowed further identification of thymic subsets of PNA + and PNA- cells [191]. The PNA 4, Ly6.2 ÷ cell population contained suppressor cells; the PNA-, Ly6.2 ÷ population contained helper cells; and the PNA ÷, Ly6.2- population contained cells of the immature phenotype, Lyt123 ÷. Terminal deoxynucleotidyl transferase TdT was first demonstrated in the calf thymus as an unusual DNA synthesizing enzyme, one which did not use a template, but added nucleotides in a random fashion [170]. PNA was used to separate cells to identify thymocyte subpopulations using TdT [192]. It was not surprising that the PNA ÷ cell population included the TdT ÷ cells, but it was surprising that TdT and TL could 682 J.R. LUMB

be induced in some of the PNA- cells by incubation with one of the thymic hormones. This raises two possibilities, that an immature cell population is among the PNA- cell populations, or that the mature PNA- cells remain capable of responding to thymic hormone influences out of the microenvironmental context of the thymus. Goldschneider et al. [218] examined proliferating cortical and medullary thymus cells by applying tritiated thymidine to the outside of the thymus and by systematic injection. They found dividing cells in the low and high thyl thymic subsets, and that all the TdT ÷ thyl low dividing cells are in the cortex, while two-thirds of the TdT- thyl low cells are in the medulla and also have high levels of H-2 antigen. Thymic education (MHC restriction) The differentiation of T cells is made more complex by the restrictions placed upon T cell function. The MHC is a gene complex which codes for a number of cell surface molecules, identified by their immunological properties as antigens, initially involved in the determination of tissue transplantation compatibility. In the mouse, this gene complex is called H-2 and in the human, HLA. Now it is clear that the several gene products of the MHC are responsible for control of the immune response. One of these is the restriction of T cell response to antigens presented in the context of self MHC determinants. The T cell antigen receptor now isolated has a dual specificity; i.e. it is specific for a particular foreign and for the appropriate self MHC antigen. This is further complicated by the existence of two classes of MHC molecules, and the fact that different subpopulations of T cells are restricted by particular classes. For example, cytotoxic T cells respond only to foreign antigens presented by cells with same class I MHC molecules as the T cell. Helper T cells are restricted to foreign antigens seen in the context of class II MHC molecules. Helper T cells are restricted to recognition of foreign antigens in the context of the Ia MHC gene product. Ia ÷ dendritic reticular cells (part of the meshwork in which the thymocytes lie) from the bone marrow are found in the medulla of the thymus [229]. Using antiserum to deplete an animal of Ia ÷ cells prevents the development of helper phenotype cells (MHC class II restricted), but not the MHC class I restricted cytotoxic cells [230-234]. The role of these bone-marrow-derived Ia ÷ accessory cells is further shown by the observation that when fetal liver cells were used to reconstitute a lethally irradiated animal instead of adult bone marrow, the MHC restriction is that of the host only, never the donor [235]. Since the cells presumably responsible for thymic education, especially in regard to MHC class II restriction, come from the bone marrow which does not begin to function until after birth, one could conclude that the Ia ÷ accessory cells may also be transferred in the bone marrow. Other mechanisms for thymic education must be present in the fetal thymus. In that context, it has been shown that the gamma T cell receptor gene expression is not required for MHC class II restricted helper T cells [236]. Non-thymic T cell development Even the assumption that T cells are all differentiated in the thymus is in question. The nude mouse, which has a congenital lack of a thymus, has some minimal development of T cell function [237, 238]. The marker thyl has been used as the definition of a mature T cell. There is question as to both the necessary and sufficient nature of this marker to T cell function. It has already been mentioned that thyl cells are found early in the fetal liver prior to the development of T cell function, and therefore, are not a sufficient condition for mature T cell function. That thyl is not a necessary condition is illustrated by recent work showing that natural killer cells have rearranged and expressed beta T cell receptor genes [239]. Natural killer (NK) cells carry neither immunoglobulin nor thyl markers and, therefore, did not fit the definitions of antigen specific lymphocytes. However, they do kill tumor cells preferentially, sparing normal cells. It will be important to determine if the alpha or gamma T cell receptor or immunoglobulin genes are rearranged in NK cells.

Summary of T cell differentiation The thymus is the first lymphoid tissue to develop. Lymphoid precursor cells migrate to the mouse fetal thymus by 14 days gestation and by birth, before the bone marrow develops, all the subpopulations are represented. In the adult, thymic precursor cells migrate from the bone marrow. Lymphocyte differentiation: the need for mathematical models 683

The subcapsular cells (dLy 1) appear to be the bone-marrow-derived emigrants because they include population(s) which will repopulate the thymus transiently in only one wave after depletion. However, there is no evidence for a self-renewing cell population in the thymus. The two major cell populations in the thymus are the major cortex cells (dividing: high thyl, low H-2, TdT ÷, PNA ÷, Lyt2 ÷, L3T4 ÷) the major medulla cells (resting: low thyl, high H-2, TdT-, PNA- and either Lyt2 ÷ or L3T.4 + but not both). Careful studies employing several markers have shown several more subpopulations. The distribution of cell subpopulations in the thymus is shown in Table 6. Cells which migrate from the thymus to the peripheral tissues have the phenotype of the medullar cells. The pathway(s) of differentiation through the thymus is(are) not fully understood at this time. In fact, the questions, (1) How many separate pathways exist in the thymus? and (2) Where do they diverge? are matters of considerable controversy. Other questions which must be resolved with quantitative information include: (3) How many cells of each type are produced? (4) At what rate do they emigrate? and (5) How many cells at each stage die in the thymus or elsewhere?

Control of T cell repertoire As with B cells, one of the most important issues is the control of the development of the T cell repertoire. In order to account for an editing step in which self-reactive T cells are destroyed, immunologists generally assume that the great majority of cells produced die in the thymus without emigrating. The earliest citation for this conclusion is Miller and Osoba [240]. In this review, the authors gave two pieces of evidence that thymocytes did not emigrate. One was an experiment in which animals, implanted with 20 thymuses, showed no increase in the level of circulating small lymphocytes. No consideration was given to host controls of cellular proliferation that would be likely to prevent the expected over-production of T cells. The other evidence given was a labeling experiment [241] which was later shown to have been flawed [242, 243]. Everett et al. [8] calculated the rate at which small thymocytes were replaced to be 1.7%/h. In 60 h, enough cells were produced to replace all the cells in the thymus or to replace all those in the peripheral lymphoid tissues. They concluded that the 95% short-lived thymocytes must die either in the thymus or after migrating into the peripheral tissues, and that the remaining 5% of the produced cells, the long-lived lymphocytes, were produced at a rate that could account for the daily increase in the number of long-lived lymphocytes in the lymph nodes, spleen, Peyer's patches, lymph and spleen. A number of studies have been done to assess the rate of cell death in the thymus. Michaelke et al. [244] studied the cellular production of the newborn mouse thymus and found that one-third

Table 6. Intrathymic cell distribution in mice Incidence Labeling Subpopulation (%) index (%) Reference No. Thymocytes (all) 100 8-11 218,225 small 85 0.6 27, 225 large 15 43 27, 225 PNA + 87 12 225 small 72 0. I 225 large 15 39 225 PNA- 13 3.9 225 Ly2 + L3T4 + 80 225 Ly2- L3T4 + I 1 225 Ly2 + L3T4- 4 225 Ly2 L3T4- 4 31 225 B2A2- 3 1.8 225 thyl + (all) 98 117 high thyl 85 9.5 218 low thyl 15 6.4 218 H-2 (all) 100 218 low H-2 78 218 high H-2 22 218 high H-2, high thyl 8 218 high H-2, low thyl 14 218 TdT + 70 218 high thyl 62 218 low thyl I 218 high thyl, high H-2 5 218 low thyl, high H-2 < 1 218 PNA + 35 192 684 J.R. LUMB

Table 7. Cell distribution in peripheral lymphoid tissues Incidence (%) Recent emigrants* Thoracic Lymph Cell type Blood duct Spleen nodes Spleen Lymph nodes Long-livedb 66 90 slg +b 17 20 36 30 thyl +b 80 80 25 70 Ly2 + L3T4 +" 3 3 0 0 Ly2- L3T4 +a 17 40 6.8 7.0 Ly2 + L3T4 -~ 8 20 3.4 4.0 Ly2- L3T4 -a 72 37 0.8 0.4 'Incidence in the spleen and lymph nodes of cells which migrated from the thymus recently and Ly2, L3T4 data [225]. bBell [277]. of the cells were lost per day. Since they found little evidence of cell death, they suggested that the cells were lost by emigration. This is similar to the turnover rate of 1.7%/h (36%/day) calculated by Everett and Tyler [7]. However, this tremendous output from the thymus is inconsistent with the well-established observation that most of the recirculating pool of lympho- cytes is long-lived (Table 7). According to calculations by Bryant [245], only 2.5% of the cells produced by the thymus contribute to the long-lived lymphocyte pool. These data all support the notion of an excess cellular production in the thymus. The obvious question, then, is, what happens to those cells produced in the thymus? Do the cells die in the thymus or do they emigrate and then die somewhere else? The histological appearance of early dying cells is one of pyknosis, the dark staining of the chromatin. Since this is easily confused with the appearance of early dividing cells, estimates of pyknosis in the thymus vary from 0.025% in the cortex and 0.073% in the medulla [244, 245] to 0.7% in the whole thymus [247]. The conclusions from these data also vary from the suggestion that the excess cells must emigrate [244, 246] to the conclusion that the pyknotic rate is almost as high as the mitotic rate as determined by histology [247]. Two studies estimated the proportion of blast cells which incorporate label but do not proceed into mitosis to be 17% [246] and 25% [245]. The reutilization of tritiated thymidine released from the dying thymocytes is a problem in these studies, tritiated thymidine has been used along with labeled iododeoxyuridine which is not reutilized [248]. It was estimated that 19-28% of the labeled thymidine was reused in the thymus, indicating that at least some of the blast cells do die in the thymus. However, this amount of cell death would be already reflected in the results of kinetic studies which determine the production of small, non-dividing lymphocytes. It is the over- production of the mature cells which is in question. Studies of the comparative counts of lymphocytes in blood prior to entrance into the thymus and following exit from the thymus give a more direct accounting of how many cells actually emigrate from the thymus. Different results are obtained depending upon the experimental animal used. In guinea pigs, only a small increase in cells was shown, not enough to account for the over-production of lymphocytes in the thymus [249-251]. In mice and rats, studies have shown an increase in the number of lymphocytes/mm 3 of blood that is sufficient to account for the emigration of all the lymphocytes produced in the thymus [252]. The difference between species was not surprising because there is histological evidence of cell death in the guinea pig thymus [252]. Guinea pigs have large cysts in their thymic medulla which contain numbers of degenerating leukocytes, while rats and mice have very few such structures until older age when the thymus involutes. The other studies which showed little evidence for cell death in the thymus involved either rats or mice. Thus, the generally accepted notion that the majority of the cells die in the thymus is probably wrong, at least for rats and mice, which have been used for most the studies on T cell differentiation. That leaves unanswered the question of what happens to all those cells produced by the thymus. Scollay et al. [253] labeled thymocytes by an intrathymic injection of fluorescein and then detected fluorescent cells in the tissues. They calculated that 1% of total thymus cells appear in the periphery each day. Again, they did not show morphologic evidence of dying cells in the thymus, but they did observe an accumulation of fluorescent material in the lungs. Using both the reutilized tritiated thymidine and the labeled iododeoxyuridine, which is not reutilized, the cellular migration from Lymphocyte differentiation: the need for mathematical models 685 the thymus was assessed in the peripheral organs [248]. The intestine and bone marrow showed the most reutilization of labeled thymidine, indicating that cells had died in these organs. However, lungs were not tested in this study. It is very difficult to detect all the lymphoid cells in an entire animal by assaying selected organs. It is known that epithelial tissues such as the skin and the lining of the intestines contain lymphocytes. The real question of the control of repertoire development is a qualitative one. What are the specificities of the T, or B cells, which die? This question has not been addressed experimentally but perhaps a theoretical approach might lead the way. Specific methods are described above (under Theoretical Approaches to the B Cell Problem).

Theoretical approaches to the thymus problem Leblond and Sainte-Marie [254] were the first to construct models of the cellular kinetics of the thymus. At that time, thymocytes were divided into three size categories, small, medium and large, and it was believed that the sequence of differentiation proceeded from the largest through to the smallest. These researchers determined the incidence and mitotic index of cells in each size category in both the cortex and the medulla. They compared the experimental data to theoretical values calculated using models which varied the number of cell divisions in each category. The model which fit the data for the thymic cortex the best was one in which the large lymphocytes divided four times, the medium lymphocytes twice and the small lymphocytes were exported. Both the cortex and medulla showed the same mitotic index for each size category except that the cell distribution was skewed heavily toward the small lymphocytes. They concluded that the additional small lymphocytes in the medulla, which could not be accounted for by the division of large and medium lymphocytes, were migrants from the cortex. They further reasoned that, since they observed diapedesis of lymphocytes (cells moving across blood vessel walls) in the medulla, and not in the cortex, that the cortical lymphocytes must be exported through the medulla to the periphery. The nucleolar morphology and the uptake of tritiated thymidine was used to determine cellular kinetic parameters in the thymus [255]. Since two nucleolar morphologies, dense nucleoli and trabecular nucleoli, were correlated with the uptake of tritiated thymidine, they were used to identify proliferating cells. Since nucleoli structures associated with proliferating cells, and those associated with resting cells, were found in all three size categories, all kinetic parameters were defined for each morphological cell type. The model they designed showed proliferating and non-proliferating compartments for each morphological category of thymus cells. The cellular transitions are: (1) from the large, proliferating lymphoblasts to non-dividing cells with the same morphology; (2) from large proliferating lymphoblasts to proliferating, medium prolymphocytes; (3) from proliferating, medium prolymphocytes to non-dividing cells with the same morphology; and (4) from proliferating, medium lymphocytes to the mature, non-dividing, small lymphocytes. Transitions (1) and (3) are hypothetically reversible and the non-dividing cells may, hypothetically, transist from to prolymphocyte to the mature lymphocyte. Since there is much more detailed information now available from studies with markers, theoretical treatments such as this one may now be brought up to date. Prothero and Tyler [256] used the analogy of specific feedback circuitry to construct a model of thymocyte proliferation. They assume that cells enter and leave G1 at random and proceed through' S, G2 and M sequentially with a defined flux time. Flow within and between sequential compartments is defined by Sl(t)=Si_l(t -- 1) and similar recursive expressions. The flow between a sequential compartment (M) and a stochastic compartment (GI) is described by the differential equation

dx/dt = Kf(t) - g(.t), where dx/dl is the rate of labeling the G1 compartment, f(t) is the fraction of labeled cells in M, g(t) is the fraction of labeled cells in G1 at time t and K is stated to be the rate of flux for G1. However, I believe that K is the flux rate for M. This equation has been analytically solved and 686 J.R. LUMB reduced to g(t) = ge -r + f(x)(1 - e-r), which is evaluated with the other recursive equations by iteration. Five different compartments were defined according to their feedback fraction and output flux: static (resting cells), expanding (both cells produced by division reenter the population), steady state (one of the cells produced by division reenters the population), input-dependent (proliferating cells which are dependent upon input from another population, not feedback) and a variable amplifier (cells which varied between the exponentially expanding and the steady state). The model constructed had four compartments, the first two having variable amplifiers representing some feedback expansion and some input- dependent flux. The third compartment was dependent only upon the input from the second of the two amplifier compartments; and the fourth was the resting population resulting from the production of the third compartment. The model results compared favorably with experimental data, which was used for verification. A more biological approach was taken in the construction of another model of thymocyte production [26-28]. The three compartments of the model (Fig. 3) were the bone marrow compartment (where the stem cells were produced to be transported to the thymus), the blood vessel compartment (where nutritional considerations were simulated) and the thymus compart- ment (where the cellular proliferation and differentiation occurred). The equations for this model are shown in Table 2. The bone marrow compartment provided a mechanism for self-renewal of the thymocyte precursor population, as has already been described (see Theoretical Approaches to the Progenitor Cell Problem). Degeneration of the thymus was accomplished in the model by the progressive increase in generation time of the stem cells as the total number of divisions a particular cell had undergone reached the 50 divisions suggested in aging studies. Since the model was to be used for the study of leukemogenesis, a nutritional compartment was needed to provide some natural control upon the proliferation of cells which have lost their inherent control. The growth of tumor cells has been shown to be dependent upon the growth of blood vessels (angiogenesis) into the tumor to provide the nutrition required for the growth of the tumor cells. These considerations were included as a gate. At each time iteration, the ratio between the number of blood vessel cells and the number of lymphocytes was tested. If there were not enough blood vessels, the lymphocytes were not allowed to divide. This introduced a discontinuity into the model. Data were compared to that of the natural thymus, according to the cellular distribution into size categories, and the rates of development and degeneration of the thymus ranging from the fetal stage to the older adult were considered. By both criteria the model adequately described the biological condition. T cell differentiation is much more complicated and elusive than B cell differentiation. Even though there has been a large amount of experimental work on this topic during the past 10 years (since the construction of the models just described) the pathway(s) of T cell differentiation has not yet been determined. A return to the theoretical approach may be helpful in unraveling this mystery. The models could be extended to cover the separate subpopulations and the various hypotheses tested quantitatively. Even if that does not completely resolve the questions, the results will clarify the experimentation needed.

LYMPHOCYTE MIGRATION From the bone marrow and the thymus, lymphocytes migrate to the peripheral lymphoid tissues, the spleen, lymph nodes, Peyer's patches in the intestine and other locations in the skin, lungs etc. The structure of these tissues and the pathways of migration through them have been reviewed [257-259]. The cells of the folloicles of the spleen and lymph nodes are generally limited to B cells. During an immune response, germinal centers appear within the follicles which represent the cellular proliferation of antigen-specific memory B cells. T cells are generally found in the areas surrounding the blood vessels in the spleen, called the periarteriole lymphatic sheath (PALS). In the lymph nodes, the T cells are found in the areas surrounding the follicles, called the paracortex. In the spleen, the marginal zone is located between the white pulp, which contains the highest concentration of lymphocytes, and the red pulp, which is responsible for red blood cell metabolism. Lymphocyte differentiation: the need for mathematical models 687

The marginal zone contains a specific subpopulation of B cells which do not recirculate. The study of prothymocyte migration is facilitated by the discovery of a marker, called B2A2 [225, 260]. This marker is found on most (95%) bone marrow cells but the cells which will repopulate the thymus are in the B2A2- population. The donor cells remain B2A2- for approx. 1 week in the irradiated host thymus. Because this is an irradiated animal, the time may not reflect the normal state, but these results do indicate that the early stage of development in the thymus may be B2A2-. Cortical thymocytes express more of the B2A2 marker than do the medullar thymocytes, but the marker persists on emigrants from the thymus. This provides a means for distinguishing between peripheral T cells and recent thymus migrants (Table 7). It has been established, however, that the migrant ceils from the thymus are of the medullary phenotype; i.e. there is no evidence of cortical cells leaving the thymus [261-264]. Hammond [265] has made a compartmental analysis of the lymphocyte recirculation data of Ford [266] and concluded that: 80% of blood lymphocytes traversed straight through the spleen without being detained (spleen retention time = 5 min); 10% of the ceils traversed through only the red pulp (spleen retention time = 2.3 h); and the other 10% of lymphocytes also entered the marginal zone and the white pulp (spleen retention time = 5.5 h). Ropke and Everett [267] showed that the short-lived lymphocytes were formed continuously in the bone marrow and thymus to replenish the populations in circulation, and that the long-lived lymphocytes recirculate and are normal residents in the bone marrow. The migration of thymus cells through lymph nodes was followed by injection of labeled cells and examination of the draining node [268]. The traversing of the node was complete within 24 h and the cells went from the subcapsular sinus, where they entered via the lymph, along the outside of the follicles and into the subnodular spaces before leaving in the efferent lymph. Since only thymus cells were injected, no cells entered the follicles which were known to contain only B cells. Migration of labeled bone marrow cells into the peripheral tissues showed that migration occurred within 2-3 days, and that the cells were distributed in the cortex of the lymph nodes (mainly in the follicles) and in the follicular areas of the spleen [269-271]. Since hydroxyurea kills dividing cells, the depletion of tissues as a function of time gives an indication of the percentage of cells in a particular tissue which have recently divided, and the time at which a tissue is maximally depleted indicates how recently these cells have divided [135]. A maximum percentage depletion was reached first in the bone marrow (85% depleted 36 h after treatment) and thoracic duct cells (70% depleted after 36 h). In the spleen, 55% of the cells were depleted after 72 h. A comparison of the migration of T and B cells showed that, for the first few hours of migration, the two types of cells were localized together at the edge of the PALS in the spleen and in the paracortex of the lymph nodes [272]. After 6 h, the B cells migrated into the follicles and the T cells migrated into the central area of the PALS of the spleen. The vessel structure through which lymphocytes migrate is called the high endothelial venules (HEV). The dynamics of cell attachment to and migration across these vessels has been studied [273,274]. The recent demonstration of a specific receptor that allows a lymphocyte to traverse through the HEV of a lymph node but not through the HEV of Peyer's patches greatly facilitates the study of lymphocyte migration [275, 276]. In summary, the migration of lymphocytes from the primary lymphocyte differentiation tissues, bone marrow and thymus, to the peripheral tissues is an extremely important point of control in the immune response. The specificity of migration involves specific lymphocyte surface molecules which interact with specific receptors on cells of the blood vessels in the peripheral tissues. The crucial question remaining unanswered is the fate of the lymphocytes produced in the primary tissues.

Theoretical approaches to the lymphocyte migration problem A theoretical treatment of lymphocyte traffic was published in 1976 by Bell [277] using quantitative data obtained for lymphocytes before the experimental techniques were available for distinguishing subpopulations. The objective of this model was to estimate the frequency of cell-to-cell interactions in vivo. The estimate of specific T-B cell interactions in the sheep was 400/h and in the mouse, 1/h. 688 J. R. LUMI!

~ ne marrow -'-.'-~:~._ ~ __ _

lted issue

Fig. 4. A schematic representation of lymphocyte migration, where all numbers indiate the number of cells x 105 that are either contained in the organ indicated or are transferred at the indicated rate. T = T cells; B = B cells.

Figure 4 is a schematic model of the migration of lymphocytes in the mouse. The migration of cells from the bone marrow to the thymus (105) is taken from the model THYMS [27]. The values given for the migration from the thymus into the blood range from the number required for maintenance of the steady-state condition (36 x 105/day), to the total number produced by the thymus (720 x 105/day), assuming no cells die there, but are disposed of somehow upon leaving the thymus. The value for the migration from the bone marrow to the blood (10 x 105) represents those surviving after approximately half the cells produced in the bone marrow die there. The total lymphocytes produced per day, using the smaller number of T cells, is 46 x 105, a number sufficient to replenish the short-lived lymphocytes of the blood and lymph (83 x 105) and to replenish a similar number of cells in the other peripheral tissues within 4 days. This is within the range of the observed turnover time of short-lived lymphocytes. The differential migration rates of T and B cells into different tissues and the relative lifetimes of each subpopulation are subjects for future study. With the information now available a budget and distribution model may be constructed for these populations. Such a model could predict the distribution of T and B cells among tissues as a check on its validity.

CONCLUSIONS An extensive search of several literature databases revealed that no theoretical treatments of lymphocyte differentiation have been published in the last 10 years. The earlier models all concern the cellular kinetics of the thymus and reflect a level of understanding from 10 years ago. Using the more detailed analyses of lymphocyte subpopulations now available, it is possible to construct more current models of lymphocyte differentiation in the fetal liver, adult bone marrow and thymus. The intent of this review was to assemble and present the information required for such models. In so doing, it became clear that the current technology could be further exploited to provide the additional quantitative data needed for model construction. For example, the fluorescence-activated cell sorter can be used for both cell cycle analysis and for immunological marker studies. These two aspects could be combined to great advantage to provide cell cycle data for all the identifiable subpopulations. One advantage of model construction is the pressure it exerts upon the experimental work to get the kind of basic quantitative data needed. Without the pressure of the needs identified by the model, experimenters may not be motivated to provide essential information. Many times, when Lymphocyte differentiation: the need for mathematical models 689 constructing models, the data used from an experimental paper is not the main point, but rather control data, or a minor point that was not of interest to the author. Therefore, it is suggested that the field of lymphocyte differentiation would benefit greatly by the resurrection of theoretical approaches. The existing knowledge in lymphocyte differentiation has been summarized to form a basis for future theoretical approaches in this area. The unresolved questions in developmental lymphocyte kinetics include:

(1) What are the identities, characteristics and cell cycle parameters of the lymphocyte progenitor cells? The earlier stages of lymphocyte differentiation are less defined and, in most cases, there is little information concerning the progenitor cells. Mathematical treatments can be used to predict the cellular kinetic parameters of these unknown progenitor cells. (2) What are the pathways to differentiation of lymphocytes? The models of the thymus constructed 10 years ago were useful to determine the internal consistency of the data and test alternative hypotheses of thymus function. The currently unresolved questions of the sequences of lymphocyte differentiation pathways can be tested in updated models, not only of the thymus, but also similar models constructed from current information on the bone marrow, fetal thymus, fetal liver and peripheral lymphoid tissues. (3) What is the fate of cells produced by the bone marrow and the thymus? How many of them die? Where do they die? Which ones die? What is the mechanism for selection? The question of the control and maintenance of the repertoire of B and T cells is one of the most exciting at this stage of immunology. The molecular understanding of the immunoglobulin and T cell receptor genes makes it now possible to consider these questions at the level of the structures of the receptors and the antigens. The optimization methods such as the iterative solution to the traveling salesman problem, control theory from economics and network theory from electrical engineering can be applied to this problem. (4) What is the migration and distribution of lymphocyte subpopulations in peripheral lymphoid tissues? The migration and distribution of lymphocyte subpopulations is another area where a theoretical treatment may be useful. In this instance, a lymphocyte budget for each of the peripheral tissues would include input, output and steady-state concentrations. Such a model could be used to test the current hypotheses of lymphocyte migration. In any consideration of mathematical models, a cautionary note must be added. Models of biological systems have a tendency to become so complex that they surpass their usefulness. It is very important for those constructing models to stay in close touch with experimental immu- nologists in order to keep in proper focus both the experimental work and the model construction. This issue has been discussed in detail elsewhere [278].

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