Modified limiting dilution assay for cell systems

N(Igllt-~

Modified Limiting Dilution Assay for Cell Systems Stefan Maier, Mario E. M a g a h a a n d R o n a l d Mohler

Department of Electrical and Computer Engineering Oregon State University Corvallis, Oregon 97331 and Stephen Kaattari*

Department of Microbiology Oregon State University Corvallis, Oregon 97331 Transmitted by F. E. Udwadia

ABSTRACT A form of mathematical analysis that permits an extended analysis of limiting dilution assays under conditions in which accessory factors or cells are limiting is presented. This is in contrast to the traditional form of analysis that permits only a single limiting parameter. Both make use of in vitro experiments to determine the number of antibody-producing cell precursors; the traditional form of analysis requires that all accessory cells or their factors be present in saturating concentrations, whereas in the analytic method employed here, not only B cells, but accessory cells (macrophages) or their factors are limiting. Analysis under nonsaturating conditions permits greater elucidation of the mechanisms of intercellular interaction.

1.

INTRODUCTION

T h e m a t h e m a t i c a l basis of limiting d i l u t i o n analysis (LDA) has been used extensively to analyze precursor frequencies u n d e r a variety of conditions *Current address: School of Marine Science, Virginia Institute of Marine Science, College of William and Mary, Gloucester Point, VA 23062.

APPLIED MATHEMATICS AND COMPUTATION78:187-195 (1996) © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010

0096-3003/96/$15.00 PII S0096-3003(96)00008-2

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[8]. In this study a mathematical model that describes the effect of two simultaneously limiting parameters on the observed response curve is constructed. The response investigated was that of the plaque-forming cell (PFC) response to trinitrophenylated-lipopolysaccharide (TNP-LPS) [7]. For each frequency determination, 6 different responder concentrations were assayed with a minimum of 60 replicate cultures per concentration. Previous studies have demonstrated that the T-independent response of trout B lymphocytes requires provision of saturating adherent accessory (macrophages) cells [1, 6, 13] or an accessory cell-derived factor [11, 13]. This requirement can be supplanted with a factor found in stimulated adherent cell supernatants and result in single-hit kinetics when applied to lymphocytes (nonadherent cells) in saturating concentrations [11, 13]. An important aspect of our model is that its construction is based upon the knowledge of certain biological processes and is presented in closed form rather than an approximation. Consequently, its interpretation is possible and practical. 2.

EXPERIMENTAL DATA

The data utilized were obtained from several experiments performed on peripheral blood leukocytes of rainbow trout [11]. In Fig. 1 the response

Lirnit~g

Dilution

Analysis

Data

10°~

|

|

~ i 0 "I "5

J :

i

i 10"

i

i ,,

2

i

i

i

3 4 6 U : t o ~ I n u m l ~ r of ~nphocytes per well

FIG. 1.

i

8

Typical experimental d a t a plots.

g

10 x 104

Dilution Assay for Cell Systems

189

curves for responder cells under saturating and nonsaturating concentrations of macrophage-derived supernatants are depicted. Comparable plots are generated with the inclusion of adherent cells only. The plots for responder lymphocytes under nonsaturating concentrations of macrophage-derived supernatant (x) or without these supernatants (0) can be divided into three distinct regions: (1) the nearly linear low-responder region, (2) an inflection or transition, and (3) another linear region at high-responder inputs. Increasing concentrations of supernatant, putatively containing IL-1 [5, 13], result in increasing slopes prior to the transition, yet no increase in post-transition slope. A close examination of the zero supernatant plot reveals that the pretransition slope is not equal to zero. This is due to experimental limitations because of the inability of the available panning techniques to remove more that 99% of all adherent cells (macrophages). Furthermore, as the amount of supernatant is increased, the transition region disappears and the resulting straight line essentially represents single-hit kinetics, as in the case of traditional LDA, where only the immune cells in question is the limiting parameter. A final note about the graph is that the abscissa represents the total number of lymphocytes (U) in the well. This is in contrast to the traditional LDA, where the abscissa represents the number of antigen-specific precursor cells (It) in the well. A final note about the graph is that the abscissa represents the total number of lymphocytes U in the well. This is in contrast to traditional LDA, where the abscissa represents the number of precursor cells u in the well specific for the given antigen.

3.

MATHEMATICAL MODEL AND I N T E R P R E T A T I O N

In this section a mathematical model for our modified limiting dilution analysis of nonsaturating accessory function is developed and an immunological interpretation offered. In general, mathematical models are either derived from observed data or from physical a n d / o r biological laws. The latter approach is based on adaptations of more general laws for a special case, whereas the former approach is not based on known laws and opens up the possibility for deriving new or perhaps modifying old ones [2~ 3, 9, 10, 12]. In our article we use the first approach and attempt to formulate certain patterns of behavior. Further distinctions have also been established for mathematical models of biological systems [4]. For example, models are classified into "correlative" and "explanatory." According to this classification, correlative models

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are those derived from data and are used for predictive purposes. On the other hand, explanatory models are derived from concepts rather than data and, in contrast to purely correlative models, can be used for prediction and interpretation rather than for prediction only. In our case, the model that we develop in this article is correlative in the sense that we start from observed data instead of concepts, although we formulate relations from our data-derived model. Therefore, our model can in a loose sense be also classified as explanatory. It is customary in the construction of a model to establish the conditions under which it will be valid. Therefore, we state the following assumptions before proceeding with the development of the model. 1. The model must show linear slopes on a semilogarithmic scale for small values of U (pretransition region slopes) whenever different concentrations of supernatant are used. 2. The model must exhibit clear changes in the slopes when different though small concentrations of supernatant are used. 3. The introduction of saturating amounts of supernatant must produce a linear plot on a semilogarithmic scale with nonzero slope; i.e., it should approach the behavior of single-hit kinetics. 4. The slopes must be the same for large values of cell input regardless of the supernatant concentration (post-transition region slopes). Let F 0 represent the fraction of nonresponding culture wells, U the total number of lymphocytes per well, and h the concentration of supernatant per culture well. Consider the decomposition of the data plots in Fig. 2. Since a semilogarithmic scale is used in the graph, it can be shown that

Yo(U) = f (U) L(U).

(1)

f l ( U ) = e- u

(2)

Let

CU

f2(U)

~/1 + "r2e 2U '

(3)

191

Dilution Assay for Cell Systems Fo(U)

,U ir~'om

of

f~(u)

supernatant

,U +t2(U) ~/__ I increase supematant

ol .U F](;.2. Decompositionofdataplots.

where the parameter ~" is associated with the location of the transition region of a plot for a given amount of supernatant concentration. To achieve different pretransition region slopes for different supernatant concentrations, Eq. (3) is modified by introducing the parameter h as

eU f2(u) =

)A

¢1 + ~2e2u

(4)

Therefore, F(} is now given by

eu E,,( u, A) = e- u

~/1 + "r2 e 2U

)A (5)

Because F 0 represents the fraction of nonresponding culture wells (a probability), it has to be initially equal to one. This is due to the fact that

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there cannot be any culture wells that respond to antigen when no lymphocytes (U = 0) are present in the wells. Therefore, regardless of the concentration of supernatant A,

Fo(O, A) =

(~ 1

= 1.

(6)

Incorporating this modification yields

p0(u,

= (

-

¢1 +

(7)

Unlike the single-hit kinetics described by traditional LDA, whenever supernatant is present in excess amounts and the precursor cells represent the only limiting parameter, our modified LDA allows one to handle two limiting parameters, that is, the total number of responding B cells and accessory factor. However, in the limit as the concentration of the accessory factor is increased to a point where it is present in excess amounts, it should approach the behavior of single-hit kinetics. Let u be the number of specific precursor cells and f be the frequency of specific precursors in the total amount of cells. Then, when the supernatant is present in limiting amounts, the following relationship is obtained: U

f(A) = -~.

(8)

This may be justified by arguing that the ability of the precursors to trigger a response depends on the amount of supernatant A present in the culture wells. Let a be a parameter such that a = f(A = 0). Then, (9)

u = fU,

and the equation for the fraction of nonresponding culture wells becomes

Fo(U, A) = () l ~ - ~ 2 A e

-aU

!/1

e.aU ) + T2e2aU

(10)

Dilution Assay for Cell Systems

193

This mathematical model for our modified LDA still does not possess the property that the location of the transition region must shift when different amounts of supernatant )t are present in the culture wells. To make the model exhibit such a behavior, a functional dependency of the parameter T on the amount of supernatant is introduced, i.e., T(h). This explicit functional relationship is yet unknown; however, it should be such that: 1. As A approaches 1, the location of the point Ut. . . . ition must approach infinity. 2. The values of Ut.... ition are relatively close together for a wide range of h, h ~ (0, 1); that is, )t lies in the interval between 0 and 1 (excluding 0 and 1). 3. As h approaches 0, the location of the point Ut. . . . ition must approach 0. Our final mathematical model is therefore given by

Fo( U, ~) =

(¢

i + ~( ~)2

e -oU

(~/i

eaU

)h

(ll)

+ ~( ~12e2a~

A brief biological interpretation of our modified LDA experiment mathematical model is now formulated. Two important elements give rise to the mathematical description of the fraction of nonresponding culture wells F0(U, )0: 1. e -aU represents a decrease in Fo(U, h) due to an increase in u proportional to U without taking into account the effects associated with h. 2. e'~;'u(1 + 72eCrU) -'~/2 is the diminishing effect on u due to the decrease of accessory factor, which results in an increase in F0( U, h). The denominator represents the fact that this diminishing effect on u has a saturation. There is another term, in addition to the previous two, that is associated with the initial condition, that is, (1 + T2) ~/2. This factor increases the overall value of F 0. In other words, the less accessory factor is present, the larger F 0 becomes because h approaches 1. Let us consider the partial derivatives of F 0 with respect to U and h, i.e.,

a~,Fo( U,~)

ouFo( U, ;,) = -aPo( U, ;') + 1 + ~ ° ~ '

(12)

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S. MAIER ET AL. O

1

-~-Fo(U , A) = ~ ln(1 + ~'2)Fo(U, A) +

A) 1 + "r2

dh

1

+ aUFo(U, A) - -~ ln(1 + z2e2aU)Fo(U, A)

Are 2aU d'r 1 + z2e 2"UF°(U' A ) - ~ .

(13)

Let us interpret Eq. (12). The partial derivative OFo/O U is the rate of change of F0(U, A) with respect to U. It is associated with an increase or decrease of F 0, depending on the sign and type of change of U. Suppose, for example, that the total amount of cells, U, is increased; then the total number of precursors u also increases. This is true because, probabilistically speaking, a large value of U implies the presence of an equivalently large number of antigen-specific precursors, u. More important, an increase in u results in a decrease of F 0, as evidenced by the term aFo(U, A). Now, if A ~= 0 (supernatant is not present in saturating amounts) the amount of specific lymphocytes u decreases as A increases and F 0 increases. This effect is represented by the term aAF0(U, A). Note, however, that this increase cannot continue indefinitely, thus the presence of the saturation term (1 + r2e2aU) -1. Equation (13) is interpreted following the same reasoning as above. 4.

CONCLUSION

The mathematical approach for the analysis of limiting dilution assays conducted under nonsaturating conditions presented in this article is the first of its kind and provides a new avenue to understanding an intrinsically complex biological assay used to determine the presence of certain immune cells in an in vitro experiment. It gives satisfactory results in several respects. In particular, (i) it fits the data very well and (ii) it can be biologically interpreted. As opposed to most models constructed for biological processes, our model is presented in closed form rather than as an approximation using a series expansion. Some special features of the model interpretation are the saturation effects due to the dependency on the number of precursor cells on the supernatant (macrophage-derived factor interleukin-1). This is what sets it apart from the traditional limiting dilution analysis.

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REFERENCES 1. M.R. Arkoosh and S. L. Kaattari, Development of immunological memory in rainbow trout (Oncorhynchus mykiss). I. An immunochemical and cellular analysis of the B cell response, Dev. Comp. Immu~ 15:279-293 (1991). 2. G. Bell, A. Perelson, and G. Pimbley, Theoretical Immunology, Dekker, New York, 1978. 3. C. Bruni, G. Doria, G. Koch, and R. Strom, Systems Theory in Immunology, Proceedings of the Working Conference in Rome, Springer-Verlag, 1978. 4. H. Gold, Mathematical Modeling of Biological Systems--An Introductory Guidebook, Wiley, New York, 1977. 5. S. Kaattari, Fish B lymphocytes: Defining their form and function, Annual Rev. Fish Diseases, 161-180 (1992). 6. S. Kaattari and R. Tripp, Cellular mechanisms of glucocorticoid immunosuppression in salmon, J. Fish Biol. 31 (Supp. A):129-132 (1987). 7. S. Kaattari and M. Yui, Polyclonal activation of salmonid B lymphocytes, Dev. Comp. Immunol. 11:155-165 (1987). 8. I. Lefkovits and H. Waldmann, Limiting Dilution Analysis of Cells in the Immune System, Cambridge Univ. Press, Cambridge, UK, 1979. 9. G. Marchuk, Mathematical Models in Immunology, Optimization Software, Springer-Verlag, New York, 1983. 10. R. Mohler, C. Bruni, and A. Gandolfi, A systems approach to immunology, Proc. IEEE, 68(8):964-990 (1980). 11. H. Ortega, Mechanism of Accessory Cell Function in Rainbow Trout (Oncorhynchus mykiss), M.S. Thesis, Oregon State Univ., Corvallis, OR, 1993. 12. A. Perelson, Theoretical Immunology, Part One, Proceedings of the Theoretical Immunology Workshop, Santa Fe, New Mexico, Addison-Wesley, New York, 1988. 13. R. Tripp, Glucocorticoid Regulation of Salmonid B Lymphocytes, Ph.D. Thesis, Oregon State Univ., Corvallis, OR, 1988.