Journal of Immunological Methods, 153 (1992)235-247
© 1992 Elsevier Science Publishers B.V. All rights reserved 11t)22-1759/92/$05.11t)
A statistical model and computer program to estimate association constants for the binding of fluorescent-labelled monoclonal antibodies to cell surface antigens and to interpret shifts in flow
cytometry data resulting from alterations in gene expression W . G . B a r d s l e y a, A. R o s s W i l s o n b, E.K. K y p r i a n o u c a n d E.M. M e l i k h o v a a a Department of Obstetrics and Gynaecology, St Mary ~ Hospital, Whitworth Park, Manchester MI30JH, UK, Departments of h Biochemistry and Molecular Biology and c Mathematics, Unit,ersity of Manchester, Oxford Road, Manchester MI3 9PL, UK, and a Institute of Applied Molecular Biology, Ministry of Health o.f the USSR, Sympheropalsky bh,d., 8, Moscow 113149, USSR
(Received 18 December 1991, revised received 19 March 1992,accepted 16 April 1992). Flow cytometry is used to obtain estimates for the distribution of fluorescent ligands bound to cell surface receptors throughout a cell sample. The equipment used provides light scattering parameters and also cell staining data in the form of dot plots and histograms of fluorescence intensities and the frequency of occurrence of particular fluorescence intensities, it is then assumed that fluorescence intensity is proportional to the number of labelled ligands bound to surface receptors. In this paper we present an outline of a statistical theory to account for the stretching and translation of such flow cytometry profiles which occur either as a result of alterations in gene expression, or from changing the sub-saturating concentration of fluorescent-labelled monoclonal antibodies or leetins used to stain the cells. We describe how the theory has been incorporated into two programs CSAFIT (cell surface antigen fit) and MAKCSA (make data to test CSAFIT). The program CSAFIT can be used to estimate two parameters, a and /3, by constrained non-linear regression analysis of the flow cytometry profiles. If the shift results from changes in the concentration of a staining agent then the estimates ~ and /~ calculated by CSAF1T are functions of the ligand concentration, the ligand type and the cell line characteristics. They quantify the stretch and translation events that are encountered in flow cytometry. So when the parameter estimates ~ and /3 are then further analysed as functions of ligand concentration, estimates for the average association constant K for the binding-site/ligand interaction can be obtained. This paper describes details of the development of programs CSAFIT and MAKCSA. We also discuss the distribution of parameter estimates calculated by CSAFIT and the overall performance of CSAF1T as assessed by simulation studies using data generated by MAKCSA. Key words: Binding constant; Binding site; Cell surface antigen; Flow cytometry; Monoclonalt~ntibody,fluore~ent; Gene dosage;
Gene expression;Lectin; Non-linear regression; Parameter estimation
Introduction Correspondence to: W.G. Bardsley, Department of Obstet-
rics and Gynaecology, St Mary's Hospital, Whitworth Park, Manchester MI3 0JH, ilK.
Flow cytometry has become an important technique to quantitate the distribution of cell surface
236 receptors throughout a population of cells (Shapiro, 1988). Usually a specific reagent such as a fluorescent-labelled monocional antibody or lectin is added to a culture of some cell line preparation and time is allowed for equilibrium to be reached between the ligand and cell surface receptors. Experimental variations on this theme are also encountered where successive layers are built up, using selected antibodies or affinity systems such as biotin/avidin. The frequency of fluorescence intensity is then measured as a function of fluorescence intensity, after selecting appropriate gates to filter out small, large, dead, clumped or other types of atypical cell. It is then generally assumed that the intensity of fluorescence is proportional to receptor occupancy, so the flow cytometry profile is usually interpreted using the concepts of gene expression and fractional saturation of sites. The ultimate aim of flow cytometry is to account for the epitopes expressed on the cell surface and detected by the ligand as gene products, in order to identify or characterise the effect of a treatment on a particular cell line. The affinity of the ligand for the receptors is of interest, as well as the specificity of the ligand and the distribution of receptors throughout the population. The aim of this paper is primarily to develop a statistical model for the fluorescence histogram as a function of iigand concentration. Then we shall describe the development of two software items, CSAFIT and MAKCSA, to put these ideas into an experimentally useful form. Finally we shall attempt to assess the rubustness of this software by analysing simulated data. SIMFIT is a set of computer programs written by one of us (Bardsley et al., 1986, 1989; Bardsley and McGinlay, 1987) for the quantitative analysis of biochemical data using subroutines from the numerical algorithms group (NAG) library. A number of SIMFIT programs such as MAKFIL (prepare a curve-fitting type file), EDITFL (edit a curve-fitting type file), MAKMAT (prepare a data matrix), EDITMT (edit a data matrix), NORMAL (test a vector of numbers for consistency with a normal distribution), MAKHL (simulate exact data for high/low affinity sites), A D D E R R (add random error to exact data) and HLFIT kilt high/low affinity sites model to data) were also
used in the development and testing of the programs MAKCSA and CSAFIT to be described. MAKCSA and CSAFIT are now included in the SIMFIT package and PC versions are available. A limited number of executable files in PC format with the software used in this study are available for distribution to interested readers who should contact one of us (W.G.B.) directly.
The binding of ligands to cell surface receptors Suppose that the fluorescence intensities, or arbitrary channel numbers, are x 0, x~ . . . . . x,, with gates A and B and that the numbers of cells with fluorescence x~_~_