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Immunofluorescent flow cytometry in N dimensions : multiplex labelling analysis, and dynamic interpretation… Buican, Tudor Nicolae 1984

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IMMUNOFLUORESCENT FLOW CYTOMETRY IN N DIMENSIONS: LABELLING ANALYSIS, AND DYNAMIC INTERPRETATION OF MULTIPLEX THE DATA By TUDOR NICOLAE BUICAN B.Sc, The University of London, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i THE FACULTY OF (Department n GRADUATE STUDIES of Physics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1984 (g) Tudor Nicolae Buican, 1984 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of PHYSICS The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 _ . 11 December 1984 Date DE-6 (3/81) i i ABSTRACT T h i s t h e s i s i n v e s t i g a t e s two new approaches to the use of m u l t i p l e antibody l a b e l s i n flow cytometry. On the experimental s i d e , we develop the M u l t i p l e x L a b e l l i n g method, which allows the number of simultaneous antibody l a b e l s to exceed the number of fluorochromes, thus overcoming the t e c h n i c a l l i m i t a t i o n s imposed by the number of a v a i l a b l e fluorochromes and f l u o r e s c e n c e measuring c h a n n e l s . On the t h e o r e t i c a l s i d e , we c o n s t r u c t a dynamic i n t e r p r e t a t i o n of immunofluorescent flow cytometry d a t a , which allows i n f o r m a t i o n on the k i n e t i c s of c e l l d i v i s i o n and d i f f e r e n t i a t i o n to be e x t r a c t e d . The f i r s t part of the t h e s i s d i s c u s s e s m u l t i p l e x l a b e l l i n g . Chapter 1.1 presents the theory of t h i s method, a r e c o n s t r u c t i o n formula on which the a l g o r i t h m f o r m u l t i p l e x l a b e l l i n g data p r o c e s s i n g can be based, and a case study i l l u s t r a t i n g the use of t h i s method in the t r i p l e l a b e l l i n g a n a l y s i s of murine thymocytes. A murine thymocyte subset not p r e v i o u s l y d e s c r i b e d by flow cytometry i s observed i n t h i s study f o r the f i r s t time. Chapter 1.2 d e s c r i b e s the Immunofluorescence Tomograph, a microcomputer-controlled d e v i c e f o r the p r e p a r a t i o n of m u l t i p l e x l a b e l l i n g s o l u t i o n s . T h i s device makes p o s s i b l e the r o u t i n e use of m u l t i p l e x t r i p l e l a b e l l i n g , by c a r r y i n g out a complicated and time consuming part of the experimental p r o t o c o l . The second part of t h i s t h e s i s d e a l s with the dynamic i n t e r p r e t a t i o n of the d a t a . Chapter 2.1 d e s c r i b e s the theory of s k e l e t a l a n a l y s i s , which i s a coarse t o p o l o g i c a l a n a l y s i s of immunofluorescent flow cytometry d a t a , concerned with the o u t l i n e s of regions where the d i s t r i b u t i o n i s s i g n i f i c a n t l y d i f f e r e n t ' from z e r o . Chapter 2.2 i n v e s t i g a t e s the f i n e r t o p o l o g i c a l d e t a i l s of the d i s t r i b u t i o n s r e s u l t i n g from d i v i s i o n and/or d i f f e r e n t i a t i o n . We show t h a t , under c e r t a i n reasonable c o n d i t i o n s , these d i s t r i b u t i o n s a c q u i r e simple forms, which can be e a s i l y analyzed and compared to a c t u a l d a t a . Chapter 2.3 presents murine thymocyte t r i p l e l a b e l l i n g data obtained by m u l t i p l e x a n a l y s i s . These i n c l u d e data on the embryonic thymus (from day 15 to day 20 of embryonic development), as w e l l as the neonate and a d u l t thymus. The methods developed i n chapters 2.1 and 2.2 are a p p l i e d , and two p a r t i a l thymocyte l i n e a g e s are d e f i n e d . One of these l i n e a g e s has not been p r e v i o u s l y r e p o r t e d . iv TABLE OF CONTENTS List of Tables v i i List of Illustrations v i i i Acknowledgements x v i i i Introduction 1 PART 1 The Multiplex Labelling Approach to Immunofluorescent Flow Cytometry in N Dimensions Ch. 1.1 Theory of Multiplex Labelling 1.1.1 - Introduction to Multiplex Labelling 8 1.1.2 - Multiplex Labelling 15 1.1.3 - The Existence and Uniqueness of the Reconstruction 20 1.1.4 - A Reconstruction Formula 22 1.1.5 - The Case N=3 27 1.1.6 - A Case Study 28 Ch. 1.2 The Immunofluorescence Tomograph 1.2.1 - General Description 37 1.2.2 - The Fluidic System 39 1.2.3 - Electronic Circuits 43 1.2.4 - Software 46 V 1.2.5 - The Operation of the Device 51 1.2.6 - A Sample Application 56 PART 2 The Dynamic Interpretation of Immunofluorescent Flow Cytometric Data Ch. 2.1 Skeletal Analysis of Single Lineages 2.1.1 - Definitions 62 2.1.2 - Path Coordinates 65 2.1.3 - Kinematics 67 2.1.4 - Equivalent Systems 71 2.1.5 - Two-Step Lineages 73 2.1.6 - n-Step Lineages 79 2.1.7 - Periodic D i v i s i o n 80 2.1.8 - D i f f e r e n t i a t i o n and Periodic D i v i s i o n 84 2.1.9 - Non-Periodic Division 91 Ch. 2.2 Flow Dynamics of C e l l Populations 2.2.1 - Definitions 101 2.2.2 - The Flow Equation without Division 103 2.2.3 - The Solution in Terms of Path Coordinates 105 2.2.4 - The Stationary Lineage 107 2.2.5 - A Non-Stationary Lineage 109 2.2.6 - The Case n = 2 112 2.2.7 - An Equivalent System 119 2.2.8 - The Flow Equation with Division 124 v i Ch.2.3 The I n t e r p r e t a t i o n of Murine Thymocyte Data 2.3.1 - I n t r o d u c t i o n to the Phenotypic A n a l y s i s of Murine Thymocytes 127 2.3.2 - Data P r e s e n t a t i o n and A n a l y s i s 130 2.3.3 - The F i r s t Lineage 137 2.3.4 - The Second Lineage 146 2.3.5 - The Wider Phenotypic P i c t u r e 153 Con c l u s i o n s 165 B i b l i o g r a p h y 169 APPENDICES 1 - Experimental Parameters and C o r r e c t i o n s 176 2 - D e r i v a t i o n of the R e c o n s t r u c t i o n Formula 181 3 - The General L i n e a r System 185 4 - C e l l P r e p a r a t i o n and L a b e l l i n g 188 5 - The R e c o n s t r u c t i o n Software 192 v i i LIST OF TABLES Table I, p. 14 Composition of the m u l t i p l e x i n g s t a i n i n g s o l u t i o n f o r a r o t a t i o n angle <X . Table I I , p. 18 Composition of a m u l t i p l e x i n g s o l u t i o n f o r N l a b e l s , c orresponding to the standard m u l t i p l e x i n g matrix . The c o e f f i c i e n t s k; represent the t o t a l c o n c e n t r a t i o n of each antibody s p e c i f i c i t y i n the s o l u t i o n . v i i i LIST OF ILLUSTRATIONS F i g u r e 1, p. 11 P r o j e c t i o n planes i n a three-dimensional l a b e l l i n g space: double l a b e l l i n g ( a ) , and t r i p l e l a b e l l i n g , where a n t i b o d i e s Ab^. and A b3 are both conjugated to fluorochrome 2 ( b ) . The three antibody s p e c i f i c i t i e s are represented by kblz,3 > a n ^ stand f o r the i n t e n s i t y of the f l u o r e s c e n c e s due to fluorochromes 1 and 2, r e s p e c t i v e l y . F i g u r e 2, p. 30 View of a constant d e n s i t y s u r f a c e of the r e c o n s t r u c t e d d i s t r i b u t i o n of a d u l t mouse (CBA/J) thymocytes. The a n t i b o d i e s used were a n t i - T h y - 1 , a n t i - L y t - 1 , and a n t i - L y t - 2 . The numbers i n the diagram represent p u t a t i v e s u b s e t s . S1 and S7 are the two s e c t i o n s shown s e p a r a t e l y i n F i g . 3 . Each a x i s has been s c a l e d up t w o - f o l d . F i g u r e 3, p. 31 Se c t i o n s 1 (a) and 7 (b) of the r e c o n s t r u c t i o n , where the outermost constant d e n s i t y contour corresponds to the constant d e n s i t y s u r f a c e d i s p l a y e d i n F i g u r e 2. The a x i s of the channel (see t e x t f o r ex p l a n a t i o n ) i s marked "+", and the p o s i t i o n of the a u t o f l u o r e s c e n c e peak "x". The contour l e v e l s are uniformly spaced. i x F i g u r e 4, p. 33 O r i g i n a l s c a l e p r o j e c t i o n s onto the L y t - 1 , L y t - 2 plane: (a) d i r e c t l y recorded p r o j e c t i o n ; (b) computed p r o j e c t i o n of the f u l l r e c o n s t r u c t i o n ; (c) computed p r o j e c t i o n of the p a r t i a l r e c o n s t r u c t i o n based on f i v e p r o j e c t i o n s r o t a t e d about the Thy-1 a x i s . The a r t e f a c t s are marked "a , 2 " , and are ex p l a i n e d i n the t e x t . The same l a b e l s are used as i n F i g u r e 3 f o r the channel a x i s and the a u t o f l u o r e s c e n c e peak. F i g u r e 5, p. 36 (a) The Thy-1,Lyt-1 phenotype of s o r t e d and r e s t a i n e d Lyt-2 low thymocytes; (b) the corresponding d i s t r i b u t i o n based on the r e c o n s t r u c t i o n ; (c) the Thy-1, Lyt-1 phenotype of unsorted thymocytes. The v e r t i c a l a x i s i n (b) and (c) was s c a l e d up by a f a c t o r of 1.25, and the h o r i z o n t a l one by a f a c t o r of 2. Fi g u r e 6, p. 38 General view of the Immunofluorescence Tomograph; (a) pumping and valve module; (b) a d d i t i o n a l valve module; (c) r e s e r v o i r u n i t , with the mixing chamber ( d ); (e) Rockwell AIM-65 microcomputer. F i g u r e 7, p. 41 Flow c i r c u i t diagram: S1, S2 - s y r i n g e s ; V1-V4 - five-way v a l v e s ; DT1, DT2 - d i s p e n s i n g t i p s . The t w o - d i g i t numbers denote the reagent r e s e r v o i r s . X F i g u r e 8, p. 42 The mixing chamber: l o n g i t u d i n a l s e c t i o n ( t o p ) , and t r a n s v e r s e s e c t i o n i n the plane of the reagent p o r t s (bottom). The dashed l i n e s represent the r i n s e medium p o r t , which l i e s o u t s i d e the plane of the s e c t i o n s . F i g u r e 9, p. 45 E l e c t r o n i c c i r c u i t diagram: SD1, SD2 - connections to the s y r i n g e d r i v e s ; VD1-VD4 - v a l v e d r i v e s ; M - AC motors; FS1, FS2 foot switches; J3 - AIM-65 expansion c o n n e c t o r . A l l r e s i s t a n c e s are i n Ohms. A l l r e s i s t o r s are 0.25W, unle s s otherwise i n d i c a t e d . A l l c a p a c i t a n c e s are i n uF. F i g u r e 10, p. 47 Schematic software flow c h a r t : . — — i n t e r r u p t c o n t r o l t r a n s f e r ; data t r a n s f e r ; IPL - Immediate P r o c e s s i n g Loop; DPL - Delayed P r o c e s s i n g Loop; IFR - i n t e r r u p t f l a g r e g i s t e r ; IRQ - i n t e r r u p t request; CLI - c l e a r i n t e r r u p t d i s a b l e f l a g ; SEI - set i n t e r r u p t d i s a b l e f l a g ; RTI - r e t u r n from i n t e r r u p t . F i g u r e 11, p. 53 Assembling of the RINSE p r o t o c o l s t e p from e f f e c t o r f u n c t i o n s . The h o r i z o n t a l dashed arrows connect higher l e v e l f u n c t i o n s to t h e i r immediate d e f i n i t i o n s (dashed boxes). The angular brackets denote e f f e c t o r f u n c t i o n s . xi Figure 12, p. 54 Operation cycle in the ROTATE mode. Operator actions are denoted by t> . The dashed box contains operations s p e c i f i c to this mode. Rjj - reagent in reservoir i j (Fig.7); <X - angle of rotation; v - t o t a l solution volume. Figure 13, p. 59 Multiplex l a b e l l i n g analysis of human peripheral blood lymphocytes: (a) rotated projections, and their true angles of rotation (the horizontal axes coincide with the axis of rotation); (b) constant density surface of the reconstructed t r i p l e l a b e l l i n g d i s t r i b u t i o n (the hatched section i s the one with the lowest l e v e l of anti-Leu-1 binding); (c) d e f i n i t i o n of the coordinate axes, projection plane, and angle of rotation. The antibodies used were OKT5, anti-Leu-1, and anti-Leu-2a. Figure 14, p. 70 The basic geometric construct (a), and i t s use in constructing the points ( n| ( n2) , with path coordinates of the form |U, =e"n,e , |>z =e " n 2 0 (b) (see the text for d e t a i l s ) . Figure 15, p. 75 Important l o c i in the skeleton for a two-step lineage, A0— A , — * - A2 : constant U^, (a), J A Z (b), j U , ^ ( c ) . Figure 16, p. 78 Transient two-step skeletons, (a) Stationary lineage, where the source at A „ i s zero for t<0. (b) Stationary lineage, where the x i i source is zero for t<0 and t>Q . The triangular region adjacent to A0 i s the transient skeleton at t=0 . (c) Non-stationary lineage, where the tr a n s i t i o n A,—»* Az i s forbidden for t< 0 . The source i s zero for t<0. The thickened l i n e segment adjacent to A0 is the transient skeleton at t= Q . The regions marked "a" and "b" contain the c e l l s introduced into the system at t>Q , and t< 8 r respectively. Figure 17, p. 83 Diagram of the fi v e states used in the derivation of Eq.(2.l6). Figure 18, p. 85 The l i m i t cycle for the system described by Eq.(2.11). This i s the only o r b i t segment such that a post-mitotic c e l l of phenotype ft reaches the phenotype 2A in a time equal to the di v i s i o n cycle period, Q . Figure 19, p. 87 The skeleton for a single-step lineage, xQ— i n the presence of c e l l d i v i s i o n : standard representation (a), and l a b e l l i n g space representation (b). The system in (b) i s of the type described by Eq.(2.11). Figure 20, p. 90 Skeletons for two-step lineages, AQ— A , — > - h2 , in the presence of c e l l d i v i s i o n : (a) the second t r a n s i t i o n i s only allowed upon d i v i s i o n ; (b) uncorrelated d i v i s i o n and d i f f e r e n t i a t i o n . The d i v i s i o n periods are the same in both x i i i d i f f e r e n t i a t i o n s t a t e s . F i g u r e 21, p. 94 Skeleton f o r a two-step l i n e a g e with d i v i s i o n , where the c e l l s undergo the second t r a n s i t i o n before t h e i r f i r s t d i v i s i o n ( a ) , 1 2 Z 3 and i t s p r o j e c t i o n s onto the planes x ,x ( b ) , x ,x ( c ) , and x',x3 ( d ) . F i g u r e 22, p. 96 The e f f e c t of (a) the d i v i s i o n mapping, D, and (b) the flow mapping, F , on an o r b i t (xG,A ) . F i g u r e 23, p. 99 The asymptotic supports of the c e l l d e n s i t y d i s t r i b u t i o n , 0"^ , and source d e n s i t y d i s t r i b u t i o n , (TOP . The system i s d e s c r i b e d by E q . ( 2 . 1 l ) , with b2>b1. The support G " ^ i s i n c l u d e d i n . F i g u r e 24, p. 106 Diagram of s t a t e s used i n the d e r i v a t i o n of Eq.(2.26). F i g u r e 25, p. 115 The three types of constant l e v e l contours of the c e l l d e n s i t y d i s t r i b u t i o n , f o r a two-step l i n e a g e with constant d i f f e r e n t i a t i o n and death r a t e s , and with an e x p o n e n t i a l s o u r c e . The contour type depends on the value of the parameter •€ (see the text f o r d e t a i l s ) . x i v F i g u r e 26, p. 116 The regime diagram i n the ^ , p l a n e . The regime depends on the values of the parameters -€• and (see the text f o r d e t a i l s ) . F i g u r e 27, p. 117 The p a t t e r n s corresponding to the s i x regimes of Fig.26. The i n d i c e s "0" and "oo " r e f e r to the behaviour of the p a t t e r n near the f i n a l a t t r a c t o r , A 2 . F i g u r e 28, p. 118 The regime diagrams i n the plane q, |3| , f o r the p a t t e r n s P2 a (a) , and P2t> ( b ) . The r e l a x a t i o n r a t e s b1 are equal to u n i t y i n the standard r e p r e s e n t a t i o n . F i g u r e 29, p. 121 The l o c i jj{ =const. and jLi2=const. i n the standard r e p r e s e n t a t i o n ( a ) , and i n l a b e l l i n g space ( b ) . The system i n (b) i s d e s c r i b e d by Eq.(2.11), with parameters b1=1, b2=3, A^-(0,0), AJ=(1,0), A j=(0.5,l). The l o c i are p l o t t e d f o r values of the path c o o r d i n a t e s ranging from 0 to 1, i n , increments of 0.1. F i g u r e 30, p. 122 T r a n s i e n t s k e l e t o n s f o r a two-step l i n e a g e , i n the l a b e l l i n g space r e p r e s e n t a t i o n : (a) s t a t i o n a r y l i n e a g e , where the source at Aj i s zero f o r t<0; (b) n o n - s t a t i o n a r y l i n e a g e , where the t r a n s i t i o n A'—*- A^ i s fo r b i d d e n f o r t<9-, and the source i s X V zero for t<0. The parameters are as in Fig.29. The standard representations of these skeletons are shown in Fig.16 a,c, respectively. The arrows indicate the direction of motion of the corresponding edges. Figure 31, p. 123 The labelling space patterns for the regimes in Fig.28. The parameters are as in Fig.29. The standard representation equivalents of these patterns are shown in Fig.27. Figure 32, p. 134 Fetal mouse (B6) thymocyte distributions for days 16 (a), and 17 (b), of embryonic l i f e . The axes of the diagrams correspond to Lyt-2 (horizontal), and Lyt-1 (vertical) labelling intensities. The Thy-1 labelling intensities are shown by the numbers to the left of each row. The numbers inside the diagrams denote the various subsets discussed in the text. Figure 33, p. 136 Fetal mouse thymocyte distributions for days 18 (a), 19 (b), and 20 (c), of fetal l i f e . The corresponding adult distribution is shown in (d) (see also the legend to Fig.32). Figure 34, p. 138 Percentage (a) and number of cells (b) in B6 thymocyte subsets 1 ( • ); 2( O ); 4( • ); 3 + 5+7( V ); and 8 ( B ) . The total number of fetal thymocytes (• ) was taken from [70]. xvi Figure 35, p. 141 Thy-1,Lyt-2 d i s t r i b u t i o n s for days 15 (a), 16 (b), and 17 (c) of f e t a l l i f e . The axes correspond to Thy-1 (horizontal) and Lyt-2 ( v e r t i c a l ) l a b e l l i n g i n t e n s i t i e s . Figure 36, p. 143 Successive f i r s t lineage d i s t r i b u t i o n s (see the text for explanations) at t=0.2 (a), and 0.6 (b). The positions of the l i m i t cycle o r i g i n s , A( and A2 , are shown in (b), while (c) shows the d i s t r i b u t i o n in the reduced skeleton generated by Ac— A , - A2 . The parameters were b1 = 1, b2 = 3, and the d i v i s i o n period 8 =1. Figure 37, p. 148 The reduced skeleton for the second lineage in the standard representation (a), and in l a b e l l i n g space (b) (see the text for explanation). The relaxation rates in (b) were b^ f 2^k"TW ' a n (^ A* XS *n t*i e Th y- 1 rLyt-2 plane. Figure 38, p. 150 Successive second lineage d i s t r i b u t i o n s (see the text for explanation) at t=1 (a), 1.2 (b), and 1.4 ( c ) . The t r a n s i t i o n A , — A4 i s only allowed for t > 1 . The two windows in thi s transient d i s t r i b u t i o n are marked "a" and "b". The parameters were b1=1, b2=3. Figure 39, p. 152 Double l a b e l l i n g d i s t r i b u t i o n s of mouse thymocytes two days x v i i a f t e r b i r t h ( a , b , c ) , and i n the a d u l t ( d , e , f ) . The l a b e l l i n g i s Thy-1,Lyt-2 (a,d); Thy-1,Lyt-1 (b,e); and Lyt-2,Lyt-1 ( c , f ) . F i g u r e 40, p. 154 The p l a c e of the two model l i n e a g e s w i t h i n the wider thymocyte d i f f e r e n t i a t i o n scheme. The dashed arrows and boxes represent those p a r t s of the scheme which were not covered by the q u a l i t a t i v e a n a l y s i s i n the t e x t . F i g u r e 41, p. 156 The reduced s k e l e t o n s f o r the two model l i n e a g e s . The numbers i n the diagram show the approximate p o s i t i o n of the subsets r e l a t i v e to these s k e l e t o n s . F i g u r e 42, p. 179 Graph r e p r e s e n t a t i o n of a b i n a r y mixture ( a ) , and of the stock and intermediate s o l u t i o n s f o r one antibody s p e c i f i c i t y ( b ) . The e x t r e m i t i e s of each arrow represent s o l u t i o n s to be mixed, and a poin t on an arrow rep r e s e n t s a m i x t u r e , the composition of which i s i n d i c a t e d by the c o o r d i n a t e shown w i t h i n b r a c k e t s (see the text f o r d e t a i l s ) . F i g u r e 43, p. 194 The format of the PARAM f i l e . xvi i i ACKNOWLEDGEMENTS In addition to my supervisor, Dr. G. W. Hoffmann, there are several people who made this work possible, by showing confidence in my approach, by allowing me free access to their f a c i l i t i e s , and, generally, by giving me their f u l l and constant support. My deep gratitude goes to Drs. G. W. Hoffmann (Physics), H.-S. Teh (Microbiology), D. Paty (Medicine), and L. F. Kastrukoff (Medicine). My interest in mouse thymocytes is a direct result of my collaboration with Dr. Teh, who also kindly provided the embryonic mouse thymocytes. I also wish to thank Dr. W. Unruh (Physics), for his valuable suggestions. Special thanks are due to Ms. Norma Morgan (Medicine) and Ms. Margaret Ho (Microbiology), for their excellent help in the lab, and to Mr. Cy Sedger (Physics), who expertly guided me in the workshop. Financial support was provided by UBC and NSERC (postgraduate fellowships), as well as by the Departments of Physics and Medicine (Division of Neurology), the Faculty of Graduate Studies, and the UBC Alumni Association. This support is gratefully acknowledged. 1 INTRODUCTION Flow cytometry (FCM) has become a w e l l e s t a b l i s h e d a n a l y t i c a l and p r e p a r a t i v e method i n s e v e r a l f i e l d s of c e l l b i o l o g y and immunology (see [ 1 , 2 ] f o r comprehensive r e v i e w s ) . Most flow cytometers can measure simultaneously s e v e r a l p h y s i c a l parameters of i n d i v i d u a l c e l l s t r a v e l l i n g i n a narrow f l u i d j e t , one or more of these parameters being f l u o r e s c e n c e i n t e n s i t i e s due to fluorochromes attached to c e l l components e i t h e r d i r e c t l y , or i n d i r e c t l y , v i a antibody m o l e c u l e s . F l u o r e s c e n t a n t i b o d i e s r e c o g n i z i n g c e l l membrane antigens are being used to i d e n t i f y and q u a n t i t a t e d i s t i n c t c e l l subpopulations (see, f o r i n s t a n c e , [ 3 ] ) , and t h i s i s , p o t e n t i a l l y , the most powerful mode of a n a l y s i s i n flow m i c r o f l u o r o m e t r y , as the e x p r e s s i o n of some molecular s p e c i e s on the membrane r e f l e c t s the d i f f e r e n t i a t i o n and p h y s i o l o g i c a l s t a t e of the c e l l s . However, out of a p o s s i b l e m u l t i t u d e of membrane i n d i c a t o r s of the i n t e r n a l s t a t e of the c e l l , only those are " v i s i b l e " which i n t e r a c t with a v a i l a b l e monoclonal a n t i b o d i e s . Thus, the chances of seeing two c e l l s ubpopulations as d i s t i n c t i n c r e a s e with i n c r e a s i n g numbers of antibody l a b e l s being used. The c o n v e n t i o n a l way of s i m u l t a n e o u s l y using s e v e r a l l a b e l s ( i n what f o l l o w s , the term " l a b e l " denotes an antibody l a b e l , whether conjugated or not) i s to a s s o c i a t e each type of antibody with one fluorochrome, the maximum number of l a b e l s being thus equal to the number of fluorochromes that can be measured independently by a given flow cytometer. T h i s approach has 2 recently been extended to quant i ta t ing three [ 4 , 5 ] , and even four l a b e l s . We have proposed in [ 6 ] multiplex l a b e l l i n g as a new approach to th is problem, by showing that the number of labels may exceed the number of fluorochromes. This approach, which i s presented in C h.1.1, allowed us to obtain t r i p l e l a b e l l i n g data before such an a n a l y s i s became possible by conventional t r i p l e l a b e l l i n g . M u l t i p l e x l a b e l l i n g i s based on two ideas drawn from seemingly unrelated f i e l d s . The f i r s t idea i s that of mult iplexing of s i g n a l s , and i t comes from communications theory. We regarded the amounts of d i f f e r e n t antibodies bound to c e l l membranes as s i g n a l s , and the fluorochromes with the corresponding fluorescence measuring devices as the physica l channels through which these s ignals are t ransmitted. It i s well known that several s ignals can share the same transmission channel, provided a coding (multiplexing) and decoding (demultiplexing) scheme i s devised which allows the s ignals to be extracted from the common transmission l i n e . The second idea i s that of reconstruct ion from rotated project ions (as used, for example, in Computer-Assisted Tomography [7]) , and i t provides a mult iplexing/demult iplexing scheme a p p l i c a b l e to immunofluorescent l a b e l l i n g . Our goal was to extract from the data the s t a t i s t i c a l  d i s t r i b u t i o n of a c e l l population in terms of the binding of N simultaneous labels (N=3 in [ 6 ] ) , rather than quantitate these labels on i n d i v i d u a l c e l l s . The d i s t r i b u t i o n s of fluorescence i n t e n s i t i e s that can be obtained in such a system by conventional double l a b e l l i n g are 2 -dimensional project ions of 3 the N-dimensional distribution onto the (2) planes generated by pairs of antibody labels. We showed in [6] that the intensity of the fluorescence due to a given label can be varied by using a staining mixture containing both the fluorescent and the non-fluorescent forms of the label, and that the fluorescence intensity of the stained cells varies linearly with the concentration of the fluorescent form relative to the total concentration of the label in the staining solution. When two or more antibody labels share the same fluorochrome (multiplexing process), the total fluorescence intensity is a linear combination of the separately recorded fluorescences. We showed in [6] that, in a two-fluorochrome system, the coefficients can be chosen in such a way (by appropriately preparing the staining mixture) as to make these linear combinations projections onto planes rotated with respect to the label axes. Staining identical samples with a sequence of staining solutions eventually gives rise to a series of rotated two-dimensional projections, from which the N-dimensional distribution can be reconstructed (demultiplexing process). We extended in [8] the argument regarding the use of two fluorochromes and three labels to the general case of M^ 2 f luorochromes and N>M. labels, and show that the complexity of the multiplex labelling procedure depends on the increase in dimensionality, N-M, rather than the over-all dimensionality, N, of the problem. Thus, the parameter M no longer sets an upper limit on the value of N, and becomes just one of two factors determining the complexity of the analytic procedure. 4 As we have al r e a d y mentioned, m u l t i p l e x l a b e l l i n g flow cytometry i s analogous to Computer-Assisted Tomography, in that a sequence of r o t a t e d p r o j e c t i o n s of a d i s t r i b u t i o n of higher d i m e n s i o n a l i t y i s obtained e x p e r i m e n t a l l y , and the o r i g i n a l d i s t r i b u t i o n i s subsequently computed from the p r o j e c t i o n s . In terms of t h i s analogy, the flow cytometer corresponds to the X-ray camera, while the person or d e v i c e p r e p a r i n g the m u l t i p l e x e d s t a i n i n g s o l u t i o n s corresponds to the r o t a t i n g stage of the tomograph. The device d e s c r i b e d i n Ch.1.2 ( a l s o i n [9]) performs p r e c i s e l y the l a t t e r f u n c t i o n , as i t produces the d e s i r e d r o t a t i o n of the p r o j e c t i o n plane i n the space of l a b e l l i n g i n t e n s i t i e s by p r e p a r i n g the a p p r o p r i a t e s t a i n i n g s o l u t i o n . Although m u l t i p l e x i n g s o l u t i o n s can be prepared by hand, the a v a i l a b i l i t y of a f u l l y automated machine makes p o s s i b l e the l a r g e s c a l e and r o u t i n e use of m u l t i p l e x l a b e l l i n g . The instantaneous d i s t r i b u t i o n of a c e l l p o p u l a t i o n i n a l a b e l l i n g space (a r e a l v e c t o r space IRM, where each co o r d i n a t e r e p r e s e n t s the amount of antibody of a given s p e c i f i c i t y , bound to the c e l l s ; see a l s o S e c t i o n 1.1.1) depends on phenomena at two l e v e l s . At the molecular l e v e l , the mechanisms which r e g u l a t e membrane antige n e x p r e s s i o n determine the way i n which i n d i v i d u a l c e l l s move i n l a b e l l i n g space as the l e v e l s of v a r i o u s a n t i g e n s on t h e i r membranes change. At the c e l l u l a r l e v e l , such processes as d i f f e r e n t i a t i o n , d i v i s i o n , and death, a l s o a f f e c t the c e l l d i s t r i b u t i o n . Thus, c e l l d i f f e r e n t i a t i o n may a l t e r the e x p r e s s i o n of p a r t i c u l a r a n t i g e n s ; c e l l d i v i s i o n p e r t u r b s the e x p r e s s i o n of c e l l membrane a n t i g e n s , by 5 d i s t r i b u t i n g them between the daughter c e l l s ; and c e l l death removes c e l l s from the p o p u l a t i o n , thus p o s s i b l y modifying the d i s t r i b u t i o n i n a non-uniform .manner .• The c e l l d i s t r i b u t i o n thus depends on parameters belonging to two l e v e l s of b i o l o g i c a l o r g a n i z a t i o n . While t h i s may be u s e f u l , as i t a l l o w s one to d e r i v e i n f o r m a t i o n about one l e v e l from a q u a n t i t a t i v e knowledge of the o t h e r , i t a l s o c omplicates the q u a l i t a t i v e and q u a n t i t a t i v e a n a l y s i s of tha d a t a . I d e a l l y , one would l i k e to separate the c o n t r i b u t i o n s of the two l e v e l s of o r g a n i z a t i o n . We show in Chapters 2.1 and 2.2 t h a t , f o r a wide c l a s s of systems, one can f i n d a mapping from the l a b e l l i n g space to a r e a l v e c t o r space of the same d i m e n s i o n a l i t y , which transforms p a r t i c u l a r antigen k i n e t i c s i n t o "standard" ones. Then, the transformed d i s t r i b u t i o n (standard r e p r e s e n t a t i o n ) becomes independent of p a r t i c u l a r k i n e t i c s of antigen e x p r e s s i o n , and i s determined e x c l u s i v e l y by the k i n e t i c s of processes at the c e l l u l a r l e v e l , such as d i v i s i o n , d i f f e r e n t i a t i o n , and death. The mapping which allows t h i s w i l l c a r r y the i n f o r m a t i o n about a n t i g e n expression which i s con t a i n e d i n the flow cytometry d a t a . If one assumes that each d i f f e r e n t i a t i o n s t a t e i s c h a r a c t e r i z e d by set values f o r the e x p r e s s i o n of each a n t i g e n , and that r e g u l a t i o n mechanisms are a c t i v e which c o r r e c t d e v i a t i o n s from these set v a l u e s , then each d i f f e r e n t i a t i o n s t a t e i s a s s o c i a t e d with an a t t r a c t o r i n l a b e l l i n g space, and c e l l s which are away from the a t t r a c t o r r e l a x towards i t . T h i s p r o p e r t y of r e l a x a t i o n towards an a t t r a c t o r , the p o s i t i o n of which depends on the d i f f e r e n t i a t i o n s t a t e , should be c a r r i e d 6 over i n t o the standard r e p r e s e n t a t i o n . We t h e r e f o r e choose a standard r e p r e s e n t a t i o n with the s i m p l e s t p o s s i b l e type of r e l a x a t i o n , i n which the o r b i t s are s t r a i g h t l i n e s converging at the a t t r a c t o r s , and with uniform e x p o n e n t i a l r e l a x a t i o n along these o r b i t s . As we show in Chapters 2.1 and 2 . 2 , the d i s t r i b u t i o n s i n the standard r e p r e s e n t a t i o n are amenable to an a n a l y s i s which i s unexpectedly s i m p l e . Indeed, very simple geometric and a l g e b r a i c methods can be used, even i n cases where one i s d e a l i n g with complex l i n e a g e s (sequences of d i f f e r e n t i a t i o n e v e n t s ) . T h i s should be q u a l i f i e d , however, by n o t i n g that t h i s type of d i r e c t a n a l y s i s i s a p p l i c a b l e to s i n g l e l i n e a g e s , that i s to d i s t r i b u t i o n s or p a r t s of d i s t r i b u t i o n s which c o n t a i n c e l l s from only one l i n e a g e . In the general c a s e , where s e v e r a l l i n e a g e s may c o n t r i b u t e to the same region of a d i s t r i b u t i o n , each l i n e a g e must be t r e a t e d s e p a r a t e l y , and the s u p e r p o s i t i o n of the r e s u l t s must subsequently be t e s t e d a g a i n s t the experimental d i s t r i b u t i o n . The t h e o r e t i c a l framework developed i n Part 2 d e a l s p r i m a r i l y with the q u a l i t a t i v e i n t e r p r e t a t i o n of the d a t a . Such an i n t e r p r e t a t i o n i s n e c e s s a r i l y based on the t o p o l o g i c a l a n a l y s i s of the d i s t r i b u t i o n s , and we approach t h i s at two l e v e l s . The f i r s t of t h e s e , which we c a l l s k e l e t a l a n a l y s i s , d e a l s with the shape of the regions i n which the d i s t r i b u t i o n i s s i g n i f i c a n t l y d i f f e r e n t from zero i n the absence of n o i s e . In the standard r e p r e s e n t a t i o n , these r e g i o n s , or s k e l e t o n s , are polyhedra which can be d e s c r i b e d and represented g r a p h i c a l l y by simple geometric means. At the second l e v e l , we d e r i v e a l g e b r a i c 7 r e p r e s e n t a t i o n s of the d i s t r i b u t i o n s , with simple t o p o l o g i c a l p r o p e r t i e s . The i n t e r e s t i n g r e s u l t we o b t a i n here i s t h a t , under c e r t a i n reasonable c o n d i t i o n s , t r a n s i e n t d i s t r i b u t i o n s can be f a c t o r e d i n t o a s t a t i o n a r y p a t t e r n , and a moving window, the l a t t e r c o i n c i d i n g with the s k e l e t o n j u s t mentioned. Thus, even r e l a t i v e l y c o m p l i c a t e d t r a n s i e n t s are amenable to t h i s kind of a n a l y s i s . The obvious qu e s t i o n i s how to c o n s t r u c t the mapping which g i v e s the standard r e p r e s e n t a t i o n . As a knowledge of the mapping i s e q u i v a l e n t to a knowledge of the k i n e t i c s of antigen e x p r e s s i o n , the d i r e c t approach would be to i n v e s t i g a t e the l a t t e r . However, t h i s would r e q u i r e pure p o p u l a t i o n s , i n which a l l c e l l s are i n the same d i f f e r e n t i a t i o n s t a t e , as w e l l as means f o r c o n t r o l l i n g d i f f e r e n t i a t i o n and p r o l i f e r a t i o n i n  v i t r o . The a l t e r n a t i v e i s to look f o r c l u e s i n the t r a n s i e n t and steady s t a t e d i s t r i b u t i o n s d e s c r i b i n g p o p u l a t i o n s which develop i n v i v o . We use t h i s approach when a n a l y z i n g the mouse thymocyte t r i p l e l a b e l l i n g data presented in the f i n a l c h a p t e r . Mouse thymocytes form a h i g h l y complex p o p u l a t i o n of d i f f e r e n t i a t i n g and p r o l i f e r a t i n g c e l l s ( S e c t i o n 2.3.1), which, while having been e x t e n s i v e l y s t u d i e d , both by immunofluorescent flow cytometry and by other methods, i s s t i l l not f u l l y understood. The advantage of using more simultaneous l a b e l s i s i l l u s t r a t e d i n the case of murine thymocytes by the r e s u l t s presented i n S e c t i o n 1.1.3 and Ch.2.3. Indeed, we were the f i r s t to d e f i n e by flow cytometry a subset of phenotype Thy-1+,Lyt-1"2" (subset 4 i n S e c t i o n 1.1.6), and to d e s c r i b e the Lyt-1 " 2 " — » - L y t - 1 ~ 2+ l i n e a g e ( l i n e a g e 1 i n Ch.2.3). 8 PART 1 THE MULTIPLEX LABELLING APPROACH TO IMMUNOFLUORESCENT FLOW CYTOMETRY I N N DIMENSIONS Chapter 1.1  Theory of M u l t i p l e x L a b e l l i n g 1 . 1 . 1 . I n t r o d u c t i o n to M u l t i p l e x L a b e l l i n g The very l a r g e number of papers based on r e s u l t s obtained by immunofluorescent flow c y t o m e t r i c a n a l y s i s and s o r t i n g bears witness to the wide acceptance gained by t h i s method i n b i o l o g i c a l r e s e a r c h . The l a b e l l i n g p a t t e r n s observed have the apparent q u a l i t y of being s t r u c t u r a l l y s t a b l e , that i s the s t r u c t u r a l d e t a i l s , such as maxima of the d i s t r i b u t i o n , as w e l l as the regions of low d e n s i t y , are c o n s i s t e n t l y observed [ 3 ] . L a b e l l i n g p a t t e r n s are probably as s t r u c t u r a l l y s t a b l e as any p a t t e r n seen under the microscope in b i o l o g i c a l samples. T h i s s t a b i l i t y suggests that b i o l o g i c a l s i g n i f i c a n c e c o u l d be attached to s t r u c t u r a l d e t a i l , as seen in l a b e l l i n g d i s t r i b u t i o n s . In f a c t , a flow cytometer co u l d be regarded as the analogue of a microscope, as i t a l l o w s one to see b i o l o g i c a l s t r u c t u r e s d e f i n e d i n terms of p o s i t i o n , shape, e x t e n t , and s i z e , i n a space where each a x i s corresponds to the e x p r e s s i o n of a given a n t i g e n i c determinant. The amount'in which a p a r t i c u l a r a n t i g e n i s present on the c e l l membrane can be measured only i n d i r e c t l y . T h i s i s done by 9 using a n t i b o d i e s which bind s p e c i f i c a l l y to the a n t i g e n , and which have been made p h y s i c a l l y or c h e m i c a l l y d e t e c t a b l e . In flow cytometry, f l u o r e s c e n t m o l e c u l e s , such as f l u o r e s c e i n e or rhodamine, are used to tag the antibody m o l e c u l e s . Given a set of N antibody s p e c i f i c i t i e s , we d e f i n e the corresponding l a b e l l i n g space as the r e a l v e c t o r space JRM , i n which we represent a c e l l by a p o i n t , the c o o r d i n a t e s of which are equal to the amounts of corresponding a n t i b o d i e s ( l a b e l s ) bound to that c e l l . Let us a l s o assume t h a t , given a c e l l p o p u l a t i o n , the b i n d i n g of one antibody s p e c i f i c i t y does not i n t e r f e r e with the b i n d i n g of the other N-1 s p e c i f i c i t i e s . Then, each c o o r d i n a t e of the r e p r e s e n t a t i v e p o i n t of a c e l l i n l a b e l l i n g space i s p r o p o r t i o n a l to the amount of the corresponding a n t i g e n on the c e l l membrane. Under t h i s assumption, each p o i n t with p o s i t i v e c o o r d i n a t e s i n l a b e l l i n g space r e p r e s e n t s a c e l l phenotype. N a t u r a l l y , our a b i l i t y to i n v e s t i g a t e t h i s space i s l i m i t e d f i r s t of a l l by the number of a v a i l a b l e monoclonal antibody s p e c i f i c i t i e s . T h i s l i m i t a t i o n can be overcome by generating monoclonal a n t i b o d i e s with new s p e c i f i c i t i e s . A second l i m i t a t i o n i s due, not to our a b i l i t y to see i n a l l the important dimensions, but r a t h e r to our i n a b i l i t y to see i n a l l these dimensions at once. One way of overcoming t h i s d i f i c u l t y i s to design i n c r e a s i n g l y complicated a n a l y z e r s , with more and more measuring channels ( i . e . more l a s e r s , d e t e c t o r s , o p t i c a l and e l e c t r o n i c components). However, any r e a l a n a l y z e r has some maximum number of f l u o r e s c e n c e s i t can measure s i m u l t a n e o u s l y , and t h i s number i s f u r t h e r l i m i t e d by the o v e r l a p between the e x c i t a t i o n and/or emission s p e c t r a of the 10 fluorochromes. Many f i e l d s of research share t h i s l i m i t a t i o n : a wide v a r i e t y of r e c o r d i n g d e v i c e s , from r a d i o t e l e s c o p e s to medical X-ray machines, p r o v i d e only one- or two-dimensional p r o j e c t i o n s of t h r e e - d i m e n s i o n a l s t r u c t u r e s (see [10] f o r comprehensive r e v i e w s ) . The problem of " i n v e r t i n g " the p r o j e c t i o n , that i s going back, from a set of p r o j e c t i o n s , to the r e c o n s t r u c t e d s t r u c t u r e of higher d i m e n s i o n a l i t y , has been solved mathematically by J . Radon [11] i n 1917, and h i s b a s i c s o l u t i o n has been a p p l i e d with great s u c c e s s , the most widely known a p p l i c a t i o n being computer-assisted tomography (CAT) i n a l l i t s v a r i a n t s . If one c o n s i d e r s a d i s t r i b u t i o n i n terms of three l a b e l s , there are three p o s s i b l e double s t a i n i n g s , each of them g i v i n g a 2-dimensional p r o j e c t i o n along one of the three axes. Were i t p o s s i b l e to o b t a i n two-dimensional p r o j e c t i o n s along other d i r e c t i o n s as w e l l , then one c o u l d o b t a i n a whole sequence of r o t a t e d p r o j e c t i o n s , from which the three-dimensional ("three-c o l o u r " ) d i s t r i b u t i o n c o u l d be r e c o n s t r u c t e d . M u l t i p l e x l a b e l l i n g p r o v i d e s a s o l u t i o n to t h i s problem, by a l l o w i n g s e v e r a l d i s t i n c t a n t i b o d i e s to share the same fluorochrome. For example, F i g . 1 a r e p r e s e n t s the t h r e e axes of the l a b e l l i n g space d e f i n e d by a n t i b o d i e s hbxz^ , as w e l l as the two axes F&(b °f the p r o j e c t i o n recorded by the a n a l y z e r . In t h i s example, Ab, i s a s s o c i a t e d with fluorochrome 1 ( f l u o r e s c e n c e Fa) , while Ab3 i s a s s o c i a t e d with fluorochrome 2 ( f l u o r e s c e n c e Fb) . The two-dimensional data recorded by the a n a l y z e r w i l l be the p r o j e c t i o n of the t h r e e - d i m e n s i o n a l d i s t r i b u t i o n onto the plane Ab.,Ab,, 11 45° Rotation F i g u r e 1 Proj.ection planes i n a thr e e - d i m e n s i o n a l l a b e l l i n g space: double l a b e l l i n g ( a), and t r i p l e l a b e l l i n g , where a n t i b o d i e s Ab2 and Ab3 are both conjugated to fluorochrome 2 (b). The three antibody s p e c i f i c i t i e s are re p r e s e n t e d by 2 3' a n d F a b s t a n < 3 f ° r t n e i n t e n s i t y of the f l u o r e s c e n c e due'to fluorochromes 1 and 2, r e s p e c t i v e l y . 12 that is the projection along the Ab2 axis. However, this need not be the case. Let us consider again the above example, with labels Ab4 and Ab3 this time sharing fluorochrome 2. One can see that both labels Ab 2 and Ab3 contribute equally to the fluorescence Ft, and that the fluorescence Fb of a given c e l l equals the sum of the fluorescences due to Aba and Ab3 individually. This can be interpreted as a rotation of 45° about the Fa and Ab, axis, together with an expansion by a factor of VT along the Ffe axis (Fig.1b). This simple example shows that i t is possible to obtain projections along directions other than those obtained through conventional double labelling. In general, the projection on a line making an angle o< with the Ab2 axis is Ffe =x2_cos«c+x^sinflf, where x 2 3 are the individual fluorescences due to Ab 2 3 , respectively. This means that the measured fluorescence, Ffa, must contain contributions from both Ab2 and Ab3, but that these contributions must be smaller than the independent fluorescences x2 f 5 associated with the two labels, by factors cosof and sinor, respectively. The fluorescence intensity due to one particular antibody can be decreased at will by using a mixture of conjugated and unconjugated antibody, where "conjugated" means either "directly conjugated with a fluorochrome", or "conjugated with a hapten (small molecule) to which a fluorochrome can be specifically attached". Let us consider a stock solution of conjugated antibody, which saturates the binding sites on the c e l l membranes. An 13 e q u i v a l e n t stock s o l u t i o n of unconjugated antibody i s d e f i n e d as one which causes the f l u o r e s c e n c e i n t e n s i t y to decrease by a f a c t o r of two when the l a b e l l i n g i s done with a mixture i n equal p r o p o r t i o n s of the two s o l u t i o n s . Then, a f l u o r e s c e n c e i n t e n s i t y equal to m.x (where x i s the f l u o r e s c e n c e i n t e n s i t y measured i n the absence of unconjugated a n t i b o d y , and m i s a number between 0 and 1) can be obtained by p r e p a r i n g a s t a i n i n g mixture c o n t a i n i n g a p r o p o r t i o n m of the conjugated stock s o l u t i o n , and 1-m of the unconjugated one. Going back to the implementation of the p r o j e c t i o n formula, one can see that a f l u o r e s c e n c e i n t e n s i t y equal to Fb i s obtained by using the s t a i n i n g mixture with the composition g i v e n i n Table I . To summarize, the data r e q u i r e d f o r the r e c o n s t r u c t i o n of the d i s t r i b u t i o n i n terms of Ab 1,1,3 can be obtained i n the f o l l o w i n g manner: ( i ) e s t a b l i s h the s a t u r a t i n g c o n c e n t r a t i o n s f o r the conjugated Ab,) Z (^ ; ( i i ) e s t a b l i s h the e q u i v a l e n t c o n c e n t r a t i o n s f o r the unconjugated A b2 3 ; ( i i i ) decide on the "angles of view" f o r which p r o j e c t i o n s are to be recorded; ( i v ) for each a n g l e , prepare s t a i n i n g s o l u t i o n s as in Table I; proceed as i n c o n v e n t i o n a l double l a b e l l i n g , with one sample and the corresponding s t a i n i n g s o l u t i o n f o r each angle o( . The r e s u l t i n g two-dimensional d i s t r i b u t i o n s are the r o t a t e d p r o j e c t i o n s , which are used as input f o r a r e c o n s t r u c t i o n a l g o r i t h m . A set of r o t a t e d p r o j e c t i o n s , and the r e s u l t i n g r e c o n s t r u c t e d d i s t r i b u t i o n , are shown i n F i g . 1 3 . In the f o l l o w i n g s e c t i o n s of t h i s c h a p t e r , we d i s c u s s i n d e t a i l the theory of m u l t i p l e x l a b e l l i n g f o r M^2 fluorochromes, and N>M antibody l a b e l s . We a l s o d e r i v e a r e c o n s t r u c t i o n 14 Table I Composition of the m u l t i p l e x i n g s t a i n i n g s o l u t i o n f o r a r o t a t i o n angle (X . Antibody Form P r o p o r t i o n Ab2 conjugated cos < X II unconjugated 1-cos (X Ab3 conjugated s i n <K n unconjugated 1-sin CK 1 5 formula, which can be used as the b a s i s f o r a r e c o n s t r u c t i o n a l g o r i t h m . 1 . 1 , 2 . M u l t i p l e x L a b e l l i n g If a p a r t i c u l a r c e l l r e g i s t e r s f l u o r e s c e n c e i n t e n s i t i e s x xH when s e p a r a t e l y s t a i n e d with one of N independent l a b e l s , then any l i n e a r combination X]€ [ 0 , 1 ] , can be achieved e x p e r i m e n t a l l y by c o n t r o l l i n g the conjugated/unconjugated c o n c e n t r a t i o n r a t i o of each l a b e l i n the s t a i n i n g s o l u t i o n ( i n t h i s c o n t e x t , "unconjugated" and "non-f l u o r e s c e n t " may a l s o mean "conjugated to a d i f f e r e n t f l u o r o c h r o m e " ) . In a two-dye system ( f l u o r e s c e n c e i n t e n s i t i e s f, 2, ) t h i s becomes with X;j^ O , and XijtXzj^ l t where Xij i s the c o n c e n t r a t i o n of l a b e l j conjugated with dye i , r e l a t i v e to the t o t a l c o n c e n t r a t i o n of l a b e l j . The f a c t that the same c o o r d i n a t e s X J can be used i n both l i n e a r combinations ( 1 . 1 ) i s not immediately apparent. In f a c t , Eqs. ( 1 . 1 ) hold only i f the r a t i o of s p e c i f i c f l u o r e s c e n c e s of the two f l u o r e s c e n t forms i s the same f o r a l l N a n t i b o d i e s . We show i n Appendix 1 how the experimental parameters can be ad j u s t e d in order to ensure that t h i s c o n s i s t e n c y c o n d i t i o n i s s a t i s f i e d , and a l s o how to c o r r e c t 16 for possible deviations of the parameters from their optimum values. The 2XN matrix L= ( \{j ) is called a labelling matrix , and relationship (1.1) defines a linear mapping *- tR2. This mapping becomes an orthogonal projection onto a plane in i f and only if N 2] ^i j ^tej = As the coefficients of L are positive, this condition can be rewritten as « X»j ^2J = ° j - ' , - - , ^ 1 . 2 b A labelling matrix whose coefficients satisfy (l.2a,b) is called a multiplexing matrix . Labelling done according to such a matrix will orthogonally project the representative point of a c e l l in the N-dimensional space of the labels onto the "plane" of the fluorescence intensities. Relationship (1.2b) implies that each column of a multiplexing matrix contains a zero element. Consequently, L divides the N labels into two groups, corresponding to the two fluorochromes. We define a standard multiplexing matrix L_ as 17 where m=1,...,N-1. A l l other m u l t i p l e x i n g m a t r i c e s with m non-zero elements i n the f i r s t row can be w r i t t e n as the product L^- f l sL^TT of a standard matrix and an operator TT permuting the components xw. . . , xN : TT ( x , = (x^.^,... , x-rCrtO where TT i s a permutation of N numbers. T h i s r e p r e s e n t a t i o n separates the choice of c o e f f i c i e n t s of the l i n e a r combinations ( 1 . 1 ) from the c h o i c e of l a b e l s i n each combination. Table II g i v e s the composition of the s t a i n i n g s o l u t i o n c orresponding to L m . The composition for a general L m i r i s obtained by permuting the s p e c i f i c i t i e s i n Table II according to IT . Two m u l t i p l e x i n g m a t r i c e s which can be transformed i n t o each other by a permutation of rows, and/or of columns belonging to the same group, are e q u i v a l e n t , i n the sense that the p r o j e c t i o n s ( f ^ f ^ ) of a p o i n t ( x , ,...,xN ) d i f f e r at the most by a permutation of the components f | ^  . We s e l e c t a r e p r e s e n t a t i ve matrix L m i r from each eq u i v a l e n c e c l a s s by r e s t r i c t i n g the range of the index m to 1,...,[N / 2 ] , where the square b r a c k e t s mean the l a r g e s t i n t e g e r l e s s than or equal to 18 Table I I Composition of a m u l t i p l e x i n g s o l u t i o n f o r N l a b e l s , c orresponding to the standard m u l t i p l e x i n g matrix Lm . The c o e f f i c i e n t s r e p r e s e n t the t o t a l c o n c e n t r a t i o n of each antibody s p e c i f i c i t y i n the s o l u t i o n . L a b e l Form C o n c e n t r a t i o n antibody 1 fluorochrome 1 klXl l fluorochrome 2 0 • unconjugated V i -Xn > antibody m fluorochrome fluorochrome unconjugated 1 2 k m lm 0 k (1- X, ) m lm antibody m+1 fluorochrome 1 0 • fluorochrome unconj ugated 2 km+1^2,m+l km+l( 1" ^ 2,m+l) antibody N fluorochrome 1 0 fluorochrome 2 kN X 2 N kN ( 1- X2N ) unconjugated 19 N/2, and the permutations T T to the set G d e f i n e d as fo r m=1 , . . . , [N/2] ,v with the IT (1 ) <TT (N/2+1 ) f o r N even and m=N/2. a d d i t i o n a l c o n d i t i o n The t o t a l number of elements i n a l l G ^ , which i s a l s o the 14 - I number of equivalence c l a s s e s , i s 2 - 1 . A l l matrices Lm F w i l l be assumed from now on to be r e p r e s e n t a t i v e . We s h a l l show that the r e p r e s e n t a t i v e m a t r i c e s thus d e f i n e d are s u f f i c i e n t , and a l l 2 -1 equivalence c l a s s e s necessary, f o r c a r r y i n g out a complete r e c o n s t r u c t i o n . Let us now choose the angles 6,, ... ,6^-2' 6j£[ 0 , "TT/2 ] , as the N-2 independent parameters i n Lm . We d e f i n e \ i j ( 0) as X21 Ce) cose-, Sin 0, 0 0 cose^ . 'oinO;-, cos©,-. .-Sin 6,-2COS • • f>m 0M-3 Sin 614-2 i =l I = r n =\, . . . ,»n = rr>+» 1.3 These c o e f f i c i e n t s s a t i s f y E q s . ( 1 . 2 a , b ) , and t h e r e f o r e ( \ i j ( © )) i s a standard m u l t i p l e x i n g m a t r i x . One can see by i n s p e c t i o n t h a t , f o r Xii s a t i s f y i n g ( ! . 2 a , b ) , there always 20 e x i s t s a s o l u t i o n 0 £ [0,TT/2] of the above e q u a t i o n s . T h e r e f o r e , any standard m u l t i p l e x i n g matrix can be w r i t t e n as 6 ) = ( >.j ( 6 ) ) , 6£[0,TT/2]H-Z . I f we r e w r i t e Eqs.O.1) f o r a standard m u l t i p l e x i n g matrix m \ z = ^2J XJ the equations of the corresponding p r o j e c t i o n plane are K j * Xij ^ , j s i , . . . , m where ^i^^tR . The o r i e n t a t i o n of the plane depends on the angles 6 through the c o e f f i c i e n t s • 1.1.3. The E x i s t e n c e and Uniqueness of the R e c o n s t r u c t i o n In order to prove that the N-dimensional d i s t r i b u t i o n § ( x ) can be uniquely r e c o n s t r u c t e d from two-dimensional p r o j e c t i o n s 3 ( f ) , we s h a l l show that the F o u r i e r Transform § (x) can be sampled everywhere. The p r o j e c t i o n theorem f o r F o u r i e r Transforms (see Appendix 2 and [12]) p r o v i d e s a d i r e c t l i n k between the N-dimensional F.T. 3 (x) and the two-dimensional F.T. S ( f ) , so that a l l that remains to be proved i s that any point IT i s contained i n a plane generated by a m u l t i p l e x i n g 21 m a t r i x . I f x0 = ( xo t ,...,xf l N ) i s an a r b i t r a r y p o i n t , l e t us rename i t s c o o r d i n a t e s y t---iy0n c o o r d i n a t e s form two groups, y Ol such that the new and O f »y om the elements of each group having the same si g n s €| 2, r e s p e c t i v e l y , or being z e r o . We now c o n s t r u c t a 2XN matrix ( XIJ ) with c o e f f i c i e n t s given by ( \u -1o\ 1 m Lo 1*i 1.4 where d e f i n e d , and only s u b j e c t to c o n d i t i o n s (1.2a,b) i f the denominators v a n i s h . The matrix ( A{j ) thus d e f i n e d i s obviously a (standard) m u l t i p l e x i n g m a t r i x , and the plane i t d e f i n e s in F o u r i e r space, ~ f i = rsft+l,..., N 1.5 f, I^R. t c o n t a i n s ( yO J , . . . , ye H )at the p o i n t of c o o r d i n a t e s 22 1.6 i n the p l a n e . T h i s can be shown by s u b s t i t u t i n g (1.4) and (1.6) i n t o (1..5). I f Lm i s given by Xij d e s c r i b e d above and TT i s the permutation operator t r a n s f o r m i n g x G i n t o yQ , then / N O x 0 i s con t a i n e d i n the plane given by Lm i r , with the same c o o r d i n a t e s (1.6) i n the p l a n e . To sum up, one can sample § at any x e i n the f o l l o w i n g manner: ( i ) c o n s t r u c t Lw i ra s above; ( i i ) prepare a s t a i n i n g s o l u t i o n f o r L n i r as d e s c r i b e d i n the pr e v i o u s s e c t i o n , and use i t to s t a i n a c e l l sample; ( i i i ) r e c o r d the f l u o r e s c e n c e d i s t r i b u t i o n ^ ' ( f ) and compute i t s F.T. at fc given by (1.6). According to the p r o j e c t i o n theorem (Appendix 2 ) , S ( xe ) i s equal to S" (f 0 ) times a constant f a c t o r . Thus, one can uniquely r e c o n s t r u c t the F.T of ^ b y m u l t i p l e x l a b e l l i n g , and t h e r e f o r e i t s e l f . 1.1.4. A R e c o n s t r u c t i o n Formula . The r e c o n s t r u c t i o n formula i s a f o r m a l i z a t i o n of the argument presented in the pr e v i o u s s e c t i o n , which allows one to compute 5* (x) s t a r t i n g from a set of r o t a t e d p r o j e c t i o n s . Apart from suggesting a convenient way of computing C> ( i t e l i m i n a t e s the need f o r f i r s t e s t i m a t i n g 3 the formula w i l l a l s o suggest a r a t i o n a l way of o r g a n i z i n g experimental d a t a . A compact d e r i v a t i o n of a formula f o r r e c o n s t r u c t i n g two-23 dimensional d i s t r i b u t i o n s from one-dimensional p r o j e c t i o n s i s given i n [12]. T h i s d e r i v a t i o n hinges upon the use of c o o r d i n a t e s n a t u r a l to the system of r o t a t e d p r o j e c t i o n s , namely the p r o j e c t i o n c o o r d i n a t e and the r o t a t i o n a n g l e . In what f o l l o w s , we s h a l l apply t h i s approach to two-dimensional p r o j e c t i o n s generated by m u l t i p l e x m a t r i c e s , where an a t l a s of o v e r l a p p i n g l o c a l c o o r d i n a t e t r a n s f o r m a t i o n s can be c o n s t r u c t e d , using the two p r o j e c t i o n c o o r d i n a t e s f . , , and the N-2 angles of r o t a t i o n . We s h a l l f i r s t c o n s t r u c t a c o v e r i n g of f R " . Let Sn c t RN , n = 0,...,N, be the subset Sn ={x€tR*1 : x , . . . ,x n *0; x « + i f . . . » xM -0), and i t s symmetrical subset by i n v e r s i o n at the o r i g i n . Let Qm , m=1,...,[N/2], be d e f i n e d as Qm = Sm U S'm U S0 U SM , and Qm i T, TT-£ Gm , as Q =TT Qm • There are 2 -1 subsets Q m- r r , forming a c o v e r i n g of IK : [HA] (J U a ^ R1 4 , »*-• which i s redundant only over S0 U SN (2 -2 times) and over a subset of zero measure. F o l l o w i n g the argument i n the previous s e c t i o n and the d e f i n i t i o n s of L ^ ( 0" ) , Lm Tr , and °- tr>Tr ' a ny x ^ QVV»TT c a n ^e w r i t t e n as where 6€ t O , T r / 2 j a nd f €JR 2 . Equation (1.7) d e f i n e s a mapping of IR xj[o,"n"/2] onto Q w i r , which i s a c o o r d i n a t e 24 t ransformation. Since the subsets Q form a covering of IR." , for any x €. K N one can f i n d m= 1 , . . . , [N/2 ], Tr-e G m , e-€[0 , i r/2]N"2 , f £ J R 2 , such that E q . ( l . 7 ) holds . We thus have an a t las on (RN , each l o c a l coordinate transformation being associated with one representative mult iplexing matrix. Let us now project the N-dimensional d i s t r i b u t i o n ^ onto the plane L (8 ). The r e s u l t i n g two-dimensional d i s t r i b u t i o n i s where M m T r ( © ) projects IRK onto the orthogonal complement of the plane defined by L m t T ( 9 ). The Radon transform of ^ on Q ^ , Hmyt . i s defined as and represents nothing more than the set of a l l projec t ions for given m and TT . F i n i t e samplings of CS") can be obtained experimentally as sets of project ions with the same m and TT (same type of r o t a t i o n ) , but with d i f f e r e n t angles of r o t a t i o n , Q . Thus, the data obtained by multiplex l a b e l l i n g N —I experiments can be organized into 2 -1 sets of rotated project ions . If we define the N-dimensional F . T . , C ^ ' H , and i t s two-dimensional version ac t ing in the projec t ion plane, > a s * n 25 Appendix 2, and i f we use the d e f i n i t i o n of rr\vr , the p r o j e c t i o n theorem can be w r i t t e n as L5 - [ T z ^ r f f l r ( 6 ) ] ( f ) e ) 1.10 (2TT)2""' where x € QyttTT > an d f » L *WTT (6-) X • The d i s t r i b u t i o n <^  C— C*"" can be obtained from E q . O . 1 0 ) by i n v e r t i n g . I f J"N i s w r i t t e n i n the new c o o r d i n a t e s ( f , Q ) , some rearrangement of the equation e v e n t u a l l y y i e l d s the r e c o n s t r u c t i o n formula (Appendix 2 ) , which can be put, by us i n g the operator n o t a t i o n i n [ 1 2 ] , i n the f o l l o w i n g form: [N/ll _, S =2 Z % § W .... Win "rr-e<A»n The r e c o n s t r u c t i o n formula (1.11) i s a sum of 2 -1 terms, each f o r m a l l y s i m i l a r to the r e c o n s t r u c t i o n formula i n [ 1 2 ] . Each term of the sum i n v o l v e s one Radon t r a n s f o r m , that i s one set of r o t a t e d p r o j e c t i o n s . Equation (1.1 1) i n c l u d e s the f i l t e r f u n c t i o n $ , which i s given by $ c ? ^  - (i - H c i1)) i r •1 ln -1 where n=N-m, and H i s the Heav i s i d e s t e p f u n c t i o n . The f i r s t f a c t o r i n (1.12) d e a l s with the redundancy of the c o v e r i n g , 26 while the second factor is a direct generalization of the f i l t e r in [12], and results from the "radial" dependence of the volume element. TYYTT" *s t n e back-projection operator in Q m T T- , and carries over the angular dependence of the volume element. Its definition is also given in Appendix 2. We shall define a partial reconstruction as a single term of (1.11), with =|f™~x *t£~% |. It is especially important for m=1, which gives the only rotations possible in systems where one of the labels is not available in both conjugated and unconjugated form. Although partial reconstructions produce a distorted version of § , important topological details are preserved, as wil l be seen in one of the following examples. The theory of N-dimensional analysis in two-fluorochrome systems can be extended in a straightforward manner to M-fluorochrome systems. Thus, if f, , . . . , fM are the fluorescence intensities for the M fluorochromes, one can identify * and use fluorochromes 1,2 to multiplex x, ,...,x . The general case of N labels and M fluorochromes can therefore be reduced to that of N-M+2 labels and two fluorochromes, which we have already discussed. This point deserves special emphasis, because i t shows that multiplex labelling can increase the 27 d i m e n s i o n a l i t y N of measuring systems of any degree of complexity M (M*2). Moreover, the complexity of m u l t i p l e x l a b e l l i n g depends on N-M, that i s on the number of a d d i t i o n a l dimensions, r a t h e r than the o v e r - a l l d i m e n s i o n a l i t y of the problem. 1.1.5. The Case N=3 The r e c o n s t r u c t i o n formula f o r N=3 c o n s i s t s of three terms, each c o n t a i n i n g p r o j e c t i o n s r o t a t e d about one of the three l a b e l axes. I f only one type of r o t a t i o n i s p o s s i b l e , the p a r t i a l r e c o n s t r u c t i o n d e f i n e d i n the preceding s e c t i o n w i l l convolute the t r u e d i s t r i b u t i o n ^ (x) by the inv e r s e F.T. of the c h a r a c t e r i s t i c f u n c t i o n of the subset swept by the p r o j e c t i o n p l a n e s . Thus, i f only r o t a t i o n s about the x3 a x i s are used, the p a r t i a l l y r e c o n s t r u c t e d d i s t r i b u t i o n w i l l be the sum of the u n d i s t o r t e d § , and the r e s u l t of c o n v o l u t i n g C5 with the f u n c t i o n -1/x,x2. The presence of the u n d i s t o r t e d d i s t r i b u t i o n i s r e s p o n s i b l e f o r p r e s e r v i n g the major peaks of the d i s t r i b u t i o n , while the d i s t o r t e d c o n t r i b u t i o n to the r e c o n s t r u c t i o n w i l l produce spurious peaks of lower magnitude ( e s p e c i a l l y i n the presence of smoothing), and an a n i s o t r o p i c r e s o l u t i o n . The spurious peaks due to the str o n g e s t maxima of w i l l i n t e r f e r e with the weaker peaks, and w i l l t h e r e f o r e be most n o t i c e a b l e i n the regions of lower d e n s i t y . These p o i n t s w i l l be i l l u s t r a t e d by d i r e c t comparison between a f u l l r e c o n s t r u c t i o n and one of i t s p a r t i a l r e c o n s t r u c t i o n s . 28 1.1.6. A Case Study We t e s t e d the r e c o n s t r u c t i o n procedure i n a t r i p l e l a b e l l i n g a n a l y s i s of a d u l t mouse thymocytes. The monoclonal a n t i b o d i e s used in t h i s experiment recognize the thymocyte d i f f e r e n t i a t i o n antigens Thy-1, L y t - 1 , and Lyt-2 ( S e c t i o n 2.3.1). The . c e l l p r e p a r a t i o n and l a b e l l i n g procedure i s d e s c r i b e d i n Appendix 4. We prepared f i f t e e n s t a i n i n g s o l u t i o n s , corresponding to f i v e e q u a l l y spaced angles of r o t a t i o n ( 0° to 90°) about each a x i s . In a d d i t i o n , we prepared c o r r e c t i o n samples f o r each r o t a t i o n by o m i t t i n g one of the m u l t i p l e x e d l a b e l s from the s t a i n i n g s o l u t i o n . T h i s allowed us to determine the true v a l u e s of the c o e f f i c i e n t s X,j (Appendix 1). The maximum angular e r r o r was found to be of the order of 5 ° , with a s c a l e e r r o r f o r the m u l t i p l e x e d f l u o r e s c e n c e of the order of 5%. The input f o r the r e c o n s t r u c t i o n program (Appendix 5) c o n s i s t e d of the f i f t e e n p r o j e c t i o n s and the set of angle and s c a l e c o r r e c t i o n s d e r i v e d from the a n a l y s i s of the c o r r e c t i o n samples. An a d d i t i o n a l sine f i l t e r ( s i n c ( x ) i s d e f i n e d as sin(TT x)/(TTx) f o r x^O, and i s equal to 1 f o r x = 0) was i n t r o d u c e d i n the r e c o n s t r u c t i o n formula, and was used on both axes of the p r o j e c t i o n s i n order to reduce n o i s e . A l i n e a r i n t e r p o l a t i o n was used when computing the b a c k - p r o j e c t i o n . The program generated a 64x64x64 matrix of d e n s i t i e s which c o n s t i t u t e s the r e c o n s t r u c t e d 3-D d i s t r i b u t i o n . A s o r t i n g and p l o t t i n g program generated 2-D s e c t i o n s through the r e c o n s t r u c t i o n , and c a r r i e d out computed s o r t i n g f o r given f l u o r e s c e n c e i n t e n s i t y ranges. F i g u r e 2 shows a constant l e v e l 29 s u r f a c e of the r e c o n s t r u c t e d three-dimensional d i s t r i b u t i o n , o b t a i n e d by generating t h r e e - c h a n n e l - t h i c k s e c t i o n s along the Thy-1 a x i s . Two of the s e c t i o n s are shown s e p a r a t e l y i n F i g . 3 . P u t a t i v e subpopulations are marked i n F i g u r e s 2 and 3, and w i l l be d i s c u s s e d i n Ch.2.3. Two t e s t s were c a r r i e d out i n order to i d e n t i f y p o s s i b l e r e c o n s t r u c t i o n a r t e f a c t s . In the f i r s t t e s t we p r o j e c t e d the r e c o n s t r u c t i o n onto the L y t - 1 , Lyt-2 p l a n e , i n order to compare the r e s u l t i n g 2-D d i s t r i b u t i o n with the d i r e c t l y recorded one. We a l s o generated p a r t i a l r e c o n s t r u c t i o n s based on only one set of p r o j e c t i o n s , and a l s o p r o j e c t e d these onto the same plane (only the r e s u l t s obtained from r o t a t i o n s about the Thy-1 a x i s are shown). The r e s u l t i n g 2-D p r o j e c t i o n s are shown in F i g . 4 . A comparison of these p r o j e c t i o n s shows that ( i ) the main t o p o l o g i c a l d e t a i l s (such as maxima and minima) of the d i r e c t l y recorded d i s t r i b u t i o n are present in a l l the r e c o n s t r u c t i o n s , and that the r e l a t i v e amplitudes of these d e t a i l s are more d i s t o r t e d i n the p a r t i a l r e c o n s t r u c t i o n than in the f u l l one; ( i i ) the s t r u c t u r e s l a b e l l e d "a,( 2. " i n the p r o j e c t i o n s of the f u l l and p a r t i a l r e c o n s t r u c t i o n s can be i n t e r p r e t e d as r e c o n s t r u c t i o n a r t e f a c t s , as they do not appear in the d i r e c t l y recorded p r o j e c t i o n ; ( i i i ) r e g i o n s of negative d e n s i t y (not shown) may be generated, due to the f i n i t e number of a v a i l a b l e p r o j e c t i o n s . As a consequence, low d e n s i t y regions at the edge of a peak may appear to have even lower d e n s i t y ( r e g i o n a ^ i n F i g . 4 b , c ) . These a r t e f a c t s are l e s s pronounced i n the f u l l r e c o n s t r u c t i o n , and become dominant only at l a r g e d i s t a n c e s from the o r i g i n , where the c e l l d e n s i t y i s low anyway due to the 30 F i g u r e 2 View of a constant d e n s i t y s u r f a c e of the r e c o n s t r u c t e d d i s t r i b u t i o n of a d u l t mouse (CBA/J) thymocytes. The a n t i b o d i e s used were anti-Thy-1, a n t i - L y t - 1 , and a n t i -L y t - 2 . The numbers i n the diagram r e p r e s e n t p u t a t i v e subsets. and are the two s e c t i o n s shown s e p a r a t e l y i n F i g . 3 . Each a x i s has been s c a l e d up two-fold. SI + F i g u r e 3 S e c t i o n s 1 (a) and 7 (b) of the r e c o n s t r u c t i o n , where the outermost constant d e n s i t y contour corresponds to the constant d e n s i t y s u r f a c e d i s p l a y e d i n F i g . 2 . The a x i s of the channel (see t e x t f o r explanation) i s marked "+", and the p o s i t i o n of the a u t o f l u o r e s c e n c e peak "x". The contour l e v e l s are u n i f o r m l y spaced. 32 F i g u r e 4 O r i g i n a l s c a l e p r o j e c t i o n s onto the Lyt-1,Lyt-2 plane: (a) d i r e c t l y recorded p r o j e c t i o n ; (b) computed p r o j e c t i o n of the f u l l r e c o n s t r u c t i o n ; (c) computed p r o j e c t i o n of the p a r t i a l r e c o n s t r u c t i o n based on f i v e p r o j e c t i o n s r o t a t e d about the Thy-1 a x i s . The a r t e f a c t s are marked "a^ 2" , and are e x p l a i n e d i n the t e x t . The same . l a b e l s are used as i n F i g . 3 f o r the channel a x i s and the a u t o f l u o r e s c e n c e peak. 34 manner in which the gain coefficients were set. A more direct test consisted of sorting out thymocytes with a low level of Lyt-2 staining, restaining them with anti-Lyt-1 and anti-Thy-1, and analyzing them again in dual parameter mode. The same operation was simulated on the reconstructed 3-D distribution. The resulting 2-D distributions are shown in Fig.5, together with the corresponding distribution for unsorted thymocytes. Both sorted distributions show the existence of Thy-1 high, Lyt-1 low, Lyt-2 low (4) and Thy-1 high, Lyt-1 medium, Lyt-2 low (5) subpopulations, separated by a region of lower density. The main difference between the two distributions is that the peak corresponding to subpopulation 5 is narrower in the reconstruction, and the valley separating subsets 4 and 5 is deeper. This is probably due to negative contributions not being exactly compensated for at the edges of the peak. Subset 8 appears to contain a larger percentage of the thymocytes in the reconstruction. Also, a subset labelled 8' appears in the directly recorded data in Fig.5a, but not in the reconstruction. Both these subsets are small (subset 8 represents 1-2% of total thymocytes in the reconstructed distribution), and probably indicate the lower limit in the ability of this particular reconstruction to detect small but dense subsets. 35 F i g u r e 5 (a) The Thy-1,Lyt-1 phenotype of s o r t e d and r e s t a i n e d Lyt-2 low thymocytes; (b) the corresponding d i s t r i b u t i o n based on the r e c o n s t r u c t i o n ; (c) the Thy-1, Lyt-1 phenotype of unsorted thymocytes. The v e r t i c a l a x i s i n (b) and (c) was s c a l e d up by a f a c t o r of 1.25, and the h o r i z o n t a l a x i s by a f a c t o r of 2. 37 Chapter 1.2  The Immunofluorescence Tomograph 1.2.1. General D e s c r i p t i o n The Immunofluorescence Tomograph i s an automatic s o l u t i o n making machine which should f u l f i l l s e v e r a l s p e c i f i c requirements: ( i ) i t should be a b l e to prepare a l l the types of s o l u t i o n r e q u i r e d i n m u l t i p l e x l a b e l l i n g ; ( i i ) i t should be able to compute the composition of the s o l u t i o n s from the f u n c t i o n a l parameters entered by the operator (e.g. the a x i s and angle of r o t a t i o n of the p r o j e c t i o n plane i n the case of a m u l t i p l e x e d s o l u t i o n ) ; ( i i i ) i t should be a b l e to handle a wide range of reagent volumes; ( i v ) i t should minimize reagent and mixture dead volumes, i n order to a v o i d wasting expensive reagents; (v) i t should minimize c r o s s - c o n t a m i n a t i o n between s u c c e s s i v e samples. In a d d i t i o n , the o p e r a t i o n of the machine should be f a s t enough to allow a l a r g e number of d i f f e r e n t s t a i n i n g mixtures to be prepared in a reasonable time. A l s o , the machine should be able to load the reagent r e s e r v o i r s a u t o m a t i c a l l y , i n order to minimize operator i n t e r v e n t i o n when s w i t c h i n g to a new set of a n t i b o d i e s . We s h a l l show that the machine d e s c r i b e d here meets a l l these requirements. A general view of the machine i s shown in F i g . 6 . I t s main u n i t s are the pump and v a l v e module, which a l s o houses most of the e l e c t r o n i c c i r c u i t r y ( a ) ; the a d d i t i o n a l v a l v e module, which a l s o c o n t r o l s the d e l i v e r y of p r e s s u r i z e d a i r to the mixing chamber (b); the r e s e r v o i r u n i t ( c ) , with the mixing chamber (d ) ; and the Rockwell AIM-65 microcomputer ( e ) . The machine I ^ T T T l 6 ,Jen^alVie1 ° f t h e I m i ™ n o f l u o r e s c e n c e Tomograph; (a) pumping and v a l v e M?? S , 5 : 1 ? ? ; 1 « a l Y e m o d u l e ' ' <c> r e s e r v o i r u n i t , w i t h the mixing chamber (d); (e) Rockwell AIM-65 microcomputer. w 39 performs a set of pre-programmed functions by using several physical devices ( e f f e c t o r s ) , which run under computer c o n t r o l . These include stepper motor driven syringes, switching valves, and a solenoid valve. A sound alarm, two foot switches, and four timers are also treated as e f f e c t o r s by the computer. 1.2.2. The F l u i d i c System A diagram of the f l u i d i c s i s shown in Fig.7. Two gas-tight syringes, S1 and S2 (Hamilton, Reno, NV), and stepper motor syringe drives (Hamilton), are used for pumping the reagents. The f u l l stroke of the syringes i s covered in 1000 steps. The maximum plunger speed is one f u l l stroke in 5s for a 100ul syringe. Each syringe is connected to a pair of five-way precision valves (Hamilton), which work under microprocessor control as a single eight-way valve. The valves are driven by AC motors, with cam-actuated switches as position transducers (Hamilton). Two of the ports of each valve pair are connected to the source of driv i n g f l u i d ( d i s t i l l e d water), and to the d r a i n , while the remaining f i v e are connected to reagent r e s e r v o i r s . The d i s t i l l e d water i s separated from the reagents by a programmable a i r gap. Thus, the reagents never come into contact with the syringes and valves, which reduces the r i s k of cross-contamination and eliminates reagent dead volumes. There are ten r e s e r v o i r s , grouped in three modules of three reservoirs (6 ml capacity each), with a larger reservoir (25 ml) housed in a fourth module. Eight of the smaller reservoirs store antibody solutions, while the ninth (reservoir 33 in Fig.7) 40 F i g u r e 7 Flow c i r c u i t diagram: S I , S2 - s y r i n g e s ; V1-V4 - five-way v a l v e s ; DTI, DT2 - d i s p e n s i n g t i p s . The two-d i g i t numbers denote the reagent r e s e r v o i r s . aSCONECT DT 2 {medium) DT 1 (mixture) F i g u r e 8 The mixing chamber: l o n g i t u d i n a l s e c t i o n ( t o p ) , and t r a n s v e r s e s e c t i o n i n the -plane of the reagent p o r t s (bottom). The dashed l i n e s r e p r e s e n t the r i n s e medium p o r t , v/hich l i e s o u t s i d e the plane of the s e c t i o n s . 43 s t o r e s the medium used f o r r i n s i n g the mixing chamber and d e l i v e r y l i n e between samples. The l a r g e r r e s e r v o i r s t o r e s the medium used i n washing c e l l samples. The r e s e r v o i r s c o n s i s t of loops of PTFE t u b i n g , connected to flow connectors (Hamilton) mounted on the f r o n t p a n n e l . The module c a s i n g s are w a t e r t i g h t , and c o l d water i s c i r c u l a t e d through them. The mixing chamber i s made of p l e x i g l a s . A l o n g i t u d i n a l s e c t i o n through i t i s shown i n F i g . 8 , together with a t r a n s v e r s e s e c t i o n i n the plane of the reagent p o r t s . There are eig h t coplanar reagent p o r t s (diameter 0.5mm), which open i n t o a c e n t r a l channel (diameter 1.8mm). A r i n s i n g medium port i s s i t u a t e d above the reagent p o r t s . The c e n t r a l channel i s connected to the p r e s s u r i z e d a i r supply at i t s upper end, and to the d e l i v e r y t i p DT1 at the lower end. The channel i s tapered towards the d e l i v e r y l i n e c o n n e ctor, to prevent the t r a p p i n g of a i r bubbles i n the l i q u i d column. The compressed a i r passes through a combination f i l t e r / m o i s t u r e t r a p , and i t s pressure can be adjusted i n the 0-10 p s i range by a bleed-type r e g u l a t o r and pressure gauge. The flow of a i r to the mixing chamber i s c o n t r o l l e d by a three-way s o l e n o i d v a l v e , i n s e r i e s with a needle v a l v e as flow l i m i t e r . 1.2.3. E l e c t r o n i c C i r c u i t s The e l e c t r o n i c c i r c u i t s (Fig.9) were designed so as to keep the hardware to a minimum, by l e t t i n g the microcomputer c a r r y out most c o n t r o l f u n c t i o n s . The e l e c t r o n i c c i r c u i t s c o n s i s t of two p a r t s : the e f f e c t o r d r i v e r s , and the computer i n t e r f a c e . The stepper motor d r i v e r s 44 F i g u r e 9 E l e c t r o n i c c i r c u i t diagram: S D l , SD2 -connections to the s y r i n g e d r i v e s ; VD1-VD4 - v a l v e d r i v e s ; M - AC motors; FS1, FS2 - f o o t switches; J3 -AIM-65 expansion c o n n e c t o r . A l l r e s i s t a n c e s are i n Ohms. A l l r e s i s t o r s are 0.25W, un l e s s otherwise i n d i c a t e d . A l l c a p a c i t a n c e s are i n uF. 46 are integrated circuits (Z5,Z6), and solid state AC relays (Z8-Z12) are used to drive the valve motors and the solenoid valve. The interface is built around two single-chip programmable interface adaptors (Z1, Z3), which are configured by an in i t i a l i z a t i o n subroutine at the beginning of program execution. These adaptors contain the timers used in controlling the speed of the stepper motors, as well as in timing air delivery and the alarm. They also do the local processing of the interrupt signals coming from the syringe drive end-of stroke (EOS) transducers, the valve drive position transducers, the foot switches FS1, FS2, and the timers. The interface is connected directly to the J3 expansion connector of the AIM-65. Only the three most significant hex digits of the address bus are used to select the interface registers. Of these, the most significant digit is decoded by the computer (CS8 output), the next digit addresses the registers in Z1, Z3 (inputs RS0-RS3), an the third digit is decoded by the 1-of-16 decoder Z2. The position-sensing switches in the valve drives VD1-VD4 are addressed through the dual 1-of-4 decoders Z4, 1 1 . This reduces the number of interface adaptor ports used for this task from sixteen to eight, and leaves a number of ports available for future expansion. 1.2.4. Software The software is written in the 6502 microprocessor language [14], and uses 10 Kbyte of RAM (random access memory). It consists (Fig.10) of two parts: the monitor, and the interrupt PARAMETERS & FLAGS INITIALIZE > MONITOR DATA I/O »l y I FR « L , -->IFR POLLING «—» HALT sbr Y ^ - r i p r -Jlag set 7, I PL Set DPL (lag R2«-R1 Reset R1 C LI R2 POLLING . . . » FUNCTIONS * TRIG sbr's Reset DPLflag > R2 F i g u r e 10 Schematic software flow c h a r t : i n t e r r u p t c o n t r o l t r a n s f e r ; data t r a n s f e r ; IPL - Immediate P r o c e s s i n g Loop; DPL - Delayed P r o c e s s i n g Loop; IFR -i n t e r r u p t f l a g r e g i s t e r ; IRQ - i n t e r r u p t request; CLI -c l e a r i n t e r r u p t d i s a b l e f l a g ; RTI - r e t u r n from i n t e r r u p t 48 p r o c e s s i n g program, to which c o n t r o l i s t r a n s f e r r e d from the monitor whenever an i n t e r r u p t s i g n a l i s r e c e i v e d from one of the e f f e c t o r s . The f u n c t i o n i n g of the e f f e c t o r s i s c o n t r o l l e d e x c l u s i v e l y by the i n t e r r u p t p r o c e s s i n g program, while the monitor d e a l s with the communication between operator and machine (input and output of parameters), and mode s e l e c t i o n . The i n t e r r u p t p r o c e s s i n g program a l s o c o n s i s t s of two p a r t s : the Immediate P r o c e s s i n g Loop ( I P L ) , and the Delayed P r o c e s s i n g Loop (DPL). The way i n which c o n t r o l i s t r a n s f e r r e d between these and the monitor i s i l l u s t r a t e d i n F i g . 1 0 . If the monitor i s e x e c u t i n g , an i n t e r r u p t s i g n a l w i l l t r a n s f e r c o n t r o l to IPL. Once t h i s segment has been executed, c o n t r o l i s t r a n s f e r r e d to DPL. I f a new i n t e r r u p t occurs while DPL i s e x e c u t i n g , c o n t r o l i s t r a n s f e r r e d back to IPL, and then r e t u r n s to the i n s t r u c t i o n where DPL was i n t e r r u p t e d . Upon completion of DPL, c o n t r o l r e t u r n s to the i n s t r u c t i o n where the monitor was i n i t i a l l y i n t e r r u p t e d . As w i l l be seen, t h i s a l l o w s . c e r t a i n e f f e c t o r h a n d l i n g s u b r o u t i n es to be executed without d e l a y , while o t h e r s , where promptness of execution i s not important, are d e a l t with as soon as the former have been completed. The two i n t e r r u p t p r o c e s s i n g loops communicate with each other through the b u f f e r r e g i s t e r R1, i n which IPL w r i t e s i n f o r m a t i o n on e f f e c t o r o p e r a t i o n s which have been completed. The i n t e r r u p t loops communicate with the monitor only through the parameter and f l a g r e g i s t e r s . The e x e c u t i o n of the monitor i s preceded by that of an i n i t i a l i z a t i o n s u b r o u t i n e , which se t s the i n i t i a l values of the parameters and f l a g s , and c o n f i g u r e s the i n t e r f a c e adaptors Z1 49 and Z3. The monitor performs the conversion and transfer of data between the parameter registers and the operator, via data input/output subroutines and keyboard, display, and printer. It also restricts the operator's access to relevant registers while effector functions are being executed. The parameters are displayed as parameter pages, each page corresponding to one mode of operation. The functioning of each effector is controlled by two subroutines: (i) TRIG, which starts the effector; and ( i i ) HALT, which stops i t . The HALT subroutines must be executed without delay upon completion of the effector function, and are therefore called by IPL. The TRIG subroutines are called by DPL, after the interrupt information has been processed. The effectors are thus free-running (not under microprocessor control), except for brief periods at the beginning and end of an operation. This allows the effectors to function quasi-independently of each other, and complex effector functions can therefore be easily programmed. The immediate processing loop polls the interrupt flag registers (IFR) of the two interface adaptors upon receiving an interrupt request (IRQ) signal, and stops the effectors whose task has been completed by calling the appropriate HALT subroutine. The HALT subroutines also reset the interrupt flags in IFR, and set the flags in R1 signifying completion of effector function. When a l l the flags in IFR have been processed, the DPL flag is checked to determine whether DPL was executing at the time of the interrupt request. If the flag is set, a Return from Interrupt (RTI) instruction is executed, 50 which transfers control to where DPL was interrupted. If the f l a g i s c l e a r , control i s transferred to the beginning of DPL, without RTI. The interrupt from the stepper motor timers i s transferred to the buffer R1 only i f the required number of steps has been completed. The execution of DPL begins by transferring the contents of R1 to R2, following which R1 i s reset, and the interrupt disable f l a g in the processor status register i s cleared by a CLI i n s t r u c t i o n . This allows new interrupts to be processed by IPL while DPL i s executing. The R2 POLLING program segment reads the interrupt flags in R2. If a set f l a g i s encountered, the f l a g i s cleared, and control passes to the FUNCTIONS block, together with the interrupt information. For specified values of the various program flags and counters, t h i s block c a l l s TRIG subroutines to start new effector operations. If a l l the flags in R2 have been cleared, the interrupt disable flag i s set (SEI i n s t r u c t i o n ) , and R1 i s checked for new interrupts. If no new interrupt has occured while DPL was executing, there i s an RTI, with control being transferred back to the monitor. If a new interrupt has occurred, control i s transferred back to the beginning of DPL. The purpose of the FUNCTIONS block i s to assemble individual effector functions into more complex functions, protocol steps, and eventually a complete protocol. While the effector subroutines and the immediate processing loop r e f l e c t the nature of the effector and driver hardware, the FUNCTIONS block encodes the s p e c i f i c function of the machine. For the 51 purpose of i l l u s t r a t i o n , we show in Fig.11 how the RINSE protocol step (rinsing the mixing chamber and the delivery line with 25ul of medium) is assembled by the FUNCTIONS block from effector functions. The intermediate level functions, such as PUMP and CONNECT, are shared by several protocol steps. They allow the functional repertoire of the machine to be easily modified or expanded. The speed of execution is increased by the simultaneous use of several effector functions, where appropriate. 1.2.5. The Operation of the Device The Immunofluorescence Tomograph has six modes of operation, which are accessed by calling the corresponding parameter pages. Each mode displays a page of parameter values, which can be set by the operator through keyboard entries. The actual operation of the machine in a given mode is triggered through foot switches, and the parameters are printed prior to execut ion. The main modes are ROTATE, CALIBRATE, and TITRATE , corresponding to the three types of staining solutions in multiplex labelling (Appendix 1). A flow chart of the operation cycle in the ROTATE mode is shown in Fig.12. This illustrates the preparation of a multiplexed solution, where the functional parameters are as follows: (i) the antibody 3 axis is the axis of rotation; ( i i ) o< is the angle of rotation, as measured between the projection plane and the antibody 1 axis; ( i i i ) fluorochrome 2 is conjugated with antibody 3 (this determines the use of antibodies 1,2 conjugated with fluorochrome 1); and 52 F i g u r e 11 Assembling of the RINSE p r o t o c o l step from e f f e c t o r f u n c t i o n s . The h o r i z o n t a l dashed arrows connect hig h e r l e v e l f u n c t i o n s t o t h e i r immediate d e f i n i t i o n s (dashed boxes). The angular b r a c k e t s denote e f f e c t o r f u n c t i o n s . -1--"! PUMP ! *• (25 ul from res.33) <PURGE (20s)> 1 I" : Volume in SS13S25UI?. CONNECT (S1 to^source) <DRIVE SYRINGE> (fill up S1) If CONNECT (S1 to res.33) I <DRIVE SYRINGE> (25 ul out) > Display ROTATE parameter page |> Enter parameters O Depress foot switch i Print R O T A T E parameter page \ Pump 25ul rinse medium into mixing chamber I Air on (20s) Rinse chamber and delivery l ine ! - j 1 Pump simultaneously j c o s a of R ( J and -^ ( l -cos«) of R | 3 I Pump simultaneously -|sino«r of R 2 I and ^ ( l - s ina ) of R 2 3 | i Pump ± of R_. L-----1 Audible warning signal I Air on (30s) Mixture expelled through delivery line F i g u r e 12 Oper a t i o n c y c l e i n the ROTATE mode. Operator a c t i o n s are denoted by > . The dashed box c o n t a i n s o p e r a t i o n s s p e c i f i c to t h i s mode. Rj_j - reagent i n r e s e r v o i r i j ( F i g . 7) ; o< - angle of r o t a t i o n ; v - t o t a l s o l u t i o n volume. 55 (iv) v is the total solution volume to be prepared. The operation cycle is performed in approximately 90s for a total mixture volume of lOOul. Each pumping operation is preceded by the valves being set so as to connect the appropriate reagent reservoirs to the syringes. When two reagents are to be pumped simultaneously, the syringe plunger speeds are set by the program proportional to the volumes to be pumped, so that both reagents are pumped into the mixing chamber over the same time interval. This ensures proper mixing of the conjugated and unconjugated forms of the multiplexed antibodies. The speed with which the rinsing medium and the mixture are expelled through the delivery line is adjusted through the air pressure regulator. The CALIBRATE and TITRATE modes differ from ROTATE only in the operations enclosed in the dashed box of Fig.12, and the parameter page displayed. The CALIBRATE mode allows one of the multiplexing antibodies to be omitted from the staining mixture, which is then used for staining calibration samples (Appendix 1), from which the real values of the multiplexing parameters can be estimated. The TITRATE mode is used in preparing the solutions required in the conjugated vs. unconjugated antibody titration (Appendix 1). Cross-contamination between successive samples is virtually eliminated by the rinsing step. The small diameter of the reagent input, channels also ensures that there is minimal uncontrolled exchange between these and the central channel. This exchange is further reduced by the geometry of the central channel, which causes the free surface of the mixture to l i e 56 below the level of the reagent ports. The use of pressurized air for solution delivery eliminates mixture dead volumes. Reagent dead volumes are also eliminated by the use of a separate driving f l u i d . The LOAD mode allows for the automatic loading of the reservoirs. The parameter page for this mode contains the volumes, to be loaded into each reservoir, as well as the volume of the air gap. The loading cycle begins by drawing into the lines the required volume of a i r , followed by drawing in the reagents. Finally, the reagents are pumped out t i l l they reach the mixing chamber input ports. Two additional modes allow the two syringes to be used as independent pumps, in conjunction with the ten reservoirs. In these modes, the machine can be used as a dual reagent dispenser. The machine can also be used for sterile work, by placing the reservoir unit inside a sterile hood. With the exception of the mixing chamber, a l l parts which come into contact with the reagents are made of PTFE or glass, and can be chemically sterili z e d in situ . The mixing chamber is easily removed from the unit, and can be autoclaved. 1.2.6. A Sample Application We illustrate the use of the Immunofluorescence Tomograph by a multiplex triple labelling analysis of human peripheral blood lymphocytes in terms of anti-Leu-2a, anti-Leu-1 (Becton-Dickinson Monoclonals, Mountainview, CA), and OKT5 (Ortho Pharmaceuticals, Raritan, NJ). Nine multiplexed staining 57 s o l u t i o n s were prepared, corresponding to p r o j e c t i o n planes r o t a t e d about the OKT5 a x i s by u n i f o r m l y spaced angles ranging from 0° to 9 0 ° . The angles of r o t a t i o n were measured between the p r o j e c t i o n plane and the anti-Leu-2a a x i s ( F i g . 1 3 c ) . As the OKT5 antibody was used s e p a r a t e l y i n t h i s experiment, i t was r e p l a c e d by medium in the p r e p a r a t i o n of the m u l t i p l e x e d s o l u t i o n s . The c e l l s e p a r a t i o n and s t a i n i n g were done as d e s c r i b e d elsewhere [ 1 5 ] . The c e l l s were f i r s t s t a i n e d with p u r i f i e d OKT5 and FITC-conjugated goat anti-mouse IgG (Cap'pel L a b o r a t o r i e s , C o c h r a n v i l l e , PA). Nine i d e n t i c a l a l i q u o t s (10s c e l l s i n 50ul medium) were subsequently incubated with the m u l t i p l e x e d s o l u t i o n s c o n t a i n i n g p u r i f i e d and b i o t i n i l a t e d anti-Leu-2a and a n t i - L e u - 1 , f o l l o w i n g which they were incubated with Texas Red-a v i d i n (Molecular Probes, J u n c t i o n C i t y , OR). Thus, the FITC f l u o r e s c e n c e a x i s corresponds to the a x i s of r o t a t i o n (OKT5), while the Texas Red a x i s i s r o t a t e d i n the Leu-2a, Leu-1 p l a n e . The samples were analyzed on a dual l a s e r (argon, krypton) FACS IV (Becton-Dickinson FACS Systems, Sunnyvale, CA), and the two. f l u o r e s c e n c e i n t e n s i t i e s of 105 c e l l s were measured a f t e r forward l i g h t s c a t t e r g a t i n g on the lymphocyte peak [ 1 5 ] . The r o t a t e d p r o j e c t i o n s are shown i n F i g . 1 3 a , together with the t r u e v a l u e s of the r o t a t i o n a n g l e s , as determined from the c a l i b r a t i o n samples. The maximum angular e r r o r was of 6 ° . The l a r g e r angular e r r o r s occurred near one end of the angle range, which i n d i c a t e s that the c o n c e n t r a t i o n s of the stock s o l u t i o n s of antibody d e v i a t e d from t h e i r optimum values (Appendix 1), and that the c o n t r i b u t i o n of the mixing process to the angular e r r o r was i n f a c t l e s s than 6 ° . A p a r t i a l r e c o n s t r u c t i o n ( S e c t i o n s 58 F i g u r e 13 M u l t i p l e x l a b e l l i n g a n a l y s i s of human p e r i p h e r a l blood lymphocytes: (a) r o t a t e d p r o j e c t i o n s , and t h e i r t r u e angle of r o t a t i o n (the h o r i z o n t a l axes c o i n c i d e w i t h the a x i s of r o t a t i o n ) ; (b) constant d e n s i t y s u r f a c e of the r e c o n s t r u c t e d t r i p l e l a b e l l i n g d i s t r i b u t i o n (the hatched s e c t i o n i s the one with the lowest l e v e l of anti-Leu-1 b i n d i n g ) ; (c) d e f i n i t i o n of the c o o r d i n a t e axes, p r o j e c t i o n p l a n e , and angle of r o t a t i o n . The a n t i b o d i e s used were 0KT5, a n t i - L e u - 1 , and a n t i - L e u - 2 a . 59 60 1.1.4 and 1.1.5) was computed from the nine rotated projections, and a constant level surface of the resulting triple labelling distribution is shown in Fig.13b. Full reconstructions can also be obtained, as the machine allows any of the three antibody axes to be chosen as the axis of rotation. This requires, however, that a l l three antibodies be available in conjugated form (Section 1.1.4). The system described here meets a l l the stated requirements for an Immunofluorescence Tomograph. Its speed of operation, high degree of automation, and f l e x i b i l i t y , have allowed us to perform triple labelling analysis on a routine basis, while using a conventional dual fluorescence FACS IV. Its accuracy is similar to that of the much slower manual procedure (Section 1.1.6), and its r e l i a b i l i t y has proved to be very high, the machine having performed consistently in over thirty experiments. This last aspect could not be overemphasized, as the alternative manual procedure is prone to operator error due to the large number of pipetting operations on the one hand, and of pipette volume adjustments on the other. The modular design of both hardware and software allows for easy expansion by the addition of new modes of operation and effectors. Multiplex labelling, as performed with the Immunofluorescence Tomograph, has proved to be an excellent alternative to conventional triple labelling, by circumventing the need for the modification of the FGN analyzer to accomodate three fluorochromes. Since the completion of the machine, we have used multiplex labelling in studies of both human and 61 murine lymphoid c e l l s . 62 PART 2 THE DYNAMIC INTERPRETATION OF IMMUNOFLUORESCENT FLOW CYTOMETRIC DATA Chapter 2. 1  S k e l e t a l A n a l y s i s of S i n g l e Lineages 2.1.1. D e f i n i t i o n s We d e f i n e a d i f f e r e n t i a t i o n s t a t e as a s t a t e of the c e l l d e s c r i b e d by given set v a l u e s of c e r t a i n c e l l m e t a b o l i t e s . T h i s i m p l i e s that the mechanisms r e g u l a t i n g the metabolism of the c e l l w i l l tend to d r i v e that c e l l toward the set p o i n t c h a r a c t e r i s t i c of i t s d i f f e r e n t i a t i o n s t a t e . We conc e n t r a t e i n our study on c e l l membrane a n t i g e n s , which are c e l l products present on the c e l l membrane, and which i n t e r a c t with s p e c i f i c a n t i b o d i e s . We thus assume that the q u a n t i t a t i v e e x p r e s s i o n of each membrane antige n i s c h a r a c t e r i z e d by a set p o i n t , which depends on the d i f f e r e n t i a t i o n s t a t e of the c e l l . As a consequence of the above assumptions, a d i f f e r e n t i a t i o n s t a t e can be represented by a p o i n t i n l a b e l l i n g space ( S e c t i o n 1 . 1 . 1 ) . T h i s p o i n t i s , on the one hand, the c h a r a c t e r i s t i c phenotype of c e l l s i n that d i f f e r e n t i a t i o n s t a t e , and, on the other hand, the a t t r a c t o r towards which c e l l s i n that s t a t e e v o l v e . In what f o l l o w s , we s h a l l sometimes use the same n o t a t i o n f o r a d i f f e r e n t i a t i o n s t a t e , and f o r the a t t r a c t o r which corresponds to i t . 63 We define an n-step lineage as a sequence of n+1 differentiation states. If these states have attractors Aj,,...,An, then the lineage is denoted by Ac—>- A , — — > • An. We c a l l the 0-th state, Ap, the source of the lineage, and the cells in that state, the (immediate) precursors of the lineage. We shall assume in what follows that, in the absence of noise, a l l cells in a given differentiation state, and which start from the same phenotype, will follow the same orbit until they differentiate, divide, die, or are exported. Then, i f the flow is well behaved, certain assumptions about differentiation and division rates will enable us to relate the topology of the distribution in a given differentiation state of the lineage to the topology of its immediate precursor, and, eventually, to the behaviour of the i n i t i a l source (earliest precursor within the population). At a f i r s t level of analysis, we shall concentrate on the support of the density distribution in the absence of noise. As the support of the density distribution describing a population of dividing and/or differentiating cells depends on the support of the transition probability functions governing these processes, we shall assume the latter to be maximal (i.e. I R * ) , and, therefore, the density distribution support wil l i t s e l f be maximal, in that i t wil l contain any particular support. Such a maximal support i s , in an obvious sense, a minimal topological description of the density distribution, which we shall c a l l a skeleton of that distribution. Additional structure is contributed by particular transition probability functions for division and differentiation, and by the diffusion process 64 a l r e a d y mentioned. As we showed i n the I n t r o d u c t i o n , the o r b i t s followed in a l a b e l l i n g space by the c e l l s of a l i n e a g e are determined by the k i n e t i c s of a n t i g e n e x p r e s s i o n . We show i n S e c t i o n 2.1.4 t h a t , under c e r t a i n c o n d i t i o n s , one can c o n s t r u c t a c o o r d i n a t e t r a n s f o r m a t i o n such that a l l the o r b i t s f o r c e l l s i n the given l i n e a g e become s t r a i g h t l i n e s , along which the c e l l s r e l a x e x p o n e n t i a l l y , with r e l a x a t i o n r a t e s which are both i s o t r o p i c and independent of the d i f f e r e n t i a t i o n s t a t e . The space d e f i n e d by the new c o o r d i n a t e s i s the standard r e p r e s e n t a t i o n space. Phenotypes are s t i l l r epresented by p o i n t s i n t h i s space, while the new c o o r d i n a t e s no longer correspond to i n d i v i d u a l a n t i g e n s . In such a system, i f the i n i t i a l source i s a p o i n t at A0 , and n s u c c e s s i v e a t t r a c t o r s A, ,...,An are independent of each other and of A0 , the maximal support, and t h e r e f o r e the s k e l e t o n , w i l l be the n-dimensional geometric simplex ( Ac ,...,An ) [ 1 6 ] . In other words, any d i s t r i b u t i o n generated by such a l i n e a g e ( i n the absence of n o i s e and c e l l d i v i s i o n ) w i l l be c o n t a i n e d i n s i d e t h i s simplex. T h i s a l r e a d y c o n s t r a i n s the topology of any such d i s t r i b u t i o n , and i s the e x p r e s s i o n of the p a r t i c u l a r sequence of d i f f e r e n t i a t i o n s t a t e s , and of the phenotypes c h a r a c t e r i s t i c of each s t a t e . We s h a l l show that geometric methods can be used f u r t h e r i n d e s c r i b i n g the s k e l e t o n of t r a n s i e n t s t a t e s due to n o n - s t a t i o n a r y sources and/or l i n e a g e s , and thus can be a p p l i e d to the study of the ontogeny of c e l l p o p u l a t i o n s . Once the standard r e p r e s e n t a t i o n of the s k e l e t o n corresponding to a given l i n e a g e has been c o n s t r u c t e d , the i n v e r s e c o o r d i n a t e t r a n s f o r m a t i o n can be used to o b t a i n the s k e l e t o n i n l a b e l l i n g 65 space. This approach i s i l l u s t r a t e d by the analysis of murine thymocytes presented in Ch.2.3. The advantage of using the standard representation l i e s in i t s independence of p a r t i c u l a r antigen k i n e t i c s , and in the ease with which the geometric features of the skeletons can be interpreted. These points w i l l be i l l u s t r a t e d in the following sections. 2.1.2. Path Coordinates Let us consider n+1 points A0 ,A, ,...,An in the n-dimensional standard representation space, JRn , with A0 corresponding to the phenotype of the source, and A, ,...,An to the phenotypes of the d i f f e r e n t i a t i o n state attractors for an n-step lineage A0—**A!—>- ...—*-An . The points A0 ,...,An are assumed to be independent, in the sense that the vectors (A, -are l i n e a r l y independent. The points x given by the vector equation n where a\ are arb i t r a r y real numbers, form an a f f i n e space [16] of dimension n, ca l l e d the span of the points A0 ,...,An . The real numbers ac ,a, ,...,an , where a0 = 1-at -,..-an , are the barycentric coordinates of x r e l a t i v e to A0 ,...,An . If a l l barycentric coordinates are non-negative, the points x form the n-dimensional geometric simplex spanned by Ac , ...,An , which we denote by (Ae ,...,An ) . The coordinates a , , ...,an w i l l , in this case, also s a t i s f y a, +...+avi - 1 . We define a new set of .66 c o o r d i n a t e s c, , . . . , cn , such that n The new c o o r d i n a t e s are simple l i n e a r combinations of a; : j » l The c o o r d i n a t e s c -( of p o i n t s i n the simplex ( A0 , . . . ,Afl ) s a t i s f y O^c, - . . . ^ cn ^ 1 . Let us now d e f i n e the path c o o r d i n a t e s ju, , . . . , ^ X x , r e l a t i v e t o the l i n e a g e Ae— * "Aj — * - ...—*"An . Let x be an a r b i t r a r y p o i n t i n (AD ,...,An ) , and xn_ j the poi n t i n (Ac ,...,An_, ) , such that x „ _ , , xn , Aw are c o l i n e a r . The path c o o r d i n a t e |Un i s such that x=A^+|kn ( x n _ , - A n ) . The poi n t x n_, re p r e s e n t s the phenotype the c e l l c u r r e n t l y at x had immediately p r i o r to the t r a n s i t i o n An_(— An . F u r t h e r , l e t xn_2 be the poin t i n (Ac , .. . ,An_2 ) such that xn_2, xn_, , An_, are c o l i n e a r . Then, jxn_x i s such that xn_( =An_( +jUn_j ( x n - i " A n - i ' • t n e other path c o o r d i n a t e s down t o ^xx can be d e f i n e d i n a s i m i l a r manner, and one has e v e n t u a l l y n X = R N + H f * \ • n ( « i - | - A i ) 2.2 i * i By d i r e c t comparison with E q . ( 2 . 1 ) , one obt a i n s c j =^U;...|in, 67 and with the path c o o r d i n a t e s f o r p o i n t s belonging to the simplex s a t i s f y i n g the c o n d i t i o n s 0 - jUj ^ 1, i=1,...,n. The path c o o r d i n a t e s f o r a po i n t x=(yk, i • • > > jU^ ) give not only the instantaneous phenotype of a c e l l at x, but a l s o the h i s t o r y of that c e l l . Indeed, the phenotype x\ of the c e l l at the moment of the A j — A , " + | t r a n s i t i o n can be immediately determined, s i n c e the f i r s t i path c o o r d i n a t e s of x} are the same as those of x, while the remaining n - i c o o r d i n a t e s are equal to 1 : x = ( JU, , . . . , JU", , 1 , . . . , 1 ) , i = 1 , .. . , n-1 . 2.1 . 3. Kinematics In standard r e p r e s e n t a t i o n , a l l c e l l s i n the i - t h d i f f e r e n t i a t i o n s t a t e r e l a x e x p o n e n t i a l l y and un i f o r m l y towards the a t t r a c t o r , A; , a c c o r d i n g to the equation of motion 2.3 which has the s o l u t i o n - S i where 2"j i s the time elap s e d i n the i - t h d i f f e r e n t i a t i o n 68 s t a t e . By comparison with the d e f i n i t i o n of the path c o o r d i n a t e jx\ , one has |Ui - e 2.4 which allows the kinematic i n t e r p r e t a t i o n of the i - t h path c o o r d i n a t e as an ex p o n e n t i a l f u n c t i o n of the time spent by a given c e l l i n the i - t h d i f f e r e n t i a t i o n s t a t e . A s i m i l a r i n t e r p r e t a t i o n can be attached to the c j c o o r d i n a t e s . Indeed, s i n c e c; = ix \ . . . lUr, , one has vx -2>o C ; * e J s i Thus, c j i s an ex p o n e n t i a l f u n c t i o n of the time elapsed s i n c e the d i f f e r e n t i a t i o n Ai_t->Aj . As w i l l be seen i n the f o l l o w i n g s e c t i o n s , the l o c i c; =constant are p a r t i c u l a r l y important i n a n a l y z i n g t r a n s i e n t s u p p o r t s , as an important c l a s s of t r a n s i e n t s r e s u l t i n c o n s t r a i n t s on the moment of one t r a n s i t i o n , and thus i n i n e q u a l i t i e s t h a t the subsequent c { 's must s a t i s f y . The f a c t that the path c o o r d i n a t e s are exp o n e n t i a l f u n c t i o n s of time has the important consequence t h a t , given the i n i t i a l p o s i t i o n s of a set of p o i n t s , and the subsequent p o s i t i o n of one of these p o i n t s , the subsequent p o s i t i o n s of a l l the other p o i n t s can be determined by p u r e l y geometric means. In the case of a p o l y h e d r a l support f o r i n s t a n c e , t h i s means that the support at any moment can be c o n s t r u c t e d g e o m e t r i c a l l y from the p o s i t i o n of one of i t s p o i n t s , and a knowledge of the 69 support at t=0. This is due to the fact that the difference between the times elapsed since various differentiation transitions for the points involved remain constant, and therefore certain ratios between path coordinates also remain constant. Since a l l the geometric constructs we shall be using involve points moving on straight line orbits, we introduce the basic construct (Fig. 14a), which, given three points X J ' , X J ,x'( on the same orbit, and with respective path coordinates jd, , JUt, jx [ , allows one to find a fourth point x'z on the same orbit and with coordinate ^-z , such that j*i/j^z = jit / JA% . The attractor in this system is represented by the point A. It is readily seen that, i f x, ,xz are the positions of two cells at some time t , and x', is the position of the f i r s t c e l l at some later time t'> t, then x^ thus constructed is the position of the second c e l l at t'. The point B in Fig. 14a is arbitrary and not on the orbit, the lines x', C and x'2 D are parallel to AB, and DF is parallel to xzA. In ^ x2AB, x', A/x^ A = BC/BD; in A BDF and Ax^AB, xtA/x2A = EF/DF; in A BDF, EF/DF = BC/BD. Therefore, x^k/x^k ~ x\ A/x^ A, and, since x, A/xzA = j^x > a n d xjA/x^A = jx'x /jUl (from the definition of the path coordinates), one has /j*t = j*\/j*?. . An application of this method is the graphical construction of the points described by internal times of the form £Tj =nj6 » nj=0,1,..., and, therefore, by path coordinates of the form jU\=e'ni^ . This is illustrated in Fig. 14b for a two-step lineage, and will be used in Section 2.1.8, which deals with c e l l populations which differentiate, while dividing with a 70 Figure 14 The basic geometric construct (a), and i t s use i n constructing the points ( n ^ , ^ ) / with path coordinates of the form U1=e~n'e , U2=e ' " n 2 e (b) (see the text for details) . 71 constant p e r i o d . 2.1.4. E q u i v a l e n t Systems The system d i s c u s s e d i n the previous two s e c t i o n s may appear to have very l i m i t e d s i g n i f i c a n c e . However, t h i s very simple system may be used i n order to d e s c r i b e a much more general c l a s s of systems. Indeed, l e t us c o n s i d e r a continuous map, m, from the standard r e p r e s e n t a t i o n space to the l a b e l l i n g space |Rn , which has an i n v e r s e , M, on the image of ( Ae ,...,An ) . Moreover, l e t m be such that i t s Jacobian m a t r i x , Jm , i s continuous and has an i n v e r s e everywhere i n (Ac ,. . ., A „ )\(A, , . . . , An) . I f A! =m(A-,), we d e f i n e the s i n g u l a r i -simplex i n l a b e l l i n g space, (A^ ,...,A'n ) , as the image of (A0,...,An) through m [ 1 6 ] . If x'=m(x), the time d e r i v a t i v e of x' i s *'=O m <x^x 2 . 5 By combining Equations (2.3) and (2.5 ) , one o b t a i n s , f o r c e l l s in the n-th s t a t e of d i f f e r e n t i a t i o n , *' = ;im<:>o<>n->0 As m has an i n v e r s e , t h i s i s an equation of motion i n l a b e l l i n g space f o r c e l l s i n the n-th s t a t e , x ' = f ^ ( x ' ) , where fM i s given by 72 fnCx') = J M ( M ( x ' ) ) [ M ( R N ) - M ( x O ] 2 . 6 - I with M=m . For cells in the i-th state of the lineage, one can similarly derive * This defines an equation of motion, x.j =f; (x- ), where { » C x ( ) - J w ( M ( x j ))[M(R;) T M C x i ) ] 2 . T As opposed to Eq.(2.6), which defines fn everywhere in ( A ; , . . . ,A'„ ) \ ( A J ,...,A^), Eq.(2.7) only defines the restriction of f\ to (A0 ,...,Aj )\(A', ,...,A'j ). Obviously, a variety of equations of motion can be obtained in labelling space by this approach. We shall c a l l the systems which can be constructed in such a way equivalent. In a l l these systems, the orbits are the images through m of the straight line orbits of the geometric model, and the skeleton of the distribution generated by a lineage Ac—*-A',—...—•-A,n is the singular n-simplex (A0 ,...,A^ ), which is the image through m of the geometric n-simplex (A0 ,...,An ). Path coordinates can be defined in such a generalized system, by using the relationship (2.4) between a path coordinate and the time spent in the corresponding differentiation state. Obviously, i f x'=m(x), both x* and x will 73 have the same path c o o r d i n a t e s . Thus, although the path c o o r d i n a t e s no longer have a simple geometric meaning i n the g e n e r a l i z e d system, l o c i d e f i n e d by c o o r d i n a t e s or products thereof being constant can be c o n s t r u c t e d by mapping the corresponding geometric l o c i through m i n t o the l a b e l l i n g space. Let us now consider the i n v e r s e problem, which c o n s i s t s of f i n d i n g the mapping which r e l a t e s a system d e s c r i b e d by a source at AQ, and equations of motion x'=f*,(x'), i = 1,...,n, to i t s standard r e p r e s e n t a t i o n . Equation (2.6) can be r e w r i t t e n as which i s a system of f i r s t order p a r t i a l d i f f e r e n t i a l equations i n the mapping m. Equations (2.7) can be s i m i l a r l y r e w r i t t e n as with x f C (AQ , .. . ,A-, ) . These, together with I I I ( A 0 ) = A Q , are the boundary c o n d i t i o n s f o r Eq.(2. 8 ) . The ch o i c e of the p o i n t s A0,...,An i s a r b i t r a r y . We present i n Appendix 3 a p a r t i c u l a r system, and the mapping r e l a t i n g i t to i t s standard r e p r e s e n t a t i o n . 2.1.5. Two-Step Lineages We s h a l l now c o n s i d e r i n d e t a i l the s k e l e t o n s of d i s t r i b u t i o n s generated by a two-step l i n e a g e . Indeed, t h i s i s the most complex l i n e a g e which g i v e s r i s e to a s k e l e t o n which does not i n t e r s e c t i t s e l f i n a two-dimensional l a b e l l i n g space. 2 . 8 2.6 74 The s k e l e t o n generated by the l i n e a g e A0—*• At A2 i s the simplex (A0 ,A, ,A2 ) . A poi n t x i n t h i s simplex has path c o o r d i n a t e s ^U,=XjA, /ADA, and j^z=x Az/x,A2, and i s given by 2.10 The c e l l at x=( ^ U,, JJLZ ) w i l l have spent "g, =-lnyU, in the d i f f e r e n t i a t i o n s t a t e A, , and J52=-lnjJ.z i n s t a t e A2 T h e r e f o r e , the time elapsed s i n c e the f i r s t d i f f e r e n t i a t i o n step (Ac—y A , ) i s £ = C, + =-ln 1*1^2. • Important l o c i i n t h i s system are jL,=const. ( l o c u s of c e l l s having spent the same time in s t a t e A, , F i g . 15a), ^ i2= c o n s t . ( l o c u s of c e l l s having spent the same time f z i n s t a t e A x , F i g . 15b), and 1*1 =const. ( l o c u s of a l l c e l l s having l e f t the source s i m u l t a n e o u s l y , F i g . 15c). The l a t t e r i s p a r t i c u l a r l y important, as the c e l l s l e a v i n g s i m u l t a n e o u s l y the source at AQ w i l l spread out on t h i s p a r t i c u l a r l o c u s . I t can be seen from (2.10) that t h i s l o c u s i s a l i n e segment p a r a l l e l to A , Az . Let us now i n v e s t i g a t e the s k e l e t o n s generated by two types of t r a n s i e n t s which we s h a l l encounter i n the d i s c u s s i o n of mouse thymocyte ontogeny d a t a . (a) We s h a l l assume that the l i n e a g e i s s t a t i o n a r y . ( i . e . a l l t r a n s i t i o n s of the l i n e a g e are allowed at a l l t i m e s ) , and that the source i s d i s c o n t i n u o u s . The c e l l s generated at the time of the d i s c o n t i n u i t y w i l l be on the l o c u s yd,jUz =e"^ , where ~g i s the time e l a p s e d a f t e r the d i s c o n t i n u i t y i n the s o u r c e . If the source i s turned on at t=0, then the d i s t r i b u t i o n i s zero f o r > z£ e ~ * , and the s k e l e t o n of the t r a n s i e n t i s d e f i n e d by 76 the i n e q u a l i t i e s - t F i g u r e 16a shows t h i s s k e l e t o n , and i t s l e a d i n g edge. If the source i s turned o f f again at t=6 , the sk e l e t o n i s d e f i n e d by shown i n F i g . 16b, together with the l e a d i n g and t r a i l i n g edges. F i g u r e 16b a l s o shows the s k e l e t o n at t= 0 . The b a s i c c o n s t r u c t of s e c t i o n 2.1.3 i s used to obt a i n from t h i s the s k e l e t o n at some l a t e r moment. (b) Assume now that the l i n e a g e i s not s t a t i o n a r y , i n that the t r a n s i t i o n At—v AL i s only allowed i f t >0 , and that the source i s turned on at t=0. The c o n d i t i o n s d e f i n i n g the sk e l e t o n are the above i n e q u a l i t i e s and Th i s s k e l e t o n i s 5, >o which r e s u l t i n -t -tt-e) 77 F i g u r e 16 T r a n s i e n t two-step s k e l e t o n s , (a) S t a t i o n a r y l i n e a g e , where the source at AQ i s zero f o r t< 0. (b) S t a t i o n a r y l i n e a g e , where the source i s zero f o r t < 0 and t>8 . The t r i a n g u l a r r e g i o n adjacent to AQ i s the t r a n s i e n t s k e l e t o n a t t=0 . (c) No n - s t a t i o n a r y l i n e a g e , where the t r a n s i t i o n A j _ — * A 2 i s f o r b i d d e n f o r t<0. The source i s zero f o r t < 0 . The th i c k e n e d l i n e segment adjacent to AQ r e p r e s e n t s the t r a n s i e n t s k e l e t o n a t t-Q . The re g i o n s marked "a" and "b" c o n t a i n the c e l l s i n t r o d u c e d i n t o the system a t t > 0 , and t < 0 , r e s p e c t i v e l y . 79 The c o n d i t i o n J ^ t j Uz= e' ^ '& J d e f i n e s the l i n e s e p a r a t i n g the c e l l s generated before t= 0 (region"b") , from those generated afterwards ( r e g i o n "a") . F i g u r e 16c shows the skele t o n f o r t h i s type of t r a n s i e n t , and the two l e a d i n g edges. As w i l l be shown l a t e r , the a c t u a l d i s t r i b u t i o n s w i l l be d i f f e r e n t in the two regions of the s k e l e t o n . 2.1.6. n-Step Lineages For the general n-step l i n e a g e , the s k e l e t o n i s the n-simplex ( AQ ,...,An ) . The generic p o i n t x of path c o o r d i n a t e s JJL, , • • •, jUn is given by Eq.(2.2). We s h a l l now c o n s i d e r t r a n s i e n t s e q u i v a l e n t to those d e s c r i b e d i n the p r e v i o u s sect i o n . (a) Let us begin with a s t a t i o n a r y l i n e a g e , and a t r a n s i e n t s o u r c e . The locu s of the c e l l s having l e f t the source at the d i s c o n t i n u i t y i s o b v i o u s l y |U, . . • jb „ = e ~ ^ > where £ i s the time elaps e d s i n c e the d i s c o n t i n u i t y . I f the source i s turned on at t=0, then the s k e l e t o n of the t r a n s i e n t i s d e f i n e d by the i n e q u a l i t i e s (b) Let us now c o n s i d e r a n o n - s t a t i o n a r y l i n e a g e , i n which the t r a n s i t i o n Aj_t->- A; i s not allowed u n t i l t= 0 . I f the source i s turned on at t=0, then the s k e l e t o n i s d e f i n e d by the 80 inequalities -t The locus _e-ci-e) separates the cells generated by the source before and after the change in the lineage at t= 0 (regions "b" and "a", respectively). More complex situations can be envisaged, which can a l l be represented by inequalities involving products of path coordinates. 2.1.7. Periodic Division We shall assume that we are dealing with one of the topologically equivalent systems described in Section 2.1.4, and that c e l l division results in the membrane antigens being divided equally between the daughter c e l l s . Thus, division causes a discontinuity in the orbits, with a jump from x to x/2. It should be noted that the discontinuity x->x/2 applies to the labelling space, rather than the space of the standard geometric representation described in the previous chapter. In this and the following two sections, we shall restrict our analysis to systems of the form x' = fe1 ( R 1 - K') 2.H 81 where x=(x' ) , A=(A* ) , with b' - constant r e l a x a t i o n r a t e s . These are the general l i n e a r systems d e s c r i b e d i n Appendix 3. Equation (2.11) has the s o l u t i o n x' C O - n1 + ( x i - A') e where xQ =(x^ ) i s the i n i t i a l phenotype. We s h a l l w r ite t h i s i n v e c t o r form as X ( " E ^ = R + C * o - R ) € 2 .1Z Let 8 be the uniform d u r a t i o n of the c e l l c y c l e , and l e t us i n t r o d u c e two i n t e r n a l v a r i a b l e s : 75" , the time elapsed s i n c e the l a s t d i v i s i o n ( " & £ [ ( ) , Q ] ) ; and n, the c u r r e n t d i v i s i o n c y c l e number. A c e l l of phenotype x, i n i t s n-th d i v i s i o n c y c l e , and at a time £ a f t e r i t s l a s t d i v i s i o n , w i l l be represented by x(n,*S> ) . A c e l l j u s t emerging from i t s n-th d i v i s i o n , x(n,0), w i l l be c a l l e d the n-th generation s o u r c e . Let x(n-1,0) be the source f o r the ( n - l ) - t h generation c y c l e , and x(n-1, 0 ) - the s t a t e j u s t p r i o r to the n-th d i v i s i o n . Then, from (2.12), and the next generation s o u r c e , x(n,0), i s 2 . 1 3 82 i -we j where o( =e / 2 , and 0( = ( <X ) . The reccurence relation ( 2 . 1 3 ) allows x(n ,0) to be determined as a function of x ( 0 , 0 ) , the i n i t i a l source: X(n,o) = <xnx(o,o) -f 0 - « o ) R or XCn,o) - ft = [ X (0,0) - R ] c<n 2.14 where R c i l ! 2 - f l - * - ' R 2.15 2 0-<0 2e-b9.| is the position of the source x(n,0) in the limit n-*oo . Let us now consider the position of the general point, x(n ,S ) . Assume the i n i t i a l c e l l (t=0) is in the state X o (no ; ^ o ) r i t s source having been x(no,0) at t = -"5"0« Let x(n ,0) be the source of the current cycle, and x(n,"g") the current state of the c e l l s . The five states (Fig. 17) are connected by the following relationships: x C " « , < 0 - f i « [ x o ( n 0 ^ e ) - R ] e b r ° x ( n , o ) - f l - [ x ( n . , o ) - fij <x n" n° 83 F i g u r e 17 Diagram of the f i v e s t a t e s used i n the d e r i v a t i o n of Eq.(2.16). 84 By e l i m i n a t i n g x(no,0) and x(n , 0 ) , one ob t a i n s o 2.16 where RC5) - ft + (A - R) e Equation (2.17) rep r e s e n t s an o r b i t with a t t r a c t o r A, and o r i g i n at A given by (2.15). Equation (2.16) shows that the c y c l e s r e l a x to the o r b i t of (2.17) as n- » o o , and, t h e r e f o r e , the l a t t e r represents a ( s t a b l e ) l i m i t c y c l e . For t h i s system, the l i m i t c y c l e (2.17) i s the only o r b i t segment such that the co o r d i n a t e s of the i n i t i a l p o i n t are doubled a f t e r a time equal to the d i v i s i o n p e r i o d ( F i g . 18). 2.1.8. D i f f e r e n t i a t i o n and P e r i o d i c D i v i s i o n The l a s t two equations of the pr e v i o u s s e c t i o n can be combined to g i v e , f o r n0 =<5'c>=0, x C n . O - n + e ( n - R ^ + e * n ( x 0 - R ) Let us now introduce the n o t a t i o n jxi=e'n^ , j*c=e~'^ t a n d bc =b+ln2/8 . Then, the above equation can be r e w r i t t e n as X - A + j U c CR - a) •+ [*t C X o - R ) T h i s i s f o r m a l l y i d e n t i c a l t o Eq.(A3.5) i n Appendix 3, f o r a two 85 b2>b' F i g u r e 18 The l i m i t c y c l e f o r the system d e s c r i b e d by Eq.(2.11). T h i s i s the onlv^ o r b i t segment such t h a t ^ a p o s t -m i t o t i c c e l l of phenotype A reaches the phenotype 2A i n a time equal to the d i v i s i o n c y c l e period,© .,s 86 step l i n e a g e Thus, a one-step l i n e a g e x0—>• A with c e l l d i v i s i o n can be t r e a t e d as the two-step l i n e a g e x0— A —*-A. It i s evident that the f i r s t s t e p of t h i s formal l i n e a g e r e p r e s e n t s the true d i f f e r e n t i a t i o n p r o c e s s , with the d i f f e r e n c e that the a t t r a c t o r A has been r e p l a c e d by A, the o r i g i n of the l i m i t c y c l e , and that the r e l a x a t i o n r a t e s are bc =b+ln2/6 , r a t h e r than b. Moreover, the path c o o r d i n a t e jU, a s s o c i a t e d with the f i r s t step can only have d i s c r e t e v a l u e s , jxx =e . The second step has A as i t s a t t r a c t o r , and b as i t s r e l a x a t i o n r a t e s . The corresponding path c o o r d i n a t e , j±c , i s subject to the c o n s t r a i n t ^ i c > e ~ ® . T h i s approach has s e v e r a l advantages: ( i ) The a n a l y s i s of a p o p u l a t i o n of c e l l s which d i v i d e and d i f f e r e n t i a t e becomes e a s i e r , as there i s no longer a formal d i s t i n c t i o n between d i v i s i o n and d i f f e r e n t i a t i o n . Indeed, the c y c l i c behaviour of d i v i d i n g c e l l s i s no longer expressed by the c y c l i c parameter "5* , but, r a t h e r , by the n o n - c y c l i c , d i s c r e t e path c o o r d i n a t e ^ttj. ( i i ) A standard r e p r e s e n t a t i o n can be obtained f o r the combined d i v i s i o n and d i f f e r e n t i a t i o n p r o c e s s e s , which f u r t h e r s i m p l i f i e s the a n a l y s i s . F i g u r e 19a i l l u s t r a t e s the standard r e p r e s e n t a t i o n of a one-step l i n e a g e with d i v i s i o n , as compared to the l a b e l l i n g space r e p r e s e n t a t i o n ( F i g . 19b) . One should note the simple geometric c o n s t r u c t which can be used to generate the s k e l e t o n i n the standard r e p r e s e n t a t i o n , as w e l l as the f a c t that t h i s c o n s t r u c t does not r e q u i r e the o r i g i n of the space to be s p e c i f i e d . I t may be c o n v e n i e n t , i n a f i r s t , approximate a n a l y s i s , to ignore the d i v i s i o n step i n the formal l i n e a g e , as w e l l as the 87 F i g u r e 19 The s k e l e t o n f o r a s i n g l e - s t e p l i n e a g e , x Q-—•-A, i n the presence of c e l l d i v i s i o n : standard r e p r e s e n t a t i o n (a), and l a b e l l i n g space r e p r e s e n t a t i o n (b). The system i n (b) i s of the type d e s c r i b e d by Eq.(2.11). 88 d i s c o n t i n u o u s c h a r a c t e r of the d i f f e r e n t i a t i o n path c o o r d i n a t e s . In t h i s reduced, continuous r e p r e s e n t a t i o n , c e l l d i v i s i o n only manifests i t s e l f by s h i f t i n g the a t t r a c t o r from A to A, and by r e p l a c i n g the d i f f e r e n t i a t i o n r e l a x a t i o n r a t e s , b, with the e f f e c t i v e r a t e s bc =b+ln2/0 . T h i s g i v e s r i s e to a s i m p l i f i e d s k e l e t o n , which i s v a l i d near the l i m i t c y c l e , where the s e p a r a t i o n between c y c l e s d e c r e a s e s , and a continuous approximation can t h e r e f o r e be used. The formalism introduced i n t h i s s e c t i o n can be extended to m u l t i - s t e p l i n e a g e s with d i v i s i o n , i f the i n t e r a c t i o n between d i v i s i o n and d i f f e r e n t i a t i o n i s known. F i g u r e 20a i l l u s t r a t e s the case of a two-step l i n e a g e , where ( i ) d i v i s i o n i s allowed in both s t a t e s 1 and 2; ( i i ) the p e r i o d i c i t y 0 i s the same i n both s t a t e s ; and ( i i i ) the second t r a n s i t i o n i s allowed only upon d i v i s i o n ( §* =0). In t h i s c a s e , the l i n e a g e A0—*• A,—*-becomes with the upper branch c o n t a i n i n g c e l l s c y c l i n g i n d i f f e r e n t i a t i o n s t a t e 1, and the lower branch, c e l l s c y c l i n g i n d i f f e r e n t i a t i o n s t a t e 2. As b e f o r e , the a c t u a l d i f f e r e n t i a t i o n process i s c o n t a i n e d i n the d i f f e r e n t i a t i o n simplex ( A0, A , , A 2 , ) , with path c o o r d i n a t e s jUt=en , e , jxz = e 'ni ^ , n,|2_ =0,1, The values of n , , n2 for s e v e r a l p o i n t s i n t h i s simplex are shown in F i g . 20a. The d i v i s i o n c o o r d i n a t e s , ^Uc, and jx^ , are subject to the c o n s t r a i n t ^tc;^e-6 . A d i f f e r e n t s k e l e t o n i s obtained i f one e l i m i n a t e s 89 F i g u r e 20 Skeletons f o r two-step l i n e a g e s , AQ—»-A-^ > * A2, i n the presence of c e l l d i v i s i o n : (a) the second t r a n s i t i o n i s only allowed upon d i v i s i o n ; (b) un- ^ c o r r e l a t e d d i v i s i o n and d i f f e r e n t i a t i o n . The d i v i s i o n p e r i o d s are the same i n both d i f f e r e n t i a t i o n s t a t e s . 91 c o n d i t i o n ( i i i ) . In t h i s c a s e , the o r i g i n s of c y c l e s are only s u b j e c t to the c o n d i t i o n f*xf*i = e > n=0,1,..., where n i s the c y c l e number. The r e s u l t i n g l o c i i n the d i f f e r e n t i a t i o n simplex (A0,A|,A2) are l i n e s p a r a l l e l to A,A2 (see S e c t i o n 2.1.5), and the s k e l e t o n c o n s i s t s of the plane regions i l l u s t r a t e d i n F i g . 20b. These are s u b d i v i d e d by the l o c i ^-t, = en»& , jji^ = e-(n-»l,)8 ^ 0 £ n , $ n (the elements of the previous s k e l e t o n , F i g . 20a), where the c e l l s i n each s u b d i v i s i o n d i f f e r i n the number of the c y c l e i n which the second t r a n s i t i o n occured. F i g u r e 21a i l l u s t r a t e s only those regions of the s k e l e t o n which c o n t a i n c e l l s having undergone the second t r a n s i t i o n d u r i n g the f i r s t c y c l e ( l > ^ i " ^ e " © ) . In order to make comparison with experimental d i s t r i b u t i o n s e a s i e r , a frame of r e f e r e n c e was d e f i n e d , and the a t t r a c t o r s were assigned the f o l l o w i n g p o s i t i o n s : AQ on the x3 a x i s , A , and A, on the x1 a x i s , and A2, Az in the x ,x p l a n e . F i g u r e s 21b, c, d show the p r o j e c t i o n s of the s k e l e t o n onto the x',xJ p l a n e s . 2.1.9. Non-Periodic D i v i s i o n The a b s o l u t e l i m i t s on the value of 75" are zero and i n f i n i t y . We s h a l l now i n v e s t i g a t e the e f f e c t of these l i m i t s on the r e s u l t i n g c e l l d i s t r i b u t i o n . Let ( x0 ,A) be the o r b i t of o r i g i n x0 and a t t r a c t o r A. T h i s i s a l s o a one-dimensional s i n g u l a r simplex, and the general p o i n t on i t i s given by Eq.(2.12), which can be r e w r i t t e n as XC^O - n + ( X0- R)yUb 2.»8 92 F i g u r e 21 S k e l e t o n f o r a two-step l i n e a g e with d i v i s i o n , where the c e l l s undergo the second t r a n s i t i o n b e f o r e t h e i r f i r s t d i v i s i o n (a), and i t s p r o j e c t i o n s onto the planes x l , x 2 (b) , x 2 , x 3 ( C) , and x 1 ^ ^ (d) . 93 95 where i s as d e f i n e d by Eq.(2.4). We d e f i n e the d i v i s i o n mapping, D, as D:x I—>• x/2. I f one assumes that the same equation of motion holds f o r any p o s i t i o n of the a t t r a c t o r , one can e a s i l y see from the l a s t equation that and that the path c o o r d i n a t e of x£ ( x0 ,A) i s the same as that of Dx -€(Dx0 ,DA). Thus, D maps a t r a j e c t o r y i n t o a t r a j e c t o r y , and p r e s e r v e s the c o o r d i n a t e ( F i g . 22a). We d e f i n e the flow mapping with a t t r a c t o r A1, F_, , , as the mapping which takes x to the p o i n t of c o o r d i n a t e jj.1 on (x,A'): F x = fl' + ( X - A') M/ b One can show that P ^ j L i ' C X o , ^ = ( Ff t/^» * ° , FR, ^( A) z.,g and that the p o i n t s x on ( xe ,A), and F , ( x on ( F ^ , ^ , x0, FR, ^ , A) have the same path c o o r d i n a t e yU . Thus, F a l s o maps an o r b i t i n t o an o r b i t , and prese r v e s the c o o r d i n a t e ( F i g . 22b). Let us now d e f i n e the n-dimensional flow simplex ( xe ,A, , . . . , An) , with v e r t i c e s x0 ,A , ,...,An , i n the same way we d e f i n e d the geometric simplex i n S e c t i o n 2.1.1, with the d i f f e r e n c e that o r b i t s are as given by (2.18), r a t h e r than / / b F i g u r e 22 The e f f e c t of (a) the d i v i s i o n mapping, D, and (b) the flow mapping, F , ., on an o r b i t (x_,A). 97 s t r a i g h t l i n e s . The p o i n t x i n t h i s simplex has path c o o r d i n a t e s jU,, . . . ,jU„, ^ J'€[0,1], and i s given by Eq. A3.5 i n Appendix 3, which becomes fe <T» b i =2 By s u b s t i t u t i n g t h i s i n t o (2.19), one o b t a i n s f o r FD x rx Fn. x = A + ( ^ . | „ ^b( x0- f l ^ 2 ( f ' i - f « f ) (flf-,ii)+fWfi) which i s a p o i n t of the (n+1)-simplex (xQ,A,,...,An,A). If one d e f i n e s Fq as FRx={Ffl x| ^ £ [0,1 ]}, t h i s l a s t r e s u l t may be w r i t t e n as which means that the e f f e c t of the mapping F^ on an n-simplex i s to c r e a t e an (n+1)-simplex by adding the vertex A to the r i g h t . We can now i n v e s t i g a t e the e f f e c t of flow and d i v i s i o n on c e l l d i s t r i b u t i o n s . Assume that the i n i t i a l source p o i n t i s x0 . The support of the 0-th g e n e r a t i o n source d i s t r i b u t i o n i s the 0-simplex Cr^ = ( x0) . The support of the d e n s i t y d i s t r i b u t i o n f o r the same generation w i l l be CT© =F^(xQ ) = ( x0, A ) . For the next g e n e r a t i o n , 9 8 and, for the n-th generation, S ( t o _A_ ft \ 0* s j ft ft n The simplices 0" n and (T„ differ from ( — — , ,.-.,-5-) fl ft and ^"^nTT ' " J r T ' * * *' ^  ^' respectively, only in the leftmost vertices, which a l l go to zero as n-*oo . Therefore, in this l i m i t , Thus, the limiting distribution of sources and cells have maximal supports independent of the i n i t i a l phenotype,xQ . These supports are infinite-dimensional simplices, the edges of which A A are orbits of the form ( , — , n>m. The support of the source distribution is included in that of the c e l l density. Let us now consider the two-dimensional projection of these simplices. One can show that a l l edges of are contained A A inside the contour generated by (0,A) and ( , - . ), n> 1, s A and a l l edges of C ^ , inside the contour generated by (O,-^—) A fl and (^o -» 2""*^ ' n ^ 2 23). Since a l l the elements of these l s contours are also elements of and O"^ , respectively, they coincide with the boundaries of the two-dimensional projections of these simplices, and therefore with the skeletons 99 F i g u r e 23 The asymptotic supports of the c e l l d e n s i t y distribution,<T<2 , and source d e n s i t y d i s t r i b u t i o n , ^ . The system i s d e s c r i b e d by Eq.(2.11), with b 2 > b 1 . The support i s i n c l u d e d i n C^, . 100 of the c e l l and source d i s t r i b u t i o n s . As these l i m i t i n g supports are independent of the starting point x0 , they behave as attractors determined j o i n t l y by flow (regulation of antigen expression) and c e l l d i v i s i o n . The results of th i s chapter i l l u s t r a t e the progression in the complexity of the attractor as r e s t r i c t i o n s on i t s representation are relaxed. Thus, while the attractor i s point-l i k e in the absence of d i v i s i o n (or in the reduced representation of d i v i s i o n - d i f f e r e n t i a t i o n ) , i t becomes a l i m i t cycle in the presence of d i v i s i o n with a fixed period, and an infinite-dimensional simplex when the d i v i s i o n process i s non-per i o d i c . It should be noted that we have treated the last two cases only in a linear system, with the simple antigen kinetics given by Eq.(A3.l), and that, even under these s i m p l i f i e d assumptions, we have not brought together d i f f e r e n t i a t i o n and non-periodic d i v i s i o n . Combining more general antigen and c e l l d i v i s i o n k i n e t i c s should be the object of further i n v e s t i g a t i o n . 101 Chapter 2.2  Flow Dynamics of C e l l Populations 2.2.1. Defi n i t i o n s The previous chapter was dedicated exclusively to the study of the motion of individual c e l l s in l a b e l l i n g space. This allowed us to study the boundaries of the regions occupied by c e l l s at a given moment. These regions were termed "skeletons", and they represent, in the absence of noise, the supports of the l a b e l l i n g d i s t r i b u t i o n s generated by given lineages. Skeletons are coarse topological descriptions of l a b e l l i n g d i s t r i b u t i o n s , and i t i s precisely t h i s coarseness that makes them useful in a f i r s t analysis of these d i s t r i b u t i o n s (Ch.2.3). A finer topological analysis must take into account the f u l l l a b e l l i n g d i s t r i b u t i o n , $ , and, therefore, the motion of the c e l l population, rather than that of individual c e l l s . Such an analysis i s the object of the present chapter. As the c e l l s giving r i s e to the l a b e l l i n g d i s t r i b u t i o n move in l a b e l l i n g space, i t is reasonable to assume that ^ s a t i s f i e s a flow conservation equation. This equation should contain, apart from the flow term, terms representing the sources (precursor d i f f e r e n t i a t i o n , c e l l d i v i s i o n , c e l l import) and sinks ( c e l l death, further d i f f e r e n t i a t i o n , c e l l export) present in the system. The flow term, -div(§ v ) , i s given by the equation of motion, v=f(x), which is also the kinetic equation of antigen expression. Thus, some of the features of $ are due to the behaviour of f (and thus to the p a r t i c u l a r antigen kinetics of the c e l l s involved), while others are due to the behaviour of 1 02 c e l l s and c e l l populations, which determines the source and sink terms. The problem of the d i s t r i b u t i o n being j o i n t l y determined by processes at antigen and c e l l l e v e l has already been discussed in the previous chapter. We adopt here the same approach, by solving the flow equation in the standard representation space (Sections 2.2.2 to 2 . 2 . 6 ) , and then transforming the solution through the mapping from the standard representation to the l a b e l l i n g space (Section 2.2.7; see also Section 2 . 1 . 4 ) . We show in Sections 2.2.4 and 2.2.5 that, i f the source of precursor c e l l s varies exponentially in time, and the death and d i f f e r e n t i a t i o n rates are stationary (do not depend e x p l i c i t e l y on the time,t), then the solutions of the flow equation can be factored into an immobile pattern, a moving "window", and a time factor which i s independent of phenotype. This greatly s i m p l i f i e s the analysis of the d i s t r i b u t i o n s , as the windows coincide with the skeletons of the previous chapter, and the finer topological analysis only involves the immobile patterns. Sections 2.2.6 and 2.2.7 i l l u s t r a t e the fact that, for constant death and d i f f e r e n t i a t i o n rates, there i s a limited number of possible pattern types, which are defined by i n e q u a l i t i e s s a t i s f i e d by the c e l l kinetics parameters. These results are applied in Ch.2.3 to the analysis of the murine thymocyte data. Section 2.2.8 deals with the flow equation in a system where there i s non-periodic d i v i s i o n . 1 03 2,2.2. The Flow Equation without D i v i s i o n Let us now consider the motion of a c e l l p o p u l a t i o n i n the standard r e p r e s e n t a t i o n space. For c e l l s i n the j - t h d i f f e r e n t i a t i o n s t a t e of a l i n e a g e , we d e f i n e the d e n s i t y d i s t r i b u t i o n , where X J and T j a r e , r e s p e c t i v e l y , the g e n e r i c p o i n t i n the simplex ( A0, . . . , A j ) , and the time spent i n the j - t h s t a t e , as d e f i n e d i n S e c t i o n 2.1.3. In the absence of noise and c e l l d i v i s i o n , the d i s t r i b u t i o n S: s a t i s f i e s the c o n t i n u i t y equation where v i s the v e l o c i t y v=x^ , <5j and a r e , r e s p e c t i v e l y , the death r a t e cSj ( £ j , t ) , and r a t e of d i f f e r e n t i a t i o n to the (j + 1)-th s t a t e , *fj ( , t ) , and S ^ (XJ,"5J ,t) i s the source of c e l l s i n the j - t h s t a t e . As we are d e a l i n g with a s i n g l e l i n e a g e Ae—*- . . . —>• A „ , the c e l l s i n the j - t h d i f f e r e n t i a t i o n s t a t e are c o n c e n t r a t e d i n the simplex ( A0, . . . , A j ) . T h e r e f o r e , the c e l l d e n s i t y d i s t r i b u t i o n i s r e s t r i c t e d to t h i s simplex. If we now d e f i n e the standard a t t r a c t o r s i n the standard te k r e p r e s e n t a t i o n as having c o o r d i n a t e s A^ =1 f o r j , and Aj =0 fo r k>j, §.' becomes a f u n c t i o n of the f i r s t j c o o r d i n a t e s , Q\ (X| ,...,X"{ ,"5": , t ) . The source d i s t r i b u t i o n S{ i s conc e n t r a t e d on the simplex ( A0 , . . . , A j _ , ) e = ( A0 ,...,Aj ) , which c o n t a i n s the c e l l s i n the ( j - l ) - t h d i f f e r e n t i a t i o n s t a t e . For the moment, we r e s t r i c t S^ to the simplex (AG ,...,Aj ) , and w i l l f u r t h e r r e s t r i c t i t to ( A0 , . . . , A j _| ) i n the next s e c t i o n . F o l l o w i n g these c o n s i d e r a t i o n s , one can s u b s t i t u t e v given 1 04 by Eq.(2.3) i n t o (2.20), which y i e l d s T h i s equation has the s o l u t i o n 2 . 2 2 where S"i Since one assumes that the c e l l s i n the j - t h d i f f e r e n t i a t i o n s t a t e are e x c l u s s i v e l y d e r i v e d by the d i f f e r e n t i a t i o n of c e l l s i n the ( j - 1 ) - t h s t a t e , the source S'j has the form S 4 * x j ' S'j ,t)=Sj (x_j,t) o " ( ^ j ) . The s o l u t i o n (2.22) thus becomes S/<Kh*4 \0«S^ 2.23 where <£>: CS.O = f j C S . B . O 2.24 k Let us now d e f i n e the d e n s i t y d i s t r i b u t i o n 1 05 oo o which does not depend on the internal time, £ j . By using the solution (2.23) of the flow equation, the d i s t r i b u t i o n becomes oo !A i © ^•C«j + Cxj-RjDe,t-e)e AjCa.O^ -Ce.O-de 2.25 o 2.2.3. The Solution in Terms of Path Coordinates Let jx± be the j-th path coordinate of the point x j , and •V that of a point X J between the source xj_( and xj ( F i g . 24). If S"j i s the time required for a c e l l to reach Xj , and the time required to reach the point x.' at V , then - C ? j - e ) V - e Q. and V = ^Uj e , d 9 = d v / v . The source Sj i s concentrated on the simplex (A0 ,... ,Aj_, ) , and, i f x j =( jxt , — , ), where sj i s the number of c e l l s d i f f e r e n t i a t i n g per unit time, per unit volume of the simplex (ka,...,Aj„4 ). Then, e « >. • • Ht • F i g u r e 24 Diagram of s t a t e s used i n the d e r i v a t i o n of Eq.(2.26). 107 We can now write the solution (2.25) as S j C f A | , . . . l / * 4 , t ) « S j ^ l j . . . , ^ . | > i + | n ^ j ) 2.26 As cells in the n-th differentiation state do not differentiate where s is the source at A0 . 2.2.4. The Stationary Lineage Let us consider the relationship between the distribution of cells in the ( j - l ) - t h state, S j - | , and that of cells differentiating per unit time to the j-th state, s j . In what follows, the death rates Sj will not depend explicitely on the time, t . Since f t ) is the transition probability per unit time, one has this is given by SjCjU.,..., = V« j - | C - | n ^ i -l >t ) S i - i ( j f i .|. . .|/ * i - i|t ) The transition Aj_l-j> Aj is said to be stationary i f the differentiation rate ^ j - i does not depend explicitely on time. 108 Then, one o b t a i n s from (2.26) the f o l l o w i n g r e l a t i o n s h i p between the d i s t r i b u t i o n s i n the ( j - l ) - t h and j - t h s t a t e s : si^ «.-./«*.t>sj.1(/.1,...1/.f.,,fi./lj^ H£iJ^  2 . 2 ^ where we use the shorthand n o t a t i o n , , f o r the v a l u e s of the r e s p e c t i v e f u n c t i o n s at *&j =-ln ^LLJ . The l i n e a g e A 0 —* . . A f, i s s a i d to be s t a t i o n a r y i f a l l t r a n s i t i o n s are s t a t i o n a r y . Then, the recurrence r e l a t i o n s h i p (2.27) can be used to determine S j r e l a t i v e to the source s at A Q : s Jc / t,,...,^i>*ct*iVl.../lJ)^TrJ^k 4-2,.,"-where P ^ ( ? ^ ) = (S\j) $y^(S"R) • The d i s t r i b u t i o n i n the l a s t s t a t e i s given by Let us now assume that the source s i s an e x p o n e n t i a l f t f u n c t i o n of time, which i s turned on at t=0, s ( t ) = s e H ( t ) . Then, 3 n c a n ^e w r i t t e n as where sn i s the time f a c t o r sn (t)=s e , Pni s the s t a t i o n a r y p a t t e r n 109 and w M i s the window Wn( | U , , — , | L n , t ) = H ( jxt . . . jj,n -e~ x ) . T h i s r e s u l t i s i n t e r e s t i n g , because i t shows t h a t , i f the source i s e x p o n e n t i a l , the r e s u l t i n g d i s t r i b u t i o n can be f a c t o r e d i n t o an immobile p a t t e r n , Pn, which i s seen through the moving window, Wn. D i r e c t comparison with the r e s u l t s i n s e c t i o n 2.1.6 shows that t h i s moving window i s nothing but the s k e l e t o n f o r a t r a n s i e n t due to a n o n - s t a t i o n a r y s o u r c e . An e x p o n e n t i a l source i s a reasonable approximation to the s i t u a t i o n both i n a s t a t i o n a r y p o p u l a t i o n , and i n a developing organ, such as the e a r l y embryonic mouse thymus, where there i s e x p o n e n t i a l growth. I f the death and d i f f e r e n t i a t i o n r a t e s can be approximated in more d e t a i l i n S e c t i o n 2.2.6. 2.2.5. A Non-Stationary Lineage Let us assume now that the t r a n s i t i o n A j -|—*• A j i s n o t allowed i f t<6 , and that the source i s turned on at t=0. Moreover, l e t us assume that the t r a n s i t i o n i s s t a t i o n a r y a f t e r t = 0 , and that a l l other t r a n s i t i o n s are s t a t i o n a r y at a l l t i m e s . The treatment of the p r e v i o u s s e c t i o n can be a p p l i e d to by c o n s t a n t s <Vj and , r e s p e c t i v e l y , the p a t t e r n Pfl becomes <\-\ 1 10 the p a r t i a l l i n e a g e s Aj_( , and A j — A j + j — ^ . ...—>• A „ . Thus, can be determined from a knowledge of s, and Sr« from s j . The d i s t r i b u t i o n s and <§0 are given by i -t^e ' ' C - ^ f * i - 0 i > 0 - l v . p l . . . ( U 4 - i (£1*,) h-i KH-*-) The r e l a t i o n s h i p between SJ and i s Then, where and are equal to 1 i n s i d e the s e c t i o n s " a " and "b," r e s p e c t i v e l y , of the s k e l e t o n d e s c r i b e d i n S e c t i o n 2.1.6, and zero o u t s i d e . The d i s t r i b u t i o n i n the window W „ „ i s i d e n t i c a l 111 with that of the p r e v i o u s s e c t i o n , and thus one has a p a t t e r n there i f the source i s e x p o n e n t i a l . However, because of the f a c t o r p j _( ( t - 0 + l n j . . . ^ ih ) , the d i s t r i b u t i o n i n W „ t i s not s t a t i o n a r y , unless i s a c o n s t a n t . In t h i s c a s e , and, assuming an e x p o n e n t i a l source l i k e that i n the previous s e c t i o n , the d i s t r i b u t i o n i n can be w r i t t e n as where the p a t t e r n Pn^ i s A i - i A* V ft J \ i-« ^ n - fTf feAk, T T frfc Afe. "2 i n-i . PhAte. T T K A and the time f a c t o r i s C O . . p , . l e p ' - V * - P ' " ' > t -Under these assumptions, the p a t t e r n i n window "a" becomes with the time f a c t o r 1 12 It The two patterns d i f f e r in the exponents of ,... , yn^ f and, therefore, the two patterns w i l l behave d i f f e r e n t l y near the attr a c t o r s AJ_( ,...,An. If the death and d i f f e r e n t i a t i o n rates are constant, the pattern Pn^ i s 4-2. n - i TT i TT 1 with s ;b( t ) - s p , . . . ^ . , / i j •••p0.,ePi"'e e ^ ' ' 0 * . A detailed analysis for n=2 w i l l be given in the next section. 2.2.6. The Case n=2 Let us begin by applying the results of the previous two sections to the part i c u l a r case we have already discussed in Section 2.1.5, that i s the two-step lineage in a two-label space. We s h a l l assume throughout that the source s is exponential, and that the death and d i f f e r e n t i a t i o n rates are 1 13 c o n s t a n t . F i g u r e s 16a,c show the supports f o r the two types of t r a n s i e n t d i s t r i b u t i o n we have d i s c u s s e d so f a r (these are the sk e l e t o n s of Ch.2.1, and the windows of the previous s e c t i o n s ) . The l a b e l s "a" and "b" i n F i g . 16c denote the type of p a t t e r n "seen" through each window; "a" o b v i o u s l y corresponds to the p a t t e r n f o r the s t a t i o n a r y l i n e a g e , while "b" corresponds to that f o r a n o n - s t a t i o n a r y t r a n s i t i o n A , — » • A^ . The p a t t e r n s f o r c e l l s i n d i f f e r e n t i a t i o n s t a t e 2 are while the p a t t e r n s f o r c e l l s i n d i f f e r e n t i a t i o n s t a t e 1 (on the (A0 ,A, ) simplex) are with the time f a c t o r s s * a CO with the time f a c t o r s 1 14 * 2* « i b ft) • e. For t< &• , the time f a c t o r s*. i s s*. (t)=se . Both p a t t e r n s P2 are of the form Pz = 1/^c, ^<2Z , o r , s o l v i n g Eq.(2.l0) f o r ^ , and ^ 2 , where "5, = }f, , = Y"2 - , . The constant l e v e l contours a r e , t h e r e f o r e , of the form with €. =_,S,/Z^ • These contours are of three t y p e s , depending on the value of •€. : type A (•€• <0), IB (()<•€ <1 ) , . and I . (-6 >1 ) . These are shown i n shown i n F i g . 25. Each type generates two regimes, depending on whether P2 goes to zero ( ^ < 0 ) or i n f i n i t y ( j f2> 0 ) as x approaches A2= ( 1 , 1 ) . Noting that tfi = ^ , + ^2. 1 o n e o b t a i n s the regime diagram i n the plane "3, »^ t which i s shown i n F i g . 26. Pa t t e r n s e x e m p l i f y i n g the s i x regimes are shown i n F i g . 27. F i g u r e s 28a, b show the regime diagrams i n the plane q, |J, f o r the p a t t e r n s P2 a and P2^ , r e s p e c t i v e l y . I t should be noted t h a t , s i n c e j S t ^ 0 r the regimes Ac , B 0 , C o o , w i l l not be a c c e s s i b l e , unless oc 2-oC,>1. T h i s 115 F i g u r e 25 The three types of constant l e v e l contours of the c e l l d e n s i t y d i s t r i b u t i o n , f o r a two-step l i n e a g e with constant d i f f e r e n t i a t i o n and death r a t e s , and w i t h e x p o n e n t i a l source. The contour type depends on the value of the parameter e (see the t e x t f o r d e t a i l s ) . F i g u r e 26 The regime diagram i n the *$T_, ^ 2 plane. The regime depends on the val u e s of the parameters € and Y*2 (see the t e x t f o r d e t a i l s ) . 117 F i g u r e 27 The p a t t e r n s corresponding to the s i x regimes of F i g . 2 6 . The i n d i c e s "0" and "oo " r e f e r to the behaviour of the p a t t e r n near the f i n a l a t t r a c t o r , A 0. 118 P i Figure 28 The regime diagrams in the plane q,P-., for the patterns P2 a (a) , and P.^ (b) . The relaxation rates bi are equal to unity in the standard representation. 119 can be achieved e i t h e r by a l a r g e death rate i n d i f f e r e n t i a t i o n s t a t e 2, or by a negative death rate i n s t a t e 1. A negative death rate may indeed be used to provide a continuous approximation to the behaviour of the system i n the presence of d i v i s i o n . I t should a l s o be noted that the regime diagrams for P^a and P^y, do not c o i n c i d e , and thus v a r i o u s combinations of regimes may be o b t a i n e d , which allows a l a r g e degree of d i v e r s i t y i n the t r a n s i e n t d i s t r i b u t i o n s a s s o c i a t e d with a t r a n s i e n t l i n e a g e . 2.2.7. An E q u i v a l e n t System Let us c o n s i d e r a system with a t t r a c t o r s A^ =(0,0), A', =(1,0), and Ajj =(^5 , ^  ), and with equations of motion for the d i f f e r e n t i a t i o n s t a t e s 1 and 2, r e s p e c t i v e l y . T h i s i s a p a r t i c u l a r case of the system d e s c r i b e d i n Appendix 3. The mapping m which r e l a t e s t h i s system to i t s standard r e p r e s e n t a t i o n can be obtained from Eq.(A3.4) by s u b s t i t u t i n g the p a r t i c u l a r c o o r d i n a t e s of the a t t r a c t o r s A'e , A', , A'2 : x1'- w1 cx)=V- o- * ' )v , l. * o - y x i - *,>b l The Jacobian determinant f o r t h i s mapping i s 120 b ' - » b 2 - l , , b ' - » b ' + b z - 2 F i g u r e 29 shows the l o c i ^U,=const. and ^4z=const. i n the standard and l a b e l l i n g space, f o r b1 =1, b2 =3, A^MO.5, 1). The l o c i ^jt, =const. are a l s o the o r b i t s , which, f o r t h i s c h o i c e of parameters, are c u b i c p a r a b o l a e . F i g u r e 30 shows the t r a n s i e n t s k e l e t o n s f o r two-step l i n e a g e s i n t h i s system. These can be compared to t h e i r e q u i v a l e n t s i n the standard r e p r e s e n t a t i o n ( F i g . 16a, c ) . The d i s t r i b u t i o n §' in l a b e l l i n g space can be obtained as S ' C x ' ) = 7 --» where M=m , from the d i s t r i b u t i o n s i n S e c t i o n 2.2.6 and the Jacobian determinant above. The regime diagrams are shown in F i g . 28, and the s i x regimes are i l l u s t r a t e d i n F i g . 31. The importance of these r e s u l t s l i e s i n the f a c t t h a t , i f experimental d i s t r i b u t i o n s may be i n t e r p r e t e d i n terms of these regimes, then i n e q u a l i t i e s may be e s t a b l i s h e d which i n c l u d e the parameters q, , ^x r and b' . Thus, i n s i g h t s may be gained as to the behaviour of c e l l a n t i g e n s ( r e l a x a t i o n r a t e s bl ) , and c e l l s ( d i f f e r e n t i a t i o n r a t e |3i , death r a t e s OC, , and e x p o n e n t i a l growth c o e f f i c i e n t , q ) . The f a c t that parameters d e s c r i b i n g d i f f e r e n t l e v e l s of o r g a n i z a t i o n are brought together i n these i n e q u a l i t i e s , opens the p o s s i b i l i t y of using flow cytometry data and independent q u a n t i t a t i v e r e s u l t s at one 121 F i g u r e 29 The l o c i |U^=const. and ^ 2 ~ c o n s t - i n t n e standard r e p r e s e n t a t i o n (a), and i n l a b e l l i n g space (b). The system i n (b) i s d e s c r i b e d by Eq.(2.11), with parameters b l = l , b 2=3, A5=(0,0), Aj_=(l,0), and A ^ = ( 0 . 5 , l ) . The l o c i are p l o t t e d f o r values of the path c o o r d i n a t e s ranging from 0 to 1, i n increments of 0.1. 122 A; F i g u r e 30 T r a n s i e n t s k e l e t o n s f o r a two-step l i n e a g e , i n the l a b e l l i n g space r e p r e s e n t a t i o n : (a) s t a t i o n a r y l i n e a g e , where the source at AQ i s zero f o r t < 0; (b) n o n - s t a t i o n a r y l i n e a g e , where the t r a n s i t i o n A-j *• A£ i s f o r b i d d e n f o r t<6 , and the source i s zero f o r t < 0 . The parameters are as i n Fi g . 2 9 . The standard r e p r e s e n t a t i o n s of these s k e l e t o n s are shown i n Fig.16a,c, r e s p e c t i v e l y . The arrows i n d i c a t e the d i r e c t i o n of motion of the corresponding edges. 123 a A, A. b A 0 A A. A, F i g u r e 31 The l a b e l l i n g space p a t t e r n s f o r the regimes i n Fig.28. The parameters are as i n Fig.29. The standard r e p r e s e n t a t i o n e q u i v a l e n t s of these p a t t e r n s are shown i n F i g . 2 7 . 124 l e v e l , i n order to estimate parameters belonging to a d i f f e r e n t l e v e l . 2.2.8. The Flow Equation with D i v i s i o n We s h a l l again assume i n t h i s s e c t i o n that the equation of motion i s of the form given i n Appendix 3 by Eq.(A3.1). We s h a l l c o n s i d e r the d e n s i t y d i s t r i b u t i o n f u n c t i o n ^ ( x , T 5 , t ) , where <5 i s the i n t e r n a l time ela p s e d s i n c e the l a s t d i v i s i o n , as d e f i n e d i n S e c t i o n 2.1.7. A l s o , l e t us introduce the s t a t i o n a r y d i v i s i o n r a t e , <f (*£ ) . i t i s b i o l o g i c a l l y reasonable to assume that ^(0)=0, that *f (IS )->• 0 as £->oo , and that the decrease to 0 i s q u i t e r a p i d i n both l i m i t s . The flow equation f o r S ( x , £ , t ) i s a d i r e c t g e n e r a l i z a t i o n of Eq.(2.21): n n where B= 2 b V r $ ( C ) i s the s t a t i o n a r y death r a t e , S ' ( x , ^ , t ) i s the " d i f f e r e n t i a t i o n " source, and S " ( x , ^ , t ) i s the " d i v i s i o n " s o u r c e . While the d i f f e r e n t i a t i o n source i s e x t r i n s i c , being determined by the behaviour of the immediate p r e c u r s o r , S" i s i n t r i n s i c , and depends on ^ as oo Y\ ^  I I O b v i o u s l y , the c e l l s c o n t r i b u t e d by the d i v i s i o n source have * £ = ( ) . The f a c t o r 2 i s the product of the f a c t o r 2 due to d i v i s i o n , and the f a c t o r 2n , due to the e f f e c t of the 125 d i v i s i o n mapping, x-*.x/2, on the volume element. The s o l u t i o n of Eq.(2.28) can be obta i n e d from the s o l u t i o n (2.22) of the equation without d i v i s i o n : x e®* ACQ) $ ( 0 ) <je where A and ^ are d e f i n e d by (2.24). T h i s s o l u t i o n i s the sum of c o n t r i b u t i o n s from the two types of source. I f the d i f f e r e n t i a t i o n process i s c o r r e l a t e d with c e l l d i v i s i o n i n such a way that the newly d i f f e r e n t i a t e d c e l l s are at the beginning of the c e l l c y c l e ( "5=0), the d i f f e r e n t i a t i o n source has the form S' ( x , - £ , t ) = cT(S* ) s ^ ( x , t ) . Then, the c o n t r i b u t i o n s from the two s o u r c e s , 3 ^ ( d i f f e r e n t i a t i o n ) , and %a ( d i v i s i o n ) , are b£ x where o Equations (2.29a,b) are "same ge n e r a t i o n " e q u a t i o n s , which d e r i v e the c e l l d e n s i t y from the sources of the same generation through the flow mapping (see S e c t i o n 2.1.9). Equation (2.30) i s a "next g e n e r a t i o n " e q u a t i o n , r e l a t i n g the c e l l d e n s i t y to the 126 d i v i s i o n source i n the next generation v i a the d i v i s i o n mapping. Equations (2.29) and (2.30) can be combined to g i v e the next generation equation f o r the d i v i s i o n source: Co o For a s t a t i o n a r y d i f f e r e n t i a t i o n source S', and a death r a t e lower than the d i v i s i o n r a t e , the f i r s t term i n the l a s t equation w i l l dominate as t —*-oo , and one o b t a i n s the f o l l o w i n g asymptotic equation f o r sc: s c c x , o = 2 f>c:s)e ACs^ c C R - t C^x-fl)e } ^ ) c k % 2.31 o The support of the asymptotic sc and ^ w i l l be w i t h i n the s 6. s i m p l i c e s a n <3 d e s c r i b e d i n S e c t i o n 2.1.9. The s o l u t i o n s of Eq.(2.3l) r e q u i r e f u r t h e r i n v e s t i g a t i o n . As we mentioned in S e c t i o n 2.1.9, the motion of c e l l s i n 0*« may be c h a o t i c , and s o l u t i o n s of t h i s equation may e x h i b i t f l u c t u a t i o n s which remain f i n i t e i n the t — * < » l i m i t . 127 Chapter 2.3  The Interpretation of Murine Thymocyte Data 2.3.1. Introduction to the Phenotypic Analysis of Murine  Thymocytes The thymus is an organ of the lymphatic system consisting of two symmetric lobes, and situated dorsal to the sternum. We follow Weiss [17] in a brief description of the histology of the thymus. The thymic lobes are compact, branched structures, consisting of lobules. The latter are separated by septa, which are extensions of the connective tissue enclosing the thymus (capsule), and through which blood and efferent lymphatic vessels, as well as nerves, connect with the thymic mass. The lobules consist mainly of epithelial cells and lymphocytes, called thymocytes. The epithelial cells form a meshwork (cellular reticullum), which is infiltrated by thymocytes. The central region of the lobule, or medulla, contains relatively fewer thymocytes, while the outer region, or cortex, is richer in thymocytes. The cortex also contains many macrophages, some of which form tight units with the surrounding proliferating thymocytes. These macrophages contain phagocytized thymocytes. The blood vessels and capillaries of the thymus have an additional, outer layer of epithelial cells (the perivascular cuff), which form the blood-thymus barrier. This barrier is tight in most of the cortex, but f a i r l y permeable to proteins in the medulla and the juxtamedullar region of the cortex. There 128 are no apparent a f f e r e n t lymphatic v e s s e l s . The thymus i s a primary lymphoid organ, where a l a r g e segment of the lymphocyte p o p u l a t i o n , the T c e l l s , o r i g i n a t e s [18-22]. The embryonic thymus [23,24], as w e l l as thymi which have been d e p l e t e d by i r r a d i a t i o n [25-28] or drugs [29-31], are populated by bone marrow d e r i v e d c e l l s , which t y p i c a l l y lack the T c e l l phenotype and f u n c t i o n . Thus, the thymus appears to be the main environment i n which thymocytes g r a d u a l l y a c q u i r e , by processes of d i f f e r e n t i a t i o n and m a t u r a t i o n , the phenotypic and f u n c t i o n a l c h a r a c t e r i s t i c s of T c e l l s . The d i r e c t evidence as to in t r a t h y m i c p r o l i f e r a t i o n [18,29,32-34], d e f i n e the thymus as a s i t e of expansion of T c e l l s from p r o g e n i t o r c e l l s . More i m p o r t a n t l y , the thymic environment determines to a great extent the s p e c i f i c i t y r e p e r t o i r e of the d e r i v e d T c e l l s [ 3 5 ] . As the expansion of the thymocyte p o p u l a t i o n i s accompanied by a high r a t e of c e l l death [29,33,36,37], some form of c l o n a l s e l e c t i o n [38] i s probably r e s p o n s i b l e f o r T c e l l r e p e r t o i r e g e n e r a t i o n , although there i s no general agreement on t h i s p o i n t . Thymocytes and thymus d e r i v e d lymphocytes (T c e l l s ) are not homogeneous i n the q u a n t i t a t i v e e x p r e s s i o n of some of t h e i r membrane a n t i g e n s . Moreover, these antigens are expressed s e q u e n t i a l l y i n embryonic thymi, and i n thymi which are being repopulated a f t e r a r t i f i c i a l l y induced thymocyte d e p l e t i o n . Such a n t i g e n s , whose q u a n t i t a t i v e e x p r e s s i o n changes as c e l l s proceed from one s t a t e of d i f f e r e n t i a t i o n to another, are r e f e r r e d to as d i f f e r e n t i a t i o n antigens . The main thymocyte d i f f e r e n t i a t i o n a n t i g e n s used i n our s t u d i e s are Thy-1, L y t - 1 , and L y t - 2 . Thy-1 [39-41] i s a 129 glycoprotein (molecular weight 19,000 to 25,000 daltons), associated with a g l y c o l i p i d . It i s a general marker of thymocytes and T c e l l s , although the le v e l of i t s expression may vary from one type of c e l l to another. It i s also found on non-T hematopoietic c e l l s [42,43], as well as in non-lymphoid ti s s u e s , such as brain and epidermal c e l l s . Lyt-1 [44,45] i s a glycoprotein with a molecular weight of 67,000 daltons, and i s mainly found on thymocytes and T c e l l s , although Lyt-1 bearing B c e l l s have been described [46,47]. P r a c t i c a l l y a l l T c e l l s seem to carry this antigenic determinant [45]. Lyt-2 [44,45,76] i s a complex, disulfide-bonded glycoprotein, consisting of two homodimers, one of 30,000 daltons, and the other of 35,000 daltons . This tetramer also c a r r i e s the Lyt~3 antigenic determinant. Lyt-2 i s present on approximately 90% of thymocytes, and 50-70% of T c e l l s in lymph nodes and spleen. The molecular complex carrying the Lyt-2 determinant has been reported to be involved in the recognition of antigen by some cytotoxic T c e l l clones [48-51,76]. Labelling patterns for one or two of these antigens have been published by several authors [45,52,53], and their expression has been related to the maturation and d i f f e r e n t i a t i o n processes taking place in the thymus [54-57], as well as to the functional properties [58-63,76] and s p e c i f i c i t y repertoires [51,64,65] of thymocytes and T c e l l s . Despite their wide acceptance as s i g n i f i c a n t markers for thymocytes, these antigens have f a i l e d to provide a detailed picture of the thymus. Indeed, of these three antigens, only 130 Lyt-2 c l e a r l y d i s t i n g u i s h e s between two thymocyte su b p o p u l a t i o n s , w h i l e , at the other extreme, Lyt-1 s t a i n i n g p a t t e r n s e x h i b i t only one d i f f u s e maximum which extends over a very wide range of f l u o r e s c e n c e s . Even when double l a b e l l i n g i s used, the d i s t r i b u t i o n appears very d i f f u s e along the Lyt-1 a x i s . We presented i n [6] an a n a l y s i s of thymocytes in terms of three antigens (M1/69, L y t - 1 , and L y t - 2 ) , which reveal e d much more s t r u c t u r e than p r e v i o u s l y seen, and showed that c e r t a i n subpopulations had w e l l d e f i n e d Lyt-1 phenotypes. 2.3.2. Data P r e s e n t a t i o n and A n a l y s i s The theory o u t l i n e d i n the pr e v i o u s two chapt e r s i s based on the f o l l o w i n g assumptions: ( i ) each d i f f e r e n t i a t i o n s t a t e i s c h a r a c t e r i z e d by one a n t i g e n i c phenotype, with i t s r e p r e s e n t a t i v e p o i n t i n l a b e l l i n g space a c t i n g as an a t t r a c t o r ; ( i i ) the k i n e t i c equations governing antigen e x p r e s s i o n i n each d i f f e r e n t i a t i o n s t a t e are of the form x ' = f ( x ' ) , and can be transfromed i n t o standard equations of the form x=A-x by a co o r d i n a t e t r a n s f o r m a t i o n . Under these assumptions, the theory allows one to c o n s t r u c t models of c e l l p o p u l a t i o n s , and to d e r i v e the consequent l a b e l l i n g d i s t r i b u t i o n s . These models c o n s i s t , on the one hand, of a model of antigen e x p r e s s i o n , and, on the o t h e r , of a model of c e l l k i n e t i c s . The former i s embodied i n the equations of motion, and, a f t e r a l i n e a g e has been p o s t u l a t e d , i n the co o r d i n a t e t r a n s f o r m a t i o n (mapping m, S e c t i o n 2.1.4); the l a t t e r i s contained i n the model l i n e a g e , and the c e l l k i n e t i c s r a t e s f o r each step of the l i n e a g e . 131 Ideally, one should begin the dynamic interpretation of data for a particular c e l l population by establishing the kinetic equations of antigen expression. Then, the coordinate transformation which maps the labelling space into the standard representation space can be constructed (Section 2.1.4 and Appendix 3). Subsequently, the standard representations of the labelling distributions can be obtained (Section 2.2.7). These standard representations are determined exclusively by c e l l kinetics, and, therefore, should be easier to interpret than the original labelling distributions. The process of interpretation can be based on a comparison with experimental distributions for c e l l populations whose differentiation and division properties are known, and also with the distributions predicted by various models. In this chapter, we adopt the second approach, as a body of standard experimental distributions for known c e l l kinetics remains to be established. Moreover, we postulate linear kinetics of antigen expression (the linear system in Appendix 3), as a direct investigation at this level remains to be done. A linear model appears to be a good approximation, in view of the similarity between predictions and experimental data for the c e l l s described in this chapter. Some of the more important topological details of the labelling distribuions are associated with local maxima. In what follows, these will be called "subsets", with the understanding that they are phenotypically, rather than physiologically homogeneous subsets. Indeed, we shall interpret several such subsets in the mouse thymus as consisting of ce l l s which belong 1 32 to at l e a s t two l i n e a g e s . Subsets w i l l be d e f i n e d by t h e i r p o s i t i o n i n l a b e l l i n g space, o r , where there i s no ambiguity, by t h e i r p o s i t i v e or negative phenotype, r e l a t i v e to a set of a n t i g e n s . The f e t a l thymus data presented i n t h i s chapter are four-d i m e n s i o n a l , with three dimensions corresponding to the l a b e l l i n g i n t e n s i t i e s f o r the three a n t i b o d i e s , while time i s the f o u r t h dimension. The four - d i m e n s i o n a l d i s t r i b u t i o n (B6 thymocytes) i s presented i n F i g u r e s 32 and 33 as two-dimensional s e c t i o n s ( L y t - 1 , Lyt-2 l a b e l l i n g ) on a two-dimensional l a t t i c e , with s e c t i o n s i n the same ( h o r i z o n t a l ) row of the l a t t i c e c o r r e sponding to the same Thy-1 l a b e l l i n g i n t e n s i t y , and s e c t i o n s i n the same ( v e r t i c a l ) column corresponding to the same age of the f e t u s . The rightmost column i n F i g . 33 c o n t a i n s the t r i p l e l a b e l l i n g d i s t r i b u t i o n f o r a d u l t B6 thymocytes. Thus, s u c c e s s i v e two-dimensional d i s t r i b u t i o n s i n each column are s e r i a l s e c t i o n s through the instantaneous l a b e l l i n g d i s t r i b u t i o n s , p e r p e n d i c u l a r to the Thy-1 a x i s . C o n t i n u i t y of s t r u c t u r a l f e a t u r e s along the Thy-1 a x i s can be e s t a b l i s h e d by comparing s u c c e s s i v e s e c t i o n s i n the same column, while s u c c e s s i v e s e c t i o n s i n the same row of the l a t t i c e w i l l r e v e a l c o n t i n u i t y i n time. The d i s t r i b u t i o n s were normalized to 105 c e l l s i n each column. The contours i n each s e c t i o n are l i n e a r l y spaced. Each s e c t i o n along the Thy-1 a x i s i s two-channel t h i c k , and the columns were cut o f f at a Thy-1 l a b e l l i n g i n t e n s i t y c orresponding to the p o s i t i o n of the Thy-1+,Lyt-1",2* maximum in the a d u l t B6 thymus, without l o s s of s t r u c t u r a l f e a t u r e s . A constant d e n s i t y s u r f a c e of the Thy-1,Lyt-1,2 133 F i g u r e 32 F e t a l mouse (B6) thymocyte d i s t r i b u t i o n s f o r days 16 (a), and 17 (b), of g e s t a t i o n . The axes of the diagrams correspond t o Lyt-2 ( h o r i z o n t a l ) and Lyt-1 ( v e r t i c a l ) l a b e l l i n g i n t e n s i t i e s . The Thy-1 l a b e l l i n g i n t e n s i t i e s are shown by the numbers to the l e f t of each row. The numbers i n s i d e the diagrams denote the v a r i o u s subsets d i s c u s s e d i n the t e x t . 16d 17d 135 F i g u r e 3 3 F e t a l mouse thymocyte d i s t r i b u t i o n s f o r days 18 (a), 19 (b), and 20 (c) of g e s t a t i o n . The corresponding a d u l t d i s t r i b u t i o n i s shown i n (d) (see a l s o the legend to F i g . 3 2 ) . 136 1 37 d i s t r i b u t i o n of a d u l t CBA thymocytes i s shown in F i g . 2. The corresponding d i s t r i b u t i o n f o r B6 thymocytes i s given by the rightmost column in F i g . 33. A d i r e c t comparison of the two d i s t r i b u t i o n s r e v e a l s that they are homologous, as s t r u c t u r a l d e t a i l s i n one d i s t r i b u t i o n have corresponding d e t a i l s i n the other (denoted by the same i n d i c e s ) , and as the r e l a t i v e p o s i t i o n s of homologous elements are s i m i l a r . As we have al r e a d y mentioned i n the I n t r o d u c t i o n , the theory of dynamic i n t e r p r e t a t i o n can be d i r e c t l y a p p l i e d only to those regions of a l a b e l l i n g d i s t r i b u t i o n which c o n t a i n c e l l s from a s i n g l e l i n e a g e . For the Thy-1,Lyt-1,Lyt-2 thymocyte d i s t r i b u t i o n s which form the o b j e c t of our s t u d y , the Lyt-2* region seems to meet, at l e a s t i n p a r t , t h i s requirement. We show in the next two s e c t i o n s that the topology of the d i s t r i b u t i o n of Lyt-2* thymocytes i n the Thy-1,Lyt-1,Lyt-2 l a b e l l i n g space can be e x p l a i n e d i n terms of two l i n e a g e s , which only o v e r l a p i n the region d e f i n e d as subset 1 i n F i g u r e s 2 and 33. On the b a s i s of the a v a i l a b l e d a t a , we do not attempt to i n t e r p r e t the d i s t r i b u t i o n i n the Lyt-2" r e g i o n , and, t h e r e f o r e , one should ignore d i s c r e p a n c i e s between p r e d i c t e d and experimental d i s t r i b u t i o n s which may occur i n t h i s r e g i o n . 2.3.3. The F i r s t Lineage Subset 1, which has a Thy-1*,Lyt-2*, Lyt-1 low or negative phenotype, i s one of the major elements i n the a d u l t thymocyte p o p u l a t i o n (>40%, F i g . 34a). An examination of s u c c e s s i v e s e c t i o n s i n the lowest row of F i g u r e s 32 and 33 shows that t h i s maximum i s present from day 17 of f e t a l l i f e onwards, and that 138 F i g u r e 34 Percentage (a) and number of c e l l s (b) i n thymocyte subsets 1 ( T ) ; 2 ( 0 ) ; 4 ( # ) ; 3+5+7 ( v ) ; and 8 ( B ) . The t o t a l number of f e t a l thymocytes ( • ) was taken from [70] . 140 i t i n c r e a s e s i n both r e l a t i v e and absolute s i z e at l e a s t up to the moment of b i r t h ( F i g . 34). The d i s t r i b u t i o n f o r day 16 shows the e x i s t e n c e of c e l l s with t h i s phenotype, although the maximum at high Lyt-2 i s absent. The e a r l y e v o l u t i o n of t h i s phenotype between days 15 and 17 can b e t t e r be understood from the Lyt-1,2 p r o j e c t i o n s shown i n F i g . 35. I t i s apparent that the phenotype o r i g i n a t e s around day 15 i n the low Thy-1, low Lyt-2 r e g i o n , the express i o n of these two antig e n s i n c r e a s e s on day 16, and a maximum i s formed by day 17. The d i s t r i b u t i o n s f o r days 15 and 16 show that subsets 1 and 4 are c l o s e l y r e l a t e d , and that the low d e n s i t y bridge connecting them i n the a d u l t i s a s i g n i f i c a n t f e a t u r e , as i t i s present on days 15 and 16, when the phenotype a s s o c i a t e d with subset 1, but not the maximum, i s alre a d y p r e s e n t . Moreover, subset 4, which accounts f o r only 6-7% of a d u l t thymocytes, appears to be an important subset i n the e a r l y f e t a l thymus, while d e c r e a s i n g i n r e l a t i v e s i z e i n l a t e r l i f e ( F i g . 34a). A comparison between the Thy-1, Lyt-2 d i s t r i b u t i o n at 15d and 16d ( F i g . 35a,b) on the one hand, and the p a t t e r n f o r the regime A i n F i g . 31b on the o t h e r , suggests that t h i s regime a p p l i e s to the e a r l i e s t L y t - 1 " , 2* c e l l s . I f one combines the A«, p a t t e r n with a type "a" window ( F i g . 30a), one ob t a i n s the su c c e s s i v e d i s t r b u t i o n s shown i n F i g . 36a,b. These, we b e l i e v e , are good q u a l i t a t i v e d e s c r i p t i o n s of the behaviour of the Lyt-1",2+ c e l l s i n the i n t e r v a l 15-I6d, and lea d us to the f o l l o w i n g c o n c l u s i o n s as to the ontogeny of subset 1 up to and i n c l u d i n g day 16 of embryonic l i f e : ( i ) There i s a s t a t i o n a r y , two-step l i n e a g e Ac (Thy-1",Lyt-Figure'35 Thy-1,Lyt-2 d i s t r i b u t i o n s f o r days 15 (a), 16 (b) and 17 (c) of f e t a l l i f e . The axes correspond t o Thy-1 ( h o r i z o n t a l ) and Lyt-2 ( v e r t i c a l ) l a b e l l i n g i n t e n s i t i e s . 142 F i g u r e 36 S u c c e s s i v e f i r s t l i n e a g e d i s t r i b u t i o n s (see the t e x t f o r e x p l a n a t i o n s ) a t t=0.2 (a), and 0.6 (b) . The p o s i t i o n s of the l i m i t c y c l e o r i g i n s , and A^, are shown i n (b), w h i l e (c) shows the d i s t r i b u t i o n generated by AQ—> A i — > ^ 2 • T n e Parameters were b^-=l, b 2 = 3, and the d i v i s i o n p e r i o d 0 =1. 1 44 1 " , 2" )—>" A j (Thy-1 *,Lyt-1 " , 2 " ) — » • At (Thy- 1 + , Ly t- 1 " , 2 *) , where the intermediate state i s included in subset 4, and the. f i n a l a t t r a c t o r , Az (Section 2.1.1), i s included in subset 1. ( i i ) The precursor population (phenotype Thy-1Lyt-1",2"), begins to d i f f e r e n t i a t e on or around day 15. ( i i i ) The tr a n s i t i o n p r o b a b i l i t y per unit time for the Thy-1+,Lyt-1",2" Thy-1+,Lyt-1" ,2* step i s approximately constant, ^f, (£",)= ^ , (see Sections 2.2.2 and 2.2.4). (iv) A stationary lineage and the regime A0 ( F i g . 31b) require the parameters to s a t i s f y the inequality q> b - (X, - |S,> b +b - <XZ ( F i g . 28a; see also Sections 2.2.4 and 2.2.6 for discussions of these parameters). Thus, i f the death rate «2 i s low in t h i s i n i t i a l stage (of2<bl+bl) , then q must be d e f i n i t e p o s i t i v e , and the source population at Ad increases exponentially . This i s an important r e s u l t , which i l l u s t r a t e s the a b i l i t y of the dynamic interpretation to extract information about c e l l u l a r processes not d i r e c t l y linked to antigen expression. The maximum c h a r a c t e r i s t i c of subset 1 appears on day 17, at lower values of Thy-1 and Lyt-2 than those already reached by day 16 ( F i g . 35b,c). This suggests that the maximum, together with most of the c e l l s at higher Thy-1 and Lyt-2, are cycling c e l l s . This assumption i s supported by the projected skeleton in F i g . 21b, where x' corresponds to Thy-1, and x2 to Lyt-2. Indeed, the skeleton expands as shown in F i g . 36, u n t i l , at t= 0 , i t has f i l l e d the triangular region AeC0C^ in F i g . 21b. Then, the c e l l s on the leading edge C0C^ di v i d e , and the region Bt B C i C J begins to be gradually occupied, the leading edge moving from B,Bj at t=0 , to C,CJ at t = 2 0 . This w i l l indeed 145 give r i s e to the r e q u i r e d maximum. It should be noted that a second l i n e a g e , which w i l l be d e s c r i b e d i n the next s e c t i o n , a l s o c o n t r i b u t e s to subset 1. If one leaves out the c e l l s o u t s i d e the dashed contour i n F i g . 35c, one obtains the reduced d i s t r i b u t i o n ( S e c t i o n 2.1.8). T h i s appears to be s i m i l a r to the p a t t e r n i n F i g u r e 27a. Indeed, the l a b e l l i n g space d i s t r i b u t i o n c o r r e s p o n d i n g to t h i s p a t t e r n i s shown in F i g . 36c, f o r a d i v i s i o n p e r i o d 0= 1. The f o l l o w i n g c o n c l u s i o n s can be drawn at t h i s p o i n t r e g a r d i n g the ontogeny of subset 1 on day 17 and a f t e r : ( i ) Subset 1 continues to be generated by the same two-step, s t a t i o n a r y l i n e a g e as on days 15 and 16. ( i i ) The ce3*ls of subset 1 are d i v i d i n g . T h i s may a l s o be true of the c e l l s i n the intermediate s t a t e (subset 4 ) . ( i i i ) The t r a n s i t i o n p r o b a b i l i t y per u n i t time can s t i l l be approximated by a constant ^ , although i t i s p o s s i b l e that i t i n c r e a s e s at l a r g e g , ( S e c t i o n 2.2.2). ( i v ) The s t a t i o n a r y l i n e a g e and regime r e q u i r e the parameters to s a t i s f y q<b^ - C K c t - P i ^1 - <*i - £>i +ln2 / 0 (see S e c t i o n 2.1.8 f o r d e f i n i t i o n s of these parameters), which s e t s an upper l i m i t on q. T h i s i s c o n s i s t e n t with the source c e a s i n g to grow e x p o n e n t i a l l y , and, e v e n t u a l l y , s e t t l i n g at a constant l e v e l i n the a d u l t thymus. 146 2.3.4. The Second Lineage Subset 2 is not well defined, in that i t lacks a central maximum, while being closely associated with a maximum which we denoted by the.separate subset 1. As will be seen, in this section, the cells of subset 2 exhibit a transient phenotype (which is why there is no central maximum), and eventually reach subset 1, where they mix with the cells generated by the lineage described in the previous section. The Thy-1 *,Lyt-1 +,2+ phenotype characteristic of subset 2 is absent on day 15, but begins to emerge on day 16 from a region of phenotype Thy-1+,Lyt-1*2~ (Fig. 32a). On day 16, the cells of subset 2 s t i l l exhibit a low level of Lyt-2, which increases by day 17 virtually to steady state level, when a maximum can be seen at high Lyt-1,2 in the top sections of Fig. 32b. In this respect, the evolution of the Lyt-2 phenotype of these cells over the interval I6d-17d parallels that of the subset 1 cells over the interval I5d-16d (see the previous section). However, there is a significant difference between the phenotype evolution of the cells of subsets 1 and 2, which will be discussed later. An examination of the cells of Thy-1*,Lyt-1+2+ phenotype reveals that they are roughly concentrated in the neighbourhood of a plane going from low Thy-1 and high Lyt-1 to high Thy-1 and low Lyt-1. This suggests that the expression of Thy-1 and that of Lyt-1 are anticorrelated. Our hypothesis is that the lineage generating subset 2 is Ae (Thy-1",Lyt-1+,2")—• A ( (Thy-1 + ,Lyt-1-,2")—*• Afc (Thy-1+ , Lyt-1-,2+), and that the cells in both differentiation states divide. Then, the geometric 2-simplex which is the reduced skeleton for this lineage has the right 14? o r i e n t a t i o n , as shown i n F i g . 37a. F i g u r e 37b shows the 2-simplex i n l a b e l l i n g space, i f one assumes that the r e l a x a t i o n r a t e s f o r the three antigens are b^|_2> b-j^ {> , with the Thy-1 l e v e l f o r A2 lower than that f o r A, . Let us now return to the d i f f e r e n c e of behaviour between the c e l l s of the two l i n e a g e s . As we showed i n the previous s e c t i o n , the c e l l s of subset 1 which f i r s t express Lyt-2 are low in Thy-1, which i s c o n s i s t e n t with a s t a t i o n a r y l i n e a g e . However, an examination of the Lyt-1*2* c e l l s on days 16 and 17 ( F i g . 32) shows that they begin to express Lyt-2 s i m u l t a n e o u s l y , r e g a r d l e s s of t h e i r Thy-1 l e v e l . The r e s u l t s of s e c t i o n 2.2.5 show that such a behaviour i s c o n s i s t e n t with a n o n - s t a t i o n a r y l i n e a g e , i n which the second t r a n s i t i o n , Thy-1 *,Lyt-1'2" —^-Thy-1*,Lyt-1j2*, i s allowed only a f t e r some time, 0 , has elapsed a f t e r the onset of the f i r s t t r a n s i t i o n , Thy-1~,Lyt-1*2~—>• Thy-1 *,Lyt-1 } 2 '. One can see from F i g u r e 30b that one of the l e a d i n g edges of the s k e l e t o n generated by such a l i n e a g e i s p a r a l l e l to the (A0,A,) s i d e . F i g u r e 38 shows a sequence of s k e l e t o n s i l l u s t r a t i n g t h i s p o i n t . The d i s t r i b u t i o n s f o r days 16 and 17 suggest a regime for the p a t t e r n i n window b. I f one assumes that the top s e c t i o n in F i g . 32b (day 17) g i v e s a s e c t i o n through the d i s t r i b u t i o n i n window "a " , the f a c t t h a t the maximum i s l o c a t e d at h i g h Lyt-2 suggests t h a t the p a t t e r n i n t h i s window i s e i t h e r A w > or IBQO • An i n s p e c t i o n of the regime diagrams f o r the two windows ( F i g . 28a,b) shows that the A « > domain f o r p a t t e r n P 2 b o v e r l a p s with both and [Boo domains f o r the p a t t e r n P2 a • T h e r e f o r e , values of the parameters e x i s t such that both regime 148 x 3 'Thy-1 F i g u r e 37 The reduced s k e l e t o n f o r the second l i n e a g e i n the standard r e p r e s e n t a t i o n ( a ) , and i n l a b e l l i n g space (b) (see the t e x t f o r e x p l a n a t i o n ) . The r e l a x a t i o n r a t e s i n .(b) were b T , „> b m, . > b T , , , and A~ i s i n the Thy-1, Lvt-2 , L y t - 2 ^ T hy-1 ^ L y t - 1 ' 2 1 149 F i g u r e 38 S u c c e s s i v e second l i n e a g e d i s t r i b u t i o n s (see the t e x t f o r explanations) at t = l ( a ) , 1.2 ( b ) , and 1.4 (c) . The t r a n s i t i o n A2 i s only allowed f o r t - 1 . The two windows are marked "a" and "b". The parameters were b l = l , b2=3. 151 combinations are p o s s i b l e . The data we have do not allow us to decide which of the two p o s s i b i l i t i e s a p p l i e s . On the b a s i s of a v a i l a b l e d a t a , i t i s a l s o d i f f i c u l t to decide whether the steady s t a t e p a t t e r n approached a f t e r day 17 belongs to the A«o or [Boo regime. T h i s may i n d i c a t e that the values of the parameters are such that the r e p r e s e n t a t i v e p o i n t in the regime diagram f o r p a t t e r n P2 a ( F i g . 28a) i s c l o s e to the boundary s e p a r a t i n g the domains Aoo and Boo • The d i s t r i b u t i o n of the Thy-1+ ,Lyt-1+,2 + c e l l s a f t e r day 17 can be compared to the s k e l e t o n i n F i g . 20a, where x1 corresponds to Thy-1, xl to L y t - 2 , and xJ to L y t - 1 . A l s o , the double l a b e l l i n g d i s t r i b u t i o n s i n F i g . 39 can be compared to the p r o j e c t e d s k e l e t o n s of F i g . 20b,c,d. I t seems reasonable to us to i n t e r p r e t subset 2 as c o n s i s t i n g of c e l l s i n the regions of the s k e l e t o n with n > 2 , that i s c e l l s which have d i v i d e d at l e a s t twice a f t e r the Thy-1 " ,Lyt-1 +,2'—>- Thy-1 * ,Lyt-1 "2" t r a n s i t i o n , and which have f u r t h e r d i f f e r e n t i a t e d to Thy-1*,Lyt-1^2* before the f i r s t d i v i s i o n . The b r i d g e connecting subsets 2 and 5 can be i n t e r p r e t e d as the r e g i o n of the s k e l e t o n where n=1, which c o n s i s t s of the c e l l s which must d i v i d e once more i n order to reach subset 2. The c e l l s of the region where n=0 may account f o r the low d e n s i t y regions marked "0" i n F i g . 39. In view of the above d i s c u s s i o n , we can reach the f o l l o w i n g c o n c l u s i o n s : ( i ) Subset 2 i s generated by a n o n - s t a t i o n a r y l i n e a g e Ac (Thy-1",Lyt-1 *2-)—*- A , (Thy-1+,Lyt-1 "2") > Az ( T h y - 1+, L y t - 1 " 2+) , where the intermediate s t a t e i s i n c l u d e d i n subsets 3 and 5, and the f i n a l a t t r a c t o r , A2, i s i n c l u d e d i n subset 1. I t i s p o s s i b l e 152 F i g u r e 39 Double l a b e l l i n g d i s t r i b u t i o n s of mouse thymocytes two days a f t e r b i r t h (a,b,c), and i n the a d u l t ( d , e , f ) . The l a b e l l i n g i s Thy-1,Lyt-2 (a,d), Thy-1,Lyt-1 (b,e), and Lyt-2 , L y t - 1 ( c , f ) . The numbers i n the diagrams show the approximate p o s i t i o n s of the subsets d i s c u s s e d i n the t e x t . 1 53 that subset 5 consis ts e x c l u s i v e l y of c e l l s in the intermediate state of t h i s l ineage . ( i i ) C e l l s in the second d i f f e r e n t i a t i o n state (subsets 1 and 2) d i v i d e . It i s poss ible that the c e l l s in the f i r s t state (subsets 3 and 5) a lso d i v i d e . ( i i i ) The precursor population (phenotype T h y - 1 " , L y t - 1 + , 2 " ) probably begins to d i f f e r e n t i a t e before day 15, and may be growing exponential ly , * ~ . ( iv) The second t r a n s i t i o n , T h y - 1 + , L y t - 1 "2"—*«• Thy-1 * , Lyt-1 ~2 + , becomes poss ible only on between days 15 and 16. The cause of t h i s i s probably e x t r i n s i c (a d i f f e r e n t i a t i o n s ignal may be absent before day 15), as the lineage appears stat ionary af ter day 16. (v) The regime A © o in window " b " , and Aw or Boo in window " a " , requires the inequal i ty £ - * b | - ° < c z - b1 + b*- + 3\nl/B in the presence of d i v i s i o n . This inequal i ty sets an upper l i m i t on the exponential c o e f f i c i e n t for source growth. Thus, one cannot decide on the basis of t h i s q u a l i t a t i v e analys is whether the source i s growing exponent ia l ly . 2 .3 .5 . The Wider Phenotypic Picture In the previous two sec t ions , we analyzed two lineages which account for part of the topology of the experimental d i s t r i b u t i o n s . It should be emphasized that these p a r a l l e l two-step l ineages only p a r t i a l l y account for the d i f f e r e n t i a t i o n and maturation processes in the thymus. This i s i l l u s t r a t e d in F i g . 40, where the dashed arrows and boxes represent regions of the thymocyte ontogeny scheme which were not touched upon by our 154 I I , 1 1 i lineagel 1 Thy-rLyt-r2_—Thy-rLyt-r2"—Thy-rLyt-1"2+-stem ceils I I I I lineage 2 I I I—•Thy-1" Lyt-f 2" —• Thy-1*Lyt-r2~ —• Thy-1* Lyt-1"2* — J I • I I I emigrant ! ! 'cells I 1 I I F i g u r e 40 The p l a c e of the two model l i n e a g e s w i t h i n the wider thymocyte d i f f e r e n t i a t i o n scheme. The dashed arrows and boxes r e p r e s e n t those p a r t s of the scheme which were not covered by the q u a l i t a t i v e a n a l y s i s i n the t e x t . 1 5 5 q u a l i t a t i v e a n a l y s i s . Thus, on the b a s i s of the t r i p l e l a b e l l i n g data p r e s e n t e d , we c o u l d not e s t a b l i s h whether the two l i n e a g e s have a common p r e c u r s o r , whether they are followed by a d d i t i o n a l d i f f e r e n t i a t i o n s t e p s , or what i s the nature of the o t h e r , p a r a l l e l l i n e a g e s . S o l v i n g these important q u e s t i o n s w i l l r e q u i r e the use of a d d i t i o n a l monoclonal a n t i b o d i e s , p o s s i b l y flow c y t o m e t r i c a n a l y s i s of higher d i m e n s i o n a l i t y , and a q u a n t i t a t i v e dynamic i n t e r p r e t a t i o n of the flow cytometry d a t a . Moreover, i n v i t r o s t u d i e s w i l l be r e q u i r e d f o r the e l l u c i d a t i o n of the i n d i v i d u a l d i f f e r e n t i a t i o n s t e p s . The reduced r e p r e s e n t a t i o n s of the s k e l e t o n s f o r the two l i n e a g e s are brought together i n F i g . 41, where the r e l a t i v e p o s i t i o n s of the v a r i o u s subsets are a l s o shown. A f u l l e r r e p r e s e n t a t i o n may be obtained by adding to the s k e l e t o n of F i g . 21a that of F i g . 21b. According to the models of the previous two s e c t i o n s , subset 1 c o n s i s t s of c y c l i n g c e l l s . However, our q u a l i t a t i v e a n a l y s i s does not exclude the p o s s i b i l i t y that subset 1 i n c l u d e s n o n - c y c l i n g c e l l s as w e l l . Indeed, small Lyt-2* thymocytes, which appear around day 17 of embryonic development [53], predominate i n the a d u l t thymic c o r t e x [53,66], and do not d i v i d e [36,67]. These thymocytes c o e x i s t with a smaller p o p u l a t i o n of l a r g e thymocytes of the same phenotype [67] which d i v i d e , and a t r a f f i c of c e l l s appears to e x i s t between these two compartments [ 3 6 ] . As both l i n e a g e s 1 and 2 c o n t r i b u t e to subset 1, f u r t h e r i n v e s t i g a t i o n i s r e q u i r e d i n order to e s t a b l i s h the source of the two compartments. Let us now review the thymocyte subsets d e f i n e d i n F i g u r e s 156 F i g u r e 41 The reduced s k e l e t o n s f o r the two model l i n e a g e s . The numbers i n the diagram show the approximate p o s i t i o n of the subsets r e l a t i v e to these s k e l e t o n s . 1 57 2, 32, and 33, i n the l i g h t of the model l i n e a g e s 1 and 2. ( i ) Subset 1, with a w e l l - d e f i n e d c e n t r a l maximum, c o n t a i n s the f i n a l a t t r a c t o r f o r both l i n e a g e s , and thus i n c l u d e s both types of c e l l s . Some of these are c y c l i n g c e l l s , which accounts f o r the shape of the maximum. ( i i ) Subset 2, adjacent to the maximum of subset 1, c o n t a i n s the t r a n s i e n t phenotypes of c e l l s i n the second d i f f e r e n t i a t i o n s t a t e of l i n e a g e 2. These c e l l s are c y c l i n g . ( i i i ) Subset 3 i n c l u d e s the t r a n s i e n t phenotypes of c e l l s i n the f i r s t d i f f e r e n t i a t i o n s t a t e of l i n e a g e 2. Other c e l l s are probably a l s o i n c l u d e d i n t h i s s u b s e t . ( i v ) Subset 4 i n c l u d e s the t r a n s i e n t phenotypes of c e l l s i n the f i r s t d i f f e r e n t i a t i o n s t a t e of l i n e a g e 1. Other c e l l s are a l s o p r e s e n t . (v) Subset 5 probably c o n s i s t s e x c l u s i v e l y of c e l l s i n the f i r s t d i f f e r e n t i a t i o n s t a t e of l i n e a g e 2. (v i ) Subset 6, which only seems to be important i n the e a r l y embryonic thymus, may c o n t a i n a p r e c u r s o r of l i n e a g e 2 c e l l s . The p o s i t i o n of t h i s subset r e l a t i v e to that of l i n e a g e 2 c e l l s i n the second d i f f e r e n t i a t i o n s t a t e , suggests that the l a t t e r may be d e r i v e d through d i v i s i o n from the c e l l s of subset 6. ( v i i ) Subset 7 probably i n c l u d e s the e a r l y p r e c u r s o r s of the c e l l s of l i n e a g e 1. I t probably c o n t a i n s other c e l l s as w e l l , some of which are p o s s i b l y mature thymocytes. ( v i i i ) Subset 8 i n c l u d e s the p r e c u r s o r of l i n e a g e 1. I t probably a l s o c o n t a i n s pro-thymocytes. The d i s t r i b u t i o n s f o r days 18 and 19 d i f f e r mainly i n the Thy-1 low, Lyt-1*2* r e g i o n , where the d e n s i t y i n c r e a s e s from day 158 18 to 19. As we showed in Section 2.3.4, this may be interpreted as a Peo regime for cells following lineage 2. It is actually possible that several functional lineages belong, phenotypically, to lineage 2, some having Aeo , and others IBoo regimes. Let us now review the published data relevant to lineages 1 and 2. Lineage 1, consisting of the transitions Thy-1",Lyt-1 - ,2" Thy-1 * ,Lyt-1 " ,2" Thy-1 + ,Lyt-1 " ,2* , has not been previously described. This is not surprising, if one considers that subset 4 (Fig.2), which contains the cells in the intermediate state of this lineage, was only recently described by flow cytometry [8,13], However, thymocytes with this phenotype (Thy-1*,Lyt-1",2") were described by immune staining of frozen microscopic sections in the fetal [68] and adult thymus [66]. These thymocytes were shown to be concentrated in the subcapsular region of the cortex, together with thymocytes of the null phenotype (Thy-1", Lyt-1",Lyt-2") characteristic of , the lineage 1 precursors. Moreover, the phenotype characteristic of the third state in lineage 1 (Thy-1*, Lyt-1",2+) is well represented in the cortex. Thus, a l l three phenotypes belonging to lineage 1 have been independently documented, and they are physically adjacent to each other. The phenotypes characteristic of lineage 2 have also been documented. The Lyt-1*,2" and Lyt-1*,2* phenotypes have been demonstrated by flow cytometry in the adult thymus by Mathieson et a l . [52], The former phenotype was shown by van Ewijk et a l . [66,68] to be preferentially localized in the medulla, where i t is accompanied by a high expression of Lyt-1, and a low 159 e x p r e s s i o n of Thy-1. The second phenotype was shown by the same authors to be c h a r a c t e r i s t i c of the c o r t e x , where the expression of Lyt-1 i s v a r i a b l e , and, i n g e n e r a l , low, while that of Thy-1 i s g e n e r a l l y h i g h . Our t r i p l e l a b e l l i n g data show the same a n t i c o r r e l a t i o n between the e x p r e s s i o n of Lyt-1 and Thy-1 in the a d u l t [8,13] and f e t a l thymus ( t h i s c h a p t e r ) , and s i m i l a r r e s u l t s were obtained f o r s p l e n i c T c e l l s by Ledbetter et a l . [ 4 5 ] . The problem of the h i s t o l o g i c a l l o c a t i o n of l i n e a g e 2 c e l l s i s more complicated than that f o r l i n e a g e 1. The obvious h y p o t h e s i s would p l a c e the p r e c u r s o r (Thy-1~,Lyt-1*,2") and i n t e r m e d i a t e (Thy-1+,Lyt-1",2") s t a t e s i n the m edulla, and the f i n a l s t a t e (Thy-1+,Lyt-1",2* ) i n the c o r t e x . As d i s c u s s e d i n S e c t i o n 2.3.4 and below, the Lyt-1" d e s i g n a t i o n of these s t a t e s r e f e r s to the phenotype these c e l l s approach as they g r a d u a l l y l o o s e the Lyt-1 antigen c a r r i e d by t h e i r p r e c u r s o r , and not to t h e i r instantaneous phenotype, which i s s t i l l L y t - 1 * . According to our model l i n e a g e 2, the l o s s of the Lyt-1 a n t i g e n begins i n the intermediate s t a t e of t h i s l i n e a g e , and c o n t i n u e s i n the f i n a l s t a t e . Thus, the instantaneous phenotypes should be predominantly high Lyt-1 i n the intermediate s t a t e , and low L y t -1 i n the f i n a l s t a t e , which i s c o n s i s t e n t with the r e s p e c t i v e medullar and c o r t i c a l l o c a l i z a t i o n of these s t a t e s . The presence of a small p o p u l a t i o n of b r i g h t Lyt-1 c e l l s i n the cortex and even i n the subcapsular region [ 6 6 ] , together with the o b s e r v a t i o n that homing bone marrow c e l l s f i r s t appear near the capsule [ 3 4 ] , r a i s e the p o s s i b i l i t y that the l i n e a g e 2 precursor i s l o c a t e d i n the subcapsular r e g i o n of the c o r t e x . A t h i r d 160 hypothesis may be formulated as a combination of the previous two, by a l l o w i n g an e a r l y Lyt-1* thymocyte to migrate from the cortex to the medulla, where the immediate pre c u r s o r of l i n e a g e 2 would be generated. Further r e s e a r c h i n t o the m i g r a t i o n of thymocytes and i t s c o r r e l a t i o n with phenotype i s r e q u i r e d before t h i s q u e s t i o n c o u l d be s e t t l e d . An i n t e r p r e t a t i o n of the Lyt-1+,2* phenotype d e s c r i b e d by Mathieson et a l . [55] i s o f f e r e d by the sequence of d i f f e r e n t i a t i o n s t a t e s which make up l i n e a g e 2. As shown i n S e c t i o n 2.3.4, the L y t - 1+, 2+ phenotype (subset 2 i n F i g . 2) does not correspond to a t r a n s i e n t d i f f e r e n t i a t i o n s t a t e , but, r a t h e r , to a t r a n s i e n t phenotype. In other words, at l e a s t at high l e v e l s of Lyt-1 e x p r e s s i o n , the c e l l s with t h i s phenotype are g r a d u a l l y l o o s i n g t h e i r Lyt-1 a n t i g e n . The p e r s i s t e n c e of t h i s phenotype appears as a consequence of the constant i n f l u x of c e l l s ' f o l l o w i n g t h i s p a r t i c u l a r d i f f e r e n t i a t o n r o u t e . As regards the d i f f e r e n t i a t i o n s t a t e to which these c e l l s belong, i t i s d e s c r i b e d i n our l i n e a g e 2 model as being L y t - 1 " , 2+, which means that the Lyt-1 a n t i g e n i s no longer being s y n t h e s i z e d , and that the phenotype these c e l l s are approaching i s Lyt-1",2*. T h i s phenotype may a c t u a l l y never be reached, as one would expect the speed of Lyt-1 a n t i g e n l o s s to decrease as the Lyt-1" phenotype i s approached, so that the c e l l s may d i e before completely l o o s i n g t h e i r a n t i g e n . One should a l s o add that a c o n t i n u i n g , low r a t e s y n t h e s i s of Lyt-1 cannot be r u l e d out at t h i s s t a g e , c o n s i d e r i n g the l i m i t e d s e n s i t i v i t y of the flow cytometer (approximately 103 antigen molecules per c e l l ) . The e a r l y n o n - s t a t i o n a r y c h a r a c t e r of l i n e a g e 2 ( S e c t i o n 161 2.3.4) pr o v i d e s a simple e x p l a n a t i o n f o r the "dramatic appearance of Lyt123* c e l l s on day 17 of g e s t a t i o n " (quotation from Mathieson et a l . [ 5 5 ] ) . According to our dynamic i n t e r p r e t a t i o n of the t r i p l e l a b e l l i n g data f o r days 16 and 17 of g e s t a t i o n , the c e l l s of l i n e a g e 2 are i n i t i a l l y trapped i n the f i r s t d i f f e r e n t i a t i o n s t a t e of t h i s l i n e a g e , where they express phenotypes ranging from Thy-1~,Lyt-1*,2" (phenotype of the p r e c u r s o r of t h i s l i n e a g e ) to Thy-1*,Lyt-1",2" (phenotype of the intermediate d i f f e r e n t i a t i o n s t a t e ) . S i n c e , i n a l l p r o b a b i l i t y , both these c e l l s and t h e i r immediate p r e c u r s o r s d i v i d e , a l a r g e number of Thy-1*,Lyt-1*,2" c e l l s have accumulated by day 16. As soon as s y n t h e s i s of Lyt-2 becomes allowed around day 16, what amounts to a r a p i d l y propagating shock wave s t a r t s moving i n the Lyt-2 d i r e c t i o n ( F i g . 38, and S e c t i o n 2.3.4). By day 17, the l e a d i n g edge of t h i s wave (where the c e l l d e n s i t y i s h i g h e s t ) has reached a h i g h L y t - 2 , L y t - 1+ phenotype, where i t i s s t a b i l i z e d by c e l l d i v i s i o n . The data on f e t a l thymocytes p u b l i s h e d by Mathieson et a l . [55] and van Ewijk et a l . [68] show that the expression of the Lyt-1*,2' precedes that of the Lyt-1*,2* phenotype. The same sequence of events i s observed by Lepault et a l . [69] d u r i n g the r e c o n s t i t u t i o n of i r r a d i a t e d a d u l t thymi with f l u o r e s c e i n -l a b e l l e d bone marrow c e l l s . Moreover, L e p a u l t et a l . show t h a t , i n i t i a l l y , Lyt-1 and Lyt-2 are expressed on separate s u b s e t s , and t h a t , as i n the embryonic thymus, the Lyt-1*,2" c e l l s precede the Lyt-1*,2* ones. These r e s u l t s are f u l l y c o n s i s t e n t with our own, and imply that there are Lyt-1*,2" [55] and L y t -1",2* thymocytes which are not d e r i v e d from the Lyt-1*,2* 162 subpopulation. Lepault et a l . discuss the p o s s i b i l i t y that the early Lyt-1",2* thymocytes they describe by immunofluorescence microscopy may in fact be Lyt-1 low, and thus belong to the Lyt-1*,2* subset. While t h i s would be consistent with results indicating that p r a c t i c a l l y a l l adult thymocytes do express Lyt-1 [45], we do not think t h i s to be an important issue from the point of view of population a n a l y s i s , as our t r i p l e l a b e l l i n g data [8,13] show that there i s a low Lyt-1 phenotype in the adult thymus which i s c l e a r l y d i s t i n c t from the higher Lyt-1 phenotype (as i l l u s t r a t e d by subsets 4 and 5 in F i g . 2). Thus, we can conclude that the Lyt-1",2* subset described by Lepault et a l . as preceding the Lyt-1*,2* one i s indeed a d i s t i n c t subset. Mathieson et a l . [55] do not discuss the expression of Lyt-2 on the Lyt-1" f e t a l thymocytes, which, as can be seen from F i g . 1b in their paper, are s t i l l c l e a r l y present on day 17 of gestation. Our results show that the phenotypic evolution of these c e l l s c l o s e l y p a r a l l e l s that of the corresponding c e l l s in the reconstituted adult thymus. Thy-1* f e t a l thymocytes were detected before Lyt-2* thymocytes in thymus sections [68], and by l y s i s with complement [70], These results are in concordance with our own data on the embryonic thymus, and may indicate that a s p e c i f i c thymic microenvironment must develop before the f i r s t d i f f e r e n t i a t i o n step of lineage 1 may occur (a similar suggestion regarding the expression of Lyt-2 i s made by Mathieson et a l . [55]), or that lineage 1 has a precursor d i s t i n c t from that of the thymocytes seen on day 14. Considering the rudimentary state of the thymus at that early stage, the f i r s t interpretation may be the correct 163 one. In experiments i n v o l v i n g the r e c o n s t i t u t i o n of i r r a d i a t e d a d u l t mice with f l u o r e s c e i n - l a b e l l e d bone marrow, Lepault et a l . [69] showed that Thy-1 and Lyt-2 are expressed on donor-derived thymocytes w i t h i n 3 hours of r e c o n s t i t u t i o n , and that Lyt-2 appears to be expressed before Thy-1. These o b s e r v a t i o n s are c o n s i s t e n t with our assumptions t h a t , f o r l i n e a g e 1 c e l l s , the Lyt-2" — L y t - 2+ t r a n s i t i o n can occur immediately f o l l o w i n g the Thy-1"—*-Thy-1+ t r a n s i t i o n , and that the r e l a x a t i o n r a t e f o r Lyt-2 i s higher than that f o r Thy-1. Indeed, although a l l c e l l s of t h i s l i n e a g e which begin to express Lyt-2 are a l r e a d y s y n t h e s i z i n g Thy-1, the higher r e l a x a t i o n r a t e f o r Lyt-2 means t h a t , on some c e l l s , the e x p r e s s i o n of t h i s antigen may reach d e t e c t a b l e l e v e l s before that of Thy-1. Notable by i t s absence from our i n t e r p r e t a t i o n i s the d i f f e r e n t i a t i o n step r e s u l t i n g i n the phenotypic change L y t -1*,2+—*.Lyt-1 + ,2~, which was i n i t i a l l y p o s t u l a t e d by Cantor and Boyse [ 7 1 ] . The lac k of hard evidence f o r i t s e x i s t e n c e has been d i s c u s s e d by s e v e r a l authors [52,55,75]. Indeed, the Lyt-1+2" phenotype i s a l r e a d y present i n the a d u l t thymus [ 5 2 ] , and t h i s phenotype precedes that of i t s assumed p r e c u r s o r i n the f e t a l thymus [55,68,69], Our p u b l i s h e d data c o n f i r m the a d u l t thymus r e s u l t s [6,8,13], while the data presented i n t h i s chapter c o n f i r m and extend the r e s u l t s r e g a r d i n g the f e t a l thymus. As was shown i n the previous s e c t i o n , the evidence p o i n t s to the o p p o s i t e t r a n s i t i o n , L y t - 1+ , 2 " — > L y t~ r , 2+, as being dominant in both the f e t a l and the a d u l t thymus. T h i s does not r u l e out the p o s s i b i l i t y that a small number of thymocytes or T c e l l s do 1 64 undergo the t r a n s i t i o n p o s t u l a t e d by Cantor and Boyse, although the l a b e l l i n g data do not provide evidence f o r t h i s p o s s i b i l i t y . As regards the other t r a n s i t i o n p o s t u l a t e d by Cantor and Boyse [ 7 1 ] , namely L y t - 1 * , 2+— » » L y t - 1 " , 2+, t h i s i s co n t a i n e d i n our l i n e a g e 2 ( S e c t i o n 2.3.4). However, the Lyt-1",2* phenotype i s a l s o produced by l i n e a g e 1 ( S e c t i o n 2.3.3), which does not i n v o l v e an L y t - 1+ p r e c u r s o r . Some of the t r a n s i t i o n s which form l i n e a g e s 1 and 2 have been demonstrated i n c e l l and organ c u l t u r e s . Thus, a subset of bone marrow c e l l s undergo i n v i t r o the Thy-1" — T h y - 1 * t r a n s i t i o n i n the presence of thymus-derived f a c t o r s [72,73], and these c e l l s were shown by Komuro et a l . [74] to be r e q u i r e d f o r the r e p o p u l a t i o n of the thymus. S e v e r a l authors [67,68,70,71] have a l s o shown that c e l l s e x p r e s s i n g Thy-1 and Lyt a n t i g e n s can be generated by organ c u l t u r e of 13-14 day o l d thymic rudiments. 165 CONCLUSIONS The thymus i s a complex system, which c o n s i s t s of s e v e r a l c e l l p o p u l a t i o n s , connected by a network of d i f f e r e n t i a t i o n p r o c e s s e s . I t i s t h e r e f o r e n a t u r a l that a r e a l understanding of the p o p u l a t i o n s t r u c t u r e and dynamics of the thymus c o u l d only be achieved through the use of methods of i n v e s t i g a t i o n which can p r o v i d e c o r r e l a t e d measurements of a l a r g e number of parameters, at the l e v e l of i n d i v i d u a l c e l l s . However, most methods used to date in the study of t h i s organ do not meet e i t h e r requirement. The most obvious example i n t h i s sense i s the use of complement mediated l y s i s of antibody l a b e l l e d c e l l s , on which the current view of thymocyte and T c e l l d i f f e r e n t i a t i o n i s s t i l l l a r g e l y based. Flow cytometry, the use of which i s c u r r e n t l y growing at a f a s t r a t e , c e r t a i n l y meets the second requirement. As regards the f i r s t requirement, the number of v a r i a b l e s of the c e l l d i s t r i b u t i o n s has been l i m i t e d by the number of measuring channels of the flow cytometer. We e s t a b l i s h e d i n the f i r s t p a rt of t h i s t h e s i s the theory and p r a c t i c e of m u l t i p l e x l a b e l l i n g a n a l y s i s , which allows the number of antibody l a b e l l i n g v a r i a b l e s to exceed that of measured f l u o r e s c e n c e s . We proved that a m u l t i d i m e n s i o n a l l a b e l l i n g d i s t r i b u t i o n can be r e c o n s t r u c t e d from f l u o r e s c e n c e d i s t r i b u t i o n s of lower d i m e n s i o n a l i t y , and d e r i v e d a r e c o n s t r u c t i o n formula on which our r e c o n s t r u c t i o n a l g o r i t h m i s based. T h i s methodology allowed us to o b t a i n the f i r s t d i s t r i b u t i o n s i n terms of three antibody l a b e l s , b e f o r e t h i s became p o s s i b l e by c o n v e n t i o n a l t r i p l e 166 l a b e l l i n g . The p r e p a r a t i o n of the complex s t a i n i n g s o l u t i o n s r e q u i r e d i n m u l t i p l e x l a b e l l i n g was g r e a t l y s i m p l i f i e d by the use of the immunofluorescence tomograph, which we designed and b u i l t . Although we have a p p l i e d i t only to t r i p l e l a b e l l i n g , t h i s methodology i s t h e o r e t i c a l l y a p p l i c a b l e to any number of l a b e l s , p r o v i d e d that at l e a s t two fluorochromes are used. We a l s o showed that the complexity of the p r e p a r a t i v e procedure depends on the d i f f e r e n c e between the number of l a b e l s , and that of fluorochromes. The importance of t h i s r e s u l t l i e s i n the f a c t t h a t , as flow cytometers with more f l u o r e s c e n c e measuring channels become a v a i l a b l e , the complexity of the m u l t i p l e x l a b e l l i n g procedure f o r the same number of a d d i t i o n a l l a b e l s does not i n c r e a s e . The only l i m i t a t i o n s t o the use of m u l t i p l e x l a b e l l i n g i n higher dimensions come from the i n c r e a s i n g number of c e l l s which have to be a n a l y z e d , and the i n c r e a s i n g l y powerful computers r e q u i r e d f o r performing the r e c o n s t r u c t i o n . These a r e , however, problems which a r i s e i n higher dimensions whether m u l t i p l e x l a b e l l i n g i s used or n o t . Our t r i p l e l a b e l l i n g d i s t r i b u t i o n s of a d u l t and f e t a l thymocytes showed that i n c r e a s i n g the number of simultaneous l a b e l s does indeed l e a d to more d e t a i l e d d e s c r i p t i o n s of complex c e l l p o p u l a t i o n s . T h i s i s w e l l i l l u s t r a t e d by the thymic subpopulation of Thy-1*,Lyt-1",2" phenotype, which we d e s c r i b e d f o r the f i r s t time by flow cytometry. The t r i p l e l a b e l l i n g data f o r the embryonic thymus, while c o n f i r m i n g p r e v i o u s s i n g l e or double l a b e l l i n g r e s u l t s obtained by other a u t h o r s , p r o v i d e d new in f o r m a t i o n on the ontogeny of c e l l phenotypes. T h i s allowed us 167 to formulate a two-lineage model of the ontogeny of Lyt-2* thymocytes, which i s much more deta i led than previous models. The formulation of t h i s model i s based on a dynamic i n t e r p r e t a t i o n of immunofluorescent flow cytometry data, which was presented in the second part of t h i s t h e s i s . We showed that , while the l a b e l l i n g d i s t r i b u t i o n s for d i f f e r e n t i a t i n g and/or d i v i d i n g c e l l populations are determined by both c e l l and antigen k i n e t i c s , the e f f e c t s of the two can be separated. Indeed, a knowledge of the k i n e t i c equations governing antigen expression allows one to construct a coordinate transformation which, in i t s turn , can be used to obtain a transformed c e l l d i s t r i b u t i o n depending only on the k i n e t i c s of c e l l d i v i s i o n and d i f f e r e n t i a t i o n . These "standard representations" have r e l a t i v e l y simple p r o p e r t i e s , which can be analyzed by geometric and algebraic means, and can be used to test models of d i f f e r e n t i a t i o n and d i v i s i o n k i n e t i c s against experimental l a b e l l i n g d i s t r i b u t i o n s . There are several d i r e c t i o n s in which the work presented in t h i s thes is could be continued. As regards multiplex l a b e l l i n g , new reconstruct ion algorithms should be i n v e s t i g a t e d . Algorithms based on the entropy maximization technique may prove to be useful in t h i s f i e l d , as the reconstructions they would provide should lack regions of negative densi ty , as wel l as a r te fac ts due to noise in the recorded p r o j e c t i o n s . However, these advantages should be weighed against the increased computational requirements of such algori thms. The dynamic i n t e r p r e t a t i o n could be improved by combining c e l l d i v i s i o n with more general k i n e t i c equations of antigen 168 e x p r e s s i o n , and by i n c o r p o r a t i n g noise i n t o the t h e o r e t i c a l framework. A l s o , i n v i t r o s t u d i e s of d i f f e r e n t i a t i n g and d i v i d i n g c e l l s should p r o v i d e general i n f o r m a t i o n on the k i n e t i c s of antigen e x p r e s s i o n i n such p o p u l a t i o n s . In p a r t i c u l a r , t h i s i n f o r m a t i o n should be u s e f u l i n f o r m u l a t i n g a d e t a i l e d q u a n t i t a t i v e model of thymocyte d i f f e r e n t i a t i o n . The simultaneous use of more than three antibody s p e c i f i c i t i e s r e c o g n i z i n g thymocyte d i f f e r e n t i a t i o n a n t i g e n s should a l s o c o n t r i b u t e to a f i n e r a n a l y s i s of thymocyte d i f f e r e n t i a t i o n pathways. We developed i n t h i s t h e s i s two t e c h n i q u e s , m u l t i p l e x l a b e l l i n g a n a l y s i s , and the dynamic i n t e r p r e t a t i o n of immunofluorescent flow c y t o m e t r i c d a t a . The j o i n t use of these t e c h n i q u e s , together with t h a t of monoclonal a n t i b o d i e s a g a i n s t d i f f e r e n t i a t i o n a n t i g e n s , provide a powerful t o o l f o r the i n v e s t i g a t i o n of complex, e v o l v i n g c e l l p o p u l a t i o n s . 169 BIBLIOGRAPHY 1. 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(1983): Generation of thymocyte subpopulations in organ culture: correlated analysis of Lyt-2 phenotype and c e l l cycle status by flow microfluorometry. J . Immunol. 131: 1085-1089 68. Van Ewijk, W., Jenkinson, E. J . , Owen, J . J . T. (1982): Detection of Thy-1, T-200, Lyt-1 and Lyt-2 bearing cells in the developping lymphoid organs of the mouse embryo in vivo and in vi t r o . Eur. J . Immunol. 12: 262-271 69. Lepault, F., Coffmann, R. L., and Weissman, I. L. (1983): Characteristics of thymus-homing bone marrow c e l l s . J . Immunol. 131: 64-69 70. Kamark, M. E., Gottlieb, P. D. (1977): Expression of thymocyte surface alloantigens in the fetal mouse thymus in vivo and in organ culture. J . Immunol. 119: 407-415 71. Cantor, H., Boyse, E. A. (1977): Lymphocytes as models for the study of mammalian cellular differentiation. Immunol. Rev. 33: 105-124 72. Komuro, K., Boyse, E. A. (1973): In vitro demonstration of thymic hormone in the mouse by conversion of precursor cells into lymphocytes. Lancet, 1: 740-743 73. Scheid, M. P., Hoffman, M. K., Komuro, K., Hammerling, U., Abbott, J . , Boyse, E. A., Cohen, G. H., Hooper, J . A., Schulof, R. S., Goldstein, A. L. (1973): Differentiation of 175 T c e l l s induced by p r e p a r a t i o n s from thymus and by nonthymic agents. The determined s t a t e of the pr e c u r s o r c e l l . J . Exp. ' Med. 138: 1027-1032 74. Komuro, K., G o l d s t e i n , G., Boyse, E. A. (1975): Thymus-r e p o p u l a t i n g c a p a c i t y of c e l l s that can be induced to d i f f e r e n t i a t e to T c e l l s i n v i t r o . J . Immunol. 115: 195-198 75. S c o l l a y , R. (1983): Intrathymic events i n the d i f f e r e n t i a t i o n of T lymphocytes: a c o n t i n u i n g enigma. Immunol. Today, 4: 282-286 76. L e d b e t t e r , J . A., Seman, W. E., Tsu, T. T., Herzenberg, L. A. ( l 9 8 l ) : L y t - 2 and Lyt-3 are on two d i f f e r e n t p o l y p e p t i d e subunits l i n k e d by d i s u l f i d e bonds. R e l a t i o n s h i p of subunits to T c e l l c y t o l y t i c a c t i v i t y . J . Exp. Med. 153:1503-1516 77. Durda, P. J . , G o t t l i e b , P. D. (1978): S e q u e n t i a l p r e c i p i t a t i o n of mouse thymocyte e x t r a c t s with a n t i - L y t - 2 and a n t i - L y t - 3 s e r a . I . Lyt-3.1 a n t i g e n i c determinants r e s i d e on separable molecular s p e c i e s . J . Immunol. 121:983-989 78. Je n k i n s o n , E. J . , van Ewijk, W., Owen, J . J . T. (1981): Major h i s t o c o m p a t i b i l i t y complex antigen e x p r e s s i o n on the e p i t h e l i u m of the developing thymus i n normal and nude mice. J . Exp. Med. 153:280-292 176 APPENDICES 1. Experimental Parameters and C o r r e c t i o n s  1.1. T i t r a t i o n s If $ ( f ) i s a (normalized) d i s t r i b u t i o n of c e l l f l u o r e s c e n c e i n t e n s i t i e s over a given c e l l p o p u l a t i o n , we s h a l l — r°° use the average f l u o r e s c e n c e i n t e n s i t y , f = \ f $(f)dt, as an o v e r - a l l measure of the i n t e n s i t y of the s t a i n i n g . The i n t e n s i t y of the f l u o r e s c e n c e coming from a s i n g l e c e l l , f , i s the sum of i t s a u t o f l u o r e s c e n c e , fa , and s t a i n i n g f l u o r e s c e n c e , f5 . The d i s t r i b u t i o n of f ^ i s c h a r a c t e r i s t i c of the c e l l p o p u l a t i o n , while that of fs a l s o depends on the s t a i n i n g s o l u t i o n . The average values of these f l u o r e s c e n c e i n t e n s i t i e s s a t i s f y the r e l a t i o n s h i p f = f ^ + fs . The s t a i n i n g f l u o r e s c e n c e i n t e n s i t y , f& , can be a t t e n u a t e d by an a r b i t r a r y f a c t o r m •€ [0,1 ] , by adding n o n - f l u o r e s c e n t antibody to the s t a i n i n g s o l u t i o n . Thus, the average t o t a l f l u o r e s c e n c e becomes If the f r a c t i o n of f l u o r e s c e n t antibody on the c e l l membranes i s equal to that i n the s t a i n i n g s o l u t i o n , then m = / fli.2 C,/d, + Co/do 177 where c, and c0 are the respective molar concentrations of the i n i t i a l fluorescent and non-fluorescent antibody solutions, and d, , dQ are the corresponding dilution factors in the mixture. Equations (A 1.1) and (A1.2) yiel d , after rearrangement, 1 i l l . . , + R>.3 where f = f& + fs After establishing a value of d , that ensures saturation of the membrane binding sites, a sequence of identical samples can be stained with solutions having a constant d, , but varying values of d0 . A plot of the l.h.s. of Eq.(A1.3) versus d, /d0 has been shown to yield a straight line [6]. This confirms that Eq.(A1.2) holds. One can see from Eq.(A1.3) that the value of d0 given by dQ=sd,, where s is the slope of the line of best f i t , reduces the average staining fluorescence to half its maximum value. Stock solutions of the fluorescent and non-fluorescent antibody, having respective dilution factors d( and d0 determined as above, are therefore said to be equivalent , and have the same (effective) molar concentration of antibody. For each of the N antibodies, one can. similarly determine the equivalent dilution factors d, , dz for the two fluorescent forms, and d0 for the non-fluorescent form. The titrations also provide the average staining intensities fft< , fs z • These 5XN parameters characterize the set of reagents and the c e l l population used in the multiplex labelling experiment. 178 1.2. Stock S o l u t i o n s In what f o l l o w s , we s h a l l use a graph r e p r e s e n t a t i o n of s o l u t i o n m i x t u r e s . I f A and B represent two s o l u t i o n s ( F i g . 42a), then any mixture thereof can be represented as a poi n t C ( r ) = rB + ( l - r ) A , where r -€[0 , 1 ] i s the p r o p o r t i o n of s o l u t i o n B i n the mi x t u r e . Let C0-, , C,-, , C2; be the i n i t i a l s o l u t i o n s of antibody i in unconjugated, fluorochrome I - , and fluorochrome Z-c o n j u g a t e d form r e s p e c t i v e l y , as used i n the t i t r a t i o n s , and l e t M be the d i l u t i o n medium. The i n i t i a l s o l u t i o n s can be d i l u t e d with medium by f a c t o r s dQ', /N, d (j /N, dzi /N, to provide the e q u i v a l e n t stock s o l u t i o n s S0; , S,j , S2j ( F i g . 42b). S o l u t i o n s Stj , S2", w i l l g ive average s t a i n i n g f l u o r e s c e n c e i n t e n s i t i e s fS | i , fS 2 jj r e s p e c t i v e l y , and can be f u r t h e r d i l u t e d with S0, to y i e l d the new stock s o l u t i o n s S,'j , S2j with average s t a i n i n g f l u o r e s c e n c e s Y",j f5, ^ , fs 2 ;j • Since the c o n c e n t r a t i o n c o e f f i c i e n t s ,^2," [0,1] are a r b i t r a r y , one can chose them such that where f ^ j = min ( f ^ j , t$2t\ ^» a n <^ g-, •€ [ 0 , 1 ] . The f i r s t of these two e q u a l i t i e s ensures that the c o n s i s t e n c y c o n d i t i o n i s s a t i s f i e d , as the f l u o r e s c e n c e i n t e n s i t i e s f o r antibody i w i l l be the same, r e g a r d l e s s of fluorochrome. The second e q u a l i t y allows one to int r o d u c e an a r b i t r a r y a t t e n u a t i o n f a c t o r gj Thus, the s e n s i t i v i t y along each antibody a x i s can be independently c o n t r o l l e d . A C(r) B • • C 0j b F i g u r e 42 Graph r e p r e s e n t a t i o n of a b i n a r y mixture ( a ) , and of the stock and i n t e r m e d i a t e s o l u t i o n s f o r one antibody s p e c i f i c i t y (b) . The e x t r e m i t i e s of each arrow r e p r e s e n t s o l u t i o n s to be mixed, and a p o i n t on an arrow r e p r e s e n t s a mixture, the composition of which i s i n d i c a t e d by the c o o r d i n a t e w i t h i n b r a c k e t s (see the t e x t f o r d e t a i l s ) . 180 The f i n a l stock solutions for antibody i are S c ; , S(* , S^j , and they are used to prepare the intermediate solutions I ti , I i i corresponding to the c o e f f i c i e n t s X i{ , XgJ in the multiplexing matrix. The appropriate intermediate solutions for the N antibodies are mixed in equal proportions to produce the l a b e l l i n g solution for a given multiplexing matrix L m 1 , (0) . 1.3. Corrections Limited accuracy in the preparation of solutions results in rotation angle and scale errors in the recorded projections. These errors can be compensated for by using the intermediate solutions for direct staining of correction samples. Given Lmtr (® ' a n c^ t^i e intermediate solutions I JJ , one can thus determine experimentally the "true" c o e f f i c i e n t s Xfj ( 8 ) . In general, these c o e f f i c i e n t s w i l l not s a t i s f y Eq.(l.2a), and M Z Xkj <•) - i - i ff f They can, however, be normalized to Xjj ( 6 ) = \;j ( 6 )/rj , and these new values can be used in Equations (1.3) to determine the true value, 0' , of the rotation angles. Thus, the recorded fluorescences become 181 t 1 -<\Zx 1 j(e ' )* j M where Xij(6') = \{j (8), and r are the s c a l e c o r r e c t i o n f a c t o r s f o r the two axes of the p r o j e c t i o n . The t r u e angles and the s c a l e c o r r e c t i o n f a c t o r s for each p r o j e c t i o n are used as a d d i t i o n a l input f o r the r e c o n s t r u c t i o n a l g o r i t h m . 2. D e r i v a t i o n of the R e c o n s t r u c t i o n Formula We d e f i n e the F o u r i e r Transform in P as / V / and that in the p r o j e c t i o n plane as (R2 Let xD€ I R be an a r b i t r a r y p o i n t , and m=1 , . . . , [N/2] , " H€ Gm , 6 •€ [ 0 , TT/2 ]N _ 2" , f ^ - C R2, such that Eq.(1.7) h o l d s : 182 S u b s t i t u t i n g t h i s i n the r . h . s . of E q . ( A 2 . l ) , one o b t a i n s [ ^ C « ] C x . i - ^ \$Me <U RZ.3 Let f=L m T T ( © ) x and =M mit: ( 0 )x be the p r o j e c t i o n s of x i n the p i ane L (0 ) and i t s orthogonal complement r e s p e c t i v e l y . Then x=L^.jr (0 )f + ( 0 ) ^ , which can be s u b s t i t u t e d i n (A2.3) to y i e l d , a f t e r some rearrangements, o r , u s i n g E q s . ( 1 . 8 ) , (1.9), and (A2.2), where the plane L ( 6 ) c o n t a i n s x0 , and f „ are the c o o r d i n a t e s of xe i n that p l a n e . T h i s i s the p r o j e c t i o n theorem as a p p l i e d to m u l t i p l e x l a b e l l i n g . Equation (A2.4) can be i n v e r t e d to give £ ( x ) . However, the i n t e g r a t i o n must be c a r r i e d out s e p a r a t e l y on each component Q m i r of the c o v e r i n g of ^M , and the redundant c o n t r i b u t i o n must be s u b t r a c t e d from the t o t a l . Thus, 183 One can now make use of the coordinates f,0 defined by Eq.(1.7). The volume element in these coordinates is where n=N-m, and for m^ 2, and This yields 184 I* ft ' " ' s f c - L L \ , 1 - ( , 1 J , X where $ c o . ( . - ^ H ( f ? , ) ) i ? ; - - ' c ' i But x . Lm n( 6 ) f = Lm i r( 0 ) x . f , and, using the i n v e r s e F . T . ^ , Eq.(A2.6) can be w r i t t e n as or where m i r a c t s on 4*(f > 6 ) as Equation (A2.7) i s the r e q u i r e d r e c o n s t r u c t i o n f o r m u l a . 185 3. The General Linear System The general linear system for an n-step lineage is described by the equations of motion x ' = fc>. ( * ; ' - X ' ) .'-I,..-,« F V 3 . I where t>j = (b*| ) are the relaxation rates in the i-th state, and A.J = (A,j') i s the attractor in that s t a t e . The system i s f u l l y defined by the relaxation rates, b( ; the a t t r a c t o r s , A|; and the source phenotype, A^. Let us consider a two-step li n e a r system, and l e t A =(0,0), A,=(1,0), and A2=(1,1) be the positions of the attractors in the standard representation. We s h a l l f i r s t solve, in guise of an example, the d i f f e r e n t i a l equations for the mapping m. We s h a l l subsequently show a more dir e c t way of finding t h i s mapping. The d i f f e r e n t i a l equations (2.8) for the mapping m become in this case — 0-**) * C - X 2 ) = b 2 ( f t i ' - n V ) U » , 2 R3.2 By using the substitutions y =.u c i - x*) the equations (A3.2) become 186 i l l + J5!. = t i .•-»,* with the general solution v(Y,i,)-r'c«r-T)*fci«it ' . - . 2 where are arbitrary functions. This leads to the following solution for (A3.2): N l-x2 ' where ^ ' =e*^  Let us consider now the boundary conditions (2.9), which become where z=ln(l-x{). The solution of this equation is H ; , ( 2 ) = C ; e , ° | 2 - ( « ! ' - R a ' ) 1-1,2 where C1 are arbitrary constants. The last equation can be substituted in (A3.3), which becomes £ x \ x M , R i ^ ( , - x ^ [ ( » i ' - « i ' ) - c ' ( - S r ) b ! J u , - z The constants C1 can be determined from the additional boundary 187 c o n d i t i o n , m(k0)=k0, which gives C =A,-A0. F i n a l l y , the mapping m becomes m;(x\x')=Ri^ci-x 2) b i[(fli'-fti')+(fli'-fl!')(7f4) b ,J flM t X The mapping m can a l s o be found by using the s o l u t i o n s of the equations of motion, ( A 3 . 1 ) . For a c e l l i n the i - t h s t a t e of d i f f e r e n t i a t i o n of the l i n e a g e AQ—>. ...—>• An , the s o l u t i o n of the equation of motion i s f f , t / v — b; S i t = I , • • • ; with the i n i t i a l p o s i t i o n , X J ' _ J , i n the simplex (A^ , . . . ,A-'(). The d e f i n i t i o n (2.4) of the path c o o r d i n a t e jU\ can be used, and one obt a i n s x^=A.' + (x;_| - A j ) ^ ' . T h i s recurrence r e l a t i o n allows us to d e f i n e the point of path c o o r d i n a t e s jJiif--'>jxn a s «,-ni + Z(fl , ' . 1 - ' » i)A^-A!;" i= i ' ' The corresponding r e l a t i o n s h i p i n the standard r e p r e s e n t a t i o n , (2.2), f o l l o w s from (A3.5) i f a l l the r e l a x a t i o n r a t e s are equal to one. Let us now choose the a t t r a c t o r s A* in the standard r e p r e s e n t a t i o n such that the f i r s t i c o o r d i n a t e s are equal t o one, and the remaining n - i are equal to z e r o . Then, Eq.(2.2) becomes 188 This can be solved for the path coordinates to give fr-I - X J I- XJ** j = i> ... ,n-i If these are subst i tuted into (A3.5) , one obtains the mapping, m, from x to x' : + ( n / . , - A i X < - x " ) b " After some rearrangement, one can see that , for n=2, t h i s resul t coincides with (A3.4) . 4. C e l l Preparation and L a b e l l i n g  4 .1. Mice In the case study of Section 1.1.6, the mice were 8 weeks old female CBA/J ( referred to as "CBA") , obtained from the Jackson Laboratory (Bar Harbor, ME). The mice used in the experiments described in Ch.2 .3 were C57B1/6J ( referred to as "B6") , bred l o c a l l y from Jackson progeni tors . 189 4.2. Reagents The c e l l c u l t u r e medium used f o r c e l l l a b e l l i n g was DME (GIBCO, Grand I s l a n d , NY) supplemented with 10% heat-i n a c t i v a t e d FCS (GIBCO), lOrnM HEPES, lOOU/ml p e n i c i l l i n , and I00ug/ml s t r e p t o m y c i n . The monoclonal a n t i b o d i e s were anti-Thy-1 (clone 30-H12, p u r i f i e d and b i o t i n - c o n j u g a t e d form), a n t i - L y t - 1 (clone 53-7, p u r i f i e d , b i o t i n - and a r s a n i l a t e - c o n j u g a t e d forms), and a n t i -Lyt-2 (clone 53-6, p u r i f i e d , b i o t i n - and a r s a n i l a t e - c o n j u g a t e d forms), and were obtained from Becton-Dickinson, Mountainview, CA). The f l u o r e s c e n t reagents were F I T C - a n t i - a r s a n i l a t e (Becton-D i c k i n s o n ) , and TR - a v i d i n (Molecular Probes, J u n c t i o n C i t y , OR). 4.3. P r e p a r a t i o n of Thymocyte Suspensions The procedure f o r o b t a i n i n g f e t a l thymocytes f o l l o w s that d e s c r i b e d i n [ 7 8 ] . Male and female B6 mice were kept together o v e r n i g h t , and checked f o r v a g i n a l plugs the next morning. The day on which v a g i n a l plugs were observed was designated as day z e r o . The mice were s a c r i f i c e d by c e r v i c a l d i s l o c a t i o n . The fe t u s e s were d i s s e c t e d from the u t e r u s , and immersed i n c u l t u r e medium. F o l l o w i n g d e c a p i t a t i o n , the t h o r a c i c t r e e s , i n c l u d i n g the thymi, were removed by using f i n e watchmaker's f o r c e p s . The f e t a l thymic lobes were separated under a d i s s e c t i n g microscope. Adult thymi were cut i n t o small p i e c e s . The thymi were g e n t l y teased through a s t a i n l e s s s t e e l wire mesh. The c e l l s were c o l l e c t e d i n 5ml c u l t u r e medium, and the d e b r i s allowed to s e t t l e f o r 5 minutes. The c e l l s were subsequently washed twice with c o l d medium, and then passed through a l a y e r of FCS. The f i n a l c e l l c o n c e n t r a t i o n was 190 ad j u s t e d to 2x107 c e l l s / m l . 4.4. Antibody S o l u t i o n s Stock s o l u t i o n s were prepared as d e s c r i b e d i n Appendix 1. The t i t r a t i o n data showed t h a t , f o r subset 1 ( F i g . 2 ) , the TR-a v i d i n f l u o r e s c e n c e due to a n t i - T h y - 1 - b i o t i n was approximately f i v e times more intense than that due to a n t i - L y t - 2 - b i o t i n . In order to prevent the d i s t r i b u t i o n from being concentrated i n a narrow r e g i o n near the Thy-1 a x i s , we d i l u t e d the stock s o l u t i o n of T h y - 1 - b i o t i n by a f a c t o r of f i v e with the e q u i v a l e n t stock s o l u t i o n of unconjugated anti-Thy-1 (Appendix 1), thus e f f e c t i v e l y reducing the gain f o r Thy-1 f l u o r e s c e n c e by a f a c t o r of f i v e . The same t i t r a t i o n data allowed us to check f o r the c o n s i s t e n c y c o n d i t i o n d e s c r i b e d i n Appendix 1. T h i s c o n d i t i o n was found to be s a t i s f i e d to w i t h i n 4%, and we t h e r e f o r e decided to compensate f o r t h i s small d e v i a t i o n by c o r r e c t i n g the d a t a , rather than by modifying the stock s o l u t i o n s of a n t i - L y t - 1 and a n t i - L y t - 2 . The t r i p l e l a b e l l i n g a n a l y s i s of a d u l t CBA thymocytes ( S e c t i o n 1.1.6) i s based on three s e t s of f i v e p r o j e c t i o n s , each set corresponding to r o t a t i o n s about one of the three axes. The angle of r o t a t i o n i n each set covered the 0°-906 i n t e r v a l ( i n equal i n c r e m e n t s ) , and thus only 12 of the 15 p r o j e c t i o n s were independent. De s p i t e t h i s redundancy, a l l 15 p r o j e c t i o n s were used i n the r e c o n s t r u c t i o n , i n order to improve the s i g n a l - t o -noise r a t i o . The t r i p l e l a b e l l i n g d i s t r i b u t i o n s f o r embryonic and a d u l t 191 B6 thymocytes (Ch.2.3) are based on projections rotated about only one of the axes (Lyt-2), each reconstruction being based on 7 projections. The small number of projections was due to the small number of thymocytes which can be recovered from the early embryonic thymus without trypsin treatment. The use of trypsin was avoided because of the sensitivity of the Lyt-2 antigen to this proteolytic enzyme [51,76,77], Previous experiments ([6,8] and Section 1.1.6) have shown that as few as five projections give reasonable reconstructions for this type of distribution, and a reconstruction based on seven projections seems to be quite adequate. 4.5. Cell Labelling The following two-step staining procedure was used in a l l experiments. Fifty microlitres of c e l l suspension were added to 50ul of the f i r s t step staining.solution (mixture of antibodies) and incubated on ice for 30 minutes. The cells were washed twice with 1.5ml cold medium. The c e l l pellet was resuspended in 100U1 of the second step staining solution (mixture of TR-avidin and FITC-anti-arsanilate) and incubated on ice for a further 30 minutes, after which the cells were washed again twice with cold medium, and finally resuspended in 200ul of cold medium. The stained cells were kept on ice until they were analyzed on the FACS-IV (Becton-Dickinson FACS System, Sunnyvale, CA). As a rule, we analyzed 105 cells in dual-fluorescence mode (projections), and 3x10" cells in single fluorescence mode (titration and calibration samples). Dead cells were eliminated by gating out cells with forward light scatter below the main 1 92 thymocyte peak. 5. The R e c o n s t r u c t i o n Software The RCONST program computes the t r i p l e l a b e l l i n g d i s t r i b u t i o n from r o t a t e d two-dimensional p r o j e c t i o n s , by a d i s c r e t e implementation of the formula (1.11). The program can accomodate one or three sets of p r o j e c t i o n s , each set corresponding to r o t a t i o n about one of the axes. For one set of p r o j e c t i o n s , the program assumes zero d e n s i t y i n the missing p r o j e c t i o n s , and the r e c o n s t r u c t e d d i s t r i b u t i o n i s a f f e c t e d by the a r t e f a c t s d e s c r i b e d i n S e c t i o n 1.1.5. Each set may c o n t a i n up to 11 p r o j e c t i o n s , and the r o t a t i o n angles and s c a l e f a c t o r s (see Appendix 1) are r e q u i r e d as input parameters f o r each p r o j e c t i o n . The f i l e PARAM c o n t a i n s input and output parameters of the r e c o n s t r u c t i o n . T h i s f i l e c o n s i s t s of two t a b l e s , one f o r g l o b a l parameters, the other f o r data about i n d i v i d u a l p r o j e c t i o n s . The g l o b a l parameters a r e : NSET - number of p r o j e c t i o n s e t s (1 or 3) NPRJ - number of p r o j e c i o n s per set (2-11) NINT - number of i n t e r p o l a t e d p r o j e c t i o n s per set (2-11) NSMT - number of smoothing c y c l e s (exponent of sine f i l t e r ) MODE - s e l e c t s f i l t e r f u n c t i o n ( n o r m a l l y , should be set to 0 f o r f u l l r e c o n s t r u c t i o n s , and 1 f o r p a r t i a l ones) THCK - number of s e c t i o n s to be r e c o n s t r u c t e d ( t y p i c a l l y 64) SR - p o s i t i o n of the o r i g i n of the r e c o n s t r u c t i o n (same s h i f t , i n number of ch a n n e l s , along each a x i s ) 193 The projection parameters are: SET - set to which the current projection belongs (1-3) PROJ - number of the current projection within its set (1 corresponds to 0 ° , numbers increase with the angle) FILE - name of the FACS f i l e containing the projection; this name is of the form N.BI, I2 I3 IA I& .T, T2T^ , where N is an arbitrary character (usually the i n i t i a l of the data owner), I, . . . Is is the FACS f i l e ID, and T, TZT^ is the FACS f i l e tag. ANGL - true angle of rotation of the projection (see Appendix 1) SH, SV - reciprocals of the scale factors r,, rz in Appendix 1 SHH, SHV - horizontal and vertical coordinates (in channel numbers) of the autofluorescence peak (origin of the projection) The format of the f i l e PARAM is shown in Fig. 43. The FACS dual parameter f i l e s containing the projections have been converted from the RT-11 format to MTS line f i l e s consisting of 512 lines of 8 hexadecimal integers (format z7). Each row of the 64X64 FACS distribution is broken down into eight successive lines. The output is written into the temporary MTS f i l e -RECON, as 256 X THCK lines of 16 integers (format 16). Normally, THCK=64. Each successive group of 256 lines represents a 64X64 section. Each successive group of 4 lines within a section represents a row of the section. A f u l l reconstruction based on 3X5 projections requires 10s of CPU time on the Amdahl 470/V8 computer at the UBC Computing 1 2 3 4 5 6 NSET NPRJ NINT NSMT MODE THCK 194 1 1 63 SET PROJ FILE 8 -9 ANGL SH 10 SV 1 1 SHH 1 2 SHV 1 3 SR 1 T . B22083 .002 0. 0 1 .00 1 .00 5. 0 2. 5 2 2 T . B22083 .003 16. 1 1 .00 1 .00 5. 0 2. 5 2 3 T. B22083 .004 27. 1 1 .00 1 .00 5. 0 2. 5 2 4 T . B22083 .005 39. 6 1 .00 1 .00 5. 0 2. 5 2 5 T. B22083 .006 49. 4 1 .00 1 .00 5. 0 2. 5 2 6 T. B22083 .007 58. 3 1 .00 1 .00 5. 0 2. 5 2 7 T . B22083 .009 68. 7 1 .00 1 .00 5. 0 2. 5 2 8 T . B22083 .010 79. 1 1 .00 1 .00 5. 0 2. 5 2 9 T. B22083 .011 90. 0 1 .00 1 .00 5. 0 2. 5 2 'igure 43 The format of the PARAM f i l e . 195 Centre. We l i s t below the references for the MTS system subroutines and library subroutines used in the RCONST program: CMD - UBC System Subroutines. System Interface. UBC Computing Centre, UBC, Vancouver, 1980, p1 CMDNOE - UBC System Subroutines. System Interface. UBC Computing Centre, UBC, Vancouver, 1980, p.3 DFOUR2 - UBC FOURT. UBC Computing Centre, UBC, Vancouver, 1980 FTNCMD - UBC FORTRAN. UBC Computing Centre. UBC, Vancouver, 1981, 71-76 MOVEC - UBC CHARACTER. UBC Computing Centre, UBC, Vancouver, 1981, 33-34 L i s t i n g of *SETPAR at 12:55:14 on OCT 15, 1984 f o r CCid=TNB. Page 1 DIMENSION NF(3),CP(3,11,4),TA(3,11),NFA(3.3,11),ISR(3) CALL CMDNOE('$SET ECHO=OFF '.14) CALL CMD('SCREATE PARAM ',14) CALL CMD('$EMPTY PARAM OK ',16) CALL FTNCMD('ASSIGN 7=PARAM;') NS=1 NP = 9 NIN=9 ISM=2 IREC=1 NT=1 DO 8 1=1,3 ISR(I)=0 DO 8 J=1,11 TA(I,J)=0. DO 5 K=1.4 5 C P U ,d,K)-0. 8 CONTINUE 17 WRITE(6,900) 900 FORMAT(1X,'PARAMETERS',/,1X,'-1 END; 00 DISPLAY; 99 SAVE') WRITE(6,1000) 1000 FORMAT(/,1X,'1 2 3 4 5 6') WRITE(6,1010) 1010 FORMAT(1X,'NSET NPRd NINT NSMT MODE THCK',/,1X,30('=')) WRITE(6,1020)NS,NP,NIN,ISM,IREC.NT 1020 F0RMAT(6(3X,12),/) WRITE(6,1030) 1030 FORMAT(10X,'7',12X,'8',4X.' 9 10 11 12 13') WRITE(6,1040) 1040 FORMAT( 1X, 'SET PROJ F I L E ' , 9X, ' ANGL SH SV SHH SHV SR' , * / , 1 X . 5 0 C - ' ) ) DO 35 I=1,NS DO 30 J=1,NP DO 20 K=1,3 20 NF(K)=NFA(K,I,J) 30 WRITE(6, 1050)1. J,NF,TA( I ,d) ,CP( I, J , 1), CP( I , J , 2 ) , CP( I, J , 3 ) , CP( I, J , 4 ) , *ISR(I) 35 WRITE(6,1045) 1045 F0RMAT(1X,50('=')) 1050 FORMAT(3X,I1,3X,I2,1X.3(A4),1X,F4.1,2(1X,F4.2),2(1X,F4.1),1X,I2) 40 WRITE(6, 1080) 1080 FORMAT(1X,'PARAMETER CHANGE? (ENTER "RETURN TO DISPLAY)') READ(5,1090)11 1090 FORMAT(12) I I I I I I I I I (II.EO.- 1) GO TO 2000 (II.EO.O) GO TO 17 (II.EQ.1) READ(5,1110)NS (II.E0.2) READ(5,1090)NP (II.E0.3) READ(5.1090)NIN (II.E0.4) READ(5,1110)ISM (II.EQ.5) READ(5,1110)IREC (II.E0.6) REA0(5,1090)NT (II.NE.13) GO TO 50 WRITE(6,1130) READ( 5 , 1110) IS H READ(5,1090)ISR(IS) VO GO TO 40 CJ\ 50 CONTINUE L i s t i n g of «SETPAR at 12:55:14 on OCT 15, 1984 for CCid=TNB. Page 2 IF(II.LT.7) GO TO 40 IF(11.GT. 12) GO TO 100 WRITE(5.1130) 1130 FORMAT(1X,'ENTER SET NO. (1,2,3)') READ(6.1110)IS WRITE(5,1140) 1140 F0RMAT(1X,'ENTER "- 1" TO EXIT,"RETURN" FOR ALL PROJECTIONS,'. */,1X,'0R PROJECTION NO..') READ(6,1090)IP IF (IP.E0.-1) GO TO 40 IF (IP.NE.O) GO TO 60 I L= 1 IH=NP GO TO 70 60 IL=IP IH=IP 70 CONTINUE DO 90 I=IL.IH IF (II.E0.7) GO TO 75 IF (II.E0.8) READ(5,1150) TA(IS.I) IF (II.E0.9) READ(5,1160) CP(IS,I,1) IF (II.E0.1O) READ(5,1160) CP(IS,I,2) IF (II.E0.11) READ(5,1170) CP(IS,I,3) IF (II.E0.12) READ(5,1170) CP(IS.I,4) GO TO 90 75 READ(5,1120)NF 00 80 K=1,3 80 NFA(K.IS,I)=NF(K) 90 CONTINUE GO TO 40 100 CONTINUE IF(11.NE.99) GO TO 40 WRITE(7,1000) WRITE(7,1010) WRITE(7,1020)NS,NP,NIN,ISM,IREC.NT WRITE(7,1030) WRITE(7,1040) DO 150 I»1,NS DO 140 J=1,NP DO 120 K=1,3 120 NF(K)=NFA(K.I,J) 140 WRITE(7,1050)1.J.NF,TA(I,J),CP(I.J,1),CP(I,J,2),CP(I.J,3),CP(I.J,4), *ISR(I) 150 WRITE(7,1045) 1110 FORMAT(11) 1120 FORMAT(3(A4)) 1150 F0RMAT(F4.1) 1160 F0RMAT(F4.2) 1170 FORMAT(F4.1) 2000 CONTINUE END —1 L i s t i n g of RCONST at 12:55:40 on OCT 15, 1984 for CC1d=TNB. Page 1 CALL CMDNOE('$SET ECHO=OFF ',14) CALL CMD('SCREATE -RECON ',15) CALL CMO('$EMPTY -RECON OK ',17) CALL FTNCMD('ASSIGN 3=PARAM;') CALL FTNCMD('ASSIGN 7=-REC0N;') DIMENSION NC(8).NF(3) C0MPLEX*1G M LOGICAL*1 IB(1G) EOUIVALENCE(IB(1),M,NF(1)) COMMON DRC(3,11,64,64) COMMON/PAR1/NP,NS,NIN,ISM,IREC.NT C0MM0N/PAR2/CP(3.11,4),TA(3,11),NT0T(3,11),ISR(3) READ(3,20)NS,NP,NIN.ISM,IREC.NT 20 F0RMAT(4(/),6(3X.I2),3(/)) DO 70 IS=1,NS READ(3,80)IC DO 70 IP=1.NP READO. 100)NF,TA(IS, IP) ,CP(IS, IP. 1) ,CP(IS. IP, 2) ,CP( IS, IP, 3) , »CP(IS.IP,4),ISR(IS) NT0T(IS.IP)=O C************ * *LOAD PROJECT IONS* ** * ************** A=16.**5 CALL MOVEC(1,' ',IB(16)) CALL MOVECO, 'WC , IB( 13)) CALL FTNCMD('ASSIGN 4=?;',0,M) DO 50 1=1,64 DO 40 J=1 ,8 READ(4,120)(NC(K),K=1.8) DO 40 K=1,8 AN=FLOAT(NC(K)) IZ=INT(AN/A) D=AN-A*FLOAT(IZ) DRC(IS;iP,(K+8*(J-1)),I )=D 40 NTOT(IS,IP)=NTOT(IS,IP)+INT(D) DO 50 0=63,64 50 DRC(IS,IP,J.I)=0. NTOT(IS.IP)=NTOT(IS,IP)-INT(DRC(IS,IP,1,1)) DO 60 1=63,64 DO 60 J=1,64 60 DRC(IS.IP,J,I)=0. 70 DRCUS, IP, 1 , 1)=0. 80 FORMAT(A4) 100 FORMAT(10X,3(A4),1X.F4.1,2(1X.F4.2),2(1X.F4.1),1X,12) 120 F0RMAT(8(Z7)) CALL CORT(X) CALL FFLT(X) CALL BAPR(X) 140 CONTINUE END C= = = = = = = = = = = = = = :: = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = SUBROUTINE INTN(I,IL,IH,S,SH,11,C) IF (I.LT.IH) GO TO 20 A = SH+FLOAT(I- 128)/S+1.E-6 GO TO 30 20 IF(I.GT.IL) GO TO 40 A=SH+FL0AT(I)/S+1.E-6 30 II'INT(A) Listing of RCONST at 12:55:40 on OCT 15, 1984 for CCid = TNB. Page 2 C=FL0AT(II+1)-A RETURN 40 11=0 RETURN END SUBROUTINE CALC(IS,IP.I,J.S,D) DIMENSION D(4) COMMON DRC(3.11,64,64) DO 10 K=1,4 10 D(K)=0. 11=1+1 JJ=J+1 IF ((I*J).E0.O) RETURN D(1)=DRC(IS,IP,I,d)/S I F ( ( I . EO . 64 ) . AND . ( J . EQ . 64 ) ) RETURN IF (I.NE.64) GO TO 20 D(3)=DRC(IS,IP,64.JJ)/S RETURN 20 D(2)=DRC(IS,IP.II,J)/S IF(J.EQ.64) RETURN D(3)=DRC(IS,IP,I,JJ)/S D(4)=DRC(IS,IP,II.JJ)/S RETURN END C = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = SUBROUTINE CORT(X) DIMENSION D1(4),D2(4) REAL*8 PC COMMON DRC(3,11,64,64),PC(3,11,128,128) COMMON/PAR1/NP.NS.NIN,ISM C0MM0N/PAR2/CP(3,11.4),TA(3,11) WRITE(6,10) 10 F0RMAT(1X,'CORRECTION') B=90./FL0AT(NIN-1) DO 70 IS=1,NS DO 70 IA=1,NIN IF ((IA.NE.1).AND.(IA.NE.NIN)) GO TO 25 IF (IA.EO.1) IP=1 IF (IA.EO.NIN) IP=NP CA=1 . GO TO 50 C = = = = = = = = "CALCULATE ANGLE 25 A=B*FL0AT(IA-1) C==========FIND PROJECTIONS FLANKING A DO 30 K=1.NP IF (A.GE.TA(IS.K)) GO TO 30 IP=K-1 CA=(TA(IS.K)-A)/(TA(IS,K)-TA(IS.IP)) IO=K GO TO 40 30 CONTINUE C==========PREPARE CORRECTIONS 40 SH2=CP(IS,I0,1) SV2=CP(IS,10.2) M SHH2=CP( IS, 10,3) i£> SHV2=CP(IS, 10,4) 'XI IL2=INT(SH2*(64.-SHH2J+1.E-6) Listing of RCONST at 12:55:40 on OCT 15, 1984 for CC id = TNB. Page 3 IH2=INT(128.5+SH2*(1,-SHH2)+1.E-6) dL2=INT(SV2*(64.-SHV2)+1.E-6) dH2=INT(128.5+SV2*(1.-SHV2)+1.E-6) S2=SH2*SV2 50 SH1=CP(IS,IP,1) SV1=CP(IS.IP,2) SHH1=CP(IS,IP,3) SHV1=CP(IS,IP,4) IL1=INT(SH1*(64.-SHH1)+1.E-6) IH1=INT(128.5+SH1*(1.-SHH1)+1,E-6) dL1=INT(SV1*(64.-SHV1)+1.E-6) dH1=INT(128.5+SV1*(1.-SHV1)+1.E-6) S1=SH1*SV1 C==========CALCULATE PC DO 70 1=1.128 DO 70 J=1,128 C==========CALCULATE INTERPOLATION PARAMETERS IF((IA.EO.1).OR.(IA.EO.NIN)) GO TO 60 CALL INTN(I,IL2,IH2,SH2,SHH2,12,CI2) CALL INTN(J,JL2,JH2,SV2,SHV2,02,C J2) CALL CALC(IS,I0,I2,J2.S2.D2) Y = (1.-CA)*(CI2«(CJ2*D2(1) + (1.-Cd2)*D2(3))+(1.-CI 2)*(CJ2*D2(2) *( 1 -CJ2)*D2(4))) PC(IS.IA.I,J)»Y 60 CALL INTN(I.IL1,IH1,SH1,SHH1,I1,CI1) CALL INTN(d.dL1,JH1,SV1,SHV1,d1,Cd1) CALL CALC(IS.IP,I1.d1.S1.D1) X=CA*(CI1*(Cd1*D1(1) + (1.-Cd1)*D1(3))+(1.-C11)*(Cd1*D1(2) + *(1.-CJ1)*D1(4))) 70 PC(IS.IA,I,d)=PC(IS.IA,I,d)+X 80 RETURN END C = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = SUBROUTINE FFLT(X) DIMENSION NDIM(2) REAL*8 DATA(128,128),PC,FILT(65,127),A,ANI,ANd,AI,Ad,F,F1, *Z,G,T2,T3 C0MPLEX*16 TRAN(65,128) COMMON DRC(3.11,64,64 ) ,PC(3,11, 128,128) COMM0N/PAR1/NP,NS.NIN,ISM.NSA.IREC COMMON/PAR2/CP(3,11,4),TA(3,11),NT0T(3,11) EQUIVALENCE(DATA,TRAN) NDIM(1)=128 NDIM(2)=128 WRITE(6,5) 5 FORMAT(1X,'FFT') C**********************CONSTRUCT FILTER************************* DO 40 1=1,65 CONTINUE IFO.EQ. 1) GO TO 10 AI=DFL0AT(I-1) ANI=AI/128.D0 F1=DSIN(ANI)/ANI GO TO 20 10 F1=1.D0 20 CONTINUE DO 40 d=2,65 dd=129-d O O Listing of RCONST at 12:55:40 on OCT 15, 1984 for CC1d=TNB. Page JK=J-1 AJ=DFLOAT(J-1) ANd=Ad/128.00 F=((DSIN(ANJ)*F1/ANJ)**ISM)*AJ FILTU, JJ) = F IF(IREC.EO.O) GO TO 30 FILT(I,JK)=F GO TO 40 30 FILTd . JK) = F/3.D0 40 CONTINUE DO 140 IS=1.NS DO 140 IP=1,NIN C******************** *TRANSFER DATA***************** DO 50 1=1,128 DO 50 J=1,128 50 DATAd.d) = PCdS,IP,I.J) CONTINUE CALL DFOUR2(DATA,NDIM,2,-1,0) c.»**•»**.».***»***«»«N0RMALI2E AND FILTER********** Z =1000.DO/OFLOAT(NTOT(IS,IP)) DO 80 1=1,65 DO 70 J = 2,128 70 TRAN(I,J)=TRAN(I,d)*FILT(I,(J-1))*Z 80 TRAN(I,1)=0.DO CONTINUE IFdREC.NE .0) GO TO 120 IF((IP.NE.1).AND.(IP.NE.NIN)) GO TO 120 CONTINUE DO 100 1=1,65 DO 100 J=66,128 100 TRAN( I,J) = TRAN(I,J)/2.DO 120 CONTINUE CALL 0F0UR2(DATA,NDIM,2.1,- 1) CONTINUE DO 140 1=1,128 DO 140 J»1.128 140 PC(IS,IP,I.d)=DATA(I,J)/16384.00 160 RETURN END 0============================== = = = = = = = = = = = = = SUBROUTINE BAPR(X) DIMENSION CA(11),SA(11),DC(64,64),NC(16) REAL*8 PC COMMON DRC(3,11,64,64),PC(3.11.128,128) C0MM0N/PAR1/NP.NS,NIN,ISM,IREC.NT COMMON/PAR2/CP(3,11,4),TA(3.11),NT0T(3,11),ISR(3) WRITE(6,10) 10 FORMAT(1X,'BACK PROJECTION') C************CALCULATE SINES AND COSINES********** A=3.14159265359/FL0AT(2*(NIN-1)) DO 60 1=1,NIN AI=FL0AT(I-1) CA(I)=COS(AI*A) 60 SA(I)=SIN(AI*A) C************CALCULATE BACK PROJ.*»**********»»»•* IF(NS.E0.2) ISR(3)=ISR(2) IF(NS.NE.I) GO TO 65 ISR(2)=ISR(1) Listing of RCONST at 12:55:40 on OCT 15, 1984 for CCid=TNB. Page 5 ISR(3)=ISR(1) 65 CONTINUE DO 170 1=1.NT II=I-ISR(1) AI=FL0AT(II-1) IF(I.LE.ISRO)) 11 = 11 + 128 DO 170 K=1,64 KK=K-ISR(3) AK=FLOAT(KK-1) IF(K.LE.ISRO)) KK = KK+128 DO 120 d=1.64 dd=d-ISR(2) Ad=FL0AT(dd-1) IF(J.LE.ISR(2)) dd=dd+128 OC(d,K)=0. DO 120 IP=1,NIN CF = 1 . IF((IP.EO. D.OR.(IP.EO.NIN)) CF=0.5 CONTINUE IF(IREC.GT.1) GO TO 80 AS=Ad*CA(IP)+AK*SA(IP)+1. IST=INT(AS) BS=AS-FLOAT(1ST) IF(IST.LT.I) IST=128+IST ISU=IST+1 IFlISU.EO.129) ISU=1 DC(d,K)=DC(d.K) + (PC(1,IP.II.IST)*(1.-BS)+PC(1,IP,11,ISU)*BS)*CF 80 CONTINUE IF((IREC.EQ.1).OR.(IREC.E0.3)) GO TO 100 AS=AK*CA(IP)+AI*SA(IP)+1. IST=INT(AS) BS=AS-FLOAT(1ST) IFUST.LT.1) IST = IST+128 ISU=IST+1 IF(ISU.EQ.129) ISU=1 DC(d,K)=DC(d,K)+(PC(2,IP.dd,IST)*(1.-BS)+PC(2,IP.dd.ISU)*BS)*CF 100 CONTINUE IF((IREC.EO.1).OR.(IREC.E0.2)) GO TO 120 AS=AI*CA(IP)+Ad»SA(IP)+1. IST=INT(AS) BS=AS-IST IF(IST.LT.I) ISTMST+128 ISU=IST+1 IF(ISU.EO.129) ISU=1 DC(d,K)=DC(d,K)+(PC(3.IP.KK.IST)*(1.-BS)+PC(3,IP.KK,ISU)*BS)*CF 120 CONTINUE DO 160 d=1,4 DO 140 L=1, 16 140 NC(L)=INT(DC((L+16*(d-1)),K)) 160 WRITE(7,10O0)(NC(L),L=1,16) 170 CONTINUE 10O0 FORMAT(16(16)) 180 RETURN END ro o to PUBLICATIONS Buican, T. N.: On the efficiency of information processing in the living c e l l . Analele Universitatii din Bucuresti (1973). Mosora, F., and Buican, T.N.: Elemente de Fizica General a si Biofizica. Handbook of biophysical experiments (in Romanian). Bucharest University Press (1973). Buican, T.N.: Biofizica. Lecture notes for biochemistry students, v o l . 1 (in Romanian). Bucharest University Press (1976). Teh, H.-S., Buican, T.N., and Takei, F.: Fine antigenic specificities of cytotoxic lymphocytes derived from mature and immature thymocytes. Behring Inst. Mitt. No. 70:32-38 (1982). Buican, T.N., and Purcell, A.: "Many Colour" flow microfluorometric analysis by multiplex labelling. Surv. Immunol. Res. 2:178-188 (1983). Buican, T.N., and Hoffman, G.W.: "Many Colour" flow microfl uorometric analysis of differentiation antigens. In "Leucocyte Typing" (Bernard, A. et a l . , eds.). Springer, Berlin (1984). Buican, T.N., and Hoffmann, G.W.: Immunofluorescent flow cytometry in N dimensions: The multiplex labelling approach. Cell Biophys., accepted for publication (1984). Buican, T.N., and Hoffmann, G.W.: The Immunofluorescence Tomograph: an automated device for the preparation of multiplex labelling solutions. Cell Biophys., accepted for publication (1984). 

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