Specialization in phagocytosis

J. theor. Biol. (1976) 57, 197-205

Specialization in Phagocytosis P. BONGRAND, C. CAPO, A. M. BENOLIEL AND R. DEPIEDS Laboratoire

d’lmmunologie, U. E. R. de Mkdecine, Boulevard Jean Moulin, 13385 Marseille, Cedex 4, France

(Received 24 March 1975, and in revisedform

27 June 1975)

When phagocytes have been incubated with a mixture of two types of particles (say A and B), a microscopic count can be made of ceils having ingested i A-particles and j B-particles, for various i and j. Simple assumptions (the validity of which has been checked in a previous work) allow derivation of formulae that give a fair account of experimental results. Only such a quantitative study can determine whether there is a specialization among phagocytic cells, which question is of great immunological interest. 1. Iatroduction Some authors have studied the phagocytosis of mixtures of different types of particles: Perkins & Leonard (1963) found that among a population of rat peritoneal macrophages some cells were only able to ingest erythrocytes of distinctly related species, whereas other cells displayed a less rigid kind of specificity. Linz & Mandelbaum (1960) reported that most guinea pig polymorphonuclear leucocytes ingested at the same time E. coli and Staphylococcus as particles, but some phagocytes ingested only one type of bacteria. Rhodes & Lind (1968) found that mouse peritoneal macrophages ingested at the same time aggregated human serum albumin and ferritin, but polymorphonuclear leucocytes engulfed only one type of particle. It appeared that the significance of these results needed a quantitative study. A mathematical procedure enabled us to study this problem more thoroughly, using basic assumptions that have been checked in a previous work (Capo, Bongrand, Benoliel & Depieds, 1974). 2. Theoretical Basis of the Method Let phagocytes ingest type A and type B particles. We assume that the “avidity” of a given macrophage is multiplied by x or y when it ingests an A particle or a B particle respectively. Thus, when a phagocyte already 191

1%

P.

BONGRAND,

C.

CAPO,

A.

M.

BENOLIEL

AND

R.

DEPIEDS

containing i A particles andj B particles meets a new phagocytable particle, the probability of ingestion is: u$yj

for a type A particle

(1)

bx’yj

for a type I3 particle

(2)

Let m and n be the numbers of A and B particles that a phagocyte will meet during an experiment. We assume that at any time t a phagocyte has met mz A particles and nz B particles, where O
dP(u, 0,~) = M.&‘-ly”P(u--1, dz

u, z)+nbx”y”-lP(u,

v-l,

z)-

-(mu + nb)x”y”P(u,

u, z),

(3)

where P(i,j,z)=O ifi
(4) Using equation (3), we may derive P(u, a, z) for various u and a. We describe the derivation of P(0, 0, z) and P(1, 0, .z) as an example. We have

df’(O, (4.4 = -(ma+nb)P(O, dz

0, z),

equations (4) and (5) yield P(0, 0, z) = exp (- cz) where c = mu + nb, equation (3) may be written as

W1, 0,~) = maP(0, 0, z)-cxP(1, dz

0, z).

(6)

Then, we define X(z) as P(l, 0, z) = X(z) exp (- cxz).

(7)

d-G) = ma exp [-(1 -x)cz].

(8)

Equation (6) yields ~

dz

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Using equation (4), we find mu exp ( - CXZ)- exp ( - cz) z) = y-l-x *

P&O,

(9)

Using similar techniques, we can derive P(u, V, z) for various u and ZI. We define P(u, V) as P(u, u) = P(u, u, 1). We find P(O,O)=exp(-c)=T

(10) nb TY-T

P(0, 1) = -

(11)

-

(12)

(2-x-y)Txy

+ (1 --X)(1 -y)(l

p(o,2)A!g [

yT

(1

p(3 , 0) = (ma)3

c3

-Y)U

-TY

-Y2> T(X”

T(Y’)

(1 -YY

+ (1 -YXl T’“”

XTX

+ (l-X)2(1-X2) P(2, 1) = (ma)znb c

1

1

-Y2>

(1 -x)(1 -x2)(1 -x3) - (1 -X)*(1-X”)

-xy)

(13) (14)

+ X3T

1

-(l-X)(1-X*)(1-X3)

(15)

x

-~~+(3-y)x~+(-y*+2y-1)~*+(y~+y-4)x+3-2y~~~~ (l-x)2(1 -y)(l -x2)(1 -xy)(l -x*y> + (-3y2+2y)x4+(y2+y-1)x3+(2yZ-y)x2-xy2T (1 -X)(1 -y)(l --x2)(1 -xy)(l -x2y)

+

X

(x+y-2xy)Tx

+ (1 -X2)(1 -y)(l

+

xTY

-xy)

+ (1 -x)(1 -y)(l T(X2)

- (1 --xx1 -YXl

-z-c*> (x+y-2)TXY

-x2) + (1 -x)2(1 -y)(l

-xy)

1

. (16)

200

P.

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M.

BENOLIEL

AND

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DEPIEDS

When y = 1, calculating the limits of the above formulae, we find lim P(0, k) = T

(17)

y=1

lim P(l, 1) = gF:i y=l

[(1 +cx)TX-(1

(ma)2nb

(a2 + 2)T’““’

lim P(2, 1) = 3

(W

(2 + cx)T

(1 -x)(1 -X2) -

c

y=l

+c)T] (cf2)xT

(1 -x)2 + (1 -x)(1 -X2) (2+2c+c2)T 2(1-x) 1.

(2+2cx+c2x2)Tx

2(1-X)

1

(19)

(20)

Further, lim

p(u, 0) = @f$f$z, .

x=1,y=l

COMPARISON

OF THEORETICAL

AND

(21)

.

EXPERIMENTAL

DATA

Rat peritoneal macrophages were incubated with a mixture of guinea pig (type A) and sheep (type B) red cells. Material and methods have been described elsewhere (Bongrand, Capo, Benoliel & Depieds, 1973, 1975). Counts were made of phagocytes having ingested u guinea pig red cells and v sheep red cells for various u and V. The data obtained in 12 separate experiments were gathered (Table 1). In a previous report, parameters x and y were determinated (Capo et al., 1974). Thus y = 1. x z l/2, Therefore, the obtained formulae yield: P(0, u) = (nb)“Tl/v! P(l, 0) = 2 y

(T1’2-

T)

2

($L4T1/2$+T1/4) pt1, 1) = F

[(~+c)T”~-(2+2c)T] 3(-~T+gT’i2-~T’i4+~T1/6) T1j4 - (8+2~)Tl’~

+ (f$+)

+ (2 + c + ‘j’)

TLi2].

T]

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TABLE 1

Number N(u, v) of rat peritoneal cells having ingestedu guinea pig red ceils and v sheepred cells after a 90 min incubation with a mixture of red cells and speciJicantibodies Experiment

1

2

3

4

5

6

7

8

g

528

386

235

485

300

969

112

1112

50 25

64 38

77 18

64 27

21 11

40 80

16 130

5 2 10

23 26 19

25 15 12

23 5 7

10 2 13

15 6 26

8 8 14

8 2 5 5

8 1 4 2

3 2 2 0

10

11

12

Total

512

595

‘701

539

1014

40 49

24 14

38 94

27 101

57 55

518 702

4 5 71

10 3 13

13 I 59

16 8 43

15 7 48

11 8 12

170 94 333

3 2 1 4

12 2 0 25

0 1 5

3 5 8 22

0 4 4 9

2 1 2 5

145 0 28 1 36 9 102

number

N(W)

W, 0) NU, 1) NO, 2)

NO,

3)

014 1 0 2

NO,

4)

0

2

5

0

0

0

9

0

6

5

1

2

30

0

6

2

1

3

0

0

2

4

3

1

0

22

N3,O)

W, 1) W, 2)

Others

9154

nb has been chosen in order to fit optimally

P(O,2) -------=-. WA 1)

MO, 2) w

1)’

WA 3) NO, 3) p(o,)=N(o,l);

Defining a “standard deviation”

nb has been calculated so as to minimize wwA

1)-NO,

wwo>

[SJ’. {N(o, 21, N(o, l>>]”

values of:

WA 4) WO, 4) a(o,=G@iTf

function:

SD. (PI 4) = (P/4)

cwh

the experimental

. the following sum:

l>l” + cm

3Y~KJ 1) - w-4 3)/NO, 1>1” + [S-D. {N(O, 31, N(O, l)}]” + lm4 4)P(O, 1) -NO, 4)lN(O, 1>]” [=‘a {N(O, 41, N(O, l)}]” -

Which gives nb = 0.96.

202 P.

BONGRAND,

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CAPO,

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M.

BENOLIEL

ma has been chosen to fit the experimental

AND

R.

DEPIEDS

value of P(2,O)/p(l,

0). We find

ma = 0.96.

The theoretical values of P(u, V) for various u and D are given in Table 2. Using the theoretical formulae iv(u, v)/N(u’, 0’) = P(u, v)/P(u’, 0’). TABLE

2

P(u, v) is the theoretical probability that a given phagocyte will ingest u type A particles (guinea pig red cells) and v type B particles (sheep red cells) during a 90 min incubation with opsonized erythrocytes of both species Parameter P&4 0)

P(l,0) Pa 1) PQ, 0) f$ ;; PC31 0) PC% 1)

Theoretical value 0.152 0.236 0.147 0.079 0.161 o-070 OGO78

Parameter m a

WA3) pa 4)

Theoretical value (0.057) 0.023 oaO54

0.041

The theoretical and experimental values of such parameters have been compared (Table 3): when populations of phagocytes having ingested the same kinds of particles (A only, or B only, or both A and B) are compared, the agreement between theory and experiment is very good: when two arbitrary parameters are chosen (ma and bn), six independent values can be accounted for with good accuracy [P(O, 2)/P(O, 1) ; P(0, 3)/P(O, 1) ; P(O,4)/P(O, 1); P(2,O)/P( 1,O); P(l, 2)/P(l, 1) and P(2, l)/P(l, l)]. The theoretical value of P(3,O) is less satisfactory. It is interesting to notice that our formulae failed to describe our experimental results when we compared phagocytes having ingested different types of particles [for example, the theoretical value of N(1, O)/N(O, 1) is significantly different from the experimental one]. To explain these discrepancies, we may assume that there exist different functional kinds of peritoneal cells, and we may define the fractions of these cells that can phagocyte A particles only (p) or B particles only (q), or both A and B (r), or neither A nor B (t): using approximate values for ma and nb (ma = O-96, nb = 0.96) (which must be considered as a new assumption) and using previously derived formulae (Cap0 et al., 1974) to calculate the probability P(u, 0) that a

SPECIALIZATION

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TABLE 3 Comparison W, WW,

of the theoretical and experimental values of the parameters v’ 1 K = P(u, WY u’, v’)]. The standard deviation S. D. is calculated using the formula 112 v’)

Theoretical value

Experimental value

S. D.

Corrected value

0.480 0.15 0.037

0.47 o-15 0.043

0.03 0.016 0908

043 0.15 0.037

0)

0.33 0.033

0.33 0.087

0.03 0.014

o-30 0.03

1) 1)

0.35 0.26

0.38 0.30

0.075 O-065

o-35 0.26

0.68

0.18

0.02

0.18

Parameter

pa WP(O, 1) w, pa

3YWO,l> 4m0, 1)

P(2, Q/W, P(3, wv,0) Vl,

w,

ww, o/w,

1

P(l,

W(1,O)

w,

mw,

0

1.10

o-13

0.014

0.13

put

wm-4

1)

1a61

0.74

0.043

o-74

phagocyte will ingest u A particles when it is unable to ingest B particles, and the probability P(0, V) that a phagocyte will ingest 2)B particles when it is unable to ingest A particles, we find (N is the total number of cells counted, N = 9154): N(1, 1) = rNP(1, 1) N(l, 0) = rNP(l,O)+p[exp (-ma/z)-2 exp (-ma)]N N(0, 1) = rNP(0, 1) + qNnb exp (- nb). Using experimental values for N(u, u) and theoretical values for P(u, 0). we find: p = 0,089; q = 0.18; r = 0.064. The corrected values of the theoretical parameters P(u, v)/P(u’, 0’) are given in Table (3). It must be noticed that the value (q+r = O-244) of the fraction of peritoneal cells that are able to ingest sheep red cells is very near to the number obtained in previous experiments (0.23-0.25, Capo et al., 1974).

204

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BONGRAND,

C.

CAPO,

A.

M.

BENOLIEL

AND

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DEPIEDS

The description of the phagocytosis of guinea pig red cells is not as good, but more numerous assumptions were needed to obtain the theoretical formulae (Cap0 et al., 1974). As an alternative explanation of the discrepancies we have just noticed, the specialization of macrophages might be considered as acquired during the process of phagocytosis: perhaps the ingestion of an erythrocyte might prevent further engulfment of any erythrocyte of another kind. Such an hypothesis could be tested by introducing type A and type B particles at different times in the incubation medium. 3. Discussion The validity of the formulae that were derived relies upon three assumptions : (1) The occurrence of digestion of ingested particles by phagocytes has been neglected. Such an approximation is entirely valid only when incubation does not last too long. (2) Such a parameter as z may be defined only when the kinetics of phagocytosis of both A and B particles are analogous, which seems to be the case when our experimental procedure is used. (3) In a first approximation, the “avidity” of a phagocyte is multiplied by a constant parameter when the cell has ingested another particle. Previous results favour the validity of this assumption, provided the total number of ingested particles per phagocyte is not too large. It appeared that only elaborate calculations might make possible a quantitative study of phagocytosis of mixtures of different types of particles. Though the use of constant parameters x and y may be questioned, the occurrence of clearcut enough discrepancies between some theoretical and experimental data [compare P(1, l)/p(O, 1) to N(1, l)/N(O, l)] favours the existence of distinct phagocyte sub-populations among peritoneal macrophages. Thus, the specific opsonins that are needed by phagocytes to ingest particles (Jenkin & Rowley, 1961; Vaughan, 1965) and that may be cytophilic antibodies (Berken & Benacerraf, 1966) must be unable to promote phagocytosis by themselves: indeed, in our experimental conditions, every erythrocyte may be considered as covered with specific rabbit opsonins. Therefore, perhaps there exists some kind of intrinsic specificity of macrophages, which would explain the specific ingestion of foreign particles by such cells as earthworm amoebocytes (Cameron, 1932) and the binding of such particles as tobacco mosaic virus to mouse peritoneal macrophages, by the means of non-immunoglobulin membrane receptors (Loor & Roelants, 1974).

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Further experiments are intended to make use of the obtained formulae, in order to check their validity using various types of particles: the main difficulty is the discrimination of A and B particles inside a given phagocyte. Then, a study of the ingestion of self erythrocytes by peritoneal macrophages, compared to the ingestion of alien erythrocytes might be carried out. The authors are greatly indebted to Dr Julian Lewis of The Middlesex Hospital Medical School, who suggested a clearer way of deriving equations (lo)-(16). REFERENCES BERKEN, A. &BENACERRAF, B. (1966). J. exp. Med. 123,119. BONGRAND, P., CAPO, C., BENOLIEL, A. M. & DEPIEDS, R. (1973). Ann. Immune. (In~r. Pasteur). 124 C, 531. BONGRAND, P., CAPO, C., BENOLIEL, A. M. & DEPIEDS, R. (1975). Ann. Zmmun. (Inst. Pasteur). 126 C, 137. CAMERON, G. R. (1932). J. Path. 35,922. CAPO, C., BONGRAND, P., BENOLIEL, A. M. & DEPIEDS, R. (1974). J. thcor. Biol. 47,177. JENKIN, C. R. & ROWLEY, D. (1961). J. exp. Med. 114, 363. LINZ, R. & MANDELBAUM, E. (1960). Ann. Inst. Pasteur. 98, 664. LOOR, F. & ROELANTS, G. E. (1974). Eur. J. Immun. 4,649. PERKINS, E. H. & LEONARD, M. R. (1963). J. Immun. 90,228. RHODES, J. M. & LIND, I. (1968). Zmmun.. 14,511. VAUGHAN, R. B. (1965). J. exp. Pathol. 46,71.