Centro h Inoest&&n Cient@ca y Educaci6n Superior 61% l%wna& B.C., M&co; P.O. Box 4844, San Yskh, CW,%nda 92073

Received 30 September 1986; revised 5 November I987

ABSTRACT

Receptors for low density lipoprotein (LDL) and other macromolecules are known to aggregate in specialized cell surface structures called coated pits prior to being internaked by the cell when the coated pit closes off. In &is paper, the implications for receptor aggregation of factors other than diffusion which inflm the movement of receptors toward coated pits are investigated. The effect of a radial flow toward coated pits in the presence of nonrandom insertion of receptors is considered. ?he correqonding effect of a transverse flow in the plane of the cell membrane under uniform insertion is estimated. It is shown that for physioiogically pkusible fkw values, the radial flow has a negligible effect on the trapping rate of receptors in coated pits, while a transverse flow may significantly reduce the trapping time, provided the coated pit is not swept along in the flow.

The research presented in this paper is related to the general theory of trapping of diffusing particles by sinks. This theory evokd fkm the methods introduced by Smoluchowski in 1917 [22] to calculate co&&M aggregation rates, and iu its more general setting deals with t&e matbemati- c& problem of determikg the time and space density of particks diffusing among sinks and the rate at wbicb they are rr_~oved. Using ideas and concepts developed irn this vein I will formulate mathematical obtain aDsweTs to questions about the biologkal process

e choles~~o~ in

‘Tl CES 89:F27 (9,9SS)

@Elsevier Science Publishing Co., Inc., 1988 52 Vanderbilt Ave., New York, NY 100017 00255564/88/$03.50

IO & &crnvmm mii

gener ocytosis refers to the process of internalization of S anCtX% h receptor-mediated endoqtosis, large protein mole&es called receptors that project from t&e cell membrane bind selectively to other m&c&s (for example, LDL and \ther nutrient- transporting pro-, insulk an% -other hormones) in the ceJYs medium.

ese external molecules are t&z% ligan%s. The bound ligand-receptor complexes aggregate iu coate% pits, specialize% structures on the cell surface ‘_ which appear in ekctron micrographs as fuzzy areas, because they are coated ok the cytoplasmic side by a lattice structure composed of a protein called da (Pearse [17]).

ts close an% form vesicles that bring the Sgand-receptor interior of the cell. 3In some experiiental systems, kYu%&g the receptors are returned to the cell surface for fuzther

endocytic cycles, an% the ligan%s are degrade% at the lysozomes. Evea though receptor-me%iate% en%ocytosis has beer, the subject of

intense expefimentaI research an% modeling, there are still conflicting hy- potheses an% even conflictiug data regarding the sites of receptor insertion, the way receptors are transporte% to coate% pits, an% &e nature of the intraceIIular vesicles forme% from coated pits. Mathematical models of the process give predictions that will depend on the kin% of hypotheses used. In particular, the different hypotheses lead to different predictions about the rate at which receptors aggregate in co&A pi.&+.

context of the mo%& to be ev&ated I wiII caMate the rate at traps (coated pits) on a two dimen- e denote% by EC, an% is also known

ard rate constant. It is connected to the mean a trap (mean capture time) through the relation

[ll] an% Goldstein, Griego, an% W~fsy $?], in 1977 [4], calculated k, coated pits is mai3_&3ke%

C~Iv\~rnW !T_mv EFErn 11

the model is 1.9X lo- lo d/s, the answer to the question is that the hypothesk of random insertion and pure diffusion of LDL receptoas to coated pits is consistent with experimental observations, but just barely.

Although the diffusion4imited forward rate constant calculated by these authors is consistent with availab!e experimental results on the rate of LDL internabzation, the experiments do not rule out the possibility that the true rate of accumulation +f ieaXptO,rS in coated pits is actually faster than predicted on the basis of the random insertion of receptors followed by pure diffusion to coated pits.

In this paper other mecha&ms that have been proposed by experimental cell biologists and that might speed the accumulation of receptors in coa*& pits are considered. The motivation for the inclusion of other relevant factors isbased~pparton~~thaO~tof3~notreinse uniformly into the plasma membrane but rather are inserted in small regions where coated pits form (Robeneck and Hest [2OD, and in part on the view of other scientists (e.g. Bretscher [5l) that the fast observed rate of internahzation is not determined by diffusion alone but is also influenced by a cell membrane flow.

A model will be considered for receptor aggregation in coated pits in the presence of a radially convective flow toward coated pits, together with nonuuiform insertion of LDL receptors in plaques where coated pits form (Robeneck and Hesz [2OD. An estimation of the effect of a transversely convective flow in the plane of the cell membrane, with a uniform reinser- tion of LDL receptors, will be provided.

In the models to be disc- the simpbfying assumption that the coated pit has an infinite lifetime will be made. Goldstein et al. [ll, 121 dealt in their models with two conflicting hypotheses about the coated pit recycling process. In either case the coated pit effectively has a finite lifetime. However, they found that the results (Berg and Purcell [4] and Adam and Delbrbck [ZJ) obtained under the assumption that sinks have infinite life- tunes -give good approximations for k, and . in the analyzd: receptors for low density lipoprotein 3x1 hum This also be true for more rapidly diffusk;, p”’ 9 surface receptors are in this category.

THE GEOMETRY OF THE MODELS

H. XNAVAii HERM

Qf the cd surface. e outer radius

B = I/srb2, (2)

frm a given trap tion amounts to

the reference mean time for a particle at

with a particular absorber to

coated pit) lies at the center of a circuhr d pit its share of the cell zitiae= aree i.e.,

1t eQmtration tars are free to diffuse in within m trap radii no = b/n].

b4 h( b/Q) 3b2 - a2 452~(b2-af) - 8

Far LDL proxirnZiti finite lifetime (G&be& Wofsy [IZD. One method lsqation (3) starts with

tolf czoncentration a distance r

tioa ie., the average receptor comma-

THE DEL FOR THE RA DM L FL0 W

r,B):agr<b, -

14 E. ECHAVARRjA HERAS

y cn~c~med with the cahlation of the mean capture s moving in the diffusion-convection environment de-

e the results assuming that the insertion of 1~. the restricted region u G r Q l IUI, in an attempt

g effects of the flow and the possible insertion of recqtors in plaques about co%ed pits. Letting r92 = b/a slllows us to “look at the effect of the flow alone, if receptor insertion is uniform over the whole

a. e velocity v of the membrane flow into the trap will be assumed to have

the form

v P z--n r

for 4 d r 6 b, n being k unit vector pointig out radially toward (r, t?) and 1% a r~&;i;l,T~c 2:sber that will be called the flow rate constant. The negative ~rgn h Equation (7) is neerfed because +Ae fkw is ,toward the trap and n

prints out. The dependence of v on r is necessary if we want the amount of membrane per unit area to remain constant.

The general steady state equation for the concentration C of particles moving under the influence of diffusion and convection is

-div(DgC)+div(CV) =S, 18)

where S denote rate of particle production in units of particles/cm2 s, andL)istbedi n coefficient of the particle under study, which will be assumed to be a positive constant. D has units of c&/s.

The boundary condition at r- u will be taken as absorbing, that is, = 0; and to generalize reflecting boundary condition at r = b, the flux across the bound r = b will be set \D zero; +tis last boundary

is expressed in terms of the flux vector J(r, 5 j by means of

(b,O)-n&=0, (9

boundary of H. The magnitude per unit time per unit

boundary; its units are particles/ems. The flux vector

e steady state concentration utions to the flux due

CONVECTIVE FLOW l2FFEm 15

Since by radial symmetq we capl assume that the concentration C depeuds on@ ok r, then using Equatious (7) and (lo), we find that the boundary condition (9) is equivaleut to

ac D-z I _b+ b-1

CcFfb) =o.

Finally, the insertion mode S(r) will be assumed to have the form

where S is a positive constant, and 1 < m < b/a. Not.& that the assumption that S(r) has the form given by Equation (12)

makes it necessary to add continuity conditions ou C: i.e., C and aC/ar are assumed to be continuous at rruz.

Hence the steady state concentration density C(r) of particles diffusing on the annulus Q under the additional inauence of a radial flow into the trap at t = a with a uniform insertion of particles re&cted to the region a < r < ma G 6, and with the total flux of particles across the boundary at r = b set at 0, can be modeled as follows:

Dv2C+f$$+S(r) =0,

C(,>andg continuousat r=m,

C(a) -0,

X. Dar r-b I +fC(b) =0,

where IL is the flow rate constant.

THE MODEL FOR THE TRANSVERSE FL0 W

To esthate the effect on the diffusion-limited forward rate constant k, of a convective

iI. ECHAVAIUUA HERAS

ely in the general situation. But we may by noticing that the concentration will , we may set Cjx, y) = C(x, - y). This the boundary condition that requires

= b2 to be zero, and the additional con- sistency condition that the couetitration is expected to be radially symmet- ric w&n the strength of the flow becomes zero, will permit the construction of a representation for C(x, y) using the method of separation of variables, which will possess the desired properties in the case of interest.

Assuming that the convective vector field has the form v = vi with i the unit vector pointing in the direction of the positive x-axis, Quatior! (8) will give the corresponding p.d.e. that C(x, y) satisfies in this case. Denoting by C( r, 8) the polar coordinate representation for C(x, y), we can see that under the substitution (Goldstein and Zimm [lOn

C( r, 6) = g( A, r, id j eirm9 (16)

the mtilel for the steady state concentration of particles C(r, 6) becomes equivalently

v’g-A2g= - D

Strp @I e-XraO ,

D is a positive number called the fundamental ratio of the is a differentiable function in n representing the rate of

pafticle insertion with units of particles/cm2 s, u is the flow rate constant in -dimensional diffusion coefficient in c&/s.

ows from the boundaq condition given by gh the botmdaq r - ,k to zero, _nnd in this case satisfies

P-

toward (r, fl). Notice also tingcase u+Ofor

CON/Em FLOW EFFECX 17

RQDIAL FLOW

On solving Equation (13) for C(r) subject to the boundary conditions (l4), (15), the diffusion-co~ection~tited forward rate constant knrp+, depending on &form particle insertion over the regiou 4 < r Q PW < 6, can be obtained by means of Equation (5). Nevertheks, k,,, could aIso be defined by

k

ZW~‘IS( r) dr

lap+ = w -’ (22)

where (C) is giveu by Equation (6), since the number of particles trapped is the same as the number inserted into the annuhts in a unit time. Iknce,

using km,+ as given by Equation (22) in Equation (l), the model yields the mean time a(~, m) for the capture of receptors by coated pits, given by

T(w+$&j (m2 -1ja2 gD

x (p+2Djm2-p-2Dm’/D+2

2PdPf2Dj (23)

~(p, m) as given by Equation (23) has finite limits, for all D 2 0, at every point of the region ~20, l<m<b/a, and in the limit as p-+0 and m -, b/a, a(p, .m) approaches T, the mean capture time cakulatcd by Berg and Purcell [4], and given by Equation (3). It also satisfies a~@, m)/ap < 0 and a%(~, rn)/@’ 2 0 @chavania PIeras [81).

TRANSVERSE FL0 W

Using the boundary conditions (18), (19), and (20) it is possible to obtain, using separation of variabks, a solution g&r,@) to Equation (17) that depends only on r in the limiting case u 4 0. Furthermore g( A, r, 0) can be written in the form (Echavarria Hems [8n

00 (24)

where g:(r) satisfies for K = @,l,. . . and A E lem

for u E C2([ Q, b]) satisfying boundmy conditions

u(a) = 0,

U’(b)+k(X)U(b) =0,

where %(A) is the fwunction defined by

YK( A) = - AI,(Xb)/I,(Xb), X # 0,

-K/b x=0, (29

with I, ( -) being the modified Bessel function of the first class and of order K for K = O,l,... and A E R. The boundary condition involving the function k(X) given by Equation (29) was motivated when eigenfunctions for the operator A, = -v2 having the form $x(r,8) = Rx(r)@(B) and satisfying the boundary conditions (ItI), (Is), and (20), were substituted in Equation (20) and the requirement of radial symmetry in the limiting case u + 0 was imposed. Notice also that the operator E”,(U) given by E!quation (26) originated also from the separation of variables (Echavarria Heras [$I).

For the operator L”,(u) defined by Equations (26)-(29) a Green’s func- tion Ki(t, t) given by

exists, when: K,sI .) stands for the modified ssel function of the second d of order K. Using formuh (9.653) iI1 Abramowitx and Stegun [I], for K,(z) in terms

of I,(z) we r:an obtain an expression for Kg(r: t) depending only on I,(z). For this Bessel function we kc

e proof of the existence of the solution &A, r, 8) in which depends continuously on the

).

CONVECTIVE FLOW EFFXTS 19

In this czise, kb+>,, t* “- diffusion-convectiori-titid forward rate constant in the presence **Jr a transverse flow and insertion m e 3(~,6), can be obtained from Equation (22) using C(r, 4) as given by Equations (16), (24). When 3(r, 8) is uniform all over Q, the corresponding mean capture time q can be obtained using k,, in Equation (1). It becomes

f /3J -l)a+l jbr( jb*K;(r,t)I,(ht) &+Jhr) dr K = 0 ry_ _ --p __-__.!!_ _,., _E__.__ ._11_1._ _ ._.. __~.__.l._.

9 (31) ‘A D(b2 - a2)

where

Q as given by Equation (31) has the property coefficient value D (Echavarria Heras [8n,

that for a fixed diffusion

That is, when the strength of the flow becomes zero, we recover the result calculated under the assumption of uniform insertion of LDL receptors followed by pure diffusion to coated pits, given by Equation (3).

4. APPLICATIONS TO RECEPT’OR~MEDL4TED ENDOCYTOSIS OF LOW DENSITY LIpOPIKYiEINS ON FIBROBLASTS

THE EFFECT OF THE RADIAL FL0 W

From Equation (23), setting m = b/a, we can obtain the dependence of thk mean capture time for the radial flow model on cr. In this case the im Mien of particles takes place uniformly all over the anntdus 3. Using the va! ues a = low5 cm, 6 =10e4 cm, and D = 4.5XlO-11 &/s, which are thg I values of the radius of a coated pit, the radius of the reference circle, and

ates when the portions of

VARRiA HERAS

Ho. 2. The dependence of T(C, b/a) on jk. The plot gives the predicted effect of a radial flow of rate constant c OA the mean capture time for LDL receptors in coatd pits of human fibroblastic ceBs, when the insertion mode is uniform over the whde annuhs H.

rate constant cr. For uniform reittsertion we must take m = b/o in e parameter values cftaracteristic of human fihbtasts at 37°C are: The

t density P = 0.31/j&, so that b ==l.Q pm; the coated pit radius u = 0.10 pm; tar difhsion coefficient D = 4.5 x 10’ t1 cnS/s, For a disatlrssjo~ of the

[al). For these parameter values and in the case considered, order to produce a substantial effect OA the mean capture

time T cakhted bs@cd OA diffusion and uniform insertion and (3); that ’ , to ha&e T, a radiiad flow of flow rate constant p,/, = 2 x 10” red. T&

wo~a hav@z a 5 ap~~o~ate~y 2m times bigger than the OAR that is physiohgi-

FE. 3. Predicted effect of plaque size on the mean capture time 7. In this LwqtOrs move under the influefw of diffusion and a Eced cerrr?er of the coated piQ, having rate amstant y. =lO’” d/mh. Receptors move anywhere on the 41 surface, but recycle back to the surface only near i.e., between the coated pit radhs 0 ad the piaqw radius nw. radhs g&n in muItipIes m

that pit radii.

2.5

2.0

1.5

1.0

0.5

0.0

I-I. ECHAVARRh HERAS

of the coated pit at a constant must transport into the region

mwmt of membrane equal to the surface mea

s rate constant co satisfying

would require a flow strength es the one that is physiologically plausible

3.0

2.a

2.6

2.4

2.2

2.3

1.8

1.6

1.4

1.2

1.

FIG. 4.

! 2

e effect of a trmsvem fbvw m the mean capture :he 7. In this model LDL mce of diffusiun and 8 iaansverw fkw. T3e reitzserticm nf

The ;r-axis is the transverse flow velocity (31) for the parameter values fibrdhst... Halving the ttx.ztt

by nation (3) mpires a

CCNVEC~ FLCW j2iwxTs 23

For a radial flow rate constant p, a n insertion function of the form given by tion (12) would radius of 3.4 coated pit radii in order to reduce ~(p,rn) to T/L In Figure 3 we have provi &pendence of the mean capture time on plaque radius PWZ, for the fixed radiai flow that was proposed.

THE EFFECT OF A TRANS?fERSE FLOW

Let’s consider a transverse flow of rate v, =10V6 cm/nun, which is equivalent to the maximum velocity for a radial flow having rate constant CO- Tkii, fOi "&65 parameter dues pertinent ti tie LDL eqptxknmti system on human fibroblastic cells, according to the transverse fl to halve the mean capture time T bxd on pure insertion, we will require a flow of velocity v = 250~~. That is, t!x-transverx flow must be approxktately 250 times greater than the maximum velocity of the proposed radial flow.

Bretscher [S] suggests the existence of a bulk flow of particular membrane components across the cell’s surface. He dkusses riments with motile fibroblasts that suggest partsofthecellm , after internalization through coated pits, are returned to the leading edge of b-ulk flow of lipids and receptors away from the front o contributes to the transport of large molecular aggregates on the cell s from the front to the rear of the cell, generating a process known as cappmg.

According to Bretschcr [S], the rate at which objects on the surface of fibroblasts are swept ba is approximately 1 pm/min, a flow es stronger than the radial flow having rate constant po.

Using this value as an estimate for the intensity of the transverse r’ow, and using the parameter values pertinent to the LDE experimental system on human fibroblastic cells (Goldstein et al. pi& the transverse I?ow modei yields q, = 0.607, where r is the mean capture time associa without flow. Figure 4 shows the dependence of the velocity of the trausve_rse f!ow+

CONCLUSIONS

24 H. ECHAYARRIA HERAS

any trap problem is repkced by a single esents the coated pit) particIes motig in an hich ass&us to the co pit its share of *dte cell ek the inucr boundary at r= a was taken as

absorbing and the tot of particles across the outer boundary at r = 6 was set to zero. condition takes into account the periodicity

system of traps that was consi . This condition does not about the particular sites of ary where membrane is is simply a device for approximating a many trap problem by a

a steady state concentration of receptors y a remsertion process which replaced in the anrruhu

Q 4 r & b the particles lost to the trap, that is, in general it was assumed that ernbrane is inse tmiformly over the cell surface except possibly in the

coated pits. Preferential reinsertion was coEsidered in where membrane can be inserted in annular regions

on of membrane is characterized by Equation msertion over the whole surface outside of coated

order to estimate the effect of a radial flow in the aggregation rate of receptors; in coated pits, it was assumed that membrane is lost to the

pits as they rrru~d up aEd bud off from the surface. The assumed rm insertiorr of membrane IoUowed by local removal sets up a mem-

velocity depends on the kinetics of coated ‘orruation and m EM be constant in time, but these kirretics

and debate. The radial flow velocity form s if the i&x&j is constant in time. This is most

allow us to get a~ idea of the

ue of symmetry, the outer botmdary terization of the flux vector at any

e convective vector field.

s IS an extensioE to

@ONvE- FLOW EFFECTS 25

second trap. The boundary cxxxliti0n given by Equation (20) is an average 0hara~.tetkati0n Of the desired boundary condition at r = b. This average boundary condition, together with the symmetry conditions (Iga) and (I%), permitted us to obtain the solution rA given by Equation (13). In the txnsverse flow model the deviations of the flow lines near coated pits were

neglected The c0ated pit is permeable to me rane, so straight line fl0w in the vicinity of the coated pit may b0 small. Ln view 0f this, the transverse flow model 0verestimated the aggregation rate of LDL receptors iu coated pits. The same conclusion can be obtained for the radial flow model, where the rate at which the flow transported receptors to coated pits was uxtstant in time.

The model presen&kd @nvective radial flow and preferential reinsertion) predicts that a radial flow which has the physiologically plausible rate constant j&o =lO -rr cnS/min, acting on uniformly inserted LDL receptors in human fibroblasts (Goldstein et al. [llD, will have a negligible effect on the mean capture time T calculated under the assumption of unif0rm insertion followed by pure eu-- ~Wusi0n of the receptors to coated pits. Further- more, under uniform insertion, t0 halve T wiU requke a flow two hundred times as strong as the one physiologically expected. As a concltion, we can not invoke the radial flow as a possible mechanism to reduce the aver time it takes newly inserted LDL receptors to reach coated pits. Even in case when the flow acts together with preferential insertion in plaques surrounding coated pits (Robeneck and Hesz [2OJ), halving T would rqti a plaque radius of 3.4 coated pit radii; since coated pits cover approximately 1% of the cell surfa= at 37OC, plaques would CCXi &bout 10% Of the 04 surface. Thus even in the presence Of the radial flow, receptor i.r~M0n should be limited to a very small region of the plasma membrane near coated pit sites, to effect a substantial decrease in the average time it takes LDL receptors to diffuse to a coated pit. This plaque radius is very cbse to

diffusion to coated pi

26 H. ECHAVARRiA HERAS

tit& it. h t&tb GS~, we w~dd need CO consider a transverse flow with a eqaSl to e v&&y of the membrane relative to the coated pit.

uch a rekive flow woutd have a much smaller velocity than the proposed transverse flow v&&y9 and consequently *the effect on the trapping time would be smdkr, In fact, no evidence has been reported one way or another on what happens so a coated pit in a membrane flow. We conclude that typical transverse flow vekcities may significantly reduce: the trapping time of receptors by coated pits providing the coated pit is not swept along in the flow.

?ltis paper contains results from my Ph.D. dissertation presenied at the Department of Mathematics and Statistics of the University of New Mexico. I want to express a great debt to my thesis director, Dr. Carla Wofv, and to Dr. Byron Go!&ein from L.os Aiamos National Laborato.ry for their advice and encouragement. Two anonymow reviewers provided valuabk crientation and criticisrm. 77teir comments are greatly appreciated.

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CON-VECTIVE FLOW EFFECTS 27

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