Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol162.Issue1-2.Nov1999:

Optimal control of receptor reinsertion in the

Low Density Lipoprotein endocytic cycle

Hector Echavarr

õa-Heras

*

Department of Ecology, CICESE Research Center Ensenada B.C. Mexico, P.O. Box 434844, San Diego, CA 92143-4844, USA

Received 3 November 1998; received in revised form 9 August 1999; accepted 19 August 1999

Abstract

On the basis of this study, it is concluded that within physiological limits the min-imun value for the mean capture time of LDL receptors by coated pits must be induced fundamentally by an optimal characterization of their insertion rate function. The corresponding steady-state surface aggregation patterns for the unbound receptors are consistent with experimental observations. The implications of the derived results for the estimation of the minimum physiological value for the referred mean capture time are also discussed. Ó _{1999 Published by Elsevier Science Inc. All rights reserved.}

AMS classi®cation:92C40; 92C05; 92C50

Keywords:Optimal control; Endocytic cycle surface receptor patterns

1. Introduction

This research pertains to theoretical aspects of Receptor Mediated Endo-cytosis (RME). Using this process large biologically active ligands bind to specialized membrane receptors and are removed from the plasma membrane. These ligand±receptor complexes aggregate in cell surface structures called coated pits which invaginate to transport the bound molecules to the sites www.elsevier.com/locate/mathbio

*

Fax: +1-617 50 545, +1-617 45 154.

E-mail address:hechavar@cicese.mx (H. EchavarrõÂa-Heras)

where they are processed. An example is the internalization cycle for Low Density Lipoproteins (LDL) in human ®broblast.

The LDL macromolecule carries about two-thirds of the cholesterol in human plasma. Following internalization, LDL is carried to lysosomes and degraded, freeing the cholesterol subunits [1]. It is considered that the LDL receptors recycle to the cell surface [2]. Cells need cholesterol because it is an essential membrane component, but cholesterol molecules also present a haz-ard since they are insoluble and can build up in arteries, provoking coronary disease, embolism and heart attacks [3,1].

In the study of the LDL endocytic cycle, one important aspect pertains to the characterization of the receptor insertion mode. Goldstein et al. [2] con-sidered uniform insertion all over the cell membrane. Contrasting these views, Robeneck and Hezs claimed [4] that insertion was restricted to regions where new coated pits will form. This paradigm followed experimental observations of receptor clusters on the cell surface which were called plaques.

Wofsy et al. [5] modelled preferential insertion considering that receptors were replaced uniformly within annular regions surrounding coated pits, and called these regions plaques. They concluded that insertion in plaques dra-matically reduces the mean capture time determined by di€usion and uniform insertion for the LDL receptors. Nevertheless the corresponding surface ag-gregation pattern was not explored. Echavarrõa-Heras and Solana [6] consid-ered the e€ect of di€usion and a general radially symmetric insertion mode. They demonstrated that insertion in plaques could not induce the observed surface aggregation patterns [4] and found that a continuous and decreasing insertion mode could be a more ecient mechanism to reduce the mean cap-ture time of LDL receptors by coated pits.

traps, induce a weaker di€usion process for the receptors [8]. Furthermore, in that case, if receptor insertion is optimal in the sense of producing the mini-mum physiological value for the mean capture time for the LDL receptors by coated pits, then the associated steady-state surface aggregation pattern will be necessarily in the form suggested in [4]. The conclusions obtained here followed the study of surface aggregation patterns generated by means of computer graphics techniques [9±11].

In Section 2, I review the theoretical methods. Applications of control theory to the receptor mediated endocytic cycle addressed here are presented in Section 3. The last section discusses the physiological implications of the ®ndings of this study.

2. Theoretical background

The rate at which di€using particles (receptors) hit traps (coated pits) on a two-dimensional surface (the cell) is known as the forward rate constant [12] and will be denoted byk‡. This rate can be found as the ¯ux of particles into a

trap divided by the mean particle concentration [13]. In the two-dimensional case for a circular sink of radiusa,k‡ is de®ned by means of the equation

k‡ ˆ

2paD C h i

o_C

o_r

rˆa

; …2:1†

whereC…r†is the steady-state radial distribution function of receptors unbound to coated pits, D>0, their di€usion coecient, and h iC , the receptor con-centration averaged over all the di€usion space [12]. The constantk‡ times the

number of traps per unit area gives the probability per unit time that a di€using particle hits the trap.

The mean time s for a particle to hit a trap (mean capture time) can be obtained [12] from the relation

sˆ 1 k‡P

; …2:2†

where P is the number of coated pits per unit area distributed on the cell surface.

Pˆ 1

pb2; …

2:3†

wherePis the number of absorbers per unit area distributed on the cell surface [12]. The receptors will be assumed to start their movement at random locations in the ring surrounding the trap. Evidence of internalization and reinsertion of LDL receptors would support the claim that a steady-state cell surface con-centration of receptors is maintained. The basis for this is the apparently un-detectable pool of receptors inside the cell during the endocytic process [16].The setting up of a steady state will require that the number of particles inserted equates the number lost to the trap. In the case where convective transport is not invoked [15] this amounts to considering an absorbing boundary condition at

rˆaand a re¯ecting boundary condition atrˆb. If radial convection is as-sumed, a ¯ux vanishing boundary condition [7] atrˆbwill be required.

Goldstein et al. [8,12,17], concluded that if the traps are sparsely distributed over the entire surface of the cell, even in the case of transient behavior the results of Adam and Delbruck [18] and Berg and Purcell [15] obtained under the assumption that sinks have in®nite lifetimes, give good approximations for the dynamics of the LDL experimental system in human ®broblastic cells. This will be also true for more rapidly di€using receptors.

The steady-state concentration densityCs…r†, of particles at a distancerfrom

the center of the trap, di€using on the annulusX ˆ …f r;h†a6r6bgunder the in¯uence of a radial ¯ow directed toward the center of the sink and inserted by a radially symmetric rate functionS…r†P0, can be modelled [7,8,11] by means of the equation

Dr2

Cs‡

l

r

o_Cs

o_r ‡S r… † ˆ0 …2:4†

with an absorbing boundary condition

Cs…a† ˆ0; …2:5†

and an outer boundary condition

DoCs

o_r

rˆb ‡l

bCs… † ˆb 0; …2:6†

following from a ¯ux vanishing requirement [7] atrˆb.

During the invagination process the radial ¯ow must transport into the coated pit an amount of membrane equal to its area. Then an invagination time of 5 min and a coated pit of radius 0:10lm determine a reference value of

l0ˆ1:610ÿ13cm2=s for the ¯ow rate constant. Willingham and Pastan [19]

claimed that at 37°_{C the coated pit lifetime could be as low as 14 s. For that} invagination time the maximum possible value for the coated pit radius produces an estimatel₁of 810ÿ12_cm2_{=s. This gives a ¯ow rate constantlof}

It is considered that the di€usion coecient of an unbound LDL receptor at 37°_{C has a reference value of D0ˆ4:510}ÿ11 _cm2_{=s [20]. Nevertheless, the} experiments do not rule out the possibility that it could be smaller thanD0.

This could be expected to happen within a coated pit. In fact, Goldstein et al. [8] estimated that if the coated pit lifetime isk, the di€usion coecient in a coated pit must be bounded above by Dcˆa2k=4 before radial convection

could e€ectively keep the receptors trapped. Usingkˆ0:14 s, and the smallest experimentally determined value for the coated pit radius one gets for Dc a

value as low asDcmˆ4:4610ÿ13cm2=s.

The expected steady-state surface aggregation pattern for the unbound LDL receptors induced by a the triplet…l;D;S…r††can be obtained by rotating the plot ofCs… †r (cf. Eq. (3.5)) about a vertical axis and then projecting by means

of the computer graphics technique of ray tracing [9±11] the tones of grey associated with the values ofCs… †r . (See Fig. 2.)

3. The control problem for the receptor mediated endocytic cycle

3.1. The objective functional for the mean capture times

ForS…r†P0 de®ne the functionhs… †z to be

hs… † ˆz

Z b

z

xS x… †dx: …3:1†

Using the polar coordinate form of the Laplacian it follows from Eq. (2.4) that

Cs…r†satis®es the di€erential equation

o_Cs

o_r ˆ ÿ

lCs…r†

Dr ‡

hs… †a Dr ÿ

hs… †a

Dr us… †r ; …3:2†

The control function us… †r will be the proportion of particles inserted in the

annulusa6z6r. Hence,

us… † ˆr

Rr

azS…z†dz

hs… †a

: …3:3†

Necessarily us…a† ˆ0;us…b† ˆ1. Furthermore, I will assume that the control

functionus… †r satisfy the admissibility constrain

us… † 2r U …3:4†

beingUa set to be estimated.

It is easy to show [11] that the responseCs…r†becomes

Cs… † ˆr rÿl=D

D

Z r

a

From the above equation we obtain

Hence the combination of equations (2.1)±(2.3), (3.1), (3.2) and (3.6) give for the corresponding mean capture timess,

ssˆ

Thenss will be minimal whenever the integral

J…us…r†† ˆ

Z b

a

a…r†us…r†dr …3:9†

obtains its maximum value. This de®nes the objective functional. The control problem will be to characterize an optimal formu_{† 2r U which produces the}

minimum physiologically expected values _{for the mean capture time of the}

receptors by coated pits. This will be equivalent to maximize the functional (3.9) over the setU.

Lets consider now two possible forms usp…r† and usq…r† for the control function generated by insertion rate functions Sp…r† and Sq…r†, respectively. Assume that the corresponding values of the functional (3.7), are respectively, s_p ands_q, and thats_p coincides withs_q. Then Eqs. (3.7) and (3.8) imply

J…usq…r†† ˆJ…usp…r††: …3:10†

On the other hand, integration by parts gives

J…us…r†† ˆ/…r† ÿ

Z b

3.2. The admissibility set U for the control functionus… †r

From Eq. (3.3) it is easy to conclude that whenever the productrSp…r†

de-creases the plot ofusp…r†will be concave down. Correspondingly for Sq…r† in-creasing,usq…r† will have a concave up plot. Since either atrˆa or atrˆb,

usp…r†andusp…r†coincide, necessarilyJ…usp…r††>J…usq…r††andsp<sq. Hence in the general case there must be functionsumin…r†andumax…r†such that the setU

of statement (3.4) can be de®ned by the inequality

umin… †r 6us… †r 6umax… †r ; …3:14†

whereus…r†is de®ned by an insertion rate functionS…r†for which the product rS…r† decreases monotonically in …a;b†. Obviously an increasing S…r† or the stepwise insertion mode considered in [5] are not included inU.

Forb2R, lets consider theb-family of functions,

Sb…r† ˆkr

ÿb 

3:15†

withkP0 a constant. The associated family of controlsus_b…r†becomes

us_b…r† ˆ

Since the derivative of the product rS1… †r vanishes, from Eq. (3.3) we have d2us_b

de®nes a mean capture times_b of

sb ˆ

The behavior ofsbdepending onbfor the parameter values associated with

Consider the casebP1 and suppose that a given decreasing functionS…r† along with S_b…r† satisfy the criteria of Eq. (3.13). Hence, either one of these insertion rate functions determine the same value for functional (3.7). Invoking the generalized form of the mean value theorem for integrals (e.g. [21, p. 128]), there exist a numberrs in …a;b† associated withS…r†such that the criteria of

Eq. (3.13) can be written equivalently

/…rs† ˆf…b†; …3:18†

where forl6ˆ2D,

f…b† ˆ 1 lÿ2D

2ÿb

4ÿb

_b2_ÿa2_a=b†2ÿb_†

2 1ÿ… …a=b†2ÿb† ‡

Dbÿl=D‡2_al=D

l ÿ

a2

2

ÿ b

l=D_ÿal=D_a=b†2ÿb_†

… †Dbÿl=D‡2_2ÿb†

1ÿ…a=b†2ÿb

… †l…l=D‡2ÿb†

0

B B @

1

C C A

…3:19†

while forlˆ2D

f…b† ˆf1…b† ‡f2…b† ‡f3…b† …3:20†

with

Fig. 1. The variation ofsbas given by Eq. (3.17) for the parameter values associated with the LDL system in human ®broblastic cells, aˆ10ÿ5 _{cm, bˆ10}ÿ4 _{cm, l}

0ˆ1:610ÿ13 cm2=s, D0ˆ 4:510ÿ11 _cm2_{=s [7,12]. We notice thats}

f1…b† ˆ ofbfor which Eq. (3.18) is satis®ed. As a conclusion, given any insertion rate functionS…r†producing a mean capture timesswe can ®nd a memberSb…r†of

theb-family (3.15) producingsb in such a way thatssandsb agree to highest

order. Furthermore, ifCs…r†andCs_b…r†are given by Eq. (3.5) forS…r†andSb…r†,

Hence forl®xed, wheneverDbecomes suciently small the plots ofCs…r†and Cs_b…r†will be alike and the surface patterns induced by either one will be

in-distinguishable in an annulusa6r62a. Consequently we can use the subset of the b-family generated by bP1 and the criteria of Eq. (3.18) to study the surface aggregation patterns of unbound LDL receptors induced by an arbi-trary decreasing functionS r… †. Fig. 2, displays examples of these patterns.

The criteria of Eq. (3.18) can provide estimations forumin…r†andumax…r†. In

fact, experimental results on the LDL system [12] lead to the inequality

k‡P2:310ÿ101:610ÿ10cm2=s; …3:22†

Using Eqs. (2.2) and (2.3) and the maximum possible value of the lower bound fork‡ one gets an upper bound forss of 1:2678 min. This value is determined

by S3:0375…r†. Hence whenever us…r†Pu3:0375…r†, then ss6s3:0375 and the experi-mentally determined lower bound for k‡ provides an empirical criteria to

chooseu3:0375…r†as a reasonable lower bound for the admissibility setU. In order to obtain an estimate for umax…r†, notice that Eqs. (3.7) and (3.9)

corresponding mean capture time. Since the kernel in integral (3.9) de®ning

J us_b… †r

lim

b!1J…usb…r†† ˆ

Z b

a

a…r†dr: …3:24†

Then for u1…r† Eq. (3.23) assigns a vanishing value for s1. Consequently,

inequality (3.14) could be formally expressed in the form

u3:0375…r†6us…r†6u1…r†. Nevertheless, experimental results indicate that the

associated mean capture time for LDL receptors is positive. Necessarily, there must exist a di€erent upper bound umax…r† which produces the minimum

physiologically plausible values _{for the mean capture times}

s. Obviously, to

ful®ll the requirement of a positives _{we must haveu}

max…r†<u1…r†.

From Eq. (3.5) it is easy to conclude that the maximum value that C…r† attains is proportional to the total number of particles inserted per unit time. Due to the steady-state assumption the greater this amount becomes, the smaller the associated mean capture time will be. Consequently, whenever a plaque is formed we expect that the smaller its outer radius, the smaller the corresponding mean capture time. Experiments with LDL-ferritin have shown that in the steady state the ratio of receptors bound in coated pits to those in Fig. 2. Examples of the surface aggregation patterns induced by the triplet (l;D;S_b… †). In the plotsr

other sites of the plasma membrane is 2.2 [12]. If receptors are inserted by the optimal reinsertion mode, S…r† which generates s_{, we expect the}

corre-sponding plaque to have the minimum possible outer radius. Since the maxi-mum density of receptors on the cell surface occurs in coated pits it is reasonable to suppose that the optimal plaque could have at most the same density of receptors. A simple calculation shows that the outer radius of that plaque cannot be smaller that 1:21a. Considering the experimental error as-sociated with the measurements ofa, the maximum possible value for the outer radius of the optimal plaque could be 1:82a. Most of the reported plaques had a radius between one and two times the average coated pit radius.

Bretscher [22] states that the transit time for an LDL receptor from its binding in a coated pit to its reappearance on the plasma membrane is less than 15 s. By virtue of the steady-state assumption for the number of unbound receptors transit times would imply the same values for mean capture times. Hence we could expect s _{to attain a smaller value than 15 s. For lˆl}

1,

DˆDcm and S15…r†, the annulusa6r61:21a contains most of the unbound

receptors and practically, the total number of receptors will be located in the annulusa6r61:82a, (see Fig. 3) while the corresponding mean capture time will have a value of 1.30 s. Since the plaques are uniquely determined by the triplet (l,D;S…r††then the criteria of Eq. (3.18) and inequality (3.21) indicate

Fig. 3. The assumption that the density of receptors within a surface plaque is bounded above by their density in coated pits, implies that the replaced receptor must be distributed in the annulus

a6r61:21a. The ratiol1=Dcm(see text for details) andS15… †r induce a form forCs… †r where the receptors will be practically depleted forrP1:21a. (see (a)), In (b) the innermost circle atrˆa

that the plaque associated with (l1;Dcm;S15…r††gives a higher order estimation for the optimal surface aggregation pattern.

In summary, the admissibility setUof statement (3.4) could be de®ned as the set of functionsus…r†which are produced by Eq. (3.3), constrained by

in-equality (3.14) and have a concave down plot. Experimental results have been invoked to show that umin…r† can be approximated by u3:0375…r†. The ratio

l₁=Dcm permits an estimate ofumax…r†by means of u15…r†. In general, given a value of the ratio l=D every function us… †r which belongs to Ucan be

rea-sonably approximated by a memberus_b…r†of theb-family (3.16) for whichbis

given implicitly by Eq. (3.18).

3.3. The maximum principle formalization

The intuitively obtained result that ss must be minimized by a decreasing

insertion rate function which producesumax… †r through Eq. (3.3) can be

for-mally obtained by means of the maximum principle. To this aim we consider the optimization problem

s_ˆmax us2U

J…us…r††

f g; …3:25†

whereJ…us…r††is given by Eq. (3.9). The Hamiltonian becomes

H_ˆk…r† h…a† ÿlCs…r†

rD

‡r…r†us…r†; …3:26†

where

r…r† ˆ a…r†

ÿk…r†h…a† rD

: …3:27†

ConsequentlyH _{will attain its maximum value ifu…r†is chosen in the form}

u…r† ˆ umax…r† if r…r†>0;

umin…r† if r…r†<0:

…3:28†

Solving the associated adjoint equation for k…r† and considering the trans-versality conditionk…b† ˆ0, from Eq. (3.27) fora6r6b we have

r… † r a… †r: …3:29†

Since by virtue of Eq. (3.8) a…r† is positive, in the whole domain flP0g f…a;b†g we conclude that whenever u… † 2r U, maximizes the Hamiltonian in (3.26) then

u…r† ˆumax…r†; …3:30†

whereumax… †r <u1… †r produces the physiologically determined minimum value

4. Discussion

Robeneck and Hesz [4] claimed that experiments with LDL particles bound to colloidal gold provided the ®rst clear demonstration of the sequential clustering of their receptors near coated pits. They concluded that this e€ect is produced when recycled LDL receptors are inserted in regions where coated pits form. Wofsy et al. [5] argued that in these experiments the LDL-gold particles were highly multivalent and thus may have bound more eciently to aggregated than single receptors. In their view, aggregation of newly inserted LDL receptors in regions around coated pits is a controversial question. Nevertheless the ability of the cell to sort receptors within speci®c targets along the endocytic pathway in order to control the internalization of speci®c ligands, [23] makes it reasonable to assume that reinsertion can be also accommodated in such a way that the mean capture time of LDL receptors could be adapted to speci®c metabolic requirements. If the cells needs to remove the LDL ligand at maximum rate, even negligible values for the mean capture time of its receptors by coated pits could be expected. The present study shows that in that case the surface plaques will be formed.

If the receptors are inserted as envisioned in [4] then their replacement oc-curs in sites were new coated pits form, and remain aggregated in plaques until the coated pit invaginates. The interaction of the cytoplasmatic tail of the re-ceptor with clathrin or other protein which makes up a developing lattice-like coat in these replacement regions could produce a weaker di€usion process for these particles. Then the convective transport induced by the formation of coated vesicles could e€ectively keep the receptors there. This e€ect could ex-plain the permanence of the observed receptor clusters. The ratiol0=D0, will

not induce the surface [4] plaques not even for the Wofsy et al. [5] insertion mode with an extremely restricted replacement annulus [10,11]. Nevertheless, from a theoretical perspective in the general situation the combination of a fast convective transport, a slow di€usion process, and a suitable insertion mode could explain the observed plaques. In general, receptor replacement in sites near coated pits will induce increased trapping rates for the receptors [5,6,11] Obviously, the smaller the mean capture time, the smaller the radius of the surface plaque formed.

Since the value for the dissociation rate of bound LDL receptors from coated pits has not been determined,s_{within physiological limits is uncertain.}

Nevertheless, the present analysis concludes that ifS_{r†produces the optimal}

proportion of inserted LDL receptors u_{r†, then whatever value its}

corre-sponding mean capture times_{attains, there will be a member of theb-family}

(3.15), withbobtained by means of the criteria (3.18) for which the associated mean capture time sb agrees with s

_{to highest order. If S…r† inserts the}

maximum density of unbound receptors which equals their density in coated pits. In that case practically the total number of unbound receptors must be located in an annulus of inner radius aand maximum outer radius of 1:82a. The estimation of the transit time of LDL receptors in [12] indicates thats_is

less than 15 s. Fig. 3 shows that for the ratiol₁=Dcm the plaque generated by

S15… †r would lie entirely within the annulusa6r61:82a. The corresponding mean capture time is 1.30 s. This means that the surface plaques could provide an experimental criteria to estimates_.

As a conclusion, the requirement of an enhanced aggregation rate of LDL receptors in coated pits could induce the plaques of Robeneck and Hesz [4]. Their paradigm for the reinsertion of receptors in sites where new coated pits form could be consistent with the strategy of the cell to assimilate the LDL ligand at the fastest rate, and the surface plaques will be an evidence of the response of the cell to these requirements. Recent experiments claim the exis-tence of preferential coated pit formation sites [24]. If the recycled receptors are sorted to these speci®c membrane sites, the preferential insertion paradigm [4] provides a reasonable explanation of the surface plaques. If more experimental results corroborate their existence, these surface aggregation patterns could provide through the analysis performed here a criteria to estimate the mini-mum physiologically expected value for the mean capture time of LDL re-ceptors by coated pits.

Acknowledgements

I acknowledge a great debt to Dr Carla Wofsy from the University of New Mexico and Dr Byron Goldstein from Los Alamos National Laboratories who introduced me to the fascinating world of mathematical modeling. Two anonymous reviewers provided valuable orientation. Elena Solana A. and Cecilia Leal R. contributed a great deal, both with technical discussions and encouragement.

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