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

Mathematical Biosciences 162 (1999) 53±67
www.elsevier.com/locate/mathbio

Optimal control of receptor reinsertion in the
Low Density Lipoprotein endocytic cycle
Hector Echavarrõa-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 minimun 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 Endocytosis (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

*

Fax: +1-617 50 545, +1-617 45 154.
E-mail address: hechavar@cicese.mx (H. EchavarrõÂa-Heras)

0025-5564/99/$ - see front matter Ó 1999 Published by Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 9 9 ) 0 0 0 4 3 - 7

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H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

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 hazard 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] considered 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 dramatically reduces the mean capture time determined by di€usion and uniform
insertion for the LDL receptors. Nevertheless the corresponding surface aggregation pattern was not explored. Echavarrõa-Heras and Solana [6] considered 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 capture time of LDL receptors by coated pits.
The invagination of the coated pits suggests the existence of a local radial
¯ow which provokes the convective transport of the receptors towards the
centers of the traps [7,8]. Using computer graphics techniques [9] , EchavarrõaHeras et al. [10] explored the surface aggregation patterns of LDL receptors

induced by di€usion, radial convection and insertion in the form considered in
[5]. They found that unless the radial ¯ow takes a strength far beyond its
physiological limit, that insertion mode would not induce the surface plaques
[4]. Solana et al. [11] extended that analysis and con®rmed that within physiological limits the radial ¯ow in combination with suitable continuous and
decreasing insertion rate functions increase signi®cantly the trapping rate of the
receptors by coated pits. They also suggested that the invoked insertion mode
could produce the observed steady-state aggregation of the receptors around
coated pits. On the basis of the present study I will con®rm that whenever the
di€usion coecient takes a value smaller than the strength of the radial ¯ow,
then a suitable reinsertion mode can produce the surface plaques. This situation could occur, for instance, if we assume that in the replacement sites envisioned in [4] the development of a clathrin coating for the forming endocytic

H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

55

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 minimum 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 by k‡ . 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 radius a, k‡ is de®ned by means of the equation

2paD oC 
k‡ ˆ
;
…2:1†
hC i or rˆa

where C…r† is the steady-state radial distribution function of receptors unbound

to coated pits, D > 0, their di€usion coecient, and hC i, the receptor concentration averaged over all the di€usion space [12]. The constant k‡ 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.
The set of coated pits can be envisioned as a system of sparse and orderly
distributed traps [2,14]. By virtue of this, the Berg and Purcell [15] approximation device can be applied. Hence we can consider a single coated pit (or
coated pit location) of radius a, with particles di€using about it in an annulus
of outer radius b, which assigns to the coated pit its share of the cell surface.
Under the assumption that traps are in®nitely long lived, the outer radius b can

be found from the relation

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H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

Pˆ

1
;
pb2

…2:3†

where P is 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 concentration of receptors is maintained. The basis for this is the apparently undetectable 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
r ˆ a and a re¯ecting boundary condition at r ˆ b. If radial convection is assumed, a ¯ux vanishing boundary condition [7] at r ˆ b will 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 Delbr
uck [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 density Cs …r†, of particles at a distance r from
 ˆ f…r; h† a 6 r 6 bg under the
the center of the trap, di€using on the annulus X
in¯uence of a radial ¯ow directed toward the center of the sink and inserted by
a radially symmetric rate function S…r† P 0, can be modelled [7,8,11] by means
of the equation
Dr2 Cs ‡

l oCs
‡ S …r † ˆ 0
r or

…2:4†

with an absorbing boundary condition
Cs …a† ˆ 0;
and an outer boundary condition

oCs 
l
‡ Cs …b† ˆ 0;
D
or rˆb b

…2:5†

…2:6†

following from a ¯ux vanishing requirement [7] at r ˆ b.
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 0:10 lm determine a reference value of
l0 ˆ 1:6  10ÿ13 cm2 =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 estimate l1 of 8  10ÿ12 cm2 =s. This gives a ¯ow rate constant l of
50 times l0 .

H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

57

It is considered that the di€usion coecient of an unbound LDL receptor at
37°C has a reference value of D0 ˆ 4:5  10ÿ11 cm2 =s [20]. Nevertheless, the
experiments do not rule out the possibility that it could be smaller than D0 .
This could be expected to happen within a coated pit. In fact, Goldstein et al.
[8] estimated that if the coated pit lifetime is k, the di€usion coecient in a
coated pit must be bounded above by Dc ˆ a2 k=4 before radial convection
could e€ectively keep the receptors trapped. Using k ˆ 0:14 s, and the smallest
experimentally determined value for the coated pit radius one gets for Dc a
value as low as Dcm ˆ 4:46  10ÿ13 cm2 =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 of Cs …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 of Cs …r†. (See Fig. 2.)

3. The control problem for the receptor mediated endocytic cycle
3.1. The objective functional for the mean capture time s
For S…r† P 0 de®ne the function hs … z† to be
Z b
h s … z† ˆ
xS … x† dx:

…3:1†

z

Using the polar coordinate form of the Laplacian it follows from Eq. (2.4) that
Cs …r† satis®es the di€erential equation
oCs

lCs …r† hs …a† hs …a†
ˆÿ
‡
ÿ
us …r†;
or
Dr
Dr
Dr

…3:2†

The control function us …r† will be the proportion of particles inserted in the
annulus a 6 z 6 r. Hence,
Rr
zS…z† dz
:
…3:3†
u s …r † ˆ a
hs … a †
Necessarily us …a† ˆ 0; us …b† ˆ 1. Furthermore, I will assume that the control
function us …r† satisfy the admissibility constrain
u s …r † 2 U
being U a set to be estimated.
It is easy to show [11] that the response Cs …r† becomes
Z
rÿl=D r l=Dÿ1
z
‰hs …a†…1 ÿ us …z††Š dz:
C s …r † ˆ
D
a

…3:4†

…3:5†

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H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

From the above equation we obtain
Z b 
Z b
 r l=Dÿ2 
1
hs …a†…1 ÿ us …r†† dr:
r Cs …r† dr ˆ
r 1ÿ
l ÿ 2D a
b
a

…3:6†

Hence the combination of equations (2.1)±(2.3), (3.1), (3.2) and (3.6) give for
the corresponding mean capture time ss ,
Z b
a…r†…1 ÿ us …r†† dr;
…3:7†
ss ˆ
a

where

a…r† ˆ

8
l=Dÿ2
†
>
< r…1ÿ…r=b†
lÿ2D

>
: r ln …b=r†
D

for l 6ˆ 2D;

…3:8†

for l ˆ 2D:

Then ss will be minimal whenever the integral
Z b
a…r†us …r† dr
J …us …r†† ˆ

…3:9†

a

obtains its maximum value. This de®nes the objective functional. The control
problem will be to characterize an optimal form u …r† 2 U which produces the
minimum physiologically expected value s for the mean capture time of the
receptors by coated pits. This will be equivalent to maximize the functional
(3.9) over the set U .
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,
sp and sq , and that sp coincides with sq . Then Eqs. (3.7) and (3.8) imply
J …usq …r†† ˆ J …usp …r††:

…3:10†

On the other hand, integration by parts gives
Rb
/…r†rS…r† dr
;
J …us …r†† ˆ /…r† ÿ a
hs …a†

…3:11†

where

/…r† ˆ

8
>
<
>
:

r2 ÿa2
2…lÿ2D†

1
2D



ÿD

bÿl=D‡2 …rl=D ÿal=D †
l…lÿ2D†

ÿ 2ÿ ÿ b
1

ÿ ÿ




r ln r ‡ 2 ‡ a2 ln ab ÿ 12

l 6ˆ 2D;
l ˆ 2D:

Hence the assumption that sp equals sq is equivalent to the criteria

…3:12†

H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

Z

b

/…r†
a

Sp …r†
dr ˆ
hsp …a†

Z

b

/…r†

a

Sq …r†
dr:
hsq …a†

59

…3:13†

3.2. The admissibility set U for the control function us …r†
From Eq. (3.3) it is easy to conclude that whenever the product rSp …r† decreases the plot of usp …r† will be concave down. Correspondingly for Sq …r†increasing, usq …r† will have a concave up plot. Since either at r ˆ a or at r ˆ b,
usp …r† and usp …r† coincide, necessarily J …usp …r†† > J …usq …r†† and sp < sq . Hence in
the general case there must be functions umin …r† and umax …r† such that the set U
of statement (3.4) can be de®ned by the inequality
umin …r† 6 us …r† 6 umax …r†;

…3:14†

where us …r† is de®ned by an insertion rate function S…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 in U .
For b 2 R, lets consider the b-family of functions,
Sb …r† ˆ krÿb

…3:15†

with k P 0 a constant. The associated family of controls usb …r† becomes
8
ln …r=a†
>
for b ˆ 2;
< ln …b=a†
…3:16†
usb …r† ˆ
…b‡2†
>
: …a=r†…b‡2†ÿ1 for b 6ˆ 2:
ÿ1
…a=b†
Since the derivative of the product rS1 …r† vanishes, from Eq. (3.3) we have
d 2 u sb
dr2

0

r…b ÿ 1†S1bÿ1 …r†S1 …r†
:
ˆ
h…a†

Hence for b > 1 the plot of usb …r† will be concave down. For usb …r† Eq. (3.7)
de®nes a mean capture time sb of
Z b
sb ˆ
G…r; l; b† dr;
…3:17†
a

where
G…r; l; b† ˆ

8
ln …r=a†
>
< a…r† ln …b=a†

†2ÿb †
>
: a…r†…1ÿ…r=b2ÿb
…1ÿ…a=b† †

forb 6ˆ 2;
for b ˆ 2:

The behavior of sb depending on b for the parameter values associated with
the LDL system is shown in Fig. 1. For instance, if b ˆ 3:0375, ub …r† produces
a mean capture time s3:0375 of 1.2678 min.

60

H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

Fig. 1. The variation of sb as given by Eq. (3.17) for the parameter values associated with the LDL
system in human ®broblastic cells, a ˆ 10ÿ5 cm, b ˆ 10ÿ4 cm, l0 ˆ 1:6  10ÿ13 cm2 =s, D0 ˆ
4:5  10ÿ11 cm2 =s [7,12]. We notice that s1 ˆ 2:54 min and that sb approaches zero whenever b
approaches in®nity.

Consider the case b P 1 and suppose that a given decreasing function S…r†
along with Sb …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 number rs in …a; b† associated with S…r† such that the criteria of
Eq. (3.13) can be written equivalently
/…rs † ˆ f …b†;

…3:18†

where for l 6ˆ 2D,

f …b† ˆ



0
 2 2

1
2ÿb
2ÿb …b ÿa …a=b† †
Dbÿl=D‡2 al=D
a2

ÿ
‡
l
2
B 4ÿb 2…1ÿ…a=b†2ÿb †
C
1
B

 C
2ÿb
A
l ÿ 2D @
…bl=D ÿal=D ……a=b† ††Dbÿl=D‡2 …2ÿb†
ÿ
…1ÿ…a=b†2ÿb †l…l=D‡2ÿb†

…3:19†

while for l ˆ 2D
f …b† ˆ f1 …b† ‡ f2 …b† ‡ f3 …b†
with

…3:20†

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H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67
2

f1 …b† ˆ

ln…b=a†…b ÿ 4† ‡ …b=a† …b=a†
2

2ÿb

ÿ1

!

…2 ÿ b†a2

2ÿb
2D…b ÿ 4†
…b=a† ÿ 1
 10
0 
1
4ÿb
b2 1 ÿ …a=b†
2
ÿ
b
A@ 
 A;
f2 …b† ˆ @
2ÿb
4D…4 ÿ b†
1 ÿ …a=b†

   
1
a
1
:
ÿ
a2 ln
f3 …b† ˆ
2D
b
2

!

;

In the case of interest where a < b, f …b† is monotonic and maps the real axis
into the interval …/…a†; /…b††. Then for every value rs in …a; b† there exist a value
of b for which Eq. (3.18) is satis®ed. As a conclusion, given any insertion rate
function S…r† producing a mean capture time ss we can ®nd a member Sb …r† of
the b-family (3.15) producing sb in such a way that ss and sb agree to highest
order. Furthermore, if Cs …r† and Csb …r† are given by Eq. (3.5) for S…r† and Sb …r†,
respectively, we must have


Dh…a† 
a l=D


:
…3:21†
1ÿ
Cs …r† ÿ Csb …r† 6
l…l ‡ D†
r

Hence for l ®xed, whenever D becomes suciently small the plots of Cs …r† and
Csb …r† will be alike and the surface patterns induced by either one will be indistinguishable in an annulus a 6 r 6 2a. Consequently we can use the subset of
the b-family generated by b P 1 and the criteria of Eq. (3.18) to study the
surface aggregation patterns of unbound LDL receptors induced by an arbitrary decreasing function S …r†. Fig. 2, displays examples of these patterns.
The criteria of Eq. (3.18) can provide estimations for umin …r† and umax …r†. In
fact, experimental results on the LDL system [12] lead to the inequality
k‡ P 2:3  10ÿ10  1:6  10ÿ10 cm2 =s;

…3:22†

Using Eqs. (2.2) and (2.3) and the maximum possible value of the lower bound
for k‡ one gets an upper bound for ss of 1:2678 min. This value is determined
by S3:0375 …r†. Hence whenever us …r† P u3:0375 …r†, then ss 6 s3:0375 and the experimentally determined lower bound for k‡ provides an empirical criteria to
choose u3:0375 …r† as a reasonable lower bound for the admissibility set U .
In order to obtain an estimate for umax …r†, notice that Eqs. (3.7) and (3.9)
produce
Z b
sb ˆ
a…r† dr ÿ J …usb …r††:
…3:23†
a

Let u1 …r† be the limit form of usb …r† when b approaches in®nity and s1 its
corresponding
mean capture time. Since the kernel in integral (3.9) de®ning


J usb …r† is bounded, we have

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H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

Fig. 2. Examples of the surface aggregation patterns induced by the triplet (l; D; Sb …r†). In the plots
portraited, the innermost circle at r ˆ a stands for the boundary of the coated pit, while the outermost at r ˆ b stands for the boundary of the reference region a 6 r 6 b. Although there is a
relative aggregation of receptors near r ˆ a in (a) and (b) the receptors are not depleted in the whole
annulus a 6 r 6 b. For the parameter values in (c) a plaque begins to form. The depletion region
begins at r ˆ 5:5a the radius of the ®rst circle surrounding the trap. A clearly depicted plaque is
shown in (d). Notice that the depletion annulus has been extended to the region r P 3a, i.e., the
second circle at r ˆ 3a encloses most of the unbound receptors.

lim J …usb …r†† ˆ

b!1

Z

b

a…r† dr:

…3:24†

a

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† 6 us …r† 6 u1 …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 value s for the mean capture time ss . Obviously, to
ful®ll the requirement of a positive s we must have umax …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

H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

63

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 corresponding plaque to have the minimum possible outer radius. Since the maximum 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 associated with the measurements of a, 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 l ˆ l1 ,
D ˆ Dcm and S15 …r†, the annulus a 6 r 6 1:21a contains most of the unbound
receptors and practically, the total number of receptors will be located in the
annulus a 6 r 6 1:82a, (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
a 6 r 6 1:21a. The ratio l1 =Dcm (see text for details) and S15 …r† induce a form for Cs …r† where the
receptors will be practically depleted for r P 1:21a. (see (a)), In (b) the innermost circle at r ˆ a
corresponds to the boundary of the coated pit, a second circle at r ˆ 1:21a surrounding the trap
de®nes the outer radius of the corresponding plaque. Practically all the recycled receptors lie in the
annulus a 6 r 6 1:82a. This characterization of the optimal plaque induces a mean capture time of
1.30 s, for the receptors by coated pits.

64

H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

that the plaque associated with (l1 ; Dcm ; S15 …r†† gives a higher order estimation
for the optimal surface aggregation pattern.
In summary, the admissibility set U of statement (3.4) could be de®ned as
the set of functions us …r† which are produced by Eq. (3.3), constrained by inequality (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 of umax …r† by means of u15 …r†. In general, given a
value of the ratio l=D every function us …r† which belongs to U can be reasonably approximated by a member usb …r† of the b-family (3.16) for which b is
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 produces umax …r† through Eq. (3.3) can be formally obtained by means of the maximum principle. To this aim we consider
the optimization problem
s ˆ max f J …us …r††g;
us 2U

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


h…a† ÿ lCs …r†
‡ r…r†us …r†;
H ˆ k…r†
rD

…3:25†

…3:26†

where
r…r† ˆ



a…r† ÿ


k…r†h…a†
:
rD

…3:27†

Consequently H will attain its maximum value if u …r† is chosen in the form

umax …r† if r…r† > 0;

…3:28†
u …r† ˆ
umin …r† if r…r† < 0:
Solving the associated adjoint equation for k…r† and considering the transversality condition k…b† ˆ 0, from Eq. (3.27) for a 6 r 6 b we have
r…r†  a…r†:

…3:29†

Since by virtue of Eq. (3.8) a…r† is positive, in the whole domain
fl P 0g  f…a; b†g we conclude that whenever u …r† 2 U , maximizes the
Hamiltonian in (3.26) then
u …r† ˆ umax …r†;

…3:30†

where umax …r† < u1 …r† produces the physiologically determined minimum value
s for the mean capture time ss .

H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

65

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 occurs 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 receptor 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 explain the permanence of the observed receptor clusters. The ratio l0 =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 if S  …r† produces the optimal
proportion of inserted LDL receptors u …r†, then whatever value its corresponding mean capture time s attains, there will be a member of the b-family
(3.15), with b obtained 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 receptors in regions where they di€use slowly, then a radial ¯ow could induce a
surface plaque. It is reasonable to expect that u …r† produces a plaque with the

66

H. Echavarrõa-Heras / Mathematical Biosciences 162 (1999) 53±67

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 a and maximum outer radius of 1:82a.
The estimation of the transit time of LDL receptors in [12] indicates that s is
less than 15 s. Fig. 3 shows that for the ratio l1 =Dcm the plaque generated by
S15 …r† would lie entirely within the annulus a 6 r 6 1:82a. The corresponding
mean capture time is 1.30 s. This means that the surface plaques could provide
an experimental criteria to estimate s .
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 existence 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 minimum physiologically expected value for the mean capture time of LDL receptors 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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