A theoretical model of the pinocytotic vesicular transport process in endothelial cells

A theoretical model of the pinocytotic vesicular transport process in endothelial cells

T. theor. Biol. (1977) 64, 619-647 A Theoretical Model of the Pinocytotic Vesicular Transport Process in Endothelial Cells B. T. RUBIN Department o...

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T. theor. Biol. (1977) 64, 619-647

A Theoretical Model of the Pinocytotic Vesicular Transport Process in Endothelial Cells B. T.

RUBIN

Department of‘ Aeronautics, Pllysiological Flow Studies Unit, Imperial College, London, England (Receit)ed 19 March 1976) A new theoretical model for vesicular transport in single endothelial cells is described using a kinetic molecular approach in which the vesicle diffusion process is coupled with the vesicle attachment/detachment process occurring at the cell plasmalemmal boundaries. Rate constants kf, k, characterizing a two stage reaction sequence in the attachment/ detachment region and the vesicle diffusion coefficient D are obtained by comparison of the theory with the results of tracer studies. For the condition of rapid vesicle loading/discharge of macromolecules it is found that the permeability of endothelial cells to macromolecules tends to be controlled by the vesicular attachment/detachment process rather than the vesicle diffusion process. The rate limiting step in the vesicle attachment/ detachment process tends to be the reaction process involving the rate at which a vesicle and the plasmalemmal membrane are brought into/ separated from intimate contact rather than that involving the rate of formation/dissolution of the membrane diaphragm of an attached vesicle. Estimated relaxation times for processes occurring in the attachment/ detachment region and in the diffusion region, the vesicle transit time in the diffusion region, and the viscosity of the cytoplasm in the diffusion region are deduced. Fair agreement is obtained between the predicted and the observed temperature dependence of the permeability.

1. Introduction The rate controlling step in the short-term transport of macromolecules in a given size range between the blood and the arterial wall is apparently located in the region of the blood wall interface (Caro & Nerem, 1973 ; Fry, 1973). Endothelial cells lining the blood vessel wall in the interfacial region regulate in part the mass transport of macromolecules into the interior of the vessel wall: ultrastructural studies indicate that large molecules are transported across the endothelial cell by pinocytosis rather than through the intercellular clefts (Stein & Stein, 1973). A detailed understanding of 41 619 T.Li.

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the pinocytotic transport process is thus required in order to assess the relative contribution of the vesicular transport process of macromolecules across endothelial cells to the overall transport of macromolecules into the vessel wall. For the case in which the endothelial cells represent the primary barrier to the mass transport of macromolecules into the wall the pinocytotic vesicular transport process is of particular importance. Several theoretical models have been proposed for the pinocytotic transport of macromolecules across endothelial cells (Green 8~ Casley-Smith, 1972; Shea, Karnovsky & Bossert, 1969; Shea & Bossert, 1973: Tomlin, 1969; Weinbaum, Lewis & Caro, 1974; Weinbaum & Caro, 1975). In most models (Green & Casely-Smith, 1972; Shea ct al., 1969; Shea & Bossert, 1973: Tomlin, 1969) the motion of vesicles in the endothelial cell cytoplasm was treated using the theories of diffusion and Brownian motion. The vesicle attachment/detachment process occurring at the cell plasmalemmal membrane was described by various statistical techniques (Feller, 1954, 1966). The underlying physical nature of the attachment/detachment process in these models has not been adequately explored. The most recent and comprehensive model that has been proposed for the pinocytotic transport process (Weinbaum & Caro, 1975) treats the dynamics of vesicle motion in terms of an interaction between hydrodynamic forces and Van der Waals molecular forces. The viscosity throughout the endotheial cell is assumed to be a function of distance. The solution for the vesicle transport in the endothelial layer provides the boundary condition for the flux of material into the arterial wall which is modelled as a two-phase system. A direct comparison of the theory with available vesicle transport data (Casley-Smith & Chin, 1971; Simionescu, Simionescu & Palade, 1973) was not attempted. In this treatment some aspects of the interaction of vesicles with the plasmalemmal membrane during the attachment/detachment process have been neglected including: the contribution of screening due to fluctuating dipoles and the contribution of dielectric constant variation in a composite dielectric system to the Van der Waals dispersion force (Israelachvili, 1972; lsraelachvili & Tabor, 1973; Langbein, 1969, 1971); the chemical forces involved in the making and breaking of bilayer membrane bonds; the electrical forces arising from double layer repulsion of charged membranes; inductive forces due to high order multipole interactions; and repulsive forces generated by orbital electron overlap. In the majority of models considered little attention has been focused on the relative contribution of the vesicle attachment/detachment process and the vesicle diffusion process in determining the overall vesicular transport process. Furthermore, the theoretical temperature dependence of the vesicular transport has not been examined.

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In an attempt to refine the current understanding of the pinocytotic vesicular transport process a new general theoretical approach is proposed. The specific objectives of this treatment are: (i) to present a unified kinetic molecular basis for the formulation of vesicle dynamics in a single endothelial cell, (ii) to elucidate the extent of coupling between the vesicle attachment/detachment process and the vesicle diffusion process and (iii) to explore the temperature dependence of the permeability of endothelial cells to certain macromolecules. The treatment is divided into the following sections. (2) General theoretical formulation. (3) Steady-state solution/permeability. (4) Short-time transient solution. (5) Comparison with experiment. (6) Discussion. (7) Extension of model: temperature dependence of permeability. (8) Limitations of model. (9) Conclusions. 2. General Theoretical

Formulation

The dynamics of the vesicular transport process are examined from a kinetic-mechanistic point of view in which the motion of a vesicle throughout an endothelial cell is modelled by a continuous free energy-distance profile (Fig. 1). For the purpose of analysis the vesicular transport system is partitioned into spatial regions R r, R, and R,. In regions R,, R3 vesicular motion is characterized by heterogeneous reaction rate constants, li+i, k%i for the vesicle attachment/detachment process. The treatment of vesicular motion in region R, is restricted to times t $ fl-’ where /3-i = m/6nrq. Here m is the particle (vesicle) mass, r is the radius, q is the viscosity of the medium. For this condition the motion of Brownian particles reduces to the elementary case of the problem of random flights in which the motion of Brownian particles is also described as one of diffusion and is governed by the diffusion equation (Chandrasekhar, 1943). In region R, vesicular motion is characterized by the diffusional velocity (D/D) where D is the vesicle diffusion coefficient and v is the diffusion length. The vesicle dynamics in the attachment/detachment process and diffusion process are now discussed in detail. (A)

ATTACHMENT/DETACHMENT

PROCESS

The vesicle attachment/detachment process is modelled by two free energy barriers in series which correspond to a two-stage reaction sequence. Each free energy barrier accounts for the interaction of a vesicle with the cell plasmalemmal membrane as a function of position. The corresponding free energies of activation associated with each barrier are AG$i, AGri”. respectively. [Refer to Fig. l(b).]

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VESICULAR

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The free energy of activation AG+i’ is the activation energy required for a vesicle and the cell plasmalemmal membrane to be brought into/separated from intimate contact without the creation or destruction of intramembrane chemical bonds. The free energy of activation AGfi has several components: AG$i = AG$+AGEL+AG$

+AG:

+AGl.

In equation (1) subscript f denotes direction with respect to the x axis and subscript i denotes the region specified; AG$ is the free energy of activation arising from Van der Waals interactions; AG& is the free energy of activation arising from electrical double layer repulsion between electrically charged membrane bilayers; AG$ is the free energy of activation arising from viscous and electroviscous effects; AG: is the free energy of activation originating from inductive interactions; AG: is the free energy of activation arising from short range repulsive forces due to orbital electron overlap. The free energy of activation AG$“ is the activation energy of the reaction associated with the formation/dissolution of the membrane diaphragm of an attached vesicle. It is relevant to point out that the precise nature of the reaction(s) associated with the formation/dissolution of the membrane diaphragm of an attached vesicle is unknown. From electron micrograph data (Palade & Bruns, 1968, see Figs 13, 14, 15, 17) the process associated with the formation/dissolution of a vesicle membrane diaphragm appears to be connected with the presence of a “central knob” (region of high electron density) located in the vesicle neck region where a high curvature of the membrane neck is usually apparent. Although the formation/dissolution of the vesicle membrane diaphragm may involve several reaction steps, it is assumed in this model that the process may be represented b!. one effective free energy barrier with activation energy AGrid. FIG. l(a). Schema of idealized endothelial cell. R1, R3 delineate reaction regions: Ra delineates the diffusion region. In R, attached vesicles with necks open to lumen have their cenlres located at x = m whereas attached vesicles with necks closed to the lumen by intact membrane diaphragms have their centres located at x = m’. Similar remarks apply to both types of vesicles in region R3. h is the vesicle neck diameter, 1 is the vesicle stalk length, r is the vesicle radius, and p = (2r -i- I -I- @ “) where Aged -~~(~1’ ~- 01) as seen from Fig. l(b). (b) Schema of one dimensional free energy distance profile. In Ii,. AGzlL’, AC: I are the free energies of activation which characterize the two step reaction sequence in the vesicle attachment/detachment process and Lrld, A:, are the respective corresponding activation energy distances; dY d, 1: are the respective distances between initial and final states for the reaction involving the rate of formation/dissolution of a double bilayer membrane diaphragm and for the reaction involving the rate at which a vesicle and cell plasmalemmal membrane are brought into/separated from intimate contact without the breaking of intramembrane chemical bonds. Analogous remarks apply to processes occurring in region R3. In Rz, IzD is the distance between successive equilibrium positions of the solvent molecule and Q’ is the corresponding activation free energy. & is the transition length about the boundary positions x 2 0, .Y : u respectively.

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The heterogeneous rate constants liki, k~i which describe the velocities in each stage of the two-stage reaction sequence are defined as: k*, = Loko,i = rQK+.,(k,T/h) exp (-AG2i/RT) (2) k:i = &‘9dk$P = l~*d~d+i(kbT/h) exp (-AGrtd/RT). (3) Equation (2) expresses the rate at which a vesicle and the cell plasmalemmal membrane are brought into/separated from intimate contact without the generation or decay of intramembrane chemical bonds. Equation (3) expresses the rate of formation/dissolution of the membrane diaphragm of an attached vesicle. In equations (2) and (3) subscript + indicates direction with respect to the x axis; subscript i indicates the particular region. In equation (2) $ is the length of the reaction zone traversed between initial and final state configurations; k~i is a reaction rate constant according to absolute reaction rate theory (Glasstone, Laidler & Eyring, 1941) and is defined in terms of the transmission coefficient K+~, the Boltzmann constant kb, the Planck constant h, the temperature T, the free energy of activation AG~i and the molar gas constant R. In equation (3) the superscript d denotes reference to the vesicle membrane diaphragm. All other symbols in equation (3) are defined similarly to those in equation (2). Utilizing the experimental evidence on the structural modulations of plasmalemmal vesicles (Palade & Bruns, 1968) a mechanistic interpretation of a single vesicle subject to the attachment/detachment process in region R, is envisaged as follows: (B)

ATTACHMENT

PROCESS

A vesicle whose centre is located at position (x = 0) is in an initial state on the free energy distance profile both for the attachment process occurring in region R1 and the diffusion process occurring in region R,. In the first stage of the attachment process, the cell plasmalemmal membrane “dimples” to the contour of the free unattached vesicle as the process approaches the activated state with corresponding energy AG!, at the distance A? 1. In the activated complex the vesicle bilayer membrane and the plasmalemmal bilayer membrane achieve a critical separation distance. An activated complex which descends the free energy barrier to the final state configuration at position (x = m’) emerges as an attached vesicle with a double bilayer membrane diaphragm. No new bilayer membrane material has been created nor has any bilayer membrane material been destroyed at this stage in the process. The first stage in the reaction sequence of the attachment process is now completed. At position (X = m’) the attached vesicle is in an initial state on the free energy distance profile both for the reverse reaction, i.e. the separation of the attached vesicle with diaphragm intact from the

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plasmalemmal membrane, and for the second reaction stage of the attachment process. In the second stage of the reaction sequence the membrane diaphragm of the attached vesicle is subject to dissolution as the process approaches the activated state with corresponding energy AGfqld at distance ;I?;d. An activated complex which descends the free energy barrier to the final state configuration at position (x = m) emerges as an attached vesicle without a diaphragm thus exposing the interior of the vesicle to the lumenal front. (C)

DETACHMENT

PROCESS

A vesicle whose centre is located at position (x = m) is in an initial free energy state for the first stage of the detachment process. In this stage of the detachment sequence membrane material is generated in the vesicIe neck region as the process approaches the activated state with corresponding energy AGf* d at distance A:* d. In the activated complex a membrane diaphragm is initiated. An activated complex which descends the free energy barrier to the final state configuration at position (x = m’) emerges as an attached vesicle with a double bilayer membrane diaphragm intact. At position (X = m’) the attached vesicIe is both in an initial state for the reverse attachment stage and for the second stage of the detachment process. In the second stage of the detachment process the region of contact between the vesicle bilayer membrane and the pIasmaIemmal bilayer membrane diminishes as the process approaches the activated state with corresponding energy AGT at distance 1;. In the activated complex the pIasmalemma1 bilayer membrane and the vesicle bilayer membrane achieve a critical separation distance. An activated complex which descends the free energy barrier to the final state configuration at position (x = 0) emerges as a free unattached vesicle. (D)

DIFFUSION

PROCESS

In the diffusion process the discrete increments in the motion of a large particle, i.e. a vesicle, in a medium consisting of small solvent particles is determined predominantly by a series of jumps of the solvent molecules from one equilibrium position to another. Following the treatment of Glasstone, Laidler & Eyring (1941) a solvent molecule adjacent to a vesicle must traverse a distance of at least XI’, where r is the vesicular radius in order that the vesicle traverses a distance approximately 1, in the opposite sense. Here An is the distance between successive equilibrium positions of the solvent molecule. The diffusion coefficient of the vesicle (larger particle) D is related to the diffusion coefficient of the solvent molecule (smaller particle) D, as D = (l,,/u*m)D,. The factor u* of order unity allows for the fact that a solvent molecule may not take the shortest path in its journey around

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the large particle. The full expression for the vesicle diffusion coefficient D is : D = i&/a*nr)D, = (&/a*nr)l~(k, T/h) edQ’IRT. (4) In equation (4) the diffusion coefficient of solvent molecules D, is defined according to reaction rate theory. Here Q” is the free energy of activation; kb is the Boltzmann constant; T is the temperature; h is Planck’s constant; R is the molar gas constant. All other quantities are defined above. (E)

TRANSITION

REGION

The cross-over region RJR, about the boundary position (X = 0) is characterized by a transition zone of length &. = (nf,+vJ,) in which the motion of a vesicle suffers a change from “reaction control” in region R, to diffusion control in region R,. v is the number of barriers traversed in the diffusion region [refer to Fig. l(b)]. Similar remarks apply to the crossover region RJR2 about the boundary position (x = v). (F)

MATHEMATICAL

MODEL

Vesicular motion is formally described in terms of a diffusion process coupled with reaction at the boundaries. The vesicular transport system is assumed to be subject to the following constraints. (i) The vesicular transport system is in thermodynamic equilibrium. (ii) The total number of vesicles in the system is constant. (iii) The free energy-distance profile in region R, is a perfect mirror reflection of the free energy-distance profile in region R,. (iv) The vesicle attachment/detachment process is reversible. (v) Any loading or discharge process of macromolecules in a certain size range, occurring in attached vesicles with necks open to lumenal and ablumenal surfaces, is fast compared to internal processes governing the intracellular vesicle dynamics of the attachment/ detachment process. From simple diffusion considerations it has been estimated (Tomlin, 1969) that the loading process in the absence of molecular sieving is complete for time t = IO-’ s. Further, it is assumed that the presence of the glycocalyx layer (Luft, 1964) in the vicinity of the lumenal surface does not significantly perturb the loading/discharge process. A convenient theoretical device for monitoring the vesicular transport process is depicted in Fig. 2. At time t = 0, an instantaneous uniform source of labelled macromolecules is introduced into the lumen. Attached vesicles with necks open to the lumen load labelled macromolecules instantaneously, whereas the presence of intact membrane diaphragms in the neck region of

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FIG. 2(a). A uniform source of labelled macromolecules (black dots) is introduced at the lumenal front at time t = 0. Attached vesicles with necks open to lumen load macromolecules instantaneously. The intact diaphragms of attached vesicles inhibit the loading process. (b) At time I > 0 the progress of labelled macromolecules through the cell i* governed by vesicular transport.

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attached vesicles inhibit the loading process [Fig. 2(a)]. For time t > 0 the progress of labelled material through the endothelial cell is determined solely by vesicular transport. The labelled macromolecules serve as a probe for exploring the fundamental features of vesicle motion. The equations for the vesicular transport system are: a+,

t>/at = D aZc(x, q/ax2.

06x 6 v

(5)

c(x, 0) = 0

(6)

(Wx, We},=, = k, c( m’, t) - k _ 1~(0, t) + D{&(x, t)/&x}, = o (aN(x, f)/f3t},= o = Ii - 3c(e’, f) - k3 c(v, t) - D(Bc(s, t)/ax},, *,

(7) (8)

N(x, r) = Ic(x, 1).

(9) Equation (5) is Fick’s second law of diffusion restricted to region R,; c(x, t) is the number of labelled vesicles per unit volume; D is the vesicle diffusion coefficient. Equation (6) is the initial condition for diffusion of labelled vesicles. Equations (7) and (8) are boundary conditions which describe the labelled vesicle flux balance at planes (X = 0) and (X = tl) respectively; N(x, t) is the number of labelled vesicles per unit area; c(m’, t), c(e’, t) are the number of labelled vesicles per unit volume at positions (x = m’) and (X = e’) respectively. The heterogeneous rate constant k,i is defined by equation (2). Equation (7) states that the increase in the number of labelled vesicles per unit area with time at the plane (X = 0), i.e. (aN/&),,,, is equal to the rate at which labelled vesicles arrive at position (x = 0) from position (X = m’), i.e. k,c(m’, t) minus the rate at which labelled vesicles depart from position (X = 0) toward position (x = HZ’), i.e. k-,c(O, t), minus the rate at which labelled vesicles diffuse into region R1, i.e. D(&/~x),~,. Comparable remarks apply to equation (8). Equation (9) relates the number of labelled vesicles per unit area to the number of labelled vesicles per unit volume via a thickness parameter A over the region of interest. For application of equation (9) at boundary positions (x = 0) and (x = v), i = 1,. 3. Steady-state

Solution/Permeability

The steady-state equations describing the variation in concentration of labelled vesicles as a function of distance across an endothelial cell are:

~a%/a~~ = 0, 0 Gx Q v o(ac/axjx,o = k-,~(o)-k,c(m’)

(11)

D{ac/ax),,,

(12)

= k-3 c(e’) - k3c(v).

(10)

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020

Xl0I)t.I

The solution to equation (10) with boundary conditions

(11) and (12) is:

c(x) = a,x+a,

(13)

where a = Ck-,k-,c(e’)-k,k,c(~‘)]l(k-,+k,)~ 1 (D/u)+

k:s Ii- J(k-

(14)

I + k,)

a/D+k,)c(m’) a2 = lk-,c(e’)+(k,k, ~-. _-. __~_._ -._ ---. (k-l-t-k,)+(k,k-,o/D)

.

(15)

The proportion

of labelled vesicles to total vesicles at any distance .Y is (16) G(X) = c(x)/c’(x) where c”(x) is the total number of vesicles per unit volume at a given s. Experimentally, cT(x) is essentially constant for all x (Casley-Smith & Chin, 197 1). The transverse permeability P of a single endothelial cell to labelled macromolecules may be derived in terms of the individual mass transport coefficients operating in series over each region. From Appendix A a convenient expression for the permeability is obtained, i.e.

where II is the average number of labelled macromolecules vesicle and a = [c(d) - c(e’)]/[c(O) - c(u)]. The following limiting cases are of interest. Case I. (D/v)

carried per (18)

% (k,/2)

Equation (17) becomes : (19)

Equation (19) indicates that the permeability P is controlled by the kinetics of the vesicle attachment/detachment process. Further, if kd, 9 k,, then equation (19) reduces to: nak,/2 p=GT However, if k, $ kd,, equation (19) reduces to P = nkd,/2.

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Case II (k,/2) >> (D/u) Equation (17) becomes (22)

Equation (22) indicates that the permeability is partially determined by the vesicle diffusion process as well as by the vesicle attachment/detachment process. If in addition k’: 9 kl, equation (22) reduces to: P = naD/v

(23)

where the condition (k,/2) 9 (D/u) has been employed following the application of the inequality k’: 9 k,. Equation (23) indicates that the permeability is controlled by the vesicle diffusion process. However, if /c, $ k’: equation (22) reduces to

In equation (24) the permeability is determined partially by the vesicle diffusion process and partially by the reaction process involved in the formation/dissolution of an attached vesicle membrane diaphragm. The temperature dependence of the permeability is obtained by substitution of equations (2), (3) and (4) into the appropriate expression for the permeability. The following selected formulae for later reference are obtained using expressions (17), (20) and (23) respectively,

P = [na0,/2(a+ P = (naO,/o)T

l)]T

e-AG1’iR1

(26)

edQPIRT

(27~

where 0, = ny Kl(kh/lf)

(28)

0; = A:, d&kb/h)

(29)

0, = (,l;/a*nr)(k,/h).

(30)

The permeability P of a homogeneous uniform monolayer of close-packed endothelial cells is simply related to the permeability per endothelial cell P and the total number of cells in the monolayer Ij as: P=iVP.

(31)

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4. Short-time

PROCESS

M0DF.I.

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Transient Solution

The concentration-time profile for labelled free vesicles in the semi-infinite case is obtained for short times. Referring to Fig. 2, labelled particles appear instantaneously at time t = 0 at the lumenal front. For t > 0, labelled free vesicles in region R’, (0 < x < co) diffuse in the increasing x direction. The short-time transient solution is valid for times preceding significant perturbations introduced by processes occurring at the ablumenal front. Referring to equations (5) to (9) and Fig. 2, the diffusion system in the semiinfinite case is given by:

aclat = D azcja.x2, 0 G .y G c%).

(32)

with initial condition: C(.Y,0) = 0

(33)

c(w, t) = 0

(34)

and boundary conditions:

:a(n,cyat),,, = k,

C(fd, t) - li- l c(0, t) + D(ac/dx},,“.

(35)

Equation (35) is obtained by substitution of equation (9) into equation (7) with the choice of ;? = LT. Introducing the dimensionless parameters <=(k-

,/D)x

(361

T = (/i”_ JD)t

(37)

,c = clcT,

(38)

equations (32) to (35) simplify to:

aG/a7= aZGlag2,0 G t G u3

(39)

c(T, 0) = 0

(40)

_c(m., T) = 0

(41)

Y(aC/aT)t,o = Kl@‘,

T)-do,

T)+(aC/a&eo

(42)

where y = k-, AT/D K, = k,/k-

(43)

1.

(44)

The solution of equation (39) subject to equations (40), (41) and (42) is obtained in Appendix B, i.e. _c(t, T) = u. J’ P,(z - U)F,(O) de, 0

0 < < < cx)

(45)

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where

F,(z--8)

= - i Alai eai2(r-e) erfc [~~(r-@“~]

(46)

i=2

F2(@ = (l/al)

erfc [ 0. (47) Equations (45), (46) and (47) determine the proportion of labelled to total vesicles as a function of dimensionless distance i and dimensionless time r in the diffusion region R;, 0 < x < m. The parameters a,, ai, Ai are some function of the constants y, K, and the constants PO, pi, p2 which arise from the coupling between diffusion and reaction regions through the function c(m’, r) appearing in boundary condition (42). Here, PO = PA- IID

(48)

,!I1 = lit/k-,

(49)

b2 =(k!!,+k,)/k-,.

(50)

In equation (48) p is the thickness of the region within R, containing attached vesicles with diaphragms intact. From a knowledge of the function r(&, r) (refer to Appendix B), the proportion ,cA(r) of labelled attached vesicles to the total number of attached vesicles in region R, is determined as a function of dimensionless time 5, i.e. c~(T) = (1/2)_c(m){1+(/I,/p2)[ I - e-‘pziDO)r])-(1/2)(c~,,//?~) / (1 -e-(Bz’80)(r-o)) {i~~ili*i

eaiZ” erfc (miol/,,>

da

(51)

In equation (51) cli is some function of the parameters tli, Ai.

5. Comparison of Theory with Experiment Although there is not yet sufficient detailed experimental evidence on the short time transient behaviour of vesicular transport in endothelial cells to rigorously test the theory, the electron micrographic tracer studies of Casley-Smith & Chin (1971) on mouse heart endothelium and diaphragmatic mesothelium at T = 37°C provide a useful starting point for comparison. Horseradish peroxidase was the most reliable of the tracers studied which did not exhibit molecular sieving effects. Uncertainties in the experimental data may arise from unknown tissue fixation times, leakage of the tracer through intercellular junctions, etc. From the theoretical standpoint one notes that the derivations presented in previous sections apply strictly to the behaviour of isolated single endothelial cells or to a uniform homogeneous monolayer of closely packed cells separating two continuous media.

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In the mouse heart endothelial system the presence of a heterogeneous non-uniform subendothelial space may generate perturbations both in the free energy profiles and the vesicle loading/discharge process at the ablumenal front. Consequently, application of the theory utilizing the assumptions of symmetry in free energy barriers and fast vesicle loading/discharge at both lumenal and ablumenal fronts is to be regarded with some caution. Employing steady state equation (14) and (16) with ki = k-i, k, 1 = ktj the ratio D/k, is deduced, i.e.: (52) D/k, = (AC-g,Ca, where A’ = c(e’)-&I’) and a, = al/cT(x). In the steady state, experimental observation indicates that virtually all attached vesicles in region R, are labelled thus theoretically requiring that c(m’) z I. Symmetry considerations with k: > k-, and fast discharge of vesicle contents at the ablumenal front suggest that g(e’) z 0. Consequently, A,c = - 1. The proportion of labelled vesicles as a function of distance across the diffusion region is displayed in Fig. 4. The slope @I is determined directly from the steady state curve a, i.e. a, = -4.63 x IO3 cm- *. The length of the diffusion zone v is estimated using Fig. 4, i.e. u = [L-(Z+r)-A!], where L is the combined length of regions R, and R,; I is the vesicle stalk length; r is the vesicular radius; ,i(: is the length of the reaction zone between initial and final state configurations for the reaction in which a vesicle and the lumenal plasmalemma1 membrane are brought into/separated from intimate contact without the generation or decay of intramembrane chemical bonds. Taking L = 5 x IO-’ cm, 1 = 1.5~ low6 cm, r = 3.5x lo-” cm, A(: = IO-” cm (see Discussion section), one finds ti = 4.3 x lo-” cm. Utilizing the values of the individual parameters obtained above in equation (52) the ratio D/k, is determined, i.e. D/k, = 8.64x 10e5 cm. It is relevant to note that since (Adal) > v by N five times, the ratio D/k, is relatively insensitive to modest variations in the value of L:. The theoretical transient behaviour of the proportion of labelled vesicles in the reaction region R, and in the diffusion region R; as predicted by equations (51) and (45)-(47) respectively is shown in Figs 3 and 4. All theoretical curves were constructed with the assumptions that k, = k- , , kj = k!. , and three fixed input parameters : p = (2r + I+ A:, d, = 9 x lo- 6 cm for 2:~’ N 5x 10V7 cm, D/k, = 8.64x 10m5 cm as obtained from steady state considerations, y = 0.25 (see Appendix C). The parameters k,, ky were then varied until the resulting theoretical curves emerged. For all predicted transients: k, = 3.4x 10w6 cm s-‘, I;: = 1 x 10e5 cm s-l. Combining this result with the known ratio D/k, one obtains the value of the vesicle diffusion coefficient, i.e. D = 2.937 x lo-*’ cm2 s- ‘.

634

B. T. RUBIN

The time interval (0, N 4 s) over which the theoretical transient curves in Fig. 3 and Fig. 4 are compared with the measured data corresponds to N 213 the vesicle transit time T = v*/D in the diffusion region. During the interval (0, -4 s) perturbations generated by vesicles originating from the ablumenal front are assumed negligible. In Fig. 3 the theoretical curve reveals good agreement with experimental points. The solid portion of the theoretical curve represents the range over which the theory rigorously applies with respect to the time axis. In Fig. 4 the theoretical curves (broken lines) at 2 s, 4 s, exhibit moderate agreement with experimental evidence. The observed data at 8 s indicates that the steady state is nearly achieved. The relative location of the theoretical curves along the x axis is sensitive to uncertainties in the choice of the reaction length 2: which determines the initial position for diffusion. The possible existence of a “plateau” region in the data for c in the range lo- ’ < x < 2 x 10s5 cm cannot be adequately inferred in view of insufficient data, the large errors associated with the available data, and the fact that similar transient tracer data (Casley-Smith & Chin, 1971) employing the non-molecular sieving ferrocyanide molecule does not exhibit such a “plateau” tendency.

I __

FIG. 3. Proportion of labelled attached vesicles ca in the attachment/detachment region as a function of time t. Solid line indicates the range of validity of theoretical curve. Experimental data, 0. (Casley-Smith & Chin, 1971: vesicles labelled with horse-radish peroxidase.)

VESICULAR

TRANSPORT

PROCESS

635

MODEI.

x (pm)

FIG.4. Proportionof labelledvesicles 6 in the diffusionregionasa functionof distanceX at varioustimes.Theoreticalcurves(c--- 2 s), (b --- 4 s). Experimentaldata, 0, 2 s: 0, 4 s; ~1,8 s; n , 3 16 s. (Casley-Smith& Chin, 1971:vesicleslabelledwith horseradishperoxidase). Curvea, steadystate.Seetext for further descriptionof parameters.

6. Discussion In order to ascertain the relative contribution of the vesicle attachment/ detachment process and the vesicle diffusion process to the permeability of an endothelial cell to macromoleculesexpression (17) is utilized. Employing the values of the relevant parameters determined in section 5 one finds that the characteristic diffusion velocity (D/u) is about four times greater than the characteristic reaction velocity (k,/2). Thus, the permeability P tends to

be controlled by the vesicle attachment/detachment process. Moreover, since k: is about three times greater than A, the rate-limiting step in the vesicle attachment/detachment process tends to be the reaction process involving the rate at which a vesicle and the cell plasmalemmal membrane are brought into/separated from intimate contact rather than the reaction process involving the rate of formation/dissolution of an attached vesicle membrane diaphragm. The corresponding relaxation times for the two reaction processes occurring in th e attachment/detachment region are given as: tlf, = ;.y. J/k; (53) tA = nyjk, . T.B

(54) 41

636

B.

T.

RUBIN

Taking A?,d N 5x 10m7 cm, $’ N 10e6 cm as a rough approximation, one obtains: ti = 0.05 s, tA = O-3 s. The result that t, is significantly greater than ti is consistent with the experimental finding of a relatively small proportion of attached vesicles with diaphragms intact over the time scale of observations. The vesicle diffusion relaxation time due to the motion of small solvent molecules may be defined as: tD = Ii/(a*mD).

(55)

Assuming in is approximately the same order as the mean O-O distance in water, i.e. ;i,, N 2.92 x lo-* cm, and using the values of Y and D from section 5 one obtains N 7.7 x 10v7 < t, < 7*7x 10e8 s for a* a constant or order unity (i.e. 1 < a* < IO). Comparison of the various relaxation times yields the result: t, < t: < t,. The dominant contribution of the vesicle diffusion process in determining the extent of the transition length R, = 2.159 x lo-’ cm is compatible with vesicle diffusion dynamics determined by multiple discrete jumps ot solvent molecules. A solvent molecule traverses a distance a*nr = u” x (I.1 x 10e5 cm) in order that a vesicle traverses a distance i, in the opposite sense. The time required for a vesicle to traverse the diffusion region is the vesicle transit time defined as: p = v2/D. (56) From the values of the parameters found in the previous section p = 6.3 s. The Stokes-Einstein equation for the diffusion coefficient of a large spherical particle diffusing in a medium of smaller particles yields the approximate formula for the viscosity of the medium (Eyring, Henderson, Stover & Eyring, 1964) ‘1 = k, T/4nrD. (57) Taking T = 310X, D and r from section 5, the viscosity of mouse heart endothelial cytoplasm in the diffusion region is q = 3.3 poise, The free energy distance profile for the reaction process in which a vesicle and cell plasmalemmal membrane are brought into/separated from intimate contact is explicable in terms of the individual free energy contributions to the energy barrier for the process. The leading term in the Van der Waals non-retarded dispersion interaction energy between a spherical vesicle of radius r and a planar plasmalemmal membrane surface immersed in a continuous fluid medium varies as a function of separation distance x and is proportional to -(-l/x) for x < r and -(-l/x3) for x 9 r. At intermediate separations where .Y and r are of comparable magnitude the expression for the dispersion interaction energy becomes complicated (Israelachvili & Tabor, 1973). The screening effect produced by fluctuating dipoles is the dominant contribution to the dispersion energy for x < r.

VESICULAR

‘TRANSPORI‘

PROC’ESS

MODLL

637

The Van der Waals interaction distance may typically extend to N 100 A. The interaction energy arising from electrical double layer repulsion of charged membranes is exponentially sensitive to (- IC’X) where K’ is the Debye screening length (Hogg, Healy & Fuerstenau, 1966; Parsegian, 1973). The interaction of electrical double layers may extend over distances up to several hundred Angstroms. The coupling of viscous and electroviscous hydrodynamic contributions to the interaction energy generated by the displacement of fluid between electrically charged vesicle and plasmalemmal membranes may yield a complicated spatial dependence which is expected to operate over the entire length of the reaction zone. The variation of the dielectric constant of the fluid medium due to the ordering effects of solvent molecules in the vicinity of charged membrane surfaces is expected to induce a spatial dependence in the viscosity which becomes particularly apparent when the separation distance x is in the range of about 10 A. The inductive contribution to the interaction energy originating from high order multipole effects becomes significant at comparatively short distances of separation whereas the interaction energy due to the repulsion of electron orbital overlap becomes dominant as the separation distance approaches molecular dimensions. A reliable estimate of the length n: thus awaits an ab-initio calculation of the free energy-distance profile for the particular reaction envisaged. Some speculative contributions to the free energy of interaction for the reaction process in which an attached vesicle membrane diaphragm undergoes formation/dissolution may involve tensile/compressive stress effects, the possible initiation of a biochemical reaction process in the vesicle neck region, and thermal fluctuation effects on the stability of the membrane diaphragm structure. Previous theories which attempted a direct correlation with experimental results have treated the vesicle attachment/detachment process statistically in terms of a reflection coefficient. Moreover, these theories have been restricted in apptication to the steady state experimental behaviour. In the present theoretical model a physical basis of the vesicle attachment/detachment process is suggested and incorporated into a general formulation which permits a direct comparison with steady state as well as with transient experimental data. 7. Extension of Model: Temperature of Permeability In this section measurements of (Siflinger, Parker the correctness of

Dependence

the theoretical treatment is applied to the permeability lz51 albumin in. isolated dog common carotid arteries & Caro, 1975). The validity of this application relies on the following general assumptions:

638

B.

T.

RUBIN

(i) The pinocytotic vesicular transport process occurring in the endothelial layer is the predominant barrier to labelled macromolecular uptake into the arterial wall. (ii) The vesicular loading/discharge process of macromolecules is rapid compared to intracellular vesicle dynamics and is independent of the type of labelled macromolecules present. (iii) The pinocytotic process in vascular endothelial cells of different species is not dissimilar. In particular, it is further assumed that the permeability to labelled albumin is essentially controlled by the vesicular attachment/detachment process and that equations (26) and (31) are applicable. Taking the ratio of the permeability P at two different temperatures T,, T2 and solving for the activation energy AC: gives : AC: = {R/Cl/T, --l/7”]) In [&-2*TI/~T,.Tz]. (58) From the slopes of experimental plots of “wall uptake” versus incubation time (Siflinger et al., 1975; see Fig. 2) at two different temperatures, the permeabilities P,, = 3.33 x IO-’ cm s-l at T1 = 310”K, Prr = 8.57 x IO-’ cm s-l at T2 = 291°K are obtained. Utilizing equation (58) one finds that AGf = 12.2 kcal/mol. Substituting this value back into equation (26) and using equation (31) with BT, at Tl = 310°K one obtains ~~crO,/2(0+ 1) = 0.43. The predicted temperature dependence of equation (31) is now compared with a different set of experimental data points (Siflinger t;t al., 1975; see Table 1) in Fig. 5. Fair agreement is indicated. It is interesting to note that the experimental interaction energy for the aggregation of red blood cells is N 5.5 kcal/mol of cells (Healy, 1974). In the above analysis the general assumptions (i) and (ii) are reasonably satisfied whereas assumption (iii) cannot be assessed because of lack of evidence. Some caution is required in the interpretation of the agreement obtained between theory and experiment manifest in Fig. 5 since equation (27) predicts a similar temperature dependence of the permeability to that obtained by equation (26) but with activation energy Q”. In either case the observed data is explicable in terms of a single activation energy. A complicated temperature dependence of the permeability is predicted from the more general relation (25) and involves activation energies: ACT*“, AC:, Q’. 8. Limitations

of the Model

The internal restrictions imposed on the theoretical treatment are set forth in constraints (i) to (v) in the theoretical section (Mathematical Model). Constraint (iii) regarding symmetry in the free energy-distance profiles is

VESICULAR

TRANSPORT

PROCESS

MODEL

639

T(T) FIG. 5. Permeability P of isolated dog common carotid artery to lzsI albumin as a function of temperature. The theoretical curve (broken line) is calculated assuming pinocytosis in the endothelial layer is the main barrier to the transport of macromolecules into the vessel wall. Experimental data, 0. (Siflinger et al., 1975.)

probably not met in the general case. The most serious defects in the model concern the omission of convective effects and the neglect of a spatial dependence of the diffusion coefficient. However, assessment of the relative contribution of these additional terms to vesicle dynamics awaits comparison with detailed experimental evidence. 9. Conclusions A theoretical model for vesicular transport in endothelial cells is described from a kinetic-mechanistic point of view in which the vesicle diffusion process is coupled with the vesicle attachment/detachment process occurring at the cell plasmalemmal boundaries. Rate constants lit, ki characterizing the reactions in the attachment/detachment region and the vesicle diffusion coefficient D are obtained from a comparison of theory with experimental data. For the condition of rapid vesicle loading/discharge of macromolecules it is found that the permeability of endothelial cells to macromolecules tends to be controlled by the vesicle attachment/detachment process rather than the vesicle diffusion process. The rate-limiting step in the vesicle attachment/ detachment process tends to be the reaction process involving the rate at which a vesicle and the cell plasmalemmal membrane are brought into/separated from intimate contact rather than the reaction process involving the

640

13. T.

KUBIlv

rate of formation/dissolution of the membrane diaphragm of an attached vesicle. Estimated relaxation times for processes occurring in the attachment/detachment region and in the diffusion region yield the sequence t, < t$ < t4. The vesicle transit time p, and the viscosity q of the cytoplasm are deduced. Application of the theory to 1251-albumin uptake studies in arteries reveals fair agreement between the predicted and the observed temperature dependence of the permeability. The author thanks the Nuffield Foundation for support of this work. The author is indebted to Professor J. R. A. Pearson, Dr K. Parker, Dr C. G. Caro and Professor A. Silberberg for useful discussions. The author thanks R. K. C. Oxenham for computing assistance. REFERENCES CARO, C. G. & NEREM, R. M. (1973). Circulation Res. 32, 187. CASLEY-SMITH, J. R. & CHIN, J. C. (1971).J. Microsc. 93, 167. CHANDRASEKHAR, S. (1943).Rev. Med. Phys. 15, 1. EYRING, H., HENDERSON, D., STOVER, B. J. & EYRING, E. M.

Statistical Mechanics and Dynamics. P. 463.New York: J. Wiley. FELLER, W. (1954).Trans. Am. Math. Sot. 77, 1. FELLER, W. (1966). An Introduction to Probability Theory and its Applications. New York:

John Wiley.

FRY, D. L. (1973).Atherogenesis: Initiating Factors, Ciba Foundation Symposium 12, 93. GLASSTONE, S., LAIDLER, K. J. & EYRING, H. (1941). The Theory of Rate Processes, pp.

l-28, 516-525.New York: McGraw-Hill.

GREEN, H. S. & CASLEY-SMITH, J. R. (1972). J. theor. Biol. 35, 103. HEALY, J. C. (1974).Biorheology, 11, 185. HOGG, R., HEALY, T. W. & FUERSTENAU, D. W. (1966). Trans. Farad. Sot. 62,1638. HUGHES, M. (1964). Physical Chemistry, pp. 58-59. (2nd revisededn.). New York:

MacMillan.

ISRAELACHVILI, TSRAELACHVILI,

J. N. J. N.

(1972). Proc. R. Sot. Lond. A331, 39. & TABOR, D. (1973). Progress in Surface und Membrane

Sciences

(J. R. Danielli, M. D. Rosenburg& D. A. Capenhead, eds.).Vol. 7, p. 1. New York: AcademicPress. J. Adhesion 1, 237. Phys. Chem. Solids 32, 1657. LUFT, J. H. (1964). J. Cell Biol. 23, 54A. PALADE, G. E. & BRUNS, R. R. (1968). J. Cell Biol. 37, 633. PARSEGIAN, V. A. (1973). Ann. Rev. Biophys. Bioengng. 2,221. SHEA, S. M., KARNOVSKY, M. J. & BOSSERT, W. H. (1969). J. theor. BioI. 24, 30. SHEA, S. M. & BOSSERT, W. H. (1973). Microvasc. Res. 6, 305. SIFLINGER, A., PARKER, K. & CARO, C. G. (1975). Cardiovasc. Res. 9, 478. SIMIONESCU, N., SIMIONESCU, M. & PALADE, G. E. (1973). J. Cell Biol. 57, 424. STEIN, Y. & STEIN, 0. (1973). Atherogenesis: Initiating Factors, Ciba FoundationSymLANGBEIN, LANGBEIN,

D. (1969). D. (1971).

J.

posium12, 165.

TOMLIN, S. WEINBAUM,

G. (1969). Biochem. Biophys. Acta 183, 559.

WEINBAUM,

S. & CARO, C. G. (1976). J. Fluid Mechanics 74, 611.

S., LEWIS. C. & CARO. C. G. (1974). NationalFoundationSbecialists Meeting _ on Fluid Dynamic Aspectsof Arterial Disease,Ohio State University, Columbus, Ohio, U.S.A.

VESICULAR

TRANSPORT

APPENDIX Derivation

PROCESS

641

MODEL

A

of the Transverse Permeability

P

The transverse permeability P of a single endothleial cell to labelled macromolecules may be derived in terms of the individual mass transport coefficients operating in series over each region: p’: = nj:/qc(m) - C(Jd)]

(Al)

p1 = njl/[4~‘)-c(0)1

CA2)

JV,lt-40)

CA3)

P2

=

- 441

p3 = W3/Cc(d -401

tA4)

p! = njdJ/[c(e’) -c(e)].

(A%

in equations (Al) to (A5) subscripts indicate the particular region specified, i.e. RI, R2 or R,, and the superscript d denotes reference to the reaction involving the formation/dissolution of an attached vesicle membrane diaphragm. Here j, is the steady state flux of labelled vesicles occurring in a given region. The corresponding steady state concentration difference of labelled vesicles driving the flux is given by the appropriate term in square brackets. The symbol II designates the average number of labelled macromolecules carried per vesicle. Solving equations (Al) to (A5) for each concentration difference and adding the resultant equations one obtains the permeability P: (A6)

l/P=[c(m)-c(e)]/~j=l/p~+I/p,+l!p,+l/p,+l/pd,

where the steady state requirement jf=jl=j2=j3=j~=j

has been employed. The various fluxes may be written as: jl: = k’: c(m) - k!. * c(m’)

(A7)

.j, = k,c(m’)-k-,c(O)

(A81

j* = -D(acpx) i

J3

=

= k3c(o)-k-3c(e’)

j”, = ti3 c(e’) - kd_3c(e)

[k,kJc(m’)-k-,k-,c(e’)]/(k-,fk,) D/v+k,k-,/(k-,+k,)

-

(A91 (AlO)

(Al 1) Introducing the

where equations (13), (14) were used in evaluating j,. assumptions of symmetry, i.e. kf = kd_i, ki = k-i* kd, 1 = kt3, k, 1 = k+ 3

642

B.

T.

RUBIN

and utilizing equations (Al), (A2), (A4, (A5) in conjunction with equations (A7), (AS), (AIO), (All), equation (A6) for the permeability reduces to: up = 2/d+2h

+llP,

641-9

where pf = nkd,

(A13)

pi = nk,

(A14)

PZ = n4DlG(k,/2)l(Dlv

(Al5)

+ kJ2)

and c7= [c(d) - c(e’)]/[c(O) - c(u)].

CA161

Substitution of equations (A13), (A14), (A15) into equation (A12) yields a convenient expression for the permeability, i.e. : nok’: k,/2

(A17)

(k’ +kl)a +

APPENDIX B

Short-time

Transient Solution

The diffusion system in the semi-infinite

case is given by:

aclat= D a2CjaX2,0 6 x G CO

(Bl)

with initial condition: c(.x, 0) = 0

O32)

c(q

033)

and boundary conditions:

(atn, cyst),=o = kl Introducing

t) = 0

c(m’, t) - k- 1c(0, t) -I-o{ac/ax),,

0.

(B4)

the dimensionless parameters 5 = tk- ,lD)x

V35)

z = (k? JD)t

036)

G = C/CT

(B7)

VESICULAR

TRANSPORT

PROCESS

643

MODFL

equations (Bl) to (B4) simplify to: aGjaT

= a5lag2,

0 G 5 G CO

G(5,O)

(W

(B9)

= 0

_c(oo,T) = 0 Y(adaT)c=O

@lo)

= Kl dm’.# T>-do,

0311)

~.)+(adat)~=O

where y = k- 1AT/D K,

Taking gives :

rhe Laplace transform

= k,/k-

1.

of equations

0312)

(Bt3) (BS), (BIO) and using (B9)

z(t, s) = B(s) e-s’“i where z(5, s) is the transform of ~(5, T); s is a complex variable; function. The Laplace transform of boundary condition (Bll) is

(Bl4) B(s) is a

(Bl5) In order to completely determine the function B(s) it is necessary to obtain an expression which links the unknown function c(m’, S) in the reaction region RI to the diffusion region R;. The function z(m‘, S) is determined as follows: From the labelled vesicle flux balance at (X = M’) the increase in the number of labelled vesicles per unit area per unit time is obtained, i.e. YS-(0,

s) = K,

-(d,

+-(o,

~,+(a-/ag)~=,.

p Bc(m’, q/at = kd,c(m)-k’f_,c(m’, t)-k,c(m’, Q-t-k-,c(O, t) (B16) where L) is the thickness of the region within R, containing attached vesicles with diaphragms intact (Fig. 2). Equation (B16) is subject to the initial condition c(x,O)=O, x>m. 0317) Employing dimensionless parameters (B6) and (B7) equations (B16) and (B 17) become

PO a&d,

T)/aT

= p1 (‘(m)-pp2 g(m’- T)+#,

_c(x, 0) == 0,

T)

x > 111

(I31 8) (B19)

where CB20)

Bo = pk-,lD PI = k;jk-

1

p2 = (kd.,+kJk-,.

The Laplace transform of equation (B18) subject to initial condition

(B21) (J322)

(B19)

644

6.

T.

RUBIN

yields the required general relation (~23) In equation (B23), r(m) is a constant denoting the proportion of labelled vesicles to total vesicles at (x = m). By the fast loading assumption c(m) = 1. Further, the validity of equation (B23) is not restricted to short times but rather extends over the entire time range. Substituting equation (B23) into equation (B15) and using equation (B14) gives : B(s) = u&s2 -t-z, s3’*+z,s+z, s”* +z‘J. (JQ4) Consequently, a<, s) = a0 e~S~~/~(s2+z1s3~2+z2s+z3.~1~2+24)

0323

where the constants MO= wIIYMI/Bok(~)

WV

Zl

=

l/r

(~27)

z2

=

P2lBo

z3

=

(B*/Bo)/r

z4

=

(P2180

(BW

+ l/Y

(~29) (B30)

-~,/B0)/%

Equation (B25) may be written as

-.~..

1

(B31)

(S1’2+CI*)(S”2fC13)(S1’2+M4)

where one has used the fact that: s2 -I- z1 s3j2 +z,s+z,s”*+z, =

(sl’*

+a,)(~“*

+B~)(s”*

+c(~)(s~‘~

+~4).

(B32)

The roots a,, u2, CI~,~1~are obtained from a solution of the equation s2+(a,+C12+C13+Cl4)S3’2+[C(1CL2+CIJa4+(CII+C12)(C13+Clq)]S+

= 0 (B33)

+[~,a2(a3f~14)+~3a4(~~+~2)]~~‘~+~1~~2~3~14

where clearly, z1 = a] +a2+c(j+a4

(B34)

z2

=

[~la2+C13C(4+(~,+~2)(~3+~4)]

(B35)

z3

=

[alcr2(cr3+a4)+c13a4(%

z4 = a1 a, cl3 a4.

+a2)]

W6)

(B37)

VESICULAR

Introducing

TRANSPORT

645

AIODEL

partial fractions equation (B3 1) becomes : a,

e-S”T A2

gf,

PRUC‘ES:,

s)

= s(s”2

+a,)

-

i

(P

~.

+-a*)

+

-2% (P+r,)

_

+

0-W

.LL (s”*+c(‘$)

>

where VW (B40) (B41) (B42) Combining

the known Laplace inverse transforms for Y-‘(cQ, e-S”T/s(s’i2+a,)j and ~-‘{~/(s~~~+c()) in accordance with convolution theorem one obtains the solution:

where

F,(T-0,) = - jJ AiRi eai2(r-e)erfc [q(~--@)~/~]

(B44)

i=2

F*(d) = (l/cc,) erfc [Q2tPi2]-(

I/K,) eat~++‘erfc

[5~20’~*+~~,8~~*] < B 0. (B45) Equations (B43), (B44) and (B45) determine the proportion of labelled to total vesicles as a function of dimensionless distance c and dimensionless time z in the diffusion region R;, 0 < x 6 co. The proportion of labelled vesicles to total vesicles at (x = m’) in reaction region R1 is obtained as follows: Equation (B38) is substituted into equation (B23) for ~(0, s). The result is:

. Utilizing

(B46)

partial fractions equations (B46) becomes

F(m, s)= (BlIPOk -t-_ (@olPo) _ > s(s+ /12/fio)

(B47)

where 111= ~2 = P3 = P4 =

-~~~2+~~3+P4) -A2/@2-Q -A3/@3-%) --A,/(~,--%).

W8)

VW (BW (B51)

646

B.

‘I-.

RUBIN

Equation (B47) is inverted via the convolution theorem using the known inverse transforms 2-l [l/s(~+fi~/j?~)] and .~Z’-~[l/(s~‘~ -t-cz)]. Thus,

EW, 4 = &/82k(N1

-4 (82’80)r]+ (cto//30) j G,(z - a)G,(o) do (B52) 0

where G,(T - a) = (&/&)[l

-e(aZ’Bo)(T-“rJ

G,(O) = - ‘JfJClicCieatzaerfc (~(,c”‘).

(B53) (B54)

i=l

Equations (B52), (B53) and (B54) yield the proportion of labelled to total vesicles at (x = m’) as a function of dimensionless time. From the experimental point of view it is of interest to know the proportion of labelled attached vesicles to the total number of attached vesicles in region R,, i.e. c(m) -I- c(m’, z) CA(T) = -(B55) 2(m) + CT(d) where cT(m), c7’(m’) are the total number of vesicles per unit volume at (x = m) and (x = m’) respectively. For cT(m) = cr(nt’) = cT requires that kf = k‘! i. In this case equation (B55) reduces to EA(4 = (1 P>C,c(~~~J +_c(~~f, 711 (B56) where equation (B7) has been employed. Substitution of equations (B52), with (B53) and (B54) into (B56) gives the desired expression. ~~(7) = (1/2)s(m){l

+(BJ/3&1

-e-'PziPo)r])

-

-ca2’ao)cz-“)} {,iPi~i

APPENDIX Rationalization

eaizoerfc (cQ~J~/‘)} da. (B57)

C

of Choice of y

From equation (43) the choice y = $ requires that & = D/4k- ,. Using kinetic molecular theory the diffusion coefficient in Fick’s first law is interpreted as (Hughes, 1964): D=206, x>O (Cl) where G is the average component of the vesicle molecular velocity due to diffusion in the direction perpendicular to the plane x = 0; 6 is the mean free path. Both 0 and 6 in the case of vesicle motion in the cytoplasm are functions of the motion of solvent molecules. The value o in the positive

VESICULAR

TRANSPORT

PROCESS

MODEL

647

x direction only is: ii = $33) x>o. The factor + arises from the equal probabilities of both forward ward directions. The system constraint of chemical reaction in necessitates the absence of diffusion. The velocity in the negative is ii = k-1, x < 0. The ratio of equations (C2) : (C3) along with equation (Cl) relation: ii/ii = 0/2k-, = D/4k-,A Consequently, select AT = &G/ii) = D/4k-,.

tc21

and backregion R, x direction (C3) yields the (C4) (W