13, 4154%

CRYOBIOLOGY

(1976)

A Membrane Model Describing the Effect of Temperature on the Water Conductivity of Erythrocyte Membranes at Subzero Temperatures R. L. LEVIN

E. G. CRAVALHO

AND

Cryogenic Engineering Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02iS9 AND

C. E. HUGGINS Low Temperature Surgical Unit, Harvard Medical School, and Blood Bank and Transfusion Service, Massachusetts General Hospital, Boston, Massachusetts 02116

Since in most cells the major fraction of cell volume is occupied by water, the amount of intracellular water present during freezing and thawing may be altered drastically by events occurring in the extracellular medium. For instance, ice may form outside the cell during freezing, thereby increasing the concentration of solutes in the extracellular medium. As a result, the chemical potential of water inside the cell will be higher than that outside, and cell water will be expressed through the membrane to maintain osmotic equilibrium. On the other hand, it is possible that the temperature of the cell suspension decreases more rapidly than the cell can respond osmotically, thereby permitting the water inside the cell to nucleate before significant mass transfer can occur. Obviously, the final intracellular water content is vastly different in these two cases, and cell survival may be likewise affected (see Fig. 1). In recent years, there have been several theoretical studies of the kinetics of this water loss during freezing (2, 8, 9, 17, 18). These models are useful in that they characterize the response of a cell to the therReceived

June 2, 1975.

modynamic events associated with the freezing process in terms of certain cell parameters, most notably the water conductivity of the cell membrane. With the aid of these models and numerical values for the cell parameters, it is now possible to predict the transport of cell water during freezing. More recently, these models have been coupled with models for the kinetics of ice nucleation to quantify from a theoretical point of view the conditions under which ice will form intracellularly (19). Because of the clinical importance of human red blood cell freezing (5) and because of the relative abundance of biophysical data for human red cells, most of these theoretical studies have utilized the human red blood cell (RBC) as a model cell system. Much of the understanding gained from these studies can be generalized to other cell systems as well without introducing significant errors. In the present work we shall also make use of the RBC in this fashion. As recognized by the early investigators, the behavior of RBCs during freezing depends primarily upon the nature of the water conductivity of the erythrocyte 415

Copyright All rights

Q 1976 by Academic Press, Inc. of reproduction in any form reserved.

416

LEVIN,

CRAVALHO

membrane. In nearly every case, the water conductivity of the membrane was assumed to depend on temperature. On the basis of data obtained by Jacobs (7), this temperature dependence has been described by an expression of the form k = lc, exp [6(T - T,)],

Cl1

AND HUGGINS

experimental difficulties associated with the method and that at cooling rates of 2”C/min chemical equilibrium is probably established by solute transport into the cell rather than water transport out of the cell, we believe the data indicate that for the higher cooling rates, B ,< - lOO”C/min, more water seems to remain within the cell than the results of Silvares or Mazur predict. In the present paper we shall attempt to develop physical models of water transport in homogeneous membranes and to determine from these models the possible effects of temperature on the water conductivity of a membrane. We shall then use these results to evaluate the kinetics of cell water loss from erythrocytes during cooling at subzero temperatures.

where k, varied between 0.15 and 5.0 p3/fiz min atm, b = 0.0325°K-1, and T, = 293°K. Some authors (2, 17, 18) have modified Eq. [l] to include the effects of the osmolality of the external medium on the water conductivity, but the temperature dependence of Eq. [l] remains intact. (See Appendix for nomenclature.) In spite of the many studies conducted on the water conductivity of human erythrocytes (for a comprehensive review see Sha’afi and Gary-Bob0 (16)), data are E$ect of Water Conductivity on Water available for only a limited range of temTransport peratures above 0°C (Jacobs (7), 0-3O”C, Let us first consider the effect of different and Vieira et al. (21), 5-37°C). Since the published values for the membrane contemperature range of interest for the ductivity on the amount of water retained freezing of biological cells is below OT, it within an erythrocyte during cooling at has been necessary in the thermodynamic subzero temperatures. For this purpose it models to extrapolate existing water conis convenient to rewrite the conductivity ductivity data to subzero temperatures. expression used by Silvares (17, 18) (Eq. Even apart from possible changes in [l]) in the following form: membrane water conductivity due to phase changes in the membrane at subzero teml = fi,kRT = ,& peratures or due to increasing concentrations of intracellular or extracellular solutes [2] X exp{ -EL(T) (+ - &)}, (1, 3, 14) as the cell suspension freezes, we believe that this procedure is of queswhere IZ”T~corresponds to the effective cell tionable validity since the value of b tends conductivity at T, = 293.15”K (20°C) and to increase as the temperature is lowered Ek(T) is the apparent activation energy (10). Furthermore, the experimental refor the permeation process. It should be sults of Watson (22), who used a photonoted that in writing Eq. [2] we have graphic technique to measure cell volume assumed that there is no dependence of during the actual freezing of RBC suspenthe conductivity on the concentration of sions on the stage of a cryomicroscope, any particular solution species. Some inindicate that over 80% of the initial intravestigators (1, 3, 14) have indicated a cellular water remains inside the cell, even dependence of the cell water conductivity when the cell is cooled at rates as low as on the extracellular solute concentration l.S’C/min (see Fig. 2). (1). We have chosen to express k as a Even though the validity of Watson’s function of temperature alone [cf. Mazur data may be questioned because of the (lo)] for two reasons. First, the expcri-

KINETICS

OF WATER

TRANSPORT

mental data concerning the dependence of g on extracellular osmolality was obtained over only a small range of osmolalities (200-500 mOsm) as compared to the range of osmolalities encountered during cell freezing (300-10,400 mOsm). Second, there is disagreement in the literature over whether the observed behavior of E on the osmolality of the extracellular solution is simply a result of the rectification of water flow by the cell membrane (1, 3) ; i.e., or Einflux < Lfflux but & # f(osmolality), an actual variation in k due to changes in the properties of the membrane caused by the concentration of the solutes in the extracellular medium. Using Silvares’ nonideal solution model for RBC cooling and assuming zero supercooling (AT, = 0), we determined the effect of assuming different values for ,&Tsand Ek on the percentage of initial cell water that remained within the cell at subzero temperatures (see Table 1). A plot of In E versus inverse t’emperature is shown in Fig. 3 for each of the above cases. The effects of these various values on the water transport during freezing can be seen in Figs. 4 and 5. In Fig. 4, the percentage of initial cell water remaining within the cell at subzero tomperatures for a constant cooling rate of B = -5OOO”C/min is plotted for Cases a through e. In Fig. 5, the percentage of initial cell water remaining within the cell at subzero temperatures is plotted as a funcTEMPERATURE

,T(“C)

-IOOO°C/min

=E

-

IN ERYTHROCYTES

417

TEMPERATURE, TW)

5

; 0.6~ g . 2

l 920”Wmin A 730’C/min 0 92’Chin 0 1.9’Wmin

0.4 -

2

-

d

' 0.2 -

"'

270

260

250

240

TEMPERATURE,T?K)

FIG. 2. Experimental values for intracellular water volume of red blood cells cooled at constant rates, as determined by Watson (22).

tion of the cooling rate (- lOO”C/min > B > - 2000”C/min) for Case f. From Figs. 4 and 5 we conclude that for a given cooling rate, the smaller the value of the water conductivity of the cell membrane at a given subzero temperature, i.e., the smaller the value of the pre-exponential factor XT, and the larger the value of the apparent activation energy Ek, the larger the percentage of cell water retained at subzero temperatures. Also, for constant values of LQ and Ek, the faster the cooling rate, the larger the percentage of cell water retained at subzero temperatures. That is, fast cooling rates and large values of the apparent activation energy enhance retention of cell water during freezing. On the basis of these observations and the experimental data of Watson (22) and Diller (2) (see Fig. 2), we believe that the apparent activation energy for the water conductivity of erythrocyte membranes at subzero temperatures is probably greater than the 3.9-6.0 kcal/g mol values reported by Vieira et al. (21) in the temperature range

5-37°C. TEMPERATURE,T(“K)

FIQ. 1. Volume of intracellular water of red blood cells cooled at constant rates without supercooling, as predicted by Silvares (17, 18) and Mazur (9).

These values for the activation energy are perfectly valid in this temperature range, but there is no reason to expect them to remain so at subzero temperatures.

LEVIN,

418

CRAVALHO TABLE

Case

AND

HUGGINS

1 Ek (kcal/g

i(lO-a

mol)

cm/see) Human RBC; osmotic JEow, Ref. (16) Human RBC; tracer difusive pow, Ref. (16)

3.9

a

17.3

b

5.3

c

5.3

See Case b

12.7

d

5.3

See Case b

14.6

e

5.3

See Case b

16.0

f

1.12

Artificial lecithincholesterol-hydrocarbon bilayer; osmotic flow, * Ref. (13)

14.6

6.0

Human RBC; osmotic &w, 537”C,5 (Ref. (21) Human RBC; tracer diffusive jlow, 5-37”C, Ref. (21) Artificial lecithincholesterol-hydrocarbon carbon bilayer; osmotic flow, 749”C, Ref. (12) Artificial lecithincholesterol-hydrocarbon bilayer; osmotic flow, flow, 20-37°C Ref. (13) Self-diffusion of water in single crystals of ice, Ref. (6) See Case d

a The apparent activation energy computed by Viera et al. has been increased by approximately 600 Cal/g mol to take into account conversion of data from cubic centimeters per dyne-second to centimeters per second. b Value for &P, has been corrected for the fact that artificial bilayer of Ref. (13) was only 45 H thick while RBC membranes are approximately 75 4 thick by assuming that the conductivity is inversely proportional to membrane thickness (see Eq. [as]).

In the following section we use a homogeneous membrane model to discuss some physical mechanisms that may explain the departure of the water conductivity from the values obtained by extrapolation of ambient temperature data to subzero temperatures. Physical Model of Water Transport (Homogeneous Membrane) The movement of water across homogeneous membranes has been treated from a number of different points of view. Triiuble (20) discussed the problem in terms of thermal fluctuations in the hydrocarbon chains of the membrane phospholipids. The thermal motion of the hydrocarbon chains results in the formation of conformational isomers, so-called kink-

isomers of the hydrocarbon chains. “Kinks” are pictured as being mobile structural defects which represent small, mobile free volumes in the hydrocarbon phase of the membrane. According to Trauble, small molecules can enter into the free volumes of the kinks and migrate across the membrane together with the kinks. Zwolinski, Eyring, and Reese (23) discussed this problem from the point of view of absolute reaction rate theory. The central idea of this theory is that the water dissolves within the membrane as discrete molecules and moves across by diffusion. In this instance the diffusion process is considered to consist of a series of successive “molecular jumps” of length X from one equilibrium position to another within the membrane, with the rate of jumping determined by three specific rate constants

KINETICS

OF WATER

TRANSPORT

defined as follows :

IN ERYTHROCYTES

rms = rmain = rmsout, where T,,,,~~and rmsoUt are the rate constants for passage through the intracellular and extracellular membrane -+ solution interfaces, respectively (see Fig. 6). The results of both these approaches are the same. Since we are concerned with possible changes in the apparent activation energy Ek and/or pre-exponential factor w,, with temperature for the water conductivity of an erythrocyte membrane, i.e., with the energetics of the molecular motions

rm = rate constant for diffusion within the membrane, r *m = rate constant for passage through the solution t membrane interface, = rate constant for passage through the rms membrane + solution interface. It has been assumed that rsrn = rsmin = rsm0Ut, where rsmin and rs moUt are the rate constants for passage through the intracellular and extracellular solution + membrane interfaces, respectively, and

Tt’C, 20

IO

I

419

0

I

-10

I

-20 I

3.9

-30 I

4.0

103/T t’K-‘1 FIG. 3. Cell membrane water conductivity.

4.1

42

LEVIN,

420 TEMPERATURE,

CRAVALHO

AND HUGGINS

unit area of membrane per second:

T(OC)

J,=----.-

1 dllwain A dt Csout

=

(2/f-smv

+

-

Csin

[(rm,/r,,)

(m/r,x)]’

II31

where m is the number of jumps required for a water molecule to move across the membrane, and Cein and Csout correspond respectively to the effective intracellular and extracellular water concentrations on either side of the membrane. Csin = $

l’.

270

.&

In ~,a in,

TV .

CsoUt= 2.&.t 260

TEMPERATURE,

250

c41 In xyaOUt.

240

TI’K)

FICA.4. Volume of intracellular water of red blood cells cooled at a constant rate of 5000”C/min for different values of cell membrane water conductivity (Cases a-e).

rather than with a detailed molecular mechanism for the permeation process, we have chosen to use the approach of Zwolinski et al. in our discussion of the water conductivity of membranes. Now in the steady state the number of molecules passing per unit time through a unit area of surface must be equal to the difference in the number of molecules entering and leaving toward the right and the number of molecules entering and leaving toward the left for any single equilibrium position within the membrane (see Fig. 6~). Consequently, for systems in which the intracellular and extracellular media are of sufficiently low viscosities, aqueous solutions, so that the main i.e., resistance to diffusion will be offered by either the solution-membrane interfaces or by the bulk portion of the membrane, Zwolinski derived the following relation for the steady state flux, J,, representing the net flow of water out of the cell per

Hence, dq,* in -=-dt

A ii, a in - qioutIn xWaoUt



(.$I :)‘i

[ (r Jrs

m)(m/r ,A)]’

“’

We can now define an effective membrane conductivity for a homogeneous membrane without any surface boundary layers by

I31 TEMPERATURE.

TL”C)

260 TEMPERATURE,

250

240

T(‘K)

Fw. 5. Volume of intracellular water of red blood cells cooled at constant rates for Case f. kmoticoow corresponds to Case a.

KINETICS

(0)

OF WATER

TRANSPORT

IN ERYTHROCYTES

INTRACELLULAR SOLUTlON T,

421

EXTRACELLULAR SOLUTION

X$”

T, Xy+

I

I

Rin sm

out Rsm

Rm

(b) I

I I

I I

I ‘ms *

_ rsm

‘ms

rsm-

I DISTANCE

FIQ. 6. Theoretical

so that the rate equation port, Eq. [S], becomes cl?jwain

dT

energy profile

for permeation

for water trans-

--k"A = -fi,B [-in In ZWoin - Q&t In XWaOut]. [7]

The functional dependence of the water flux J on the driving force (Csout - Cain) is similar to Ohm’s Law for the flow of electrical current driven by a difference in electrical potential (voltage). Thus the denominator on the right-hand side of Eq. [3] is the resistance offered by the membrane system to the flow of water. The electrical analog for this water circuit is shown in Fig. 6b. In this situation, the resistor R, represents the resistance of the

through

a homogeneous

membrane.

bulk portion of the membrane to the flow of water while the resistors Rsmin and RsmoUt represent, respectively, the resistances of the intracellular and extracellular solution-membrane interfaces to the flow of water. Since AC,

Jw = R in + R, + R,,noUt *m for this equivalent network, comparison with Eq. [S] allows us to write the following relations for R,, Ramin,and R,,,,o"~.

Rsmin= Rsnlout = l/r& and

i31

LEVIN,

422

CRAVALHO

AND HUGGINS

(0) DIFFUSION RATE-LIMITED

(b) ABSORPTION

RATE-LIMITED

DISTANCE

I/T (cl SURFACE

RATE-LIMITED

FIG. 7. Theoretical energy profile and cell membrane conductivity for (a), diffusion rate-limited permeation; (b), absorption rate-limited permeation; and (c), surface rate-limited permeation through a homogeneous membrane.

, Hence, the inverse of the effective membrane conductivity, l/E (Eq. [S]), can be thought of simply as resulting from the effective membrane resistance to the flow of water. By definition, the diffusion coefficient,

membrane interface

Di, is given by

Eq. [S] for the effective membrane ductivity can be rewritten as

Di = riX2,

PI

and the distribution or partition coefficient, K =, is defined as the ratio of the rate con-

stants for passage through

the solution-

K,

= rsm/rms.

WI

If the membrane thickness is given by

Cl11

LsmX,

l/E = (2X/D,,)

+ (L/Pm),

conCl21

where D,, is the diffusion coefficient associated with the passage of water across the

KINETICS

solution-membrane

OF WATER

TRANSPORT

interface,

TEMPERATURE,TW

D BIII= r,,x2,

Cl31

and P, is the permeability constant passage through the membrane itself,

Pm = K,(r,X2)

= K,D,.

Also, Eq. [S] can be rewritten

423

IN ERYTHROCYTES

for

Cl41

as

R Srnin = R 8nlout = X/D,, and

250

Cl51 R, = LIP,.

Furthermore, according to the theory of absolute reaction rate processes (see Glasstone et d. (4)), a specific reaction rate constant for any process is given by an expression of the form

ri =

Kj

gh exp(-

AFi/RT),

[lS]

240

TEMPERATURE, T(‘=K)

FIG. 9. Volume of intracellular water of red blood cells cooled at constant rates for surface rate-limited cell membrane water conductivity where &irr = 1.58 X 102exp(-6.O/RT) cm/set and &,a = 3.94 X lo6 exp( - IO/RI’) cm/set. &osmoticnow corresponds to Case a.

is the transmission

coefficient,

and RT/

Nob is a frequency factor. Since AF; = AH; - TA&,

where AFi is the change in free energy, Ki

Cl71

where AHi and A& are, respectively, the apparent enthalpy and entropy of activation for the ith process. Then from Eq. [9] the diffusion coeflicient can be written as

T (“C)

Di = n($$)k2exp(%) X exp

. WI

If we assume that the properties of the membrane do not vary greatly with temperature, then both AHi and ASi can be considered to be essentially constant, i.e., independent of temperature. Therefore, the differential of Eq. [lS] with respect to inverse temperature becomes

I03/T

(“K-‘)

FIG. 8. Surface rate-limited cell membrane water conductivity _for &irr = 1.58 X lo2 ex (-6.0/M’) cZ/ Cm/SeC and k&s = 3.94:X lo5 exp( - lO/RT) sec.

or, if we define an apparent activation energy for the ith process such that

Ei = AH< -I- RT, then

d In Dc/a(l/T)

= - EJR.

I391

LEVIN,

424

CRAVALHO

Since AHi - 3-20 kcal/g mol and RT - 0.6 kcal/g mol, let us assume that Ei is approximately independent of temperaafter integrating the ture.’ Therefore, above expression,

where D; is a constant independent temperature (4). Consequently, exp(-EE,,/RT),

of

[2la]

where E., is the apparent activation energy associated with the “absorption” of a water molecule at the membrane surface; i.e., it is the apparent activation energy for the passage of a water molecule between a position just on the outside of the membrane surface, the adlayer, and a position just on the inside of the membrane surface, the ablayer ; and D,

= D,o exp(-ED,/RT),

[21b]

where ED m = E, i&he apparent activation energy associated with the diffusion of a water molecule through the interior region of the membrane, i.e., it is the apparent activation energy associated with the passage of a water molecule through the bulk of the membrane from the ablayer on the intracellular side of the membrane to the ablayer on the extracellular side of the membrane. l For D = 5.0 X lo+ cmz/sec at T = 293.15”K with AH = 4.0 kcal/g mol and E = 4.6 kcal/g mol, then, 1. At 2’ = 273.15”K, D(Eq. [lS]) D(Eq. [20]) 2. At T = 253.15”K, D(Eq. [lS]) D(Eq. [20])

HUGGINS

Similarly,

(Eq. [lo]) K,

the partition coefficient can be written as

= (~>-p(Assm

= 2.818 X 10e6 cm2/sec, = 2.805 X 10d5 cme/sec. = 1.459 X 10-b cme/sec, = 1.436 X 10-s cm2/sec.

Although the difference between the values of D predicted by Eqs. [lS] and [ZO] increases with decreasing temperature, it is still less than 2% as the eutectic point for an aqueous sodium chloride solution is approached.

i

X exp -

WI

Di = D; exp (- Ei/RT)y

D., = D,c

AND

K,

“-‘>

AH,,- All,, RT

>

or K, where KC and

= K,Oexp(-EkJRT), is independent

[22]

of temperature,

E Km - Es, - En,, = AH,,

- AH,,

[23]

is the enthalpy of solution associated with the partition of a water molecule between the aqueous intra- or extracellular solution and the interior of the membrane. Hence, the permeability constant (Eq. [14]) can be written as P,

= Kmo+D,oexp

Earn

+

RT

ED, >

or P,

= P,Oexp(-EEp,/RT),

[24]

where EP,

=

Earn

= AH,,

+

ED,

- AH,,

-I- AH,

-I- RT

C251

and Pm0 = K,o.D,o.

WI

The theoretical energy profiles for permeation through a homogeneous membrane without any surface boundary layers is shown in Fig. 6 where the apparent activation energies for the various steps comprising the overall permeation process are plotted versus distance. For natural membranes bathed by aqueous solutions it should be noted that although the partition coefficient K, for most substances is much less than unity [K, = (r,,/r,,) < 11, the heat of solution EIC~ can be positive or negative (see Eq. [23]). If the enthalpy of solution associated with the partition of a molecule be-

KINETICS

OF WATER

TRANSPORT

tween the aqueous solution and the interior of the membrane is positive (EKE = Es, - E,, > 0), the partitioning process is endothermic, and the amount of water in solution within the membrane will decrease with decreasing temperature. On the other hand, if the heat of solution is negative (ERR = E,, - E,, < 0), the partitioning process is exothermic, and the amount of water in solution within the membrane will increase with decreasing temperature (see Eq. [ZZ]). Figure 6c shows the partitioning.of a water molecule between the aqueous solution and the membrane interior as an endothermic process. This is the case for the solubility of water in bulk hydrocarbons [Schatzberg (15)] such as those used in the artificial lipid bilayers of Price and Thompson (12) and Redwood and Haydon (13). In any event, whether the heat of solution EII,,, is positive or negative, Ep, and ED, will both be positive definite (EP,,, > 0 and ED, > 0) for the passage of a water molecule through a membrane. Two limiting cases for water permeation through a homogeneous membrane immediately present themselves. In the first case, diffusion within the membrane itself is the rate-limiting step. In this instance rm < rsm or

Pm < D,,,

c271

and the membrane conductivity (Eq. [12]) is given by an expression of the form

X exp(-EEp,/RT).

[28]

An energy profile curve illustrating diffusion-limited permeation is presented in Fig. 7a. In the second case, the rate-limiting step is the passage of water across the solution + membrane interface, and the permeation process is absorption ratelimited. For this case rm>>rBm

or

Pm>> D,,

I291

425

IN ERYTHROCYTES

and the membrane conductivity (Eq. [12]) is given by an expression of the form

X exp(-EE,,/RT).

[30]

When the transport of water is rate limited by the absorption process, the conductivity will be found to be independent of the partition coefficient K, and the thickness of the membrane L. An energy profile curve illustrating absorption-limited permeation is presented in Fig. 7b. For a given set of data for the membrane conductivity, it is impossible to determine which of these two physical mechanisms is operative. If for the moment we disregard any temperature-induced phase changes within the membrane itself, a linear relationship between In k and l/T prevails for both the diffusion-limited and the absorption-limited permeation processes (see Figs. 7a and b). In each case the slope of the straight line is proportional to the apparent activation energy of the rate limiting process : (i) Diffusion-limited

permeation,

from

Eq. Pf41, a(ln @/a(l/T) (ii) Absorption-limited

= - Ep,/R. permeation,

from

Eq. C301, a(lnE)/a(l/T)

= - E,,/R

Thus if experimental data of In k vs l/T exhibit a linear relationship, it is impossible to determine whether the water transport was diffusion rate-limited or absorption rate-limited solely from the slope (activation energy) alone. Discrimination between these two mechanisms is possible only after we determine the dependence of the membrane conductivity on the partition coefhcient Km and membrane thickness L but, unfortunately, this determination is extremely difficult, if not impossible, for water permeation through natural membranes.

426

LEVIN,

CRAVALHO

Between these two limiting cases there also exists an intermediate case for which

AND

HUGGINS

(4) report 3 A < X < 10 A with X = 5 A as a representative value. Tracer diffusion studies (16, 21) yield values for Ep, of rm 2 rms or P, 2 D,. c311 6.0 kcal/g mol and P,o/L of 1.58 X lo2 In this instance, the membrane conduccm/see. With K, = 0.003 (Naccache and tivity is given by Eq. [12] and the permeaSha’afi (11)) and from the definition of tion process is said to be surface ratethe permeability coefficient (Eqs. [24] limited. An energy-profile curve illustrating and [26]), Dmo = 3.94 X 1O-2 cm2/sec. this situation is presented in Fig. 7~. Therefore, if we assume D, m” - D,“, For this intermediate case, the situation Eq. [32] becomes is somewhat more tractable than the two 1 ; = limiting cases. The relationship between In & and l/T is nonlinear and does not k possess a unique value for the activation energy. In effect, the activation energy 1 + f c331 varies over the temperature range of 3.94 X lo5 exp interest, but the plot of Ink” vs l/T still has a negative slope (first derivative) where we have assumed E,, = 10 kcal/g throughout the range of temperatures. From the energy profile curve for such a mol. Figure 8 shows the results of submembrane, it is clear that E,, > Ep, (cf. stituting values for the temperature into Eq. [33]. Note that as l/T -+ 0, L -+ &dirr, Fig. 7c) otherwise the membrane will and that as l/T -+ cc, K”+ Labs. always be diffusion rate-limited. This As an example of the influence of a means that for typical values of the prevariable activation energy on the cell exponential terms, the second derivative of In x vs l/T is always negative, and the water volume during freezing, we have slopes of the tangents to this curve for used Eq. [33] to describe the temperature high and low temperatures represent the dependence of the water conductivity in our thermodynamic model. The results activation energies for the rate-limiting of this analysis are shown in Fig. 9 where processes dominant at these temperature we have plotted the relative cell water extremes. Figure 7c shows schematically volume as function of temperature during that for this model the transport of water freezing for various cooling rates. Note is diffusion rate-limited at high temperatures and absorption rate-limited at low that in the early stages of freezing, water transport is diffusion rate-limited and the temperatures. This latter point can be presented more results are similar to Fig. 1. As the temclearly in terms of a typical example. If perature decreases, the membrane conductivity becomes absorption rate-limited we substitute Eqs. [2la] and [24] into and in effect shuts off the flow of water. Eq. [12], we obtain For a cooling rate of 5000”K/min, the 1 mX cell water volume at 250°K is 48’% of its k= = pm0 eXp(-EEp,/RT) initial value for this model versus 15% for the case of simple osmotic flow. Obviously, 2x L-321 the cell water volume is strongly dependent +D *=” exp(-E,,/RT) on the activation energies assumed in Eq. or [33], but the possibility that the water conductivity of a membrane exhibits a l/l = (l/Wdiff) + (l/~abs)temperature dependence, such as that Typically for human red cells L N mX shown in Fig. 8, poses questions about the is on the order of 75 A, and Glasstone et al. validity of simply extrapolating to subzero

KINETICS

OF WATER

TRANSPORT

temperatures permeation data obtained at higher temperatures as has been done by many investigators. The results of this simple surface-limited permeation model for homogeneous membranes also provide a possible physical explanation for our conclusion that the apparent activation energy for the water conductivity of erythrocyte membranes at subzero temperatures is probably greater than the values obtained at temperatures above 0°C.

IN ERYTHROCYTES APPENDIX

Nomenclature A B b

E SUMMARY

Thermodynamic models show that the loss of intracellular water from human erythrocytes during freezing depends heavily upon the water conductivity of the erythrocyte membrane. These calculations, which are based on the simple extrapolation of ambient conductivity data to subzero temperatures, show that more than 9570 of cell water is transferable during freezing, whereas experiments show that at least 20% of cell water is retained. A study of the effects of different published values for the membrane water conductivity on cell water retained during freezing shows that this discrepancy may be a consequence of the simple extrapolation procedure. For a homogeneous membrane system, absolute reaction rate theory was used to develop a surface-limited permeation model that includes the resistance to the flow of water not only through the interior region of the membrane but also across possible rate-limiting barriers at the solutionmembrane interfaces. The model shows that it is unlikely that a single ratelimiting process dominates water transport in the red cell as it is being cooled from ambient to subzero temperatures. The effective membrane conductivity at subzero temperatures could possibly be much lower than a simple extrapolation of existing data would predict. With the aid of this model analytical predictions of intracellular water during freezing are more consistent with experimental observations.

427

ED EK Ek EP F

H h J K k rz” k, kc L m No P R Ri r s T T, t VW X

AT. K

=

surface area = cooling rate (‘C/min) = permeability temperature COefficient = concentration diffusivity (cm2/sec) diffusivity pre-exponential constant activation energy (kcal/g mol) diff usivity activation energy partition coefficient activation energy cell membrane conductivity activation energy (kcal/g mol) permeability activation energy free energy enthalpy Planck’s constant flux (g mol/cm2 set) partition coefficient cell membrane permeability (g mo12/p5 atm min) cell membrane conductivity (cm/see) cell membrane permeability at temperature T, cell membrane conductivity at temperature Tg cell membrane thickness number of molecular jumps within membrane Advogadro’s number permeability (cm2/sec) universal gas constant (1.987 Cal/g mol “K) resistance for ith process rate constant entropy absolute temperature (“K) reference temperature (293.15’K) time partial molar volume of water = mole fraction = degree of supercooling = transmission coefficient

428

x TI,

LEVIN,

CRAVALHO

= molecular jump distance within membrane = number of molecules (g-mol) = osmotic coefficient

Subscripts

abs diff i in m ms

= = = = = =

out

= = surface = sm = S

W

=

absorption rate-limited diffusion rate-limited ith process intracellular membrane interior membrane ---) solution face extracellular solute surface rate-limited solution --) membrane face water

inter-

inter-

Superscripts

in out a!

= intracellular = extracellular = liquid phase REFERENCES

1. Blum, R. L., and Forster, R. E. The water permeability of erythrocytes. Biochim. Biophys. Actu 203, 419-423 (1970). 2. Diller, K. R., Crsvalho, E. G. and Huggins, C. E. An experimental study of intracellular freezing in erythrocytes. Presented at the Fifth International Cryogenic Engineering Conference, May 7 to May 10, 1974, Kyoto, Japan; Med. Biol. Eng., in press. 3. Farmer, R. E. L., and Macey, R. I. Perturbation of red cell volume: Rectification of osmotic flow. Biochim. Biophys. Acta 196. 53-65 (1970). 4. Glasstone, S., Laidler, K. J., and Eyring, H. “The Theory of Rate Processes.” McGrawHill, New York, 1941. 5. Huggins, C. E. Practical preservation of blood by freezing. In “Red Cell Freezing,” A technical workshop presented by Committee on Workshops of the American Association of Blood Banks, pp. 31-53, 1973. 6. Itagaki, K. Self diffusion in single crystals of ice. J. Phys. Sot. Jap. 19, 1081 (1964).

AND

HUGGINS

7. Jacobs, M. H., Glassman, H. N., and Parpart, A. K. Temperature coefficient of hemolysis. J. Cell. Comp. Physiol. 7, 197-225 (1935). G. Ali. Kinetics of water loss from 8. Mansoori, cells at subzero centigrade temperatures. Cryobiology 12, 34-45 (1975). 9. Mazur, P. Kinetics of water loss from cells at subzero temperatures and the likelihood of intracellular freezing. J. Gen. Physiol. 47, 347-369 (1963). 10. Mazur, P. The role of cell membranes in the freezing of yeast and other single cells. Ann. N. Y. Acad Sci. 125, 658-676 (1975). 11. Naccache, P., and Sha’afi, R. I. Patterns of non-electrolyte permeability in human red blood cells membrane. J. Gen. Physiol. 62, 714-736 (1973). 12. Price, H. D., and Thompson, T. E. Properties of liquid bilayer membranes separating two aqueous phases : Temperature dependence of water permeability. J. Mol. Biol. 41, 443457 (1969). 13. Redwood, W. R., and Haydon, D. A. Influence of temperature and membrane composition on the water permeability of lipid bilayers. J. Theor. Biol. 22,1-8 (1969). 14. Rich, G. T., Sha’afi, R. I., Ronaldez, A., and Solomon, A. K. Effect of osmolarity on the hydraulic permeability coefficient of red cells. J. Gen. Physiol. 52, 941-954 (1968). of water in several 15. Schatzberg, P. Solubilities normal alkanes from CT to Cics. J. Phys. Chem. 67, 776-669 (1962). 16. Sha’afi, R. I., and Gary-Bobo, G. M. Water and nonelectrolytes permeability in mammalian red cell membranes. In “Progress in Biophysics and Molecular Biology” (J. A. V. Butler and D. Noble, Eds.), Vol. 26. Pergamon Press, Oxford, 1972. 17. Silvares, 0. M. “A Thermodynamic Model of Water and Ion Transport Across Cell Membranes during Freezing and Thawing: The Human Erythrocyte,” Sc.D. Thesis. Dept. of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., 1974. E. G., Toscano, 18. Silvares, 0. M., Cravalho, W. M., and Huggins, C. E. The thermodynamics of water transport for biological cells during freezing. J. Heat Transfer, Trans. ASME 97, 582-588 (1975). 19. Toscano, W. M., Cravalho, E. G., Silvares, 0. M., and Huggins, C. E. The thermodynamics of intracellular ice nucleation in the freezing of erythrocytes. J. Heat Transfer, Trans. ASME 97, 326-332 (1975).

KINETICS

OF WATER

TRANSPORT

20. Trtiuble, H. The movement of molecules across lipid membranes: A molecular theory. J. Membrane Riol. 4, 193-208 (1971). 21. Vieira, F. L., Sha’afi, R. I., and Solomon, A. K. The state of water in human and dog red cell membranes. J. Gen. Physiol. 55, 451-466 (1970). 22. Watson, W. W. “Volumetric Changes in Human

IN ERYTHROCYTES

429

Erythrocytes during Freezing at Constant Cooling Velocities,” S. M. Thesis. Dept. of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., 1974. 23. Zwolinski, B. J., Eyring, H., and Reese, C. E. Diffusion and membrane permeability. J. Phys. Chrm. 53, 1426-2453 (1949).

A membrane model describing the effect of temperature on the water conductivity of erythrocyte membranes at subzero temperatures.

13, 4154% CRYOBIOLOGY (1976) A Membrane Model Describing the Effect of Temperature on the Water Conductivity of Erythrocyte Membranes at Subzero Te...
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