13, 147-152

CRYOBIOLOGY

On

the

N.

Problem

S. PUSHKAR,

(1976)

of Dehydration during Freezing Y. A. ITKIN, AND

Instit&e

for

Problems

of Cryobiology

and intracellular of Cell Suspensions1

V. L. BRONSTEIN, Y. V. KOZMIN

and Cryomcdicine,

The analysis of physical-chemical phenomena accompanying the low temperature preservation of biological systems is important in studying the effects of different freezing conditions on their viability. The majority of investigat.ors (1, 9, 10, 12, 14) believe that extra- and intracellular crystallization, and the process of cell dehydration that is closely connected with it, is one of the most important factors strongly influencing the success of cryopreservation. Mazur (9) has constructed a mathematical model of the process of cell dehydrat,ion during freezing that has made great contributions to the development of concepts of the condit,ions of cell freezing. The kinetic equation obtained in this work was analytically investigated by Ling and Tien (5) who found asymptotic solutions of this equation for some practical cases. Since Ling and Tien (5) made only a mathematical investigation of the equa-

147 0 1976 by Academic Press, Inc. of reproduction in any form reserved.

GORDIYENKO,

Street,

Kharkov

93, USSR

tion they failed to advance the physical understanding of the phenomenon in comparison with t.he work by Mazur, and especially in the problem of the causes of intracellular crystallization. Mazur’s model does not describe the thermodynamic state of protoplasm, but changes in cell volume: This has led to the concept of intracellular ice growth through the membrane as a cause of intracellular crystallization. Theoretical consideration of ice permeation through the cell membrane cited in works by Ling and Tien (5) and Viad (15) is problematic, since there is no adequate model of the cell membrane (4) and is not necessary for an explanation of the phenomenon of intracellular crystallization (1). The penetration of extracellular ice into protoplasm is most likely to be accompanied by membrane destruction (6,7). This work develops the idea of intracellular crystallization being a result of protoplasm supercooling that causes homogeneous ice nucleation in cells (1, 9). The study of protoplasm supercooling on the basis of Mazur’s equation (9) is quite a complex task, requiring the use of computers and is valid only in case of the approximation of small concentrations. But such a study may be simply conducted on the basis of a somewhat more general kinetic equation (Eq. [ll]). In the work by Riggs (13) the assumption about the absence of extra- and intracellular diffusion allows for the description of

Received June 2, 1975. 1 Abbreviations and symbols : 1, time; l’, temperature; P, P’, intraand extracellular pressure; p, p’, water chemical potential inside and outside the cell; fi6, ice chemical potential; no, number of cell solute molecules; N, number of water molecules in the cell; c, total concentration of solution in the cell; vg, water volume per one molecule; r, cell radius; p, heat of crystallization of water; K, constant characterizing the membrane permeability to water molecules; 7, freezing rate; S, cell surface area; K, Boltzmann’s constant; A?‘, degree of protoplasm supercooling; D, water diffusion coefficient in protoplasm.

Copyright All rights

J$ Skorohod

E. A.

Crystallization

PUSHKAR

14s

dehydrat’ion in cells with high permeability coefficients. According to our evaluations such a neglect is not admissible for animal cells with a membrane permeability coefficient for water molecule K > 10-j cm/ secebar. This inequality is unjust for erythrocytes (9), the peculiarities of their intracellular crystallization will be considered in the present paper. THEORETICAL

CONSIDERATION

In the process of freezing a cell suspension, ice crystals appear at first in the extracellular solution. Their growth leads to an increase in salt concentrations and to cell dehydration, the protoplasm being in the supercooled state (1, 9). In the qualitative aspect it is clear that if cooling is slow enough to permit equilibration between extra- and intracellular solutions, the supercooling of protoplasm will be minor and the contents of cells will not freeze. Practically, the degree of protoplasmic supercooling will be determined mostly by two rival factors, cooling rate and dehydration rate (9). To obtain the kinetic equation let us formulate simplifying assumptions similar to those admitted in Mazur’s work (9). I. In the period of freezing the cytoplasic membrane is intact. II. The membrane is permeable for water molecules only. III. Spatial gradients of temperature are small. IV. The chemical potential of water at the external surface of membrane is equal to the chemical potential of ice. V. There is an absence of kinetic processes in the cell. CONSTRUCTION

OF KINETIC

EQUATION

According to assumption II the chemical potential of water in protoplasm is a function of one variable only, the number of water molecules in the cell, However, it is convenient to regard as an independent variable not the magnitude N, but the

ET AL,

total concentration

of intracellular

c = no/W

solution

+ no).

Cl1

If temperature, pressure, and concentration are chosen as independent variables, the chemical potential of water in the cell CL= P(P, T, 4.

L-21

Assuming a linear relation between the water flow from the cell and the force causing it (2), we have

dN/dt = KS(p’ - /.L)/v~~. According

to assumption

c31

IV,

P’ = ccs(P’, T).

Cdl

Then

dN -= tit

fj Cc(s(P’,T) - 4P, T, ~11

+ ~cc,(f",T) - A(P, T)].

C5l

Let us define c?(P,T) to be the total concentration of solute theoretically required to be in the cell [that is actual concentration (c) plus that required to be added (cl)] to bring the chemical potential of intracellular water to that of ice at the same pressure and temperature. Thus c1=c--F and

rs@‘,T) = PIP, T, W, VI.

I31

Since the chemical potential of ice depends on pressure very little, the magnitude ps(P’, T) - ps(P, T) in Eq. [S] may be neglected. In the result we obtain the following equation

dN -= dt

KS

2102 CPS- PI

= 5 CPU’,T, 3 - /4’, T, 41 = -_ KS dp(P, T, F) Cl.

%I2

6%

r-71

DEHYDRATION

AND INTRACELLULAR

CRYSTALLIZATION

+ no)] - F - cl = 0.

[S-J

We shall obtain

dN -zzz--dt

Ks a~ I-vo2ac P

N N +

- C(P, T) no

1

ac/aT = - tj/KT2,

d(C + Cl)

&

(C+ c1y

KS’ a/.~ =v,zJp

*

=S+ tit or

dC

->

dT + (N ;1”Q

KT.

[id]

P K1’2

= 0.

[15-j

Equation [la] differs from Mazur’s equaCon only by the denominator of the item in parentheses, this difference being inessential for small concentrations. Let us evaluate the degree of protoplasmic supercooling. Assuming dc,/dt - 0 in Eq. [ll], we have

For human lymphocytes and leucocytes K - 10-6-10-7 cm/secebar (3), r - 5.10e4 cm. Taking into consideration that no/c? - N - S,/3vo and assuming E - l%, we receive for t,hese cells AT - 10 set f (18). DISCUSSION

OF ADMITTED

[13]

ASSUMPTIONS

A complete discussion of assumptions I, II, III, and IV has been conducted in the work by Mazur (9). Let us consider the field of application of assumption V. The neglect of mass transport within the cell is possible provided that the intracellular concentration drop, AC, is much less than the effective concentration drop on the membrane cl. The concentration drop in the cell may be assessed by using the condition of the continuity of water flow, i.e., D(Ac/‘r)

dN dT’

-

Cl11

Equation [11] is a linear differential equation of the first order and is mostly convenient for practical calculations. The conducted consideration of the kinetics of dehydration is more general than that of Mazur’s work (9), since this equation is admissible even in cases where the concentration of solutions in cells cannot be considered small. Mazur’s equation can be easily obtained from Eq. [9] in the case of small concentrations. For this purpose let us differentiat’e Eq. [9] with respect to time.

=-

t,hat in case of =

L-101

In the deduction of Eqs. [S], [9], and [lo), the magnitude cl was considered to be small, that is, the protoplasmic supercooIing was supposed to be small, since otherwise intracellular crystallization would occur. Taking int)o consideration that cl is small, let us rewrite Eq. [lo] nit,h precision to members of cl order : dCl KS ap _- al? I_-z = dt + novo2 ac E6.

ap/aC

149

Substituting Equation [14] for aq’aT and &/dc in Eq. [13], we obtain

PI rr 0

FREEZING

Taking into consideration small concentrations

From Eq. [7] we can easily obtain equations for N and cl using the identity 1 - [iV/(N

DURING

-

where D = water

(dN/dt) (vo/S), diffusion

coefficient

Cl91 in

PUSHKAR

150

ET AL.

FIG. 1. Erythrocyte from the sample frozen at lO”C/sec to -196°C in the presence of polyethylene oxide, MW 400. The samples for electron microscopic studies were prepared by the method of substitution in the frozen state at - 110°C. Magnification, 9000.

protoplasm. we obtain

Substituting AC

D----Q

r

K 8~ v.

AC -- ‘v ;$c.;<< Cl

hence

Eq. [C;] into [19],

PO1

ac 1;

WI

where C = concentration change rate at the internal surface of the membrane (8). The degree of protoplasm supercooling in the cell center relative to the periphery is

AT-?----

ac/dT

c

1.2

ac/aT D'

WI

In the case of erythrocytes, AC >> cl, the result of the abnormally high permeability : 10-j cm/set. bar, [22] K<
voD

DEHYDRATION

AND

INTRACELLULAR

CRYSTALLIZATION

DURING

FREEZING

151

FIG. 2. Erythrocyte from the sample frozen at lO”C/sec to -196°C in t.he presence of polyethylene oxide, MW 400. The samples for electron microscopic studies wel’e prepared by t,he method of substitution in the frozen state at - 110°C. Rlagnification, 20,000.

samples frozen at +lO”C/sec. In the photographs (Figs. 1 and 2) we have noticed an increase in the dimensions of crystals formed in erythrocytes from the periphery to the center that is a direct proof of the correctness of our conclusions. When comparing formulae [lS] and [25] one can see that human erythrocybes are likely to freeze at cooling rates about 100 times as high as those for lymphocytes and leucocytes and slower than those supposed from the evaluation of Mazur (9). SUMMARY

The kinetic equation of the process of cell dehydration during freezing has been obtained. It is used to assess the degree of protoplasmic supercooling as a function of the cooling rate and cell parameters. The suggested model of dehydration cannot be applied t’o cells with permeability coefficients for water molecules more than lo-” cm/sec.bar, in particular to erythrocytes.

The peculiarit,ies of intracellular crystallization in red cells have been studied. The results show that red cells are likely to start freezing at cooling rates slower than those supposed from calculations of Mazur (9). ACKNOWLEDGMENTS Our sincerest thanks and gratitude to Dr. D. G. Dolgopolov from the Institute for Low Temperatures in Kharkov for his advice and to Dr. D. E. Pegg from the Clinical Research Centre in Harrow for his useful comments and help in preparing the article for publicat,ion. REFERENCES 1. Diller, K. R., Cravalho, E. G., and Huggins, C. E. Intracellular freezing in biomaterials. Cryobiology 9, 429-440 (1972). 2. de Groot, S. It., and Mazur, P. “Non-equilibrium Thermodynamics.” North-Holland, Amsterdam, 1962. 3. Hempling, H. G. Heats of activation for exosmotic flow of water across the membrane of leucocytes and leucemic cells. J. Cell. Physiol. 1, l-9 (1973).

PUSHKAR 4. Katchalsky, A. Membrane permeability and the thermodynamics of irreversible processes. In “Membrane Transport and Metabolism” (A. Kleinreller and A. Kotyk, Eds.), Vol. 4, pp. 69-86 Academic Press, London, 1961. 5. Ling, G. R., and Tien, C. L. Analysis of cell freezing and dehydration. ASME Paper No. 69-WA/HT-31, 1969. 6. Lozina-Lozinsky, L. K. “Essays on Cryobiology.” Nauka, Leningrad, 1972. 7. Luyet, B. J., and Gehenio, P. M. Life and death at low temperatures. Biodynamics 3, 33-69 (1940). 8. Lykov, A. V. ‘(Theory of Thermoconductivity.” Vysshaya Shkola, Moscow, 1967. 9. Mazur, P. Kinetics of water loss from cell at subzero temperatures and likelihood of intracellular water. J. Gen. Physiol. 47, 323-353 (1963). 10. Meryman, H. T. The exceeding of minimum tolerable cell volume in hypertonic suspension as a cause of freezing injury. In “The

ET AL.

11.

12.

13.

14.

13.

Frozen Cell,” pp. 51-64. Ciba Fountlaiion Symposium, 1970. Nedyelsky, G. T., Roahdestvensky, M. A., Mikhnovich, E. P., Gujko, E. I., and Malkov, L. S. The working-out of optimal regimens for erythrocytes cryopreservation. In “Urgent Problems of Cryobiology and Cryomedicine,” pp. 165-167. Naukova Dumka, Kiev, 1974. Rapatz, G., and Luyet, B. Effects on leucocytes of plasma concentrated by freezing at high subzero temperatures. Biodynamics 11, 53-37 (1971). Riggs, D. R. “The Mathematical Approach to Psychological Problems,” pp. 168-192. M.I.T. Press, Cambridge, Mass., 1963. Sherman J. K. Effects of size of int.racellular ice on consumption of oxygen and nuclear alteration of mouse kidney cells. Anat. Rcs. 149, 591--604 (1964). Viad, P. R. Freezing of the water content, of porous membrane. Cryobiology 9, 231-239 (1972).