The point of maximum cell water volume excursion in case of presence of an impermeable solute

Cryobiology 44 (2002) 193–203 www.academicpress.com

The point of maximum cell water volume excursion in case of presence of an impermeable soluteq Igor I. Katkov1 Celltronix, 9263 Pebblestone Ln, San Diego, CA 92126, USA Received 20 December 2001; accepted 25 April 2002

Abstract A relativistic permeability model of cell osmotic response (Cryobiology 40:64–83; 41:366–367) is applied to a two-solute system with one impermeable solute. The use of the normalized water volume (w), and the amount of intracellular permeable solute (x), which is the product of the water volume and intracellular osmolality (y), as the main variables allowed us to obtain a homogeneous differential equation dxD =dwD ¼ f ðxD =wD Þ, where wD ¼ w  wf , xD ¼ x  xf , and f refers to the final (equilibrium) values. The solution of this equation is an explicit function, wD ¼ gðxD Þ, which is given in the text. This approach allows us to obtain an analytical (exact) expression of the water volume at the moment of the maximum excursion (water extremum wm ). Results are compared with numeration of basic osmotic equations and with approximation given in (Cryobiology 40:64–83). Assumption that, dw=dt  0 gives good approximations of the kinetics of water and permeable CPA after the point of maximum volume excursion (the slow phase of osmotic response). Practical aspects of the relativistic permeability approach are also discussed. Ó 2002 Elsevier Science (USA). All rights reserved. Keywords: Membrane; Permeability; Modeling; Osmotic response; Volume excursions; Two-solute systems; Cryoprotectants

Prevention of osmotic injury during addition and removal of cryoprotective agents (CPAs) is an important factor and in many situations a crucial aspect of maintaining the cell viability after cryopreservation. Extensive literature has been dedicated to the modeling of processes of movement of water and solutes and development of q Part of this work was preliminary presented at the 37th Annual Meeting of the Society for Cryobiology, Boston, Massachusetts, July 30–August 1, 2000. E-mail address: [email protected]. 1 This work is dedicated to the memory of my mother Rosalia Katkova who was very supportive in my endeavors.

optimized protocols, such as [1–3,8–10,12]. Traditionally, ‘‘classical’’ permeability approaches used in Cryobiology mostly consider one-solute systems with an impermeable component, or twosolute systems with one permeable CPA and impermeable ‘‘salts,’’ such as osmotic buffers, and/or components that assure extracellular protection during cooling and freezing. As a result of such simplification, prediction abilities of the theory are usually limited and/or it needs time-consuming and costly software simulations. Recently [4–6], we have proposed a new, more general approach for membrane permeability modeling, which allows a better understanding of the pattern of cell osmotic response if any number

0011-2240/02/$ - see front matter Ó 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 1 1 - 2 2 4 0 ( 0 2 ) 0 0 0 2 9 - 9

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of permeable and/or impermeable solutes are present in the system. This approach also allows substantial simplification of basic calculations that are now not required for software simulations. This approach, which we call ‘‘relativistic,’’ is based on several postulates, namely: (1) Not the absolute values of cell membrane permeabilities, but their relative (to water) permeability terms Crpi ¼ Pi =ðLp RT Þ matter for characterization of the pattern of osmotic response. (2) An impermeable component is considered as a solute, whose relative permeability is negligibly small when comparing with the permeabilities of other solutes and water, so the intracellular amount of this solute remains practically constant during the time of the experiment. (3) Variables and parameters in the osmotic equations are normalized (relative) to the isotonic characteristic such as iso-time, iso-tonic water volume, and iso-concentration (osmolality). (4) There is no interaction between solutes and solvent during passage through the cell membrane unless otherwise it is proved by direct biophysical measurements. Previously, we obtained approximations for two-solute systems [4] and in general case for multisolute systems with all components being permeable and when impermeable solutes are present [5,6]. In this paper, we will demonstrate how the use of this approach and optimal choice of parameters allow us to obtain an exact (analytical) solution for determination of the cell water volume at the moment of maximum volume excursions (wm ) in an important from the practical point of view case of a two-solute system when one solute is considered as impermeable. We will also briefly analyze the kinetics of movement of water and the permeable solute after the point of maximum volume excursion (MVE), i.e., during the slow phase of osmotic response. Table 1 presents abbreviations and definitions of the main parameters.

Basic derivations In general, if N solutes are present in the medium, the osmotic response of the cell is described by N þ 1 differential equations: one for water movement and N for diffusion of the solutes [5]. However, if one solute is considered as impermeable, then the last equation can be substituted with the Boyle–van’t Hoff relationship that states that the amount of impermeable solute inside the cell remains constant. Such a scenario is common in a variety of cryobiological protocols when one

solute is a permeable CPA and the other solute(s) are impermeable at least during the time of observation. For such systems, the osmotic equation can be written as follows [4,5]: dW =dt ¼ Lp RTA½ðYf þ Zf Þ  ðY þ ZÞ;

ð1WÞ

dX =dt ¼ PCPA AðYf  Y Þ;

ð1YÞ

WZ ¼ constant ¼ W0 Z0 ¼ Wf Zf ¼ Wiso Ziso ;

ð1ZÞ

where W is the cell water volume, t is the time, Lp is the membrane hydraulic conductivity, R is the gas constant, T is the temperature, A is the surface area, Y and Z are the osmolalities of permeable and impermeable solutes, respectively, X ¼ WY is the intracellular amount of permeable CPA and PCPA is the solute membrane permeability. Subscript ‘‘iso’’ refers to the isotonic values, ‘‘0’’ indicates the values at time zero, and ‘‘f’’ refers to the final (equilibrium, extracellular) values. Let us now introduce normalized (relative) parameters as it was done in (5): Crp PCPA =ðLp RT Þ; siso Wiso =ðLp RTAMiso Þ; q t=siso ; z Z=Miso ;

w W =Wiso ;

crp Crp =Miso ;

y Y =Miso ;

x WY =ðWiso Miso Þ:

Such introduction substantially simplifies Eq. (1W), (1Y), (1Z), namely: dw=dq ¼ ½mf  ðy þ zÞ;

ð2wÞ

dx=dq ¼ crp ðyf  yÞ;

ð2yÞ

wz ¼ w0 z0 ¼ wf zf ¼ 1;

ð2zÞ

where mf yf þ zf is the total final (extracellular) normalized osmolality. We now express normalized osmolalities of solutes as follows: y x=w ðon definitionÞ;

ð3yÞ

z ¼ 1=w ¼ wf zf =w ðfrom ð2zÞÞ:

ð3zÞ

Substituting (3y) and (3z) to (2w), (2y), (2z) reduces the system to two basic equations: dw=dq ¼ ½mf  ðx=w þ 1=wÞ ¼ ½mf  ðx=w þ wf zf =wÞ;

ð4wÞ

I.I. Katkov / Cryobiology 44 (2002) 193–203

195

Table 1 Definition of symbols and abbreviations Symbol

Meaning

CPA BVH 0 f iso m D A Lp R T P W, w t siso q Crp; crp Y;y Z; z M; m Miso X;x a1 ; a2 A; B

cryoprotective permeable agent (also used as a subscript) Boyle–van’t Hoff relationships (e.g., formulas (1Z) and (2z)) subscript: initial intracellular at time zero subscript: final (extracellular, equilibrium) subscript: iso-osmotic subscript: value at the point of MVE subscript: refers to intermediate variablesa cell area hydraulic conductivity universal gas constant absolute temperature solute permeability cell water volumeb ;c timed characteristic ‘iso-osmotic’ time siso timeb normalized to siso relative membrane permeability termb;d osmolalityb;d of the permeable solute (CPA) osmolalityb;d of the impermeable component total osmolalityb;d of the solutes (M X þ Y ) iso-osmolality (assumed as 290 mOsm in calculations) intracellular amountb;d of the permeable solute (X WY ) parameters (formulas (12.1) and (12.2)), roots of Eq. (11) parameters (formulas (15A) and (15B)) used for the analytical solutions (formulas (14.1), (14.2), (14.3) and (19.1), (19.2), (19.3), (19.4), (19.5) parameter (formulas (19.1)) used for the analytical solution (19.2) permeability additive osmolalityb;d (formulas (19.5)) (20), (D Crp þ Y þ Z) cell volumee

k D; d V a

sD ¼ s  sf , where s ¼ w or x; and yD ¼ xD =wD . Actual values have no subscripts. c Values normalized to Wiso are non-capitalized. d Values normalized to Miso are non-capitalized. e Subscripts: ‘‘b’’ for osmotically inactive ‘‘ballast,’’ ‘‘s’’ for the volume of the solutes, osmotically active volume has no subscript, (V ¼ W þ RVs;i ), and ‘‘tot’’ refers the total volume (Vtot ¼ Vb þ V ). b

dx=dq ¼ crp ðyf  x=wÞ:

ð4xÞ

Solution of the basic system We now can divide (4x) by (4w). As we indicated elsewhere [5], we have to be careful when assuming dw=dq 6¼ 0. We will address this issue below, in the discussion of the formula for wm . Now, after dividing we have the following: dx=dw ¼ crp ðyf  x=wÞ=½mf  ðx=w þ 1=wÞ: ð5Þ To solve (5), we introduce intermediate parameters wD and xD as follows:

wD w  wf ;

ð6wÞ

xD x  xf wy  wf yf :

ð6xÞ

These parameters reflect the difference between the current and the final (equilibrium) values of x and w. Note that wf ¼ 1=zf (from (2z)) and xf 4wf yf (on definition (3y)). The reverse relationships will be the following: w wD þ wf ;

ð7wÞ

x xD þ xf xD þ wf yf :

ð7xÞ

Taking advantage of the fact that dx=dw ¼ dxD =dwD and substituting (7w) and (7x) into (5), we obtain the following equation:

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dxD =dwD ¼ crp ðyf wD  xD Þ=ðxD  mf wD Þ:

ð8Þ

where

Eq. (8) is a homogeneous differential equation. To solve it, let us introduce another intermediate parameter as follows:

A ¼ ðmf  a1 Þ=ða1  a2 Þ;

ð15AÞ

B ¼ 1  A ¼ ða2  mf Þ=ða1  a2 Þ;

ð15BÞ

yD xD =wD ðyw  wf yf Þ=ðw  wf Þ:

and ‘‘0’’ as usual refers to the initial values:

ð9Þ

Note that in this case yD is not the difference between the actual and the final values of y, but a more complex variable. Taking advantage of the fact that dxD =dwD ¼ wðdyD =dwD Þ þ yD and dividing the right part of Eq. (8) by w, we have the following:

wD;0 w0  wf ;

ð16wÞ

xD;0 x0  xf ðw0 y0  wf yf Þ=ðw0  wf Þ;

ð16xÞ

yD;0 xD;0 =wD;0 ðx0  xf Þ=ðw0  wf Þ ðw0 y0  wf yf Þ=ðw0  wf Þ:

wðdyD =dwD Þ þ yD ¼ crp ðyf  yD Þ=ðyD  mf Þ;

which after simplification gives an equation with separable variables: wðdyD =dwD Þ ¼ 

½yD2

 ðmf  crp ÞyD  crp yf =

ðyD  mf Þ:

ð16yÞ

ð10aÞ

ð10bÞ

Before separation of variables in (10b), we have to find the roots of the quadratic equation present in the numerator of its right part, namely:

If w0 ¼ wf , then wD;0 ¼ 0. Taking advantage of the fact that A þ B ¼ 1, Eq. (14.2) simplifies to: jxD  a1 wD jA  jxD  a2 wD jB   jxD;0 j ¼ 1:

ð14:3Þ

If crp ! 0, then a1 ¼ mf , a2 ¼ 0, A ¼ 0, B ¼ )1, and Eq. (14.2) simplifies to: wD yD xD ¼ wD;0 yD;0 xD;0 ¼ const:

ð14:4Þ

ð11Þ

Eq. (14.4) means that x ¼ xD þ xf remains also constant, i.e., indicates the BVH relationship for a quasi-impermeable solute. The same results can be obtained after simplification of Eq. (13).

ð12:1Þ

Determination of the point of maximum volume excurison

yD2  ðmf  crp ÞyD  crp yf ¼ 0: The roots are: a1 ¼ fðmf  crp Þ þ ½ðmf  crp Þ2 þ 4crp yf 1=2 g=2;

a2 ¼ fðmf  crp Þ  ½ðmf  crp Þ2 þ 4crp yf 1=2 g=2: ð12:2Þ It is easy to see that a1 is always > 0, while a2 is always < 0. For crp ! 0, a1 ¼ mf and a2 ¼ 0. Now we can separate the variables: dwD =wD ¼ ðyD  mf ÞdyD =½ðyD  a1 ÞðyD  a2 Þ: ð13Þ Integration of (13) from the initial to the actual values gives us the following solution: jwD =wD;0 j ¼ ½jyD  a1 jA  jyD  a2 jB =½jyD;0  a1 jA  jyD;0  a2 jB ;

ð14:1Þ

which takes advantage of the fact that A þ B ¼ 1 and (9) can be re-written as follows: jxD  a1 wD jA  jxD  a2 wD jB  ¼ jxD;0  a1 wD;0 jA  jxD;0  a2 wD;0 jB ;

ð14:2Þ

Eq. (14.1) is an explicit expression of relationships between wD and yD , which can easily be transferred to the relationships between w and y. This equation does not present the kinetics of the osmotic processes, so it has limited applicability per se. However, we can use this equation for determination of the cell water volume at the moment of transient maximum volume excursion (extremum). Here and below, subscript ‘‘m’’ determines the values of variables at this moment. As we have shown in [4,5], at this point, the first derivative of water on time dw=dq becomes equal to 0. We have formally excluded this point from our consideration because we divided (4x) by (4w) and the latter one at this point equals 0. However, similarly as we discussed for a general theory [5,6], the formulas (14.1) and (14.2) are applicable at this point as well. Let us use the fact that in the point of the maximum extremum dw=dq ¼ 0, which when combining with (4w) gives the following:

I.I. Katkov / Cryobiology 44 (2002) 193–203

mf  ðxm =wm þ wf zf =wm Þ ¼ 0;

ð18xÞ

However, if the situation is opposite, and impermeable component is inside only (z0 6¼ 0, and zf ¼ 0), for example, when an intact cell is placed in a medium containing glycerol only, the cell may undergo initial transient shrinkage, but will eventually swell infinitely (or more correctly, until it reaches its upper volume limit and the cell membrane ruptures). Mathematically, it means that wf ! 1. In this case, formulas above are inapplicable. It can be proven (paper in preparation) that the water volume at the moment of MVM can be calculated as follows:

ð18yÞ

wm ¼ w0 d0 =df þ ðcrp =df2 Þ ln jcrp =½w0 df ðyf  y0 Þ  yf j;

or xm ¼ wm mf  wf zf :

ð17Þ

Now we have to determine the intermediate variables xD and yD at the point of the extremum; we will do it using (17) and definitions (6x) and (9), so: xD;m xm  xf ¼ wm mf  wf zf  wf yf ¼ mf ðwm  wf Þ ¼ mf wD;m ; yD;m xD;m =wD;m ¼ mf :

We can easily find wm from Eqs. (14.1), (18y), and definitions (6w), (9), (16w), and (16y): wm ¼ wf  ½jmf  a1 jA  jmf  a2 jB =k;

197

ð19:1Þ

where k ¼ ½jx0  xf  a1 wD;0 jA  jx0  xf  a2 wD;0 jB :

ð19:5Þ where d ðcrp þ y þ zÞ is the normalized relative permeability additive osmolality [5]. Note that the transient minimum can (but not always) be observed during addition of the permeable solute, while during dilution the cells will swell only (see also conditions (24a) and (24b) below).

ð19:2Þ If w0 ¼ wf , then wD;0 ¼ 0; k ¼ jx0  xf j1 , and Eq. (19.1) simplifies to: wm ¼ wf  ½jmf  a1 jA  jmf  a2 jB   jx0  xf j: ð19:3Þ The signs of the modules in (19.1) and (19.3) are ‘‘þ’’ for a maximum and ‘‘’’ for a minimum. If crp ! 0, then a1 ¼ mf ; a2 ¼ 0, A ¼ 0; B ¼ 1, 00 ! 1; k ¼ jx0  xf j1 , so (19.1) transfers to: wm ¼ w0 ðy0 þ z0 Þ=ðyf þ zf Þ;

ð19:4Þ

which is the Boyle–van’t Hoff relationship assuming y-solute to be quasi-impermeable as well. Let us now consider a situation when z0 ¼ 0 and zf 6¼ 0. Such an ‘‘exotic’’ case may occur when the cells are electroporated so internal ions and other osmotically active molecules can leave the cells, while the external medium contains big and still non-permeable molecules, such as raffinose or trehalose. In this case, in accordance with (2z), wf ¼ 0; xf ¼ 0, z ¼ 0, and all formulas above are applicable. Note that eventually all water will leave the cells (collapse), a case that we discussed in [5] for a one-solute system, and will be discussed for the two-solute case in detail elsewhere. As the result, a transient maximum can (but not always) be observed only during dilution of the permeable component while during addition of the permeable solute, the cell will shrink only.

Comparison of analytical expression with numeration and approximation In previous works [4,5], we gave an approximation for calculation of the value of cell water at the point of maximum volume excursion as the following: wm  w0 ðcrp þ y0 þ z0 Þ=ðcrp þ yf þ zf Þ ¼ w0 d0 =df : ð20Þ We also compared the accuracy of approximation for almost 100 different combinations of initial and final conditions, and permeability crp , and in all cases, the accuracy was satisfactory. Note that (20) is exactly equal to (19.4) if crp ! 0. Approximation (20) also gives very good accuracy if crp  1 [4,5]. However, in some cases (crp  1), the discrepancy was relatively high, up to 14% in our simulations. We did not investigate the maximum possible deviation; probably for some very specific cases, it might be higher. Now we consider the same conditions presented in ([5], Fig. 6). We assume that the cells are transferred from an isoosmotic buffer (Y0 ¼ 0; y0 ¼ 0; Z0 ¼ 290 mOsm; z0 ¼ 1) to a freezing medium containing a cryoprotective agent (CPA) with osmolality Yf ¼ 800 mOsm (yf ¼ 2:76) and an iso-osmotic non-permeable buffer (Zf ¼ 290 mOsm; zf ¼ 1). The comparison between simulation, approxima-

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so we can use the Boyle–van’t Hoff (BVH) simplification (2z). However, ifthe second solute is permeable (crpz 6¼ 0), then we have to introduce the third equation for z similar to (2y). It makes the task much more difficult: the exact (explicit) solution does not exist. At the same time, approximation (20) is still valid as long as crpz  crpy . In cases when crpz 6¼ 0, however, we cannot use the BVH relationship and expression for the final water volume would be different [5]: wf ¼ w0 ð1 þ y0 =crpy þ z0 =crpz Þ=ð1 þ yf =crpy þ zf =crpz Þ; ð21Þ which is an exact formula. Note that Eq. (21) transforms to a BVH relationship if crpz ! 0 [5].

Kinetics of movement of water and permeable CPA after the point of MVE Fig. 1. Cell water volume at the moment of maximum volume excursion (wm ) for a two-solute system containing a permeable and a non-permeable component as a function of the relative membrane permeability term normalized to the iso-osmotic value (crp PCPA = ðLp RTMiso ). The cells are transferred from an iso-osmotic buffer (Y0 ¼ 0; y0 ¼ 0; Z0 ¼ 290 mOsm; z0 ¼ 1) to a freezing medium containing a cryoprotective agent (CPA) with osmolality Yf ¼ 800 mOsm (yf ¼ 2:75) and a non-permeable buffer (Zf ¼ 290 mOsm; zf ¼ 1). Filled diamonds represent the values of wm obtained by simulations of Eq. (2w), (2y), (2z), the dashed curve with circles are the values of wm calculated using approximation (20), and the solid line is the value obtained from the analytical expressions (19.1)–(19.3).

tion, and analytical solution is shown in Fig. 1. Filled diamonds represent the values of wm obtained by simulations of Eq. (2w), (2y), (2z), the dashed curve with circles are the values of wm calculated using approximation (20), and the solid line is obtained from analytical expressions (19.1), (19.2), (19.3). In contrast to the approximation, the analytical expressions completely coincide with the simulation (numeration) of osmotic Eq. (2w), (2y), (2z). The discrepancy is less than 0.001% and is determined by capability of the author to find the exact point of wm on the simulation curve. The same complete equivalence of simulations and the analytical solution were also obtained for other osmotic conditions (not shown). Note that the exact solution for wm given in this work is applicable only with an assumption that the second solute is completely impermeable

The majority of work on the relativistic permeability approach we published so far was dedicated to the determination of the key parameters of osmotic response at the points of MVE ðwm ; ym Þ and at the equilibrium (wf ). Explicit expression of the kinetics of water movement for one-solute systems was given in detail in [5]. We can estimate the time frame of the points of MVE and the last (equilibration) phase of osmotic response for twoand multi-solute systems, but the exact kinetics still needs simulations. However, for the case of permeable CPA þ impermeable solute discussed in this paper, we can obtain a good approximation of the kinetics of movement of water and the CPA. Still, the fast phase of Eq. (2w), (2y), (2z) cannot be obtained analytically. However, from a practical point of view for the majority of cases, this phase (shrinkage during addition of CPA and swelling during removal of the solute) occurs very fast (in a range of seconds for such big a cell as the rat embryo and much faster for spermatozoa or erythrocytes), so it is practically impossible to control it in real experimental conditions. In early publications, when simulation software was not widely available, investigators considered the CPA quasi-impermeable during the fast phase [10], so the point of maximum volume excursion was calculated on the base of Boyle–van’t Hoff relationship (19.4). This implies that the amount of CPA during the fast phase remains constant, so the intracellular concentration of the permeable solute at the point of maximum volume excursion can be calculated as follows:

I.I. Katkov / Cryobiology 44 (2002) 193–203

ym  yBVH ¼ w0 y0 =wm ¼ y0 ðyf þ zf Þ=ðy0 þ z0 Þ: ð22Þ For low-permeable to CPA membranes such an assumption is valid; in many cases, however, the cells are moderately, and even, as it was shown for rat embryos [7], are highly permeable to the solute. As the result, substantial amount of CPA penetrates the cell membrane during the fast phase, so the intracellular osmolality of the solute at the moment of MVE deviates from Eq. (22). It can be calculated by taking advantage of the fact that at the point of MVE the derivative of the water volume equals exact 0 because the water movement changes its direction to the opposite. So, assuming dw=dq ¼ 0 at the point of MVE, we can easily calculate the solute osmolality from (2w) and (2z), as follows [4]: ym ¼ mf  1=wm :

ð23Þ

At the point of MVE, intra- and extracellular osmotic pressures are equalized (it is clear from (23) taking into account the fact that zm ¼ 1=wm for impermeable ‘‘salts’’) and gradient of total osmolality m y þ z across the membrane is equal to zero. After this point, permeable solute continues to penetrate the cells driven by the gradient of the solute concentration and slightly disturbing the osmotic equilibrium. However, such a deviation is quickly and almost completely equalized by the movement of water in the same direction (in contrast to the fast ‘‘osmotic’’ phase when water and the solute move in opposite directions). The time frame of this ‘‘diffusional’’ phase is determined by diffusion of CPA and, in the majority of cases, it is much slower than the fast (‘‘osmotic’’) phase. Note that bi-phasic pattern of osmotic response not necessary occurs in any condition [5,6]. Sometimes, we can observe mono-phasic shrinkage or swelling only (though two distinct in kinetics sub-phases are usually also pronounced, details will be published elsewhere). We can predict whether the ‘‘classical’’ two-phase pattern will be observed from following estimations [5]:

dw=dq ¼ m  mf  0

ð25Þ

during the slow phase. That leads to the conclusion (see Eq. (23)) that: y  mf  1=w

ð26yÞ

or w  1=ðmf  yÞ

ð26wÞ

At the same time, x wy on definition, so the equation for the solute movement (2y) can be written as: ydw=dq þ wdy=dq ¼ crp ðyf  yÞ;

0

ð2y Þ

which, taking into account (25), (23), and mf ¼ yf þ zf on definition, can be simplified to: ðdy=dqÞ=ðmf  yÞ ¼ crp ðyf  yÞ:

ð27Þ

After separation of variables and integration from the point of MVE (qm ; ym ) to the actual values q and y, it gives the following relationship: ðyf  yÞ=ðmf  yÞ ¼ ½ðyf  ym Þ=ðmf  ym Þ ð28Þ

ð24aÞ

Substituting y and ym from (26y) and (23), respectively, Eq. (28) can be re-written as the following:

ð24bÞ

y ¼fyf  mf ð1  zf wm Þ exp½crp zf ðq  qm Þg= ½1  ð1  zf wm Þ exp½crp zf ðq  qm Þg;

w0 < w0 ðcrp þ y0 þ z0 Þ=ðcrp þ yf þ zf Þ > wf ðfor a maximumÞ:

librium protocols when it is important to estimate concentration and amount of intracellular CPA to ensure intracellular protection during freezing or vitrification. At the same time, over-exposition in the CPA can lead to excessive and often lethal swelling during removal of CPA after thawing. Traditionally, investigators use either an empiric approach, or software simulations, which as we mentioned above, is not always convenient, time consuming, and needed educated personnel. Fortunately, relativistic permeability equations allow the calculation of kinetics of the movement of water and CPA, thus, concentration and amount of CPA and concentration of impermeable intracellular solutes, using a pocket calculator. As we mentioned earlier (5), after equilibration of the gradient of total osmotic pressure at the moment of MVE, it remains very small when comparing with the gradients of the solutes, so we can with good accuracy assume that:

 exp½crp zf ðq  qm Þ:

w0 > w0 ðcrp þ y0 þ z0 Þ=ðcrp þ yf þ zf Þ < wf ðfor a minimumÞ;

199

Kinetics of the slow diffusional phase (t > tm ; q > qm ) may be crucial for a variety of non-equi-

ð29yÞ

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I.I. Katkov / Cryobiology 44 (2002) 193–203

which as an explicit equation of kinetics with the permeable solute as a function of time. Kinetics of water movement can be obtained by substituting y from (26y) to (29y) as follows: w ¼ ½1  ð1  zf wm Þ exp½crp zf ðq  qm Þg=zf : ð29wÞ

considered if high solute concentrations are used. We also have to consider the total osmotically active volume including the volume occupied by the solutes, which is not negligible now, so the diffusion equations must be transformed to the solute volume movements making them symmetrical to the water equation, which we will discuss elsewhere (see also below).

Amount of intracellular CPA (x ¼ wy) can be easily estimated as: x ¼ fyf  mf ð1  zf wm Þ exp½crp zf ðq  qm Þg=zf : ð29xÞ Fig. 2 represents the kinetics of normalized to isotonic cell water volume (w, squares), intracellular tonicity (y, diamonds), and amount of the permeable CPA (x, triangles) during the slow phase of osmotic reaction (t > tm ) during addition of Yf ¼ 800 mOsm of CPA. Other osmotic conditions are also identical to Fig. 1. Parameters of the cell are similar to those of the rat blastocyst assuming the cell as a uni-membrane sphere: cell diameter 2r ¼ 75 lm, osmotically non-active volume vb ¼ 0:3, membrane hydraulic conductivity Lp ¼ 1:0 lm=atm= min, and temperature T ¼ 300 K. That gives siso around 1 min, so the axis scale reflects both q t=siso (non-dimensional) and t (expressed in minutes). The solute membrane permeability is presumed PCPA ¼ 10:4 lm/ min, that gives Crp ¼ 282 mOsm, and crp ¼ 0:971. The point of MVE occurs at time qm ¼ 0:423 ðtm  25 sÞ. Solid lines with filled symbols represent simulation of Eqs. (2w), (2y), (2z), dashed lines with empty symbols are obtained by calculation of approximations (29w)–(29x) for time t P tm . Solid curve without symbols is the normalized reversed gradient of the total tonicity (dw=dq ¼ mf  m) given illustration of the fact that dw=dq is negligibly small during the slow phase. As we can see, the highest, but still a relatively small deviation of approximations from simulations occurs at the largest dw=dq near the point of MVE. In general, the accuracy of the approximations given by Eq. (29w)–(29x) is good not only for these particular but for other experimental conditions as well. Note that both equations for exact solutions (19.1)–(19.5) and approximations (29w)–(29x) are applicable under assumption that one of the conditions (24a) or (24b) is satisfied so the transient extremum (the point of MVE) exists. Some adjustments of Eqs. (1W), (1Y), (1Z) such as Scatchard’s corrections [10] should be

Discussion: practicability of the relativistic permeability approach Practically all cryobiological procedures involve addition of permeable and impermeable cryoprotective agents before freezing and removal (dilution) of CPAs after thawing. Freezing media are often hyperosmotic and introduction of cells in such hypertonic environment causes extensive initial or permanent dehydration and shrinkage of the cells. This shrinkage can either be transient (if the impermeable component is isotonic) or dehydration is permanent if the impermeable component is hypertonic as well. Cells can tolerate a certain level of dehydration, which is specific for each species. The lower tolerable limit the cells can withstand without substantial lost of their viability is determined as the lower volume limit (LVL [3]). If osmotically sensitive cells with high LVL, such as mammalian spermatozoa [3,11], are abruptly placed in hypertonic conditions, the cell viability can be substantially impaired; thus, they should be treated in a way that precludes excessive shrinkage. Similarly, after cells are loaded with permeable protector(s), frozen, and thawed, the permeable CPA(s) should be removed by washing in (usually) isotonic, often a cultural medium. However, such an abrupt introduction of cells loaded with an osmolite to an isotonic medium causes the swelling of cells and substantial, often lethal, damage due to hypotonic shock if the level of swelling exceeds the upper volume limit (UVL [3]). In this case, we also have to impede abrupt volume excursions of the cells. There are several basic techniques to prevent osmotic damage. One way is to place cells in a diffusion chamber [12], so the components of the freezing media would gradually permeate into the cell suspension, and as a result, the cells will have enough time to equilibrate extra- and intracellular osmolalities of the permeable CPA. The method is attractive, but has some disadvantages such as time of the processing and cost of the equipment. Another basic and widely used approach is to add

I.I. Katkov / Cryobiology 44 (2002) 193–203

the medium by portions (step-wise). In this case, we have to be sure that during any step of addition, the maximum cell volume excursion Vm during shrinkage does not exceed the lowest tolerable limit VLVL , i.e.: Vm;kðaddÞ P VLVL ;

ð30aÞ

where Vm;kðaddÞ is the minimum cell volume at the moment of maximum volume excursion during the kth step of the addition. Similarly, removal (dilution) of CPA(s) from the cells can be, and often is, performed by stepwise addition of the washing medium. In this case, we have to be sure that the maximum transient swelling that occurs initially at each step of the process would not exceed the upper tolerable limit of the cell, which mathematically means that:

Vm;kðdilÞ 6 VUVL :

201

ð30bÞ

We used this method for development of multistep addition and dilution of glycerol for such osmotically sensitive cells as mouse spermatozoa [11]. Because we speculated that long contact with glycerol might also cause some level of chemical damage (and it turned to be the case), we needed to develop a simple and fast method of addition and dilution of glycerol. Before, the most popular method was to add equal (fixed) volumes of the media to the cell suspension (FVS method [3]). However, such a scheme produced non-equal changes in the difference in osmolality of intra- and extracellular compartments. As result, the main damage occurred at the first step of addition or dilution and investigators needed an excessive number of steps to prevent the initial damage. Gao

Fig. 2. Kinetics of normalized to isotonic cell water volume (w, squares), intracellular tonicity (y, diamonds), and amount of the permeable CPA (x, triangles) during the slow phase of osmotic reaction (t > tm ) during addition of Yf ¼ 800 mOsm of CPA. Other osmotic conditions are also identical to the Fig. 1. Parameters of the cell are similar to those of the rat blastocyst, assuming the cell as a uni-membrane sphere: cell diameter 2r ¼ 75 lm, osmotically non-active volume vb ¼ 0:3, membrane hydraulic conductivity Lp ¼ 1:0 lm=atm= min, and temperature T ¼ 300 K. That gives siso around 1 min, so the axis scale reflects both q t=siso (non-dimensional) and t (expressed in minutes). The solute membrane permeability is presumed as PCPA ¼ 10:4 lm= min that gives Crp ¼ 282 mOsm and crp ¼ 0:971. The maximum volume excursion occurs at time qm ¼ 0:42 (tm  25 s). Solid lines with filled symbols represent the simulation of Eqs. (2w), (2y), (2z), and dashed lines with empty symbols are obtained by calculation of approximations (29y), (29w), (29x) for time t P tm . Solid curve without symbols is the normalized reversed gradient of the total tonicity (dw=dq ¼ mf  m) given for the illustration of the fact that dw=dq is negligibly small during the slow phase. The highest deviation of approximations from simulations occurs at the peak values of dw=dq.

202

I.I. Katkov / Cryobiology 44 (2002) 193–203

and colleagues [3] reported another method, in which the difference in external molarity of the permeable CPA between two consecutive steps was kept constant (FMS method [3]). Such a method caused lower initial volume excursions at the first step of addition, but still, the first step led to the highest level of shrinkage. Moreover, during dilution, the last step produced the most excessive swelling. Thus, still additional excessive steps (though in a lesser amount than for the ‘‘traditional’’ method) were needed. We assumed from the relativistic permeability theory that the level of shrinkage with good accuracy is proportional not to the difference of osmolarities, but to the ratio of the relative permeability additional terms d0 =df , formula (20), so if we keep that ratio constant (within the tolerable osmotic limits of the cells) at each step of the addition or dilution procedure, the level of shrinkage during addition or swelling during dilution would be equal for each step. We called it fixed shrinkage–swelling (FSS) method of step-wise addition and dilution [4,5]. It can be shown (briefly explored in [5] and will be published in detail elsewhere) that the FSS method would need the least amount of steps when comparing with any other methods of step-wise addition and dilution (like a square with equal sides has the minimum perimeter among any rectangles of the same area). If permeability coefficients are not known, it is safe to assume that crp ¼ 0, so we can use the ratio of initial internal and external (final) total osmolalities m0 =mf to be kept fixed [5]. We showed that this method was harmless to mouse spermatozoa, but still some chemical damage occurs, so the time of contact with glycerol prior and after freezing should be minimized [11]. The relativistic approach operates mostly with water volume. However, permeable solutes also contribute to the changes in osmotically active volume of cells. We can consider the total volume of the cells consisting of three basic parts: (1) the constant osmotically inactive (ballast) volume Vb that includes insoluble components, bound water, and for simplicity, we can include the volume of impermeable solutes, (2) the volume of osmotically active (‘‘free’’) water W, and (3) the volume of osmotically active (permeable) solutes P Vs ¼ W Mi ti where Mi is the intracellular osmolality of the ith permeable solute, and ti is the partial (apparent) molar volume of the ith permeable solute. If we use glycerol as the only permeable CPA present, then for a wide range of concentrations we used (0–2000 mOsm), and the partial molar volume can be assumed as a con-

stant value tCPA ¼ 0:071 Osm1 . The total cell volume Vtot can be calculated as the following: Vtot ¼ Vb þ W ð1 þ Y tCPA Þ:

ð31Þ

The osmotically inactive volume Vb is a characteristic of cell species and with good accuracy it can be estimated by using volumetric equilibrium measurements of cell volume in the presence of different osmolalities of impermeable solutes (BVH-plot). The isotonic water volume Wiso can be easily calculated if Viso and Vb are known. Theoretically, the moment of maximum excursion of the total cell volume does not coincide with the P moment of MVE of water volume, especially if Mi ti ;  1 or higher. However, for concentrations used for the majority of conventional stepwise protocols, Yf does not exceed 2 Osm, and in such a situation the moments of Vm and Wm almost match. For the example of addition of glycerol, we considered in Fig. 2 (Yf ¼ 800 mOsm; Crp ¼ 282 mOsm), the point of the maximum water volume excursion ðwm ¼ 0:475Þ occurs at time qðwm Þ ¼ 0:42 siso . At this moment, the total osmotically active volume V (Wm ) would be 0.4908 of Wiso . The cell would reach its minimum total volume (Vm ¼ 0:4907 of Wiso ) slightly earlier, at qðvm Þ ¼ 0:41. As we can see, these two values of V are practically indistinguishable, so we can use the moment of MVE of the water volume as the key parameter for estimation of regimes for optimal addition. Similar can be said for dilution. If a cell preloaded with glycerol on an isotonic buffer is transferred to an isotonic buffer without glycerol, it reaches the maximum amount of water (wm ¼ 2:030 of Wiso ) at the moment qðwm Þ ¼ 1:76 siso . The total osmotically active volume at this moment V ðWm Þ ¼ 2:0475. The maximum of Vm , however, occurs slightly earlier, qðvm Þ ¼ 1:72, but the value of the maximum (Vm ¼ 2:0477 of Wiso ) is practically the same as that at the moment of MVE of water volume. Taking advantage of the fact that at time of MVE of water volume, the intracellular solutes, osmolality Ym ¼ Mf  Miso Wiso =Wm , we can easily estimate the total cell volume at this moment from (20) and (31) as follows: Vtot;m ’ Vb þ Wiso ½wm ð1 þ Mf tCPA Þ  Miso tCPA : ð32Þ While in general cryobiologists appreciate the importance of osmotic processes and permeability modeling, the overwhelming majority of work has been performed using empirical approaches. One of the possible reasons is that simulation software

I.I. Katkov / Cryobiology 44 (2002) 193–203

is either not easy available, or its use needs a certain level of programming proficiency. In contrast, the relativistic permeability approach leads to simple equations that can be easily incorporated into a simple Microsoft Excel sheet. We particularly used elements of this approach for development of the method of Fixed shrinkage-swelling (FSS) and successfully applied this design for addition and dilution of glycerol in mouse spermatozoa [11]. Another advantage of the method is that it can be applied to multi-solute systems used in a variety of vitrification cocktails. Moreover, the method predicts some interesting effects such as ‘‘supersaturation’’ and ‘‘squeezing’’ of the permeable CPA, which we used for optimization of vitrification of rat blastocysts (briefly discussed in [7], will be published in detail elsewhere). It shows the practical potential of the method and we encourage our colleagues to contact us and obtain examples of Microsoft Excel sheets with a brief and simple instruction for the users.

[4]

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References [11] [1] R.P. Batycky, R. Hammerstedt, D.A. Edwards, Osmotically driven intracellular transport phenomena, Philos. Trans. R. Soc. London 355 (1997) 2459–2488. [2] A. Filkenstein, Water Movement through Lipid Bilayers, Pores and Plasma Membranes: Theory and Reality. (1987), Wiley, New York, 1987. [3] D.Y. Gao, J. Liu, C. Liu, L.E. McGann, P.F. Watson, F.W. Kleinhans, P. Mazur, E.S. Critser, J.K. Critser, Prevention of osmotic injury to human

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spermatozoa during addition and removal of glycerol, Human Reprod. 10 (1995) 1109–1122. I.I. Katkov, Water volume and intracellular concentration of permeable cryoprotectants at points of maximum volume excursion during addition and dilution of CPA, Cryobiology 37 (1998) 441–442. I.I. Katkov, A two-parameter model of cell membrane permeability for multi-solute systems, Cryobiology 40 (2000) 64–83. I.I. Katkov, A relativistic permeability model: summary of development, Cryobiology 41 (2000) 366–367. Katkov II, Isachenko V, Critser JK. Application of the relativistic permeability approach for optimization of vitrification of rat embryos. Cryobiology 43 (2001) 345–346. F.W. Kleinhans, Membrane permeability modeling: Kedem–Katchalsky vs a two-parameter formalism, Cryobiology 37 (1998) 271–289. R.L. Levin, A generalized method for the minimisation of cellular osmotic stresses and strains during the introduction and removal of permeable cryoprotectants, J. Biomech. Eng. 104 (1982) 81–86. P. Mazur, S.P. Leibo, R.H. Miller, Permeability of the bovine red cell to glycerol in hyperosmotic solutions at various temperatures, J. Membr. Biol. 15 (1974) 107–136. P. Mazur, I.I. Katkov, N. Katkova, J.K. Critser, The enhancement of the ability of mouse sperm to survive freezing and thawing by the use of high concentrations of glycerol and the presence of an E. coli membrane preparation Oxyrase to lower the oxygen concentration, Cryobiology 40 (2000) 187– 209. J.J. McGrath, A microscope diffusion chamber for the determination of the equilibrium and nonequilibrium osmotic response of individual cells, J. Microsc. 139 (1985) 249–263.