WATER RELATIONS IN PLANT CELLS
443
3. The dehydrating catalysts, alumina and thoria, are effective activators of copper for the hydrogenation of benzene; this would not be predicted from Medsforth’s explanation of promoter action. The authors express their thanks to Mr. W. J. Cerveny for much of the experimental work, and to Dr. W. C. Pierce, of the University of Chicago, for the spectroscopic analyses of the copper-chromia and copper-alumina series. REFERENCES (1) ARMSTRONG ASD HILDITCH: Proc. Roy. SOC. (London) 102, 27 (1922). (2) CORSON AND IPATIEFF: J. Phys. Chem. 46, 431 (1941). (3) DEWARAND LIEBMASN:U. S.patent 1,268,692 (1918). (4) I P A T I E FCORSOS, ~, AND KURBATOV: J. Phys. Chem. 43, 589 (1939). ( 5 ) IPATIEFF, CORSON, AND KURBATOV: J. Phys. Chem. 44, 670 (1940). (6) JPLIARD:Bull. SOC. chim. Belg. 46, 549 (1937). (7) JULIARD AND HERBO: Bull. SOC. chim. Belg. 47, 717 (1938). (8) LIEBXANN: British patent 12,981 (1913). (9) MEDSFORTH: J. Chem. SOC.123, 1152 (1923). (10) ROGERS:J. Am. Chem. SOC 49, 1432 (1927) (11) T ~ Y L OASD R STARKWEATHER: J. Am. Chem. SOC.62, 2314 (1930).
.
A THERMODYKAMIC FORMULATION O F T H E WATER RELATION3 I N AN ISOLATED LIVIKG CELL P. S. TANG
AND
J. S. WANG
The Physiological Laboratory and the DepaTtment of Physics, National Tsing Hua University, Kunming, China Received August 9, 1940
I The simple osmometer concept has been applied with advantage to studies on permeability to water in animal cells (7) and to studies on the water relations in plant cells (9). Although the concept has proved useful for the purposes mentioned, the ambiguity arising from the use of the terms “turgor pressure,” “suction pressure,” “wall pressure,” “osmotic pressure,” etc. (cf. 9) in the analysis of the water relations in plant cells indicates that the water relations of living cells may perhaps be treated with more lucidity in other ways. In this account an analysis of the movement of w a k r through an isolated living cell is made with the aid of certain relations in thermodynamics. We shall consider an isolated spherical vacuolated plant cell which has been plasmolyzed until the protoplasmic mass is detached from the cell
444
P. 5. TANG AND J. 5. WANG
wall, but is still capable of recovery. When this cell is placed in a large quantity of a hypotonic solution it will take up water, causing the protoplasmic mass to swell to a volume VI when the protoplasniic membrane just begins to be stretched. We shall designate this as stage 1. This swelling of the protoplasm will continue so long as the external solution is hypotonic with respect to the vacuolar sap. In the animal cell devoid of a cell wall, the swelling will continue until (1) the vacuolar sap is isotonic with the external solution, (2) the entrance of water is counteracted by the pressure developed inside of the cell. and (3) the cell is cytolyzed. In the plant cell, the swelling will usually continue until the protoplasmic membrane just touches the cell wall. This we shall designate as the second stage of swelling, which is generally known as “incipient plasmolysis”. If conditions are such that water continues to enter the cell after the second stage, the pressure which is developed inside the cell will cause the relatively rigid cell wall to yield sluggishly, and the total volume of the cell gradually increases until the final maximum volume is reached, when the cell is said to be fully turgid. This is the third stage of swelling in a plant cell. We shall proceed to analyze the tendency, @, for the passage of water into and out of the cell a t each and any of these stages.
I1 To apply the theory of thermodynamics to a living cell suspended in solution as described above, we shall follow the usual practice of treating three sets of equilibrium conditions separately. These equilibrium conditions are commonly referred to as thermal, mechanical, and chemical. The condition of thermal equilibrium requires that the whole system, including the cell and its surrounding solution, shall have a uniform temperature throughout (reference 3, page 21). Thus the first set of equilibrium conditions is easily disposed of. We shall proceed t o obtain the more complicated conditions of mechanical and chemical equi1ibrium.l During the first stage of swelling, the protoplasmic membrane is unstretched, and the condition of mechanical equilibrium requires that the pressure p inside the cell should be equal to the pressure outside, which by definition is of a constant value PO,so that
P = Po
(1)
1 The possibility that “active secretion” may take place in the protoplasm which governs the water relations of plant cells, as suggested by Bennet-Clark et al. (1) and supported by Mason and Philia (8),is not considered here, for unleas the nature of the “active secretion” does not obey the laws of thermodynamics, the proceae will be cared for by the three sets of equilibriuni conditions.
WATER RELATIONS IN PLANT CELLS
445
I n the second stage, when the protoplasmic membrane is under tension, the pressure p inside the cell now becomes greater than the pressure of the external solution (reference 3, page 169) by p - p o = j j 27 in which R is the radius of the spherical cell and y is the surface tension of the protoplasmic membrane. The radius R is of course greater than R1, which is the maximum radius of the cell when the membrane is unstretched, that is,
The surface tension of the protoplasmic membrane (y) is a function of R and may be ascertained experimentally (2, 5). Later we shall assume a reasonable form of the function for the discussion of the equilibrium stage. In the third stage of swelling the wall is stretched; hence it is under a surface tension y’, and equation 2 is replaced by p-Po=-
2(Y
+ 7’)
R
Here R is greater than Rz, which is the maximum radius of the cell wall when the latter is under no tension, that is,
In the formulation of equations 2 and 4, the thickness of the wall and of the protoplasmic membrane was neglected for the sake of simplicity. The fact that the volume of the protoplasm may actually occupy 30 to 50 per cent, of the total volume of the cell (8) does not affect the argument in our analysis. The conditions of mechanical equilibrium (equations 1, 2, and 4) determine a t each stage the pressure p inside the cell as a function of the volume
of the cell. The pressure inside the cell, p , is the so-called “turgor pressure.” We shall next investigate the condition of osmotic, or chemical, equilibrium, which is defined as the state in which the net tendency of water
446
P. S. TANG AND J. S. WANG
passage into or out of the cell is zero. This condition is most easily obtained from the condition of minimum Gibbs’ free energy G, defined by
G = U - TS
+ pV
(7)
where U is the internal energy, S the entropy, and T the temperature on the absolute scale. The total Gibbs’ free energy is the sum of the free energy G inside the protoplasm and the free energy G’ outside. The condition of chemical equilibrium is
6G
+ 6G’ = 0
(8)
for variations a t constant temperature T and pressures p and po. If the quantity of water inside the protoplasm is M , that outside is M’, and if the quantity of solute inside the cell is m, then G is a function of T , p , M , and m, and G’ is a function of T , PO, and M’. For variations a t constant T , p , and PO,we have
aG 6G = - 6M aM and
6G’ =
aG’ a-, x
6M’
The quantity, m, of the solutes in the cell is assumed to be invariable. Substituting into equation 8 we have
aM
6M
+ aM -aG’ ’ 6M’= 0
Since the total quantity of water is constant,
6M
+ 6M’= 0
(8b)
Then equation Sa becomes
or P =
P’
where
The quantities I.( and p’ are called the chemical potentials of water in the cell and outside of the cell, respectively. The condition of osmotic equilibrium for water is that the chemical potential of water inside the cell should be equal to that outside.
447
WATER RELBTIONS IN PLANT CELLS
If the external solution is pure water, then G’ is proportional to M’, and
is independent of M I , and is a function of T and po only. Otherwise p’ is a function of T , PO, and M’. But in general the quantity M’ is much greater than M , and the dependence of p’ on M’ can therefore be ignored. The chemical potential, p, of water inside the cell is a function of T, p , and the ratio M / m . The equation of osmotic equilibrium (equation 9), together with the equations of mechanical equilibrium 1, 2, and 4, give iM/m as a function of T and pn. Since the quantity m is fixed, the value of M as determined by the ratio M / m is the maximum amount of water that can pass into the cell at temperature T and outside pressure pn. We may now examine the physical significance of equation 9. If the amount of water inside the cell has not attained its maximum value, equation 9 mill not hold, and we must have in order that for 6.V
> 0 we have 6G
+ 6G’ = p6M + p‘6M‘ = (M
- p1)6M C 0
that is to say, under such conditions, the flox of water into the cell is a natural process which diminishes the total Gibbs’ free energy G G’. The difference between the chemical potentials
+
@ = f i l - p
can therefore be regarded as a measure for the tendency of water to pass into and out of the cell, which in the customary terminology corresponds to the “suction pressure” or “suction force” and “diffusion pressure deficit” (IO, 9). The process of water flowing into the cell will continue as long as the chemical potential in the external solution is greater than that inside the cell. The statement is equally true, mutatis mutandis, for the case of water passing from the cell to the external solution. The chemical potential as a function of temperature, pressure, and quantity of water can be determined by suitable measurements (6). One way of obtaining the chemical potential of water in a solution is to measure the partial vapor pressure of water in equilibrium with the solution. The condition of chemical equilibrium applied to the system solution-vapor yields the result that the chemical potentials of water in the liquid and in
448
P. S. TANG AND J. S. W A N 0
the vapor phase are equal. The chemical potential of water in the vapor phase is expressed in terms of its partial vapor pressure p* in the form (reference 3, page 66) : p =
Jc,dT
- TJc,d
log T
+ CT log p*
(12)
where c, is the specific heat of water vapor at constant pressure and C = R/[M] = 0.1102 cal., [MI being the molecular weight of water and R the universal gas constant ( R = 1.986 cal.). This form of the chemical potential is valid when the vapor can be regarded as a perfect gas. With this form of the chemical potential the tendency ip becomes ip
= CT log (p*’/p*)
(13)
in which p* and p*’ &re the partial vapor pressures of water in the cell and in the external solution, respectively.
I11 Since the customary treatment of water relations in the living cell is in terms of pressures and by means of volume-pressure diagrams, we shall so interpret the above relations in this section. Equation 9 may be regarded as an equation determining the ideal equilibrium pressure, p , of the cell as a function of T, pol and M. Instead of M we may use V , the volume of the cell. It is clear that V is a single-valued function of T, p , and M . Thus we may regard equation 9 as an equation determining p as a function of T, p a , and V . The variation of p with M is obtained from equation 9 by differentiation, viz.
or, since
is the partial specific volume of water in the cell,
The general thermodynamic condition of stability of equilibrium requires (reference 3, page 24) that
-ab> o aM
The quantity
WATER RELATIONS IN PLANT CELLS
449
is always positive, so that by equation 14
by the cwdition of thermodynamic stability (reference 3, page 23). Hence
The curve p versus V is therefore of the form shown in figure 1, which gives the ideal eyuiljbrium pressure p as a function of the cell volume. When the state of equilibrium has been attained, the pressure p in the cell is much greater than the pressure PO in the external solution. The pressure difference, p - p o , is called the osmotic pressure.2 It is seen that if M can be indefinitely increased? the cell sap will ultimately approach pure water, p will approach po, and the osmotic pressure of the cell will tend to become zero. Hence the p - V curvc is given by the line in figure 1, approaching asymptotically the straight line p = PO. If a t 5,given volume, V , of the cell, the pressure required by the condit o n s of mechanical equilibrium 1, 2, and 4 is less than the ideal equilibrium pressure given by equation 9, i.e., by figure 1, the water from the external solution tends to pass into the cell. The effect of the increasing amount of water in the cell causes on the one hand an increase in the actual pressure inside the cell, and, on the other hand, a decrease in the ideal equilibrium pressure, p, as given in figwe 1. The state of osmotic equilibrium is reached when the pressure required by the conditions of mechanical equilibrium is equal to the ideal equilibrium pressure of figure 1. Conversely, when the cell is a t a volume when the pressure required by the conditions of mechanical equilibrium is greater than the ideal equilibrium
* ‘t’lr:, term “osmotic prcssure” has bceri indiscriminately applied to the diflerence, p - PO, between the ideal equilibrium pressure of the cell and t h e pressure outside, cvc’n when t h e cell is not in equilibrium w i t h t h e outside solution.
450
P. S. TANG AND J. S. WANG
pressure, the process of water passage is reversed until the two sets of pressure are a t equilibrium, and the above analysis, mutatis mutandis, is also applicable in that case.
IV In order to make the above discussions clearer, and in conformity with the customary practice in the analysis of the water relations in plant cells, a graphic presentation of the pressure relationship is presented below. The pressure, p , of the cell in mechanical equilibrium is plotted in figure 2 as a function of V. This pressure is the so-called “turgor pressurC‘. The first part ( A B ) is given by equation 1, p = PO,and corre-
P
v
v
FIG.I
FIG.2
FIG.1. The ideal equilibrium pressure, p , inside of the cell is plotted as a function of the cell volume, V . The dotted line is for p = p o . FIG. 2. The pressure, p , inside of the cell due t o mechanical equilibrium, as a function of cell volume, V . T h e light lines C’D’and D’E’are for “wall pressure.”
sponds to the first stage. The second part (BC) is given by equation 2, and corresponds to the second stage. The third part ( C D ) is given by equation 4,and corresponds to the third stage. The last part (DE) is a straight line V = V, and corresponds to the case when the volume of the cell ceases to increase. It is seen in our treatment that the pressure of the cell wall does not come in; only the turgor pressure inside the cell is relevant. Although the wall pressure is irrelevant in our theory, its value can easily be determined a t every stage, and it is perhaps worthwhile to state the results. The pressure p’ inside the cell wall when it is unstretched during the first and second stages of swelling is equal to the pressure outside, that is
P’ = Po
(15)
451
WATER RELATIONS IN PLANT CELLS
for V 2 Tiz. During the third stage of swelling, the volume of the cell exceeds V z ,the wall is stretched, and the pressure p' inside becomes p' = p o
+27 R
(for R
> Rz)
(16)
This pressure ( p ' ) exerted on the inside of the wall is a t the same time the pressure exerted on the outside of the protoplasmic membrane. It should be noted that the turgor pressure and the wall pressure are not equal. They differ by the amount p - p ' = _ y2 R except in the first stage of swelling. The relation p' with V is plotted in figure 2 in light lines, indicated by C'D' and D'E'.
I
I
I
P I
I
I
I
1
I
I
I I
I I I
I
I
I
v,
v
,V
v
FIG.4 FIG. 3. The same as figure 2, but for the animal cell devoid of wall FIG.4. Combined diagram from figures 1 and 2, showing how the final volume, V,, of the cell may be determined by the two sets of equilibrium conditions.
FIG.3
If the cell has a maximum volume V mless than V z ,or, as in the case of the animal cell, is devoid of a cell wall, the third stage of swelling is absent and the p-V curve has the form shown in figure 3. In figures 1 and 2 the first part A B and the last part DE are straight lines. The parts BC and CD may roughly be represented by straight lines inclined a t different angles with the axis. The actual form of the curves can of course be determined experimentally. If BC is a straight line represented by the equation p - po = a(V
- VI)
Air
= - a(R3 -
3
R:)
(18)
452
P. 8. TANG AND J. S. WANQ
then equation 2 gives y =
aR(R3 - R:)
(19)
Similarly, the assumption that CD is a straight line p
- po = 'p(V - Vz)
=
4s - p(R8 - R:) 3
(20)
gives y
+ y' = 2r 7 pR(Ra - R:)
Combining the two results (equations 19 and 21), we obtain
The forms of the equations 19, 21, and 22 are, however, not of the familiar type. It might be more reasonable to assume that y is proportional to the increase of surface area, by analogy with Hooke's law of elasticity. It may be of interest to mention the observations of Harvey (4) that such is actually the case in the eggs of Arbacia. With this assumption, the values of y and y' are and where a and @ are new constants of proportionality differing from those defined by equations 18 and 20. From equations 2 and 23 we obtain the equation of the curve BC
Similarly, equations 4, 23, and 24 give the equation of the curve CD p
- PO = R-2 ( ( a + p)R2 - aRi
- pR!)
From the Considerations in this section we are in a position to obtain graphically the final volume of the cell as a resultant of the mechanical
WATER RELATIONS IN PLANT CELLS
453
and chemical forces. To do this we only need to plot the curves of figure 1 and figure 2 in a single graph. This is done for a given combination of T,p , po, and m in figure 4,by way of illustration. The point of intersection of the two curves gives the final volume, Vi, of the cell as required by the conditions of mechanical and chemical equilibrium. From that diagram, it may be observed that when a eel1 has a volume V I smaller than the final volume V,, the conditions of mechanical and chemical equilibrium demand that water enter the cell, swelling the cell volume to V I . Conversely, the cell with volume VZwill lose water and shrink, until the two sets of equilibrium conditions are satisfied a t V,. SUMMARY
The water relations of an isolated living cell suspended in solution are formulated on the basis of certain relations in thermodhamics. With the aid of equations 1 , 2 , 4 , and 9 in the text, the tendency of water to pass into or out of the cell may be defined a t any stage of swelling of the cell. The physical significance of the equations is interpreted according to the customary pressure terms, such as “osmotic pressure,” “turgor pressure,” ‘(wall pressure,” and “suction force”. REFERESCES (1) BENNET-CLARK, T. A., GREENWOOD, A. D., AND BARKER,J. W.: New Phytologist 36, 277-81 (1936). (2) COLE,KENNETHS.: J. Cellular Comp. Physiol. 1, 1-9 (1932). (3) GUGGENHEIM, E. A. : Modern Thermodynamics. Methuen and Company, Ltd., London (1933). (4) HARVEY,E. NEWTON:Biol. Bull. 62, 141-54 (1932). (5) HARVEY,E. NEWTON:Trans. Faraday SOL33, 943-6 (1937). (6) LEWIS, G. X., AND RANDALL, M.: Thermodynamics and the Free Energy of Chemical Substances. McGraw-Hill Book Company, Inc., New York (1923). (7) LUCK&BALDWIN, AND ~ ~ C ~ U T C H E O MORTON: N, Physiol. Rev. 12, 68-139 (1932). (8) MASON,T. G . , AND PHILIS,E.: Ann. Botany [N.S.] 3, 531-44 (1939). (9) MEYER,B. S.: Botan. Rev. 4, 531-47 (1939). (10) URSPRIJNG, A . : Plant Physiol. 10, 115-33 (1935)
