132 MUSCLE ACTION ENERGETICS AND MOLECULAR MECHANISMS IN MUSCLE ACTION PART 11-STATISTICAL THERMODYNAMICAL TREATMENT OF CONTRACTILE SYSTEMS BY TERRELL L. HILL Naval Medical Research Institute, Bethesda, Md. Received 29th April, 1952 Several possible models of the fundamental elastic element in muscle are illustrated. It is suggested that phase changes combined with relatively small alterations in the state of charge (or ionic strength) may account for the " razor edge " nature of muscle con- traction. However, it does not appear necessary to invoke phase changes to obtain con- siderable alterations in length of an elastic fibre at constant tension, as a result of a reason-TERRELL L . HILL 133 able change in the state of charge of the fibre. Various alternative methods of calculating the electrostatic free energy are examined.The different methods give results of the same order of magnitude in the example chosen. 1. INmoDucTIoN.---In an earlier paper 1 several models of an elastic fibre were discussed. Our purpose here is to illustrate by a few numerical examples some of the properties of the models, in particular charge effects and phase changes. Also, an analysis of the computation of the electrostatic free energy is given which is somewhat more detailed than in the earlier paper.1 It should be stated explicitly that the models considered are not supposed to be conceivable models (nor represent the behaviour or properties) of a muscle fibre but rather conceivable models on a molecular level of the fundamental elastic element of a muscle fibre.This elastic element must presumably " drag " with it water, other plastic and elastic material, etc. Also, the " elastic " element itself may exhibit non-equilibrium behaviour (e.g. in the a-/3 model 2), which we have not included. 2 . MODELS OF ELASTIC FIBREs.-In this section we consider four models, giving first an analytical treatment followed by a few examples and discussion. Deriva- tions of many equations have already been given 1 and will not be repeated here. (i) Network with free lateral swelling.-Consider a swollen cross-linked network in the form of a cylinder of radius R and length 1. There is no restraint on the lateral swelling of the cylinder. There are charges along the molecular chains so that the net charge is equivalent to n* charges E .A solution of electrolyte with Debye-Huckel constant K surrounds and penetrates the cylinder. We restrict ourselves here to the case where R > 1 / ~ and the linear Debye-Hiickel approximation can be used. Then one finds 1 that the length-tension relation is given by where V(M) is obtained from and (a) ANALYTICAL. t = a - (l/V.2), l/cc = ( 4 2 ) + q [ v / ( l - v ) ] 2 - Z [ ( K v * / 2 ) + v + In ( 1 - v)], (1) (2) t = do/vkT, tc = 1/10, q = 2nn*W/D~2vVokT, T = tension (strictly, force), 10 = rest length of unswollen (" dry ") polymer, v = number of molecular chains (between cross links) in network, D = dielectric constant of solvent, 2 = number of statistical units in a molecular chain, PVO = volume of a statistical unit, vo = volume of a solvent molecule, v = volume fraction of polymer, K = van Laar " heat " of mixing constant.VO = Zvofh = volume of unswollen polymer, In the next two models we shall find that for sufficiently large values of K (" poor " solvent for the polymer) a phase change can occur, which appears as a " loop " in the length-tension curve. This is a consequence of the property of polymer solutions with large3 K to split into two phases. We merely remark here that for large K (putting q = 0 for simplicity as the phase change is not an electrostatic effect), v in eqn. (2) is practically independent of a and is given ap- proximately by (recall that 2 is of the order of 100-1000) - In (1 - v) = ( K f ? / 2 ) $- V . (3) Thus, for large K, z, in eqn. ( 1 ) is essentially constant and no loop occurs in the length-tension curve.134 MUSCLE ACTION (ii) Network with fixed radius R*.--This is the same model as above except that the radius is kept constant, for example by rigid rings attached at intervals along the outside of the cylindrical network, or by some other means.In this case length changes are accompanied by proportional volume changes, and the thermodynamic work term is 1 (4) where Y is the volume of the cylinder and F the Helmholtz free energy. The two terms contributing to 7, combined, make a phase change possible. The first term is the usual deformation tension increasing monotonically with length while the second term is proportional to the chemical potential of the solvent 1,4 (to increase the volume, solvent must be taken in).The second term thus increases (q = 0 for simplicity again) with length, starting from negative values, until it passes through a positive maximum,4 and then falls, approaching zero asymptotically. Instead of eqn. (1) and (2), one fmds 1 rdl = [(3F/bl)T, y + nR*2(3F/3 V)T, Jdl, where 6* = R*/Ro and Ro is the " rest radius " of unswollen polymer. the phase change are related by From 3t/3a = 32t/3ci2 = 0, the values of 2, K and a at the critical point for Critical constants are given in table 1 for q = 0, 6* = 1 and q = 0, 6* = 2. (iii) The " accordion " modeL-This model consists of a number of cylindrical t n FIG, 1 .-The " accordion " model (3). In unit (a) the chains are ordinary " random flight" chains. In unit (b), the entire path must lie within a single unit.units, arranged longitudinally (fig. l), each one of which contains v polymer chains. The cross-linking takes place only in mathematical planes at the top and bottom of each unit (fig. 1). Other polymer molecules or structures, not participating in the elasticity of the system, must run laterally in these boundary planes and participate in the cross-linking in order to keep the cylinder with a constant radius R. R is taken as large enough that R > 1 / ~ and also that edge effects may be neglected (or a mem- brane around the cylinder could prevent " bulging '' of the polymer molecules). Parts of a given chain may lie inside of neighbouring units (fig. la), but by cancellation the average polymer density in a unit will be unaffected by this. An alternative model with essentially the same properties would result if the boundary planes were considered physically impenetrable and only configurations of the type shown in fig.lb counted. We need consider the thermodynamic properties of only one unit. The mixing and electrostatic free energy expressions obtained previously 1 are applic- able with V/Vo = 1/10 = a. For the deformation free energy we use the usual statistics of a single polymer chain based on a probability of occurrence of a given end to end distance I of the form P(1) = const. exp (- 312/2ZL2), (8)TERRELL L . HILL 135 TABLE 1 Critical constants, q = 0, model (2) - LT 1 1-5 2 2.5 3 3-5 4 4.5 5 6 7 6 * = 1 6* = 2 Z 12-50 43.3 110 233 436 750 1208 2710 5310 K 2-7 20 2.057 1.759 1.592 1 -484 1.410 1.356 1 -28 1 1.232 t Z 31.5 181.2 3.59 612 4.63 1559 5-64 3330 6.67 6300 7.67 8-67 9.71 11-72 13.67 K 1 -524 1.291 1.202 1.154 1-125 1.105 t 1.75 2-82 3-86 4.86 5-88 6.76 where L is the length of a statistical unit. for large extensions.) Then where Also, According to eqn.(8), 1 = 0 is the most probable value of I, but the incompress- ibility of the dry polymer (volume Yo) fixes the actual initial length at lo. It is reasonable to assume that a is of the order of magnitude of unity, for example, a = 2 (i.e. this would be the value of a if the chains are formed and attached to top and bottom of units with their most probable three-dimensional end-to-end length). The value of 20 and hence a will be reduced if the chains are formed in the above way in the presence of solvent.As in the preceding model, the volume is not an additional independent external variable. On combining the various free energies,l from r = (3F/31)T we find (Eqn. (8) can, of course, be modified AFdef. = vkTa(cc2 - l)/2, (9) a = 3102/ZL2. Vo = ZV/&I~ = 7cR210. 4 K 1 t = a u - - Eqn. (10) is similar to eqn. (5) with 6* = 1, as might be expected, and has similar properties including a phase change. The critical point is given by Z = 3cc2(a - 1)2 a - - [ (a - 2q 1)4 1 ' K = l + - 1 a3 2q cc - 1 + 2[" + m 3 - J Critical constants are given in table 2 for q = 0. TABLE 2 Critical constants, q = 0, model (3) LT Zla K 2.0 2.5 3.0 3.5 4.0 4.5 5-0 5.5 6.0 12.0 42.2 108 230 432 744 1 200 1840 2700 2.667 2.037 1.750 1-587 1.481 1.408 1.354 1-313 1.280 tla 3.68 4-70 5.7 1 6.72 7.72 8-73 9.73 10-74 11.75I36 MUSCLE ACTION Incidentally, as regards the critical point in the Flory-Huggins theory and in eqn. (7) and (12), K - 2q Z(1 - v)3 plays the role with q > 0 that K plays with q = 0.(iv) The a-/3 model.-For present purposes we need add nothing further to the earlier analysis,l but we describe the model for the benefit of the reader. Sup- pose we have a sheet of C polypeptide chains. In the a structure 5 each amino acid residue forms a hydrogen bond with its xth neighbour above and below in the same chain. According to Pauling and Corey,6 x = 3 for myosin and keratin. In the relatively extended p structure 5 a given residue forms a hydrogen bond with each of its two nearest neighbour residues on adjacent chains. In intermediate states we assume that in a given chain residues occur in groups or units of x, all residues a or all p within a unit, and that a and ,B units are scrambled statistically along the chain.This model can now be treated by a modification of the well-known quasichemical approximation in statistical mechanics, taking into account hydrogen bonds between vertical nearest neighbour a units and horizontal nearest neighbour p units. Steric contributions to the interaction energies between units and the complication of imperfect lateral alignment owing to /3 units being twice the length of a units can also be included in the analysis. With reasonable hydrogen bond energies a phase change (a loop in the length-tension curve) is predicted as is common in two and three dimensional nearest neighbour interaction problems in statistical mechanics. This is in agreement with the experimental measurements of H.J. Woods 5 on the length-tension relation in once-dried myosin strips, in which at a certain tension there is observed a sudden large increase in length at practically constant tension. This represents a change, as shown by X-rays,s from vertical a-a crystallization to horizontal P-p crystallization. (b) EXAMPLES AND DIscussIoN.-The principal objective here is to see whether reasonable changes in the state of charge (by adsorption or desorption of ions 1) of an elastic fibre are sufficient to cause considerable changes in length at constant tension, as required for example in the Morales-Botts 7 theory of muscle action.The earlier work of Riseman and Kirkwoods and Kuhn, Katchalsky and co- workers 9 should of course also be mentioned in this connection. The first three models will be considered first since the electrostatic free energy calculation is the same in these cases. For a single symmetrical electrolyte 1 with ions of charge f E and molar concentration c1, where there is a charge E at every X-th statistical unit along the chain. If there are y monomers (e.g. amino acid residues) per statistical unit, then it is easy to see that we also have where 2’ = Zy is the number of monomers in a chain, P’VO = /3vo/y is the volume of a monomer and there is a charge E at every X’ = yX-th monomer. That is, the value of q does not depend on the size of the statisticaz unit. As an illustration of the magnitude of q suppose we take (with a myosin chain in mind) X’ = 9, @‘ = 4.7,Z’ = 300 (mol.wt. N 35,000), vo = 18 ml/mole and c1 = 0.15 M. Then q = 72.9. By varying Z’, c1 and X’, values of q in the range 0-100 or higher are easily obtained. The limits of the Debye-Huckel linear approximation must of course be kept in mind 1 (see also For values of K we turn to the work of Rowen and Simha.10 From adsorption isotherms of water on proteins, near the saturation pressure, they estimate K = 2-3 for silk, 2.1 for wool, 1.8 for serum albumin, 1.6 for collagen and 0.8 for salmine. These values may be compared with those in tables 1 and 2. It will be noted that phase changes are to be expected. If K represented a pure heat effect, positive values of K would correspond to an endothermic heat of mixing water and protein (with highly polar sites already filled).ls 11 Actually, from temperature coefficients, q = z’/4c1vop‘xJ2, (14) 3).TERRELL L .HILL 137 Rowen and Simha show that the heat part of K is exothermic but that there is also an entropy contribution making the net value of K appear endothermic as a " heat " of mixing. Fig. 2 shows several length-tension curves for the " accordion " model 12 with 2 = 459 and K = 0 and 1.59. The top curve is a critical curve (table 2). Charges (q = 10,20) act against the tendency for a phase change to occur. With K = 0, only a small change in length occurs at constant tension for Aq = 20. The effect is larger with K = 1.59 and is especially large just below the critical tension where the length almost doubles.Note the much greater swelling (e.g. at t = 0) in the K = 0 case. It should perhaps be pointed out that since the elastic element can be surrounded by water, the fraction of protein in the element may be quite different than in the muscle fibre as a whole, and also the element can increase in volume I I -6 - v - 4 FIG. 2.-Length-tension curves according to model (3), with q constant. on stretching (taking in water) without changing the volume of the muscle fibre containing it. Fig. 3 shows a series of curves on both sides of the critical curve, for 2 = 400 and K = 2 (q crit. = 296). Near t = 0, Aq = 40 gives an increase in length of 55 %, but at higher tensions a very small Aq can give an extension of a factor of four or more.In principle, this is possible for a single unit 13 with an infinitesimal Aq, but with many units, not all exactly alike, the flat region will become somewhat smoothed out. In fig. 2 and 3, q has been kept constant along a length-tension curve. A more complicated case of greater interest 1 is to consider explicitly the adsorption from solution of an ion with charge opposite to that on the fibre. We then require, for example, a length-tension curve at constant concentration c of the ion in solu- tion. The value of q will change along such a curve. As an example, suppose in fig. 3 that the q = 40 curve is due to fixed charges + E on the fibre at c = 0. Let the ion being adsorbed have a charge - 4~ (e.g. ATP) and suppose there is Langmuir adsorption of these ions on sites just sufficient in number so that with all sites filled (c = a), q = 0 (the net charge n* is used 1 to determine q-see also E138 MUSCLE ACTION 9 3).Let 8 be the fraction of adsorption sites filled. Then in eqn. (10) we put q = 40 (1 - @12, where @(a) is determined from the adsorption isotherm1 The constant 7.695 is fixed by the choices of parameters already mentioned and one additional: we take X(S = 0) = 9.618 (this gives q =-: 40 with /3 = 10 and r ~. I \ I \ I \ \ \ \ \ 72 I \ I \ Z =400, K ‘2, o = 2 FIG. 3.-Length-tension curves according to model (3), with q constant. c1 = 0.1 5 M). The constant f depends 1 on the partition function for adsorption of an ion on a site and on the standard free energy of the ions in solution. In fig. 4 we show curves for fc = 0 (q == 40), 0.07 and CCI (q = 0).Fig. 3 and 4 are FIG. 4.-Length-tension curves (compare fig. 31, with concentration c of adsorbed ion qualitatively similar. In fig. 5, calculated for this same case from eqn. (10) and (15), the length of the fibre at constant tension is plotted against adsorbate ion concentration. At a certain “ critical ” concentration (estimated from fig. 4) there occurs a sudden threefold contraction of the fibre. By working on either side of such a discontinuity (even though smoothed out a little) it is possible for very small changes in, for example, ATP concentration (or pH, Mg2+, Ca2+, salt concentration-since q depends on K, etc.), in the neighbourhood of the ‘‘ critical ” concentration appropriate to a given tension, to bring about sudden constant.Variation in t3 along a length-tension curve is included.139 and very large contractions and extensions of the elastic element. (The elastic element, we recall, must presumably carry along with it considerable extraneous material.) These considerations may give a clue to the rather obvious “razor edge ” nature of muscle contraction.14 Fig. 6 shows the adsorption isotherm accompanying fig. 5. TERRELL L. HILL FIG. 5.--Ciange of length with concentration c of adsorbed ion, at constant tension, corresponding to fig. 4. Fig. 7 contains several other illustrations, all for 2 = 100 and q = 0 or 72.9 (see above). Curves la-d show Aq effects according to the first model (eqn. (1) and (2)). The first pair refer to K = 2 and the second pair to K = 0.With FIG. 6.-Amount of ion adsorbed as a function of ion concentration, at constant tension corresponding to fig. 4. K = 2 there is, for example, a respectable 90 % increase in length at f = 0, while this increase is only 20 % for K = 0. Curves 2a and 2b pertain to the second model (eqn. (5)) with 6* = 1 and K = 2. Curves 3a and 36 have been computed from the third model (eqn. 10)) with K z= 2. Very large charge effects are predicted in these latter two models. The a-13 model, for different reasons, shows in many ways the same general type of behaviour as the second and third models so we give two illustrations only.1 40 MUSCLE ACTION In fig. 8, la and 2a are typical length-tension curves 1 according to the a-/3 model, in the absence of charges.The location Curve l a happens to be a critical curve. FIG. 7. - Length-ten- sion curves with q con- stant. Curves 1 abcd- model (1); curves 2 ab-model (2) ; curves 3 ab-model (3). =0, K.2 ~72.9, K = 2 '0, K + O = 7 P q K = O of t' = 0 depends on the value of J a ~ , which in turn is determined by the relative intrinsic stability of a and /3 units (t'='~(Zp- la)/ CkT, where Is and 1, are lengths of /3 and a units, respectively). Curves 16 and 2b show, as an example, the result of introducing a charge E at every ninth residue, with c1 = 0.15 M and T = 310" K, using the dimensions in Pauling and Corey 6 and eqn. (102) of ref, (1) for the electrostatic term. As before, a re- latively large effect occurs near or in a phase change region (la -+ lb). We may note two differences between the cc-p model (4) and models (2) and (3).First, because of Jap, the phase change region in model(4) can occur at either low or high ten- sions instead of at rather high tensions only (unless a is small ; see tables 1 and 2). This could be an important advantage of this model in a muscle, if phase changes are actu- ally made use 0f.15 In fact, by changing the order of side chains, and hence Jao (through steric effects, etc.), it is conceivable that a group of elastic elements could be available with the proper J a ~ to give the desired sensi- tivity in each range of tension. Second, the maximum change in length in the a-p model is about a factor of two, so that a phase change region would almost be a necessity in 2.0 1"" LL J.0 order to give a change in length approaching FIG.8.-Length-tension ac- this order of magnitude (as a result of a 'Ording to (4), with amount of charge constant. l a and 2a without charge; l b and 26 with change in the state of charge). Although we have emphasized in this = fraction of units in c( paper the interesting possibilities of phase form. changes, it should be pointed out that model (1) appears adequate (recall, for example, the 90 % change in length in l a --f lb,TERRELL L. HILL 141 This agrees with the conclusion of fig. 7) without invoking a phase change. other investigators.7-9 compare, in a single example, several different calculations of the electrostatic term in eqn. (2), (5) and (10) and in similar equations in ref. (1). We restrict the discussion here to macroscopic (dimensions very large compared to 1 / ~ ) and iso- tropic (density of polymer segments uniform) systems-the case of interest in the present problem.A single polyelectrolyte molecule in solution is an example of a non-macroscopic, non-isotropic system for which similar but modified considera- tio ns apply.1 Also, for present comparative purposes, we omit correcting the electrolyte concentration and dielectric constant for the volume fraction of poly- mer,l and simply consider that the charges “ on the polymer ” are distributed uniformly in a macroscopic region (the “ fibre ” or “ gel ”) of the electrolyte solu- tion. These charges and the ions of the electrolyte are treated as point charges. (a) ANALYTICAL.-(i) Donnan equilibrium-linear.-This amounts 1, 16y l 7 simply to expressing the neutrality in the gel, 3.CALCULATION OF ELECTROSTATIC FREE ENERGY.-In this S&iOn We wish to Outside of this region is electrolyte solution only. where i refers to ions of the electrolyte, and the net charge is equivalent to n* charges E in the volume V of gel. On linearizing eqn. (16), solving for $, calculating the work W of charging in the presence of electrolyte and then p’ G - 3 W/3 V, one obtains L15 p’ = = n*zkT/4cl Vz (single 5 E electrolyte), or p = (4q V$/n*2kT)p’ = u2. (17) (eqn. (1 7) defines p ) . without linearizing leads to 1 (ii) Donnan equilibrium-non-linear.-For a single 41 E electrolyte, eqn. (1 6 ) Eqn. (17) and (18) are the results used in the earlier paper 1 and in 5 2. (iii) Fixed charges in a lattice-linear.-If there are n charges E at fixed points in the gel and we linearize the Poisson-Boltzmann equation for this problem, the potential $ at any point inside or outside the gel is (because of linearity and point charges) the sum over c[exp (- KY)]/DI“ for all the n charges, where r is the distance of a given charge from the point. Consider a lattice with Zi i-th neigh- bours a distance ri from a given lattice point.Then one finds where the yi are geometrical factors determined by yi3 = nri3/V. An example with two kinds of charges is included below. There is, incidentally, here a term - E ~ K M / ~ D in W, owing to the potential of the ion atmosphere at each charge, which does not appear in the Donnan method. However, this term is independent of V and does not contribute to p .(iv) Fixed charges smeared uniformly-linear.-Instead of a lattice, suppose the n fixed charges are smeared with uniform density. After integrating c[exp (- ~r)]/Dr over the macroscopic gel to obtain $, one is led again to eqn. (17), as might be expected. The same new term - e2~n/2D occurs again of course in W.142 MUSCLE ACTION (v) Charges part of " electrolyte "--linear.-In the present application charges are attached to polymer chains. One limiting model is to assume the charges are immobile, as in (3) and (4) above. The opposite limiting model, which we consider here, is to assume that the charges behave as species of electro- lyte ions (restricted to the inside of the gel). The correct situation is intermediate and inseparable from the polymer configuration part of the problem.Consider the Poisson- Boltzmann equation for the potential $ in the neighbourhood of a particular j-th charge when all (polymer) charges (the electrolyte ions are kept fully charged) are in the state of charging h (0 < h < 1 ) . Far from the particular charge let $ + $0 and average neutrality obtains (the potential outside the gel is $ = 0) : Suppose there are nj charges mje in the volume V. Putting $' = $ - $0 and linearizing, where v2$' = K A ~ $ ' , It will be noted that KO is a function of V. j-th charge is * n - d d = *o(h) K A ~ f K2 + Ko2h2, The non-self potential at the particular - (mj€hKn/ 0). (25) The work W of charging all the (polymer) charges simultaneously is then and the term in braces = 1 + 1 &(?) 4 + .. . The first term in eqn. (26a) is the same 1 as in (1) and (4) above. The second term may also be compared with (4) (the leading term in the series agrees with (4)). Finally - !(z)z{((:)2 3 + 1)" - 111;- (27a) Note that retaining only the leading term in the series does not give just p = v2.TERRELL L. HILL 143 An analogous discussion can be given using eqn. (20) in nonlinear form (to obtain z,ho) but linearizing eqn. (22). The results are algebraically complicated and will be omitted. (b) EXAMPLES AND mscussIoN.-In fig. 9 we compare p(l/er) calculated in the different ways above, in a special case : a charge E on every ninth residue of myosin (see 9 2), T = 310" K and c1 = 0.15 M. Curve 1 is for eqn. (17), curve 2 for eqn. (18), curve 3 for eqn.(19) and curve 5 for eqn. (27a). In eqn. (19), zi and yi were chosen for a simple cubic lattice.18 The summation was carried through the first four neighbour shells (32 charges) and integration (with a smeared, uniform density) was used for the remaining charges. In eqn. (27a), n = no = n* and ( K O / K ) ~ = n/2clV. It will be seem from the figure that the simple eqn. (17) gives the largest electrostatic termp but that the order of magnitude is the same in all cases. The qualitative results of 3 2 would not be affected by using an alternative electrostatic computation instead of eqn. (1 7). Eqn. (17) is least satisfactory when several types of charges are present. As an example, consider the previous case with the same total number of charges but FIG.9. - p = - (3W/3 V ) x const. plotted against l/v = V x const., according to computation methods (l), (2), (3) and (5) in a special case. Curves 111 and V (- p ) calculated from methods (3) and (5) with net charge = 0 but total charge + 0. la1 numbers of + E and - E. Then 25 rz* = 0 and p' = 0 for the Donnan methods (1) and (2) (also (4)). But p' < 0 for methods (3) and (5) because of attraction between oppositely charged ions.19 In the simple cubic lattice we assume alternate positive and negative ions, sum over the first four shells and integrate over the rest (separate integrations for positive and negative charges are required as different numbers of the two types of charge are included in the first four shells). p must be redefined in both (3) and (5) (since n* = 0) by replacing n* in the defining eqn.(17) by no. Then eqn. (27a) becomes (n* = 0) Curves I11 and V in fig. 9 give - p ( l / v ) . may be negative if n&* is large enough. more negative on neutralizing the charge. Even with a considerable net charge (n* =I= 0), according to eqn. (27a), p' But in this case it will in general become144 MUSCLE ACTION In the a-8 model the charges are smeared (n* thus comes in) on a surface and the non-linear Poisson-Boltzmann equation solved. A two dimensional lattice (see (3) above) could be used here but only with the linear differential equation (i.e. thesum over E [exp (- ~ r ) ] / D r is used again). In conclusion we may indicate the type of calculation of Wused in other work. Hermans and Overbeeck 16 used a linearized Poisson-Boltzmann equation based on eqn.(16) for spheres, not necessarily large. The linear Donnan method (1) was given as a special case (large spheres). Kimball et al.17 suggested that in the Hermans and Overbeeck problem a better approximation than linearizing is to put 021) = 0 in the Poisson-Boltzmann equation, thus obtaining eqn. (16) (with n*/ V a function of r in general) for spheres, not necessarily large. Our use 1 of eqn. (16) (non-linear Donnan) thus amounts to a special case of the method of Kimball et al. Katchalsky et a120 do not indicate in their preliminary short note on gels how the electrostatic calculation was carried out, but one may surmise from an earlier paper by Katchalsky 21 that the electrostatic problem was coupled explicitly into the polymer configuration statistics, by using E [exp (-.KY)]/DP for pairs of charges in a single polymer chain of the network. This type of coupling is of course a desirable feature, but in a gel, consideration of interactions within a singular molecular chain only is not. The writer is indebted to Dr. M. F. Morales, Dr. K. J. Laidler and Dr. S. L. Friess for very helpful suggestions. 1 Hill, J , Clzem. Physics, 1952 (in press). The problem of the size and shape of poly- electrolyte molecules in solution is treated by the same methods in Hill, J . Chem. Physics, 1952 (in press). 2 Alfrey, Mechanical Behaviour of High Polymers (Interscience Publishers, New York, 1948) ; Burte and Halsey, Textile Res. J., 1947, 17, 465. See also, for application to muscle, Butchthal and Kaiser, Dan.Biol. Medd., 1951, 21, no. 7 ; Polissar, Amer. J. Physiol., 1952, 168, 766. 3 In the Flory-Huggins theory of polymer solutions, Huggins’ p = K/2. 4 Flory and Rehner, J. Chem. Physics, 1943, 11, 521. 5 Astbury, Pruc. Roy. SOC. B, 1947, 134,303 ; Astbury and Dickinson, Proc. Roy. SOC. 6 Pauling and Corey, Proc. Nat. Acad. Sci., 1951,37,261,729 ; Nature, 1951,168, 550. 7 Morales and Botts, Arch. Biochem. Biophys., 1952 (in press). 8 Riseman and Kirkwood, J . Amer. Chem. SOC., 1948, 70,2820. 9 Katchalsky, Kunzle and Kuhn, J. Polymer Sci., 1950,5,283 ; Katchalsky, Experientia, 1949, 5, 319 ; Kuhn, Experientia, 1949, 5, 318 ; Kuhn, Hargitay, Katchalsky and Eisenberg, Nature, 1950, 165, 5 14. B, 1940, 129, 307. 10 Rowen and Simha, J . Physic. Chem., 1949, 53, 921. 11 Hill and Rowen, J, Polymer Sci., 1952 (in press). 12 Models (2) and (3) are very similar in properties. Since model (3) seems more realistic, most examples concern it. 13 A single unit must be at one end or the other of the phase change (in an approxi- mate theory of the present type). With many units in a chain, intermediate (average) lengths are possible. 14 One gets the superficial impression that a muscle fibre is a very delicately balanced system ready for a sudden, precipitous change (contraction) as a result of some small alteration in environment. Szent-Gyorgyi uses the term “ razor-edge ” in a similar connection (actin-myosin combination), Science, 1949, 110, 41 1. 15 A small Aq can give a large A1 at low or high tensions on model (4), but only at fairly high tensions on models (2) and (3) (unless a is small). But a large Aq can give a large A1 at low or high tensions on models (2) and (3) (see, for example, fig. 7). Astbury 5 believes, of course, that the #3 --f cc transition (model (4)) itself, though a property of myosin, is not involved when a muscle fibre contracts. Pauling and Corey,6 on the other hand, assume that this transition is involved. 16 Hermans and Overbeeck, Rec. trav. chim., 1948, 67, 761. 17 Kimball, Cutler and Samelson, J . Physic. Chem., 1952, 56, 57,TERRELL L. HILL 145 18 Slater, htroduction to Chemical Physics (McGraw-Hill, New York, 1939), p. 386. 19 The linear Donnan method overestimates p both with charges on the gel and with charges neutralized by opposite charges. The error in A1 will thus tend to cancel. 20 Katchalsky, Lifson and Eisenberg, J . Polymer Sci., 1951, 7, 571. 21 Katchalsky, J. Polymer Sci., 1951, 7, 393.