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Equilibrium Adsorption of Polar Molecules on Proteins as a Surface-Reconstruction Process† G. F. Cerofolini* EniChemsIstituto Guido Donegani, 28100 Novara NO, Italy Received September 29, 1995. In Final Form: December 7, 1995X The problem of protein folding-unfolding is age-old and not fully solved yet. An amino acid sequence alone is insufficient to specify completely the local configuration, because nonspecific long-range forces tend to impart to the protein a different structure from what would result from covalent bonds only. The adsorption of polar molecules on proteins involves an energy of the same order as the one which stabilizes the native configuration, so that the solvation resulting from the adsorption is expected to modify progressively the protein configuration. The surface-reconstruction model proposed by Landsberg for the Elovich adsorption kinetics is modified to describe the adsorption equilibrium on reconstructable surfaces. In this model adsorption produces a progressive exposure of the protein skeleton to the atmosphere. The model can be worked to account for denaturation and the formation of the molten globule.
1. Protein Folding Rough or fractal surfaces are usually characteristic of dispersed solids prepared in strongly nonequilibrium situations (e.g., zeolites or crushed glasses1); their nonequilibrium configuration can be preserved only if the surface is hard, with an energy landscape with few minima separated by high barriers. Since surface reconstruction requires an adsorption energy higher than, or at least comparable with, the energy stabilizing the surface configuration, rough surfaces can undergo reconstruction only through the formation of chemical bonds between adsorbent and adsorbate. Soft surfaces, i.e., surfaces with an energy landscape with several minima separated by low barriers, cannot preserve rough or fractal configuration, even when prepared in such states. In general, soft surfaces are characteristic of molecules whose configuration is produced by secondary forces or even by entropic factors, rather than by chemical bonds. Examples of soft adsorbents are linear polymers like polyethylene or lightly reticulate polymers like proteins. Though smooth, soft surfaces undergo easily surface reconstruction via the formation of secondary bonds, like those associated with the adsorption of polar molecules on their polar sites. Though the forthcoming considerations are expected to apply to all soft adsorbents, the attention will hereafter be concentrated on proteins because of their practical and conceptual relevance. Most of proteins have a globular structure. Any globular protein “is essentially a one dimensional system folded into a three dimensional structure”.2 With the protein synthesized in an aqueous medium, due to the need for minimizing the protein-water interface energy during synthesis, most of polar sites will be at the protein “surface”, while the hydrophobic arms of the chain will be in the protein “bulk”.3 The protein bulk has a fractal * E-mail:
[email protected]. † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. X Abstract published in Advance ACS Abstracts, September 15, 1996. (1) Avnir, D.; Farin, D.; Pfeifer, P. J. Chem. Phys. 1983, 79, 3566; Nature 1984, 308, 261. (2) Frauenfelder, H. In Structure and Dynamics: Nucleic Acids and Proteins; Clementi, E., Sarma, R. H., Eds.; Adeninine Press: Guilderland, NY, 1983; p 369. (3) Perutz, M. F. Proteins and Nucleic Acids; Elsevier: Amsterdam, 1962.
S0743-7463(95)00814-6 CCC: $14.00
dimension D slightly less than 3,4 so that proteins can be viewed as collapsed polymers. This view is consistent with the surface fractal dimension of globular proteins, which is typically in the range 2.1-2.2, not much greater than the value D ) 2 characteristic of smooth surfaces.5,6 The globular structure is preserved by the -S-Scovalent bridges between different segments of the protein backbone and by electrostatic interactions between polar sites on the backbone or in the side-chain groups. In general, only few disulfide bridges are present in a protein so that most of the structure results from a balance between nonspecific folding, produced by electrostatic pairing of heteropolar sites, and directional folding, produced either by covalent bonds along the backbone (which impart R-helix configuration to the protein) or by interchain hydrogen bonds (which impart β-helix configuration). The contrast between these driving forces will result in a partial unpairing of heteropolar sites. Unpaired polar sites are available to adsorb small polar molecules. The amount of water (considered as representative of small polar molecules) bound to hemoglobin (considered as representative of globular proteins) is 0.3 g of H2O/g of hemoglobin;3 assuming that one water molecule is bonded to each exposed polar group and that each amino acid residue contains on the mean three polar sites, approximately 65% of the polar sites are covered by water. This amount may be assumed as a measure of the exposed sites. The remaining sites are most presumably present as ionic pairs Bδ+‚‚‚A- (δ, < 1) and are available to adsorb water only when the solvation energy ∆Hsolv gained in the hydration process +H2O
+H2O
Bδ+‚‚‚A- 98 Bδ+‚‚‚OH2‚‚‚A- 98 Bδ+‚‚‚OH2 + OH2‚‚‚A- (1) is greater than the electrostatic pairing energy, ∆Hes. For sufficiently polar sites (δ or close to 1), the energy involved in electrostatic pairing is quite high (say, ∆Hes = 5 eV), while the hydration energy is much lower (say, ∆Hsolv = 1 eV/H2O molecule). This implies that reaction 1 is thermodynamically impossible. Elementary electrostatic considerations show, however, that the reaction becomes possible when a large polar molecule is inserted between (4) Dewey, T. G. J. Chem. Phys. 1993, 98, 2250. (5) Fedorov, B. A.; Fedorov, B. B.; Schmidt, P. W. J. Chem. Phys. 1993, 99, 4076. (6) Dewey, T. G. Heterog. Chem. Rev. 1995, 2, 91.
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Bδ+ and A-. This process is facilitated by any increase of the Bδ+‚‚‚A-distance, so that the swelling resulting from adsorption softens the adsorbates and renders new sites available to adsorption. Hydrogen-exchange experiments on native cytochrome c in low concentrations of denaturing agents have indeed shown the existence of a sequence of metastable, partially unfolded forms holding free energy levels between the ones of the native protein and of the fully unfolded state.7 Several decades ago Landsberg proposed a model, based on surface reconstruction, able to explain the Elovich equation observed in chemisorption kinetics;8 this model can now be viewed as a model able to describe the kinetics of surface reconstruction.9 This work is devoted to formulate a model where the progressive exposure of surface sites resulting from the equilibrium adsorption of a polar molecule on a protein is described in terms of surface reconstruction, in a fashion which is reminescent of Landsberg’s kinetic model.
one reproduces the Langmuir model for first-order chemisorption on homogeneous surfaces. Landsberg, however, did not accept assumption 3 and postulated (L2); in so doing he postulated that chemisorption has two contrasting effects: on one hand it deactivates one site, but on another hand it is also responsible for the creation of newly exposed area and hence of new adsorption sites. This can be restated by assuming that the active area lost per unit time and unit geometric area is bdN/dt, where b represents the net lost area per chemisorbed molecule. If the reconstruction mechanism is such that (L3) is satisfied, the number of available sites per unit geometric area evolves with time as
2. The Surface-Reconstruction Model in Kinetics
Equation 4 can be integrated by separation of variables and gives
Many isothermal kinetics (including chemisorption, desorption, and oxidation) are described by a reaction extent which increases with time t as ln t. This timelogarithm law, usually referred to as the Elovich equation, was usually accounted for in terms of fixed energy heterogeneity as well as of induced energy heterogeneity of the adsorbent.10 Landsberg, however, proposed an alternative model of the Elovich equation in terms of surface reconstruction during chemisorption.8 Landsberg postulated the following: (L1) Chemisorption occurs via the collision of impinging gas-phase molecules onto free surface sites. (L2) Chemisorption is responsible for surface reconstruction via generation of new sites at a rate proportional to the chemisorption rate. (L3) Each newly exposed zone has the same topography as the surface from which it was generated. The Landsberg model can be formalized as follows. Assumption L1 reads
p dN ) Ns*aσ dt 2πmk x BTg
(2)
where p is the gas pressure, m is the molecular mass, kB is the Boltzmann constant, Tg is the gas temperature, a is the cross section of each site, σ is the sticking coefficient, and N and Ns* are the numbers of chemisorbed molecules and available chemisorption sites, respectively, both referred to the unit geometric area. The total number of sites Ns is given by Ns ) N + Ns*. Defining the fraction Θ of covered sites, Θ ) N/Ns, one has
-
Ns*(0) Ns*(t)
dNs* dN )b N dt dt s* p ) abσN2s* 2πmk T x B g
)1+
Ns* ) Ns(1 - Θ) and putting
Ns ) constant
(3)
(7) Bai, Y.; Sosnick, T. R.; Mayne, L.; Englander, S. W. Science 1995, 269, 192. (8) Landsberg, P. T. J. Chem. Phys. 1955, 23, 1079. (9) Cerofolini, G. F. In Adsorption on New and Modified Inorganic Sorbents; Dabroski, A.; Tertykh, V. A., Eds.; Elsevier: Amsterdam, 1995; Chapter 2.4, p 435. (10) Porter, A. S.; Tompkins, F. C. Proc. R. Soc. London 1953, A217, 529.
x2πmkBTg
Ns*(0)abσt
(5)
where Ns*(0) is the density of free surface sites at time t ) 0. Inserting eq 5 into eq 2 one gets a differential equation which can be solved by variable separation:
N(t) )
1 t ln 1 + bNs*(0) b τ
(
)
(6)
with
τ ) x2πmkBTg/paσ The increase with time of the number of chemisorbed molecules is independent of b in the linear regime
t f 0 w N ∼ Ns*(0)t/τ and varies inversely with b in the long-time regime
t f +∞ w N ∼
1 t 1 ln bNs*(0) ∼ ln t b τ b
(
)
For b > 0 the surface reconstruction leads to a smooth increase with t of the chemisorbed amount, in such a way that the number of chemisorbed molecules diverges only for t f +∞. Though Landsberg considered only the case b > 0, the case of negative b is interesting too: For b < 0 the solution of eqs 2 and 4 reads
N(t) ) N ) NsΘ
p
(4)
t 1 ln 1 - |b|Ns*(0) |b| τ
(
)
(7)
so that surface reconstruction is sudden and leads to a divergence of N at a finite time t∞, where t∞ ) τ/|b|Ns*(0). The meaning of N ) +∞ is not straightforward. It is better understood if instead of N one considers Ns. The number of adsorption sites is indeed a measure of the area of the adsorbing surface, and an infinite area is a welldefined mathematical concept able to mimic a physical situation. Space-filling surfaces have indeed an infinite area. Of course, a space-filling surface is a mathematical, rather than physical, concept because of the limitations imposed by the atomistic structure of matter. These limitations hold true in other situations too, like for fractal surfaces, whose scale invariance is usually valid in a quite
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restricted scale range.11 In view of the atomistic nature of matter the maximum allowed value of Ns is given by the condition
(1 + ν)Ns ≈ FbL
3. Surface-Reconstruction Model for Equilibrium Adsorption of Polar Molecules on Proteins Adsorption of polar molecules takes place on the polar sites of the protein; on another hand, protein folding is stabilized by the electrostatic attraction between heteropolar sites. This means that only a fraction of the existing polar sites is available to adsorption. While adsorption proceeds, however, the protein undergoes a certain unfolding that allows new sites to be exposed to the atmosphere and to adsorb further polar molecules. Adsorption at room temperature is not exhausted with the formation of a monolayer but proceeds up to multilayer formation. In this case it is more convenient to use the relative pressure x, x ) p/psat (where psat is the saturation pressure at the adsorption temperature), instead of the partial pressure p. Denoting with Θ(x) the average coverage of exposed sites, when the relative pressure is varied by an amount dx, the number of adsorbed molecules varies because of two factors: (a) the change dΘ of equilibrium coverage and (b) the change dNs of exposed sites. This implies that in the adsorption equation
(9)
the number of sites is not assigned, but varies because of folding-unfolding. In analogy with the Landsberg model, the following relationship will be assumed:
dNs dN ) |b| N dx dx s ) |b|Θ′(x)Ns2
(10)
Equation 10 can be solved by separation of variables
1 Ns(x) ) Ns(0) 1 - |b|Ns(0)Θ(x)
(11)
where Ns(0) is the number of exposed sites at x ) 0. Though formally Ns can increase indefinitely, it is however limited by the total number N0 of polar sites. Inserting eq 11 into eq 9, one has the following differential equation
dΘ dN ) Ns(0) 1 - |b|Ns(0)Θ
(12)
which applies to adsorption on reconstructable (|b| * 0) proteins as well as on unreconstructable proteins |b| ) 0). For the trivial case, |b| ) 0, eq 12 gives N(x) ) Ns(0)Θ(x). (11) Pfeifer, P.; Avnir, D. J. Chem. Phys. 1983, 79, 3558.
N(x) ) -
(8)
where ν is the number of atoms in each adsorbed molecule, Fb is a typical atomic density in bulk condensed matter (Fb ≈ 1023 cm-3), and L is the thickness of the adsorption region. On another side, the complete exposure of the adsorbent, with atomic density F to the gas atomsphere implies Ns ) FL. Combining this equation with (8), one gets the density conditions for space-filling surfaces: F ) Fb/(1 + ν). Typically ν ≈ 3, so that space-filling surfaces cannot be obtained under extreme dispersion.
dΘ dN ) Ns dx dx
Otherwise, i.e., for |b| * 0, the solution of eq 9 is
1 ln(1 - |b|Ns(0)Θ(x)) |b|
(13)
Adsorption on reconstructable proteins occurs in the low coverage limit with the same law as on unreconstructable proteins
Θ , 1/|b|Ns(0) w N(x) = Ns(0)Θ(x) whatever the expression of Θ(x). At high coverage, however, different phenomena can occur. 4. High-Coverage Phenomena While the above discussion does not require any detailed knowledge of the isotherm Θ(x), understanding the phenomena occurring at high coverage requires a specification of Θ(x). Multilayer adsorption in restricted geometries is conveniently described by the following modification of the BET isotherm
Θn(x) )
n n+1 Cx 1 - (n + 1)x + nx 1 - x 1 + (C - 1)x - Cxn+1
(14)
where C is a parameter related to the difference of binding energies for adsorption in the first and upper layers and n is the maximum number of layers which can be formed in the restricted geometry.12 When n is allowed to go to +∞, one obtains the standard BET equation
Θ∞(x) )
1 Cx 1 - x 1 + (C - 1)x
(15)
Because of the partially rough surface structure of globular proteins (D ) 2.1-2.2), adsorption is expected to be described better by eq 14 than by eq 15, which is instead expected to describe adsorption either on truly smooth proteins (D ) 2) or on completely unfolded proteins. The study of eq 14 shows that the maximum coverage Θnm, attained at x ) 1, is given by12
Θnm )
n + 1 Cn 2 1 + Cn
This value represents a milestone of the following discussion. A protein will be said completely reconstructable when
|b|Ns(0)Θnm g 1 - N0/Ns(0)
(16)
otherwise it will be referred to as incompletely reconstructable. Incompletely reconstructable proteins are expected to belong to the class of hard proteins, i.e., globular proteins with a high degree of conformational stability and a low degree of flexibility and able to resist large irreversible changes in conformation upon adsorption at interfaces. Completely reconstructable proteins are expected to belong to the class of soft proteins, i.e., relatively flexible globular proteins able to modify their tertiary structure to facilitate adsorption at interfaces.13-15 Hard proteins are usually characterized by many intrachain -S-S- bridges, a few superficial polar sites, and low foamability; conversely, (12) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974. (13) Arai. T.; Norde, W. Colloids Surf. 1990, 51, 1. (14) Norde, W.; Anusiem, C. I. Colloids Surf. 1992, 66, 73. (15) Norde, W.; Favier, J. P. Colloids Surf. 1992, 64, 87.
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Figure 2. Adsorption isotherms for adsorption on an incompletely reconstructable protein for a few values of |b|Ns(0). The parameters of the BET isotherm are n ) 3 and C ) 100. Figure 1. Adsorption isotherm on an incompletely reconstructable protein, with |b|Ns(0) ) 0.2, Ns(0) ) 0.6N0, n ) 3, and C ) 100, is compared with the BET isotherms with the same n and C on an unreconstructable protein (∀x: Ns(x) ) Ns(0)) and on a fully reconstructed protein (∀x: Ns(x) ) N0).
soft proteins have many superficial polar sites and high foamability.16 4.1. Incompletely Reconstructable Proteins. When condition (16) is not satisfied, the adsorption produces a progressive exposure of polar sites, but this process is terminated before their exposure has been completed, so that the protein will maintain its original globular structure. The adsorption isotherm will be given by eq 13 with Θ(x) given by eq 14. Figure 1 compares the adsorption isotherm on an incompletely reconstructable protein with Ns(0)/N0 ) 0.60 (to be compared with Ns(0)/N0 ) 0.65 expected for hemoglobin) with those on an unreconstructable protein and on a hypothetical fully reconstructed protein characterized by the same values of C and n. Figure 2 shows the function N(x)/Ns(0) calculated for a few values of |b|Ns(0) for a BET isotherm with n ) 3 and C ) 100. 4.2. Completely Reconstructable Proteins. When condition (16) is satisfied, there exists a relative pressure xc at which all polar sites are exposed to polar molecules. In this situation unfolding is complete and most presumably the constraint on n disappears. Accordingly, for x g xc the description of adsorption requires the substitution of eq 15 for eq 14
N(x) )
{
-
1 ln(1 - |b|Ns(0)Θn(x)) for x < xc |b| N0Θ∞(x) for x g xc
Figure 3. Adsorption isotherm for adsorption on a completely reconstructable protein, calculated by assuming |b|Ns(0) ) 0.2, Ns(0) ) 0.65N0, n ) 4, and C ) 100. For a comparison the ideal adsorption isotherms on an unreconstructable adsorbent (n ) 4 and ∀x: Ns(x) ) Ns(0)) and on a fully reconstructed adsorbent (n ) +∞ and ∀x: Ns(x) ) N0) are also shown with a short- and a long-dashed line, respectively.
(17)
Figure 3 shows the adsorption isotherm N(x)/Ns(0) for a globular protein with Ns(0) ) 0.65N0, n ) 4, and C ) 100. Figure 4 shows the adsorption isotherms N(x)/Ns(0) calculated for a few values |b|Ns(0) using a BET isotherm with the same values of C and n as in Figure 3. 5. Applications As it stands, the model hitherto formulated is too vague for practical applications. Different polar molecules are
in fact expected to have different unfolding effects. The effects of some adsorbates can be predicted a priori: i. Small polar molecules, like H2O, will be characterized by high n and by a modest ability to reconstruct the adsorbate (low |b|). ii. Large polar molecules, like CO(NH2)2, will form multilayers (n as high as for molecules of type i) and unfold the protein (high |b|). Both n and |b| are expected to increase with the dipole moment of the molecule. (16) Tripp, B. C.; Magda, J. J.; Andrade, J. D. J. Colloid Interface Sci. 1995, 173, 16.
Adsorption of Polar Molecules on Proteins
Figure 4. Adsorption isotherms for adsorption on completely reconstructable proteins, calculated for a few values |b|Ns(0) for a protein with Ns(0) ) 0.65N0, n ) 4, and C ) 100.
iii. Molecules formed by a small polar head and a large nonpolar tail, like CH3(CH2)2OH, will form only one layer (n ) 1) with a modest surface reconstruction (low |b|). The biological consequences of these behaviors will be considered separately. 5.1. Small Polar Molecules. According to the previous sketch, the adsorption of small polar molecules is expected not to produce large unfolding of the protein, thus being described by adsorption isotherms like those sketched in Figures 1 and 2. It is just because of this reason that the crystalline state of proteins in vitro is a good model for the functional behavior of hydrated proteins in solution in vivo. The model could be validated by determining, via X-ray diffraction, the protein structure under different hydration equilibrium levels resulting after exposure to atmosphere at certain relative humidities x. However, I am not aware of such experiments except for studies of different crystallographic configurations of proteins (like the triclinic and tetragonal forms of lysozyme17) with their proper hydration levels. 5.2. Large Polar Molecules. The adsorption of large polar molecules is expected to produce large unfolding of the protein, thus being described by adsorption isotherms like those sketched in Figures 3 and 4. 5.2.1. Denaturation. Since large polar molecules are known to be protein denaturing agents, the transition from the folded configuration to the completely unfolded one can manifestly provide a model for denaturation. Denaturation can be considered as a phase transition, characterized by a jump discontinuity of the adsorption isotherm, between native-like state (with bulk fractal dimension D = 3 and surface fractal dimension D ) 2.12.2) and the denatured state (in which the protein can be seen as an excluded-volume polymer with D ) 5/3)6. (17) Joynson, M. A.; North, A. C. T.; Sarma, V. R.; Dickerson, R. E.; Steinrauf, L. K. J. Mol. Biol. 1970, 50, 137.
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Denaturation is therefore associated with a sudden variation of the fractal dimension of the protein. 5.2.2. The Molten Globule. The protein picture hitherto considered is oversimplified and does not consider that different parts of the protein have different hardness with respect to the adsorbate. The tertiary structure of a protein is indeed characterized by a a distribution of R helices and β strands (the so-called folding pattern, determined uniquely by the primary structure) interconnected by loops and endowed with side chains. It was long believed that the protein could admit only two states with deep energy minima: the native-like state and the denatured one. From the above picture, however, it follows that it should be possible to unfold progressively first the side chains, then the loops, and eventually the β strands and R helices. In such a way one should obtain, by suitable choice of environment (adsorbate and pressure regime), many folding states. It is worth noticing that at least one such intermediate state, the molten globule, has been identified. The molten globule (first hypothesized as kinetic intermediate and only later discovered to exist as equilibrium configuration) is a protein state characterized by (a) the same folding of R helices and β strands as in the native state, (b) a complete unfolding of loops, and (c) an open configuration of side chains (see the short review of ref 18). 5.3. Molecules with Small Polar Head and Large Nonpolar Tail. This situation is interesting because the distribution of polar sites at the smooth surface of the native protein is covered in such a way that, after monolayer completion, the protein is coated with a nonpolar (hydrophobic) layer, which can be seen as a kind of Langmuir-Blodgett cover of the protein. The adsorption of a second layer is therefore very difficult, its stabilizing energy being negligible. The expected effects of this coating are the following: (a) a reduction of the protein solubility in aqueous ambient; (b) a reduced interaction among different proteins. The reasons for (a) are trivial; (b) is due to the fact that the strong electrostatic interaction between proteins (dominated by enthalpic effects) is replaced by the weak interaction between the hydrophobic protein envelopes (dominated by entropic effects). It is therefore not fortuitous that the addition of alcohols to aqueous solutions of proteins reduces their solubility and that crystalline proteins without mutual interactions are obtained via this route. 6. Conclusions A model has been formulated which describes the protein unfolding resulting after adsorption of polar molecules in terms of surface reconstruction. The structure of the adsorbed molecule plays an important role in determining the evolution of the solvated protein: If the adsorbate is a molecule with small polar head and large nonpolar tail, adsorption will stop with the formation of a monolayer and its effects will be a reduction of solubility in water. If the adsorbate is a large polar molecule, the protein will undergo large surface reconstruction resulting in new folding states with respect to the native one; two of them have already been identified (the denatured state and the molten globule), but others are predicted. If the adsorbate is a small polar molecule, the protein will undergo unfolding without attaining the denatured state; this is the expected behavior of water (proteins synthesized in an aqueous environment are expected not to denature after water adsorption). (18) Ptitsyn, O. B. Trends Biochem. Sci. 1995, 20, 376.
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Among all polar molecules, water plays a special role because of its biological relevance. The surface-reconstruction model is extremely rough and cannot per se provide an adequate description of water adsorption on proteins: the energetic heterogeneity of adsorption sites and hysteresis in adsorption-desorption cycles have not been considered, while allostericity in folding-unfolding has been introduced in a phenomenological way through eq 17. Direct evidence for heterogeneity is provided by relaxation-time studies of hydrated proteins: both myoglobin and (especially) cytochrome c show the existence of an extremely broad distribution of relaxation times;19 hysteresis is always observed in the adsorption on porous surfaces, but is described neither by eq 15 nor eq 14. The first model for water adsorption on proteins taking into account heterogeneity, allostericity, and hysteresis effects was probably formulated by this author and his sister 15 years ago20 and formed the basis for a picture of water in biosystems as a partially ordered phase adsorbed on (19) Green, J. L.; Fan, J.; Angell, C. A. J. Phys. Chem. 1994, 98, 13780.
Cerofolini
macromolecules.21 This model has not been used as a starting point for the description of adsorption of polar molecules because it does not allow the transition from the native state to the denatured one to be considered as a phase transition. At last, it is noted that the sudden, in most cases reversible,22 transition between the native state and an unfolded state like the fully denatured one offers an exciting possibility to model switching phenomena in molecular terms. Acknowledgment. I wish to thank Dr. G. Ranghino (EniChem) for helpful discussions and Langmuir’s reviewers for their valuable comments. LA9508147 (20) Cerofolini, G. F.; Cerofolini, M. J. Colloid Interface Sci. 1980, 78, 65. (21) Cerofolini, G. F. Nuovo Cimento 1983, D2, 763, 1156; Adv. Colloid Interface Sci. 1983, 19, 103. (22) Mahler, H. R.; Cordes, E. H. Basic Biological Chemistry; Harper and Row: New York, 1968.
