Surface pressure of protein adsorption layers: application to molecular mass and molecular area measurements

Colloids and Surfaces A: Physicochemical and Engineering Aspects 149 (1999) 285–290

Surface pressure of protein adsorption layers: application to molecular mass and molecular area measurements R. Douillard *, V. Aguie´-Be´ghin Biochimie des Macromole´cules Ve´ge´tales, UPBP, INRA, CRA, 2 Esplanade R. Garros, BP 224, 51686 Reims, cedex 2, France Received 26 August 1997; accepted 24 November 1997

Abstract In the study of protein adsorption at fluid interfaces, the relation between the surface concentration and the surface pressure is of general interest and is usually quantitated by state equations. In the dilute range, the ideal gas law is a good approximation which allows the calculation of the molecular mass of the molecule. It may be extended on the increasing concentration side by the two-dimensional solution (2D solution) approximation which allows the calculation of both the molecular mass and the molecular area of the protein at the interface. When the surface concentration increases beyond a critical value, the interfacial layer enters a semi-dilute regime where a scaling law approach gives a good approximation of the structure and of the properties of the layer but which only allows a ‘‘rough’’ calculation of either the molecular area or the molecular mass from the other parameter. Moreover, in the scaling law approach, the semi-dilute regime is connected to the gas-like regime by an angulous transition which is not observed in the experimental isotherms. In this communication, a mixed approach is proposed in which the state equation is the 2D solution law below the critical surface concentration and the scaling law beyond this value where both the surface pressure and its derivative with respect to the surface concentration are equal when expressed from one law or from the other. Simple relations are found between the critical surface concentrations calculated from either the 2D solution, the scaling law or the mixed approaches. Within that framework, it is possible to define precisely the range of data belonging to the 2D solution regime and to know whether the molecular mass and molecular area calculations are performed by fitting the right equation to the right data or not. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Protein adsorption; State equation; Scaling law; Molecular mass; Molecular area

1. Introduction The equation of state of proteins adsorbed at the air/buffer interface has been approximated by the ideal gas law in the gaseous regime [1] or by laws assuming that proteins are non-penetrable objects defined by a constant surface when deviation from the ideal case is too large [2,3]. These * Corresponding author.

models give a good fit to the experimental data only at moderately low surface concentrations. For a few years, the adsorption layers of proteins have been modeled, in a scaling law approach, as multiblock copolymer layers [4–6 ]. The fit to the experimental data is good when the previous models do not hold any more. The ideal gas law is a limiting case of the constant surface area model but the scaling law description cannot reduce to the other models when the surface con-

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centration is lowered. In this communication, a ‘‘mixed model’’ is proposed where the adsorbed molecules behave either as constant area solids at relatively low concentrations or as multiblock copolymers at higher values. Molecular masses of proteins have been calculated from the isotherms assuming an ideal gas law or a 2D solution behavior [1,2,7,8]. However, when the experimental data belong to the multiblock copolymer regime, they cannot fit the previous models and the molecular mass measurement may be erroneous. In this communication, an evaluation of that systematic error is made, as a function of the relevant parameters and some general features of the mixed model are addressed.

2. Theoretical background The two-dimensional (2D) equation of state of a protein with a molecular area v and a surface concentration C may be approximated by [2,3]: p=k TC/(1−vC ) (1) B where p is the surface pressure, k the Boltzman B constant and T the absolute temperature. In this case, the adsorption layer is described as a 2D solution. This model will be referred to as the 2D solution model. When vC%1, Eq. (1) reduces to: p=k TC (2) B This is the ideal gas law. In the case of proteins, it has been shown that in the concentration range where these molecules behave like polymers in the semi-dilute regime at the air/buffer interface (between 0.3 and 1.2 mg m−2 in the case of b-casein [9]) the equation of state can be: p=k k TCy (3) p B where k is a proportionality constant (often purp posely ignored ) and where the exponent y is characteristic of the solvent quality. For instance, the value of y may be 2, 3, 8 or 2 when the polymer chain is respectively: stretched, in good solvent, in q solvent or in poor solvent conditions. The current polymer theories based on scaling law concepts [4] most often consider that the state

equation is given by Eq. (2) in the gas regime, at low C values, and by Eq. (3) in the first semidilute regime at higher C values. The surface concentration C at which the transition between s the two regimes occurs is of course given by the equality between Eq. (2) and Eq. (3) ( Fig. 1): C =k Cy or C =k1/(1−y) (4) s p s s p However, angulous points have never been observed in the isotherms of proteins at a fluid interface. Thus the previous approximation is, even qualitatively, a bit rough. In the present model, it is assumed that (1) the equation of state is given by Eq. (1) at low surface concentrations; (2) it is given by Eq. (3) at higher values of C; (3) there is no angulous point in the isotherm. Thus the critical value of the surface concentration C where the transition between the two regimes c occurs is defined by the equality of the surface pressures: k C(y−1) (1−vC )=1 p c c and by the equality of the slopes dp/dC:

(5)

k yC(y−1) (1−vC )2=1 (6) p c c Combining Eqs. (5) and (6), it follows that: y(1−vC )=1, or c C =(1−1/y)/v (7) c It can also be calculated from Eqs. (5) and (6) that the proportionality constant of the scaling law, k is: p k =yC(1−y) (8) p c with C given by Eq. (7).This model, which comc bines the 2D solution model and the scaling law model, will later be called the ‘‘mixed model’’. Returning to the values of the critical overlap concentrations, it can be seen, by combining Eqs. (4) and (8) that: C =C y1/(1−y) (9) s c These various values are presented in Fig. 1. Thus, it is obvious that the critical overlap concentrations are significantly different when the usual scaling law model or the mixed model are used. As an illustration, it can be calculated that the value of

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Fig. 1. Schematic representation of the various equations of Table 1. The numerical values have been chosen according to the experimental data obtained by Bull [10] with b-lactoglobulin. The inset is the log–log plot of the same data. The mixed model is built up of the 2D solution model below C and of the scaling law model beyond C . c c

the y term in Eq. (9) is 0.5, 0.58, 0.74 or 1 when the polypeptide chain is, respectively, stretched, in good solvent, in q solvent or in poor solvent conditions. The various models and the main results are summarized in Table 1.

the interface of area A) and which can then be written:

3. Determination of the molecular mass of spread proteins

pA=k Tn+pvn (10) B The molecular mass is then Mm=m/n. Obviously, the results can be relevant only if the data are gathered in the 2D solution regime. Using the ‘‘mixed model’’, it can be seen in Fig. 1 that the critical surface concentration beyond which the 2D solution regime does not hold is:

In the standard method, a known mass, m, of protein is spread on the surface of a trough which is then compressed while the surface area, A, and the surface pressure are recorded. The determination of the molecular mass is based on Eq. (1) where the surface concentration is expressed as C=n/A (n is the number of molecule spread on

C =(1−1/y)/v (11) c The numerical value of C can be calculated c from the value of v determined using Eq. (10) and from the value of y estimated in a log/log representation of the semi-dilute regime ( Fig. 1, inset). If the calculated value of C is equal to, or larger c than, the largest value used for the calculation in

Table 1 State equations of proteins at a fluid interface Model

State equation

Ideal gas Scaling law

p=k TC B p=k TC B p=k k TCy p B p=k TC/(1−vC ) B p=k k TCy p B p=k TC/(1−vC ) B

Mixed 2D solution

Critical concentration

CC s CC c

no critical value C =k1/(1−y) s p C =1/v (1−1/y) c C=1/v

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the 2D regime, then the molecular weight calculation is likely to be relevant. On the contrary, if the upper boundary of the 2D solution regime has been overestimated, some data are from the scaling law regime and the molecular weight determination is probably erroneous. These various cases have been illustrated by simulating an isotherm from experimental data of Bull [10] and by performing the calculations of ‘‘y’’ according to Eq. (3), of the molecular mass according to Eq. (10) and of C according to Eq. (11). These calculations have c been done using a range of data identical to that used by Bull [10] to fit equation Eq. (10) on his data (Fig. 2). As expected from the general shape of the simulated isotherm (Fig. 1), the calculated molecular weight decreases steadily when the amount of data coming from the scaling law regime increases (Fig. 2). Moreover, it can be seen that when the data include some values larger than C , the calculated value of this parameter is c within the range of the data. Thus the criterion previously indicated ( Eq. (11)) is not fulfilled and a wrong adjustment may be suspected. These possibilities are now illustrated in the cases of b-lactoglobulin and of a lipid transfer protein (LTP). In the case of the first protein, we

Fig. 2. Simulation of the calculation of the molecular mass ) of the ‘‘y’’ exponent (———) and of C (- - -). The ( c data are from the mixed model of Fig. 1. Values were ranging from C to C such that C =1.6C , as in the experimental data l h h l of Bull [10]. Mm and ‘‘y’’ were calculated from linear regressions according to Eq. (10) and Eq. (3) and C according to c Eq. (7). The correlation coefficients of the regressions were in no instance smaller than 0.95.

have used data of Bull [10]. The value of y=8.86 was determined from the isotherm data plotted on a log/log scale (Fig. 3), and the value of v= ˚ from the plot of the data used by the 7430 A author to calculate the molecular weight (Fig. 3, inset, and Eq. (10)). The calculated value of C is c 0.91 mg m−2, a value larger than those of Fig. 3 (inset). Accordingly, the calculated value of Mm is 42 000 while the true value is very close to 37 000. It can be concluded, in that case that there is no bias in the molecular weight calculation due to ‘‘fitting the good equation on wrong data’’. The case of ‘‘wrong adjustment’’ is probably illustrated by the determination of the molecular mass of the human plasma LTP [8]. The original data of the isotherm and of the pA against p plot have been redrawn in Fig. 4 and it seems very likely that both sets of data belong to the semidilute regime modeled by a scaling law. In fact, the value of C , calculated from Eq. (11) is c 0.72 mg m−2 while C in the pA against p plot ranges from 0.67 to 0.75 mg m−2. Thus the criterion of Eq. (11) is not fulfilled. In the same line, the calculated molecular mass is close to 48 000 while the value determined from electrophoresis is

Fig. 3. Double logarithmic plot of the state equation of b-lactoglobulin determined by Bull [10]. The ‘‘y’’ value was determined in the linear part of the plot ( y=8.86). The data in the inset in the C range 0.75–0.9 mg m−2 were plotted according to Eq. (10). Both sets of data were obtained independently and do not fit completely well on either plot. It can be calculated ˚ 2 and thus, according to Eq. (7): that v=7430 A C =0.91 mg m−2, a value larger than C . It can be concluded c h that the calculation was correctly performed and indeed, the Mm measured was 42 000 while the true value is very close to 37 000.

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directly calculated from Eq. (10) at the same time as the molecular mass is calculated. In the scaling law approach, the molecular area is simply: A =1/C (12) 0s s provided that the molecular mass is known and that the surface concentration can be expressed as a number of molecule per unit surface area. From Eqs. (7) and (9), it can be calculated in that case that:

Fig. 4. Double logarithmic plot of the state equation of the LTP [8]. The open symbols were redrawn from the compression isotherm and the closed rhomb from the plot of Eq. (10) in the original paper (C ranging from 0.67 to 0.75 mg m−2). It can be seen that all the data fit a single straight line with a slope (the ‘‘y’’ exponent) close to 12.5, suggesting that they are from the scaling law rather from the 2D solution regime. The rhombs were fitted by Eq. (10) and C was calculated to be c 0.72 mg m−2, a value confirming that the rhomb points are not all in the 2D solution regime. The calculated value of Mm was 48 000 while measured by electrophoresis it was found to be 69 000. It can be concluded that the Mm calulation was biaised by the choice of the data which were, in part, out of the 2D solution range.

close to 69 000. This can be a direct consequence of the use of data of the scaling law regime to calculate the molecular mass as it is shown in Fig. 2. It can be concluded in this case that the Mm calculation was biased by the choice of data which were, in part, out of the 2D solution range.

4. Determination of the molecular area of the protein In the 2D solution approach, the molecular area, A , of a molecule is equal to v which can be 0

A =vy1/(y−1) 1/(1−1/y) (13) 0s The values of Table 2 show that the molecular area is overestimated when calculated from the scaling law approach and that this overestimate increases when the Flory exponent also increases. The order of magnitude of the overestimate is close to a factor two in usual solvent conditions. Nevertheless, as long as the mixed model holds, it provides a mean to calculate the value of v since the coefficient of proportionality between A and 0s v is only a function of y, the exponent of the scaling law between the surface pressure and the surface concentration.

5. Discussion A crucial point in the current mixed model is the validity of the state equations which are used below and above the critical overlap concentration. Above that concentration, the scaling law approach has proven to be rather convenient [11]. In the dilute range, the 2D solution description is valid only when the polymer molecules interact through elastic shocks, a condition which may be fulfilled in special conditions, by example on a

Table 2 Relations between the values of the exponents n and y and the bias in the molecular area calculation Conditions

Flory exponent n

Scaling law exponent y

A /v 0s

Extended polypeptide chain Good solvent q conditions poor solvent

1 3/4 4/7 1/2

2 3 8 2

4 2.6 1.54 1

The Flory exponent is defined by: R $Nn where R is the two-dimensional radius of a polymer composed of N statistical units. y F2 F2 is defined by y=2n/(2n−1) [5]. A and v are linked by Eq. (13). 0s

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high salt concentration solution. A more general description of the gas-like state is provided by the virial development which may account for attractive or repulsive interactions between the adsorbed macromolecules. In the general case, the state equation, in the dilute range can be symbolized as: p=k Tf (C ) (14) B assuming that a scaling law holds above C and c using the same argument as for Eqs. (5)–(7), to calculate C , it follows that: c d[ f (C )] yf (C )=C (15) c c dC Cc Thus the calculation of C involves only mathec matical expressions related to the form of the state equation in the gas regime, without reference to the power-law form of the state equation in the semi-dilute regime. As an example, when the state equation below C is taken as a virial expansion c until the third degree:

A

B

p=k TC(1+A C+A C2) (16) B 2 3 Then, using Eq. (15), it can be easily shown that C can be calculated from: c C2 A ( y−3)+C A ( y−2)+y−1=0 (17) c 3 c 2 Since the molecular area is not included in the state equation, it cannot be calculated directly from the adjustment of the equation to the data but the value of A is certainly better approximated 0 by 1/C from Eq. (17) than by 1/C from Eq. (4). c s

6. Conclusion The mixed model introduced in this study provides some means to check whether the calculation

of the molecular mass and of the molecular area of a protein adsorbed at a fluid interface have been performed on the right data. Even though the present study has been focused on proteins, it is clear that the same analysis may be done with any polymer which adsorbs at a fluid interface. Finally, it may be noticed that the procedure used to built up the ‘‘mixed’’ model is probably of interest when scaling laws and other kinds of physical laws have to be connected at critical values of the variables.

Acknowledgments Thanks to B. Monties for his interest and encouragements in the study of protein surface properties and to M. Daoud for long discussions on the scaling law approach of polymer properties.

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