Denaturated proteins: Draining effect and molecular dimensions

Physica B 405 (2010) 3572–3575

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Denaturated proteins: Draining effect and molecular dimensions A. Dondos n Department of Chemical Engineering, University of Patras, 26504 Patras, Greece

a r t i c l e in f o

a b s t r a c t

Article history: Received 9 February 2010 Received in revised form 30 April 2010 Accepted 15 May 2010

Using equations derived from the synthetic macromolecules, we calculate the dimensions in solution of the denaturated proteins. For these calculations, we use a value for the Flory’s parameter F obtained from an equation established for the polymers presenting a draining effect, and which is lower than the value of 2.6  1023 (cgs) generally used. The obtained values for the dimensions of the denaturated proteins (end to end distance, statistical segment length and relation from the end to end distance and the number of residue) using the method proposed here are in good agreement with the values obtained from Flory and co-workers. On the contrary, the values obtained in this work are different from the values proposed by other authors who do not take into account the draining effect and use a value for F equal to 2.6  1023. & 2010 Elsevier B.V. All rights reserved.

Keywords: Denaturated proteins Chain dimensions Draining effect Random coil behavior Flory’s parameter ^

1. Introduction The quantitative estimation of coil dimensions of denaturated proteins has been the subject of intensive studies during the sixties [1–7]. The estimation is based on the calculation of the restriction to the rotation along the polypeptide backbone [1–4], on the intrinsic viscosity [5–7] and on light scattering [7,8]. Modern works on the stiffness of the denaturated protein chains have been limited [8,9]. In this article, we do not offer a new method for the study of denaturated proteins, but we will try to explain the difference observed in the results obtained by Flory and coworkers [1–4] and a number of researchers later on [5,6,9]. We calculate the dimensions of denaturated proteins, based on the measurements of intrinsic viscosities which have been performed in guanidine hydrochloride (GuHCl) solution, and applying the methods proposed for the synthetic polymers. A graphical method [10,11] will be applied which is adequate for the polymers presenting a relatively low molecular mass, and which do not present a high expansion coefficient. These conditions are presented by the denaturated proteins (proteins polypeptides) dissolved in a concentrated solution of GuHCl, where these polymers present a moderately non-ideal behavior. From this graphical method, we will obtain the value of the unperturbed dimensions parameter of these polymers, and from the value of this parameter we will calculate their statistical segment length. In this calculation, we need the value of the

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Flory’s parameter F. The value of this parameter, which here is not a constant, will be obtained from a relation proposed for polymers presenting a draining effect [12] and which value is lower than the value of 2.6  1023 (cgs) generally used. The dimensions obtained with the proposed graphical method [10,11], and using the value of F given for the polymer presenting a draining effect, shed light on the existing puzzle about the chain dimensions dependence of denaturated proteins on the number of the residues of the chain or about the statistical segment length of these chains. For the Kuhn statistical segment length of the denaturated ˚ while Flory and proteins, we obtain in this work a value of 24.4 A, co-workers have obtained theoretically and experimentally 23 A˚ [1–3]. On the contrary, many authors obtain lower values for the statistical segment length of the denaturated proteins. More precisely, Zhou [9] obtains an affective bond length 6 A˚ (c.a a ˚ and Lapanje and Tanford [5] obtain an statistical length 12 A) effective bond length of 7.1 A˚ (c.a. a statistical segment length ˚ 14.2 A).

2. Theoretical and procedure The dimensions of the macromolecular chains in the unperturbed state (Y conditions), are expressed by the end to end distance of the chain, ry, or by the radius of gyration of the chain, Ry, (ry2 ¼6R2y ) or even by the parameter Ky of the unperturbed dimensions (Ky ¼[Z]y/M1/2), where [Z]y is the intrinsic viscosity of the polymer in Y conditions and M the molecular mass of the polymer. The parameters ry and Ry are related to the Ky parameter

A. Dondos / Physica B 405 (2010) 3572–3575

3. Results and discussion

by the relation ry2 6R2y ¼ ¼ M M

 2=3 Ky

F

ð1Þ

where F is the Flory’s ‘‘constant’’. We have shown that the parameter F does not have a constant value in the case of native proteins, [13,14] in the case of polyelectrolytes [15,16] and in the case of wormlike polymers [12,17]. In all these cases, the macromolecules present a draining effect. Let us indicate that a constant value of F ¼2.6  1023, is observed for the synthetic linear and flexible polymers dissolved in non-aqueous solvents. We have proposed [12] the following empirical equation in order to obtain the value of F for a polymer presenting a draining effect in a given solvent.

F ¼ 0:52  1023 a2:32

ð2Þ

In the above equation, a is the exponent of the Mark– Houwink–Sakurada (MHS) equation ½Z ¼ kM a

ð3Þ

That is, the slope of the curve obtained plotting log[Z] versus log M for a homologous series of fractions (fractions of a polymer with different values of M and [Z]). Eq. (2) has been obtained a theoretical support [18,19]. The value of the unperturbed dimensions parameter, Ky, has been mainly obtained from the values of intrinsic viscosities obtained in non-ideal solvent for the polymers (far away from Y conditions). The non-ideality has been corrected applying different graphical methods based on proposed equations. The following equation proposed by Stockmayer–Fixman–Burchard (SFB) [20,21] is the most frequently used. ½Z=M 1=2 ¼ K y þ0:51FBM 1=2

ð4Þ

Nevertheless, this equation has not a good applicability in the domain of relatively low molecular masses. On the contrary, the following equation, proposed by Dondos and Benoˆıt (DB) [10,11] has a good applicability in the domain of low and medium molecular mass domain. 1=2 1=½Z ¼ A2 þ K 1 y =M

In Fig. 1, we present the variation of log[Z] as a function of log M, for a series of denaturated proteins, according to the MHS equation (Eq. (3)), considering that the different denaturated proteins constitute a homologous series, as it is considered by many researchers. The viscometric results come from the articles of Tanford et al. [5,6], Corbett and Roche [22] and Olander et al. [23]. We can see in this figure that the obtained points lie on two straight lines with different slopes. In the region of low molecular mass the slope of the straight line, or the exponent a in Eq. (3), is equal to 0.69 and in the region of the higher molecular masses we have a¼0.59. Let us indicate that Tanford [24], with a lower number of points, has plotted only one straight line with a slope equal to 0.666. From Eq. (2) with a ¼0.69, we obtain F ¼1.23  1023 (in CGS) and with a ¼0.59, F ¼1.77  1023. We have already observed that the possibility to plot, in different molecular mass regions, straight lines with decreasing slopes when increases the molecular mass, characterize the polymers presenting a draining effect [12,17,25]. Among these polymers are the DNA, and the poly-pphenylene terephthelamide. With the same viscometric results of Fig. 1 we have applied Eq. (4) (SFB equation [20,21]), plotting in Fig. 2 [Z]/M1/2 versus M1/2. The points do not lie on a straight line but rather on a curve the extrapolation of which to zero molecular mass, in order to obtain the value of the unperturbed dimensions parameter Ky, gives a value for this parameter completely erroneous. Let us indicate that the SFB equation has not a good applicability in the low molecular mass regions, as we have already mentioned (in general, we have no applicability of SFB equation for M1/2 o 200, but here even in the region of M1/2 4 200 it is not possible to plot a straight line). We now apply, with the same viscometric results of Fig. 1 or 2, Eq. (5) (DB equation [10,11]). The obtained results are presented in Fig. 3. We can see that it is possible to plot two straight lines, as in Fig. 1. The inverse of the slopes of the two straight lines, according to Eq. (5), give us the values of the unperturbed dimensions parameter Ky. In the low molecular mass region, we obtain Ky ¼ 9.7  10  2 CGS and in the higher molecular mass

ð5Þ

The parameter A2 is related to the quality of the solvent as the parameter B in Eq. (4) and the slope of the curve obtained by plotting 1/[Z] versus 1/M1/2 is equal to 1/Ky. In order to obtain the value of the statistical segment length of a macromolecule, A, (Kuhn statistical segment) we use the following equation in which ML is the mass per unit length of the chain  2=3 Ky

F

ML

ð6Þ

In the case of denaturated proteins, the unperturbed dimensions are given, as we have already mentioned, by the end to end distance, ry, of the chain, which is related to the molecular mass of the chain by the relation ry2 ¼ Dn

10

ð7Þ

where n is the number of residues of the chain and D the parameter relating ry2 to n. Having n¼M/M0, where M0 is the mass of each residue, we obtain ry2 =M ¼ D=M0

100

[η] (cm3/g)

A¼

3573

ð8Þ

The value of D lies between 130 and 70 as it was obtained theoretically and experimentally [1–7].

10000

100000

M Fig. 1. MHS plot for different denaturated proteins. The viscometric results come from Refs. [6,22–24].

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A. Dondos / Physica B 405 (2010) 3572–3575

25

containing entirely of residues with a CH2 or CH3 groups in the

b–C position. The presence in the chains of glycine and proline

[η]/M½ × 102

20

15

10

5

0

0

100

200

300

400

500

M½ Fig. 2. SFB plot for the denaturated proteins. The proteins are the same as in Fig. 1.

20

1/[η] × 102

15

10

5

0 0

5

10

15

20

1/M½ × 103 Fig. 3. DB plot for the denaturated proteins of Fig.1 or 2.

region we obtain Ky ¼14  10  2 CGS. The two molecular mass regions are the same as the regions in Fig. 1 and from the values obtained for the exponent a in the two regions, we have calculated the values of the parameter F (Eq. (2)) as we have presented before. In the low molecular mass region, with Ky ¼9.7  10  2 cm3  3/2 g mol1/2, F ¼1.23  1023 mol  1 and ML ¼29  108 g mol  1 ˚ cm  1, we obtain from Eq. (6), A¼24.3  10  8 cm or A¼24.3 A. In the high molecular mass region with Ky ¼14  10  2 cm3 g  3/2 mol1/2, F ¼ 1.77  1023 mol  1 and ML ¼29  108 g mol  1 cm  1 we ˚ These results indicate that obtain A¼24.5  10  8 cm or 24.5 A. the value of the Kuhn statistical segment length is the same for the entire molecular mass region studied here. The value of ML has been obtained by dividing the molecular mass of the chain residue, M0 (mean value 110 [24]) by the length of the residue which is equal to 3.8 A˚ (or 3.8  10  8 cm). From Eq. (1), with Ky ¼9.7  10  2 cm3 g  3/2 mol1/2 and F ¼ 1.23  1023 mol  1 (low molecular mass region), we obtain ry2 /M¼ 85  10  18 cm2 g  1mol and from Eq. (8) we obtain D ¼93.5  10  16 cm2 or 93.5 A˚ 2. A comparable value for D (D ¼87  10  16) we obtain with Ky ¼ 14  10  2 and F ¼1.77  1023 (high molecular mass region). The mean value of D is 90.25 and Eq. (7) becomes ry2 ¼ 90:25n:

ð9Þ

Flory and his collaborators [1,2], after their theoretical calculations, give ry2 ¼130n for the randomly coiled polypeptides

provokes a contraction of the polypeptide chains and the theoretical calculations give ry2 ¼90n [3]. The denaturated proteins investigated here contain 6% to 15% proline plus glycine [24] and this content is taken into account in the theoretical calculations [3] in order to give the value of D equal to 90 in Eq. (7). This value is about the same to the value obtained here (Eq. (9)). If we consider that the denaturated proteins do not present a draining effect, we must use in all our calculations the value of 2.6  1023 for the Flory’s parameter F. With this value we obtain D ¼54 in Eq. (9). This value of D is very low and it is not proposed by any researcher. Tanford et al. [6], using the values of intrinsic viscosities obtained with a relatively large number of denaturated proteins and applying the SFB equation (Eq. (4)) have taken ry2 ¼(70 715)n. The high uncertainty to the D value must be attributed to the nongood applicability of the SFB equation in the case of denaturated proteins, as we can see in Fig. 2. Nevertheless, if we consider that these products present a draining effect as we have already done in our preceding calculations, from the esults obtained by Tanford et al. [6], we obtain ry2 ¼(96715%)n. This value of D is obtained with a value of F equal to 1.3  1023 instead of F ¼ 2.1  1023 used by these authors (the value of F equal to 1.3  1023 is obtained from Eq. (2) with a ¼0.67 [6]). Another result which claims in favor that the denaturated proteins present a draining effect comes from the applicability of the modified universal calibration [26] of the gel permeation chromatography (GPC) on these macromolecules, as we have done in a previous article [14]. According to this method, we present log([Z]M/F) versus the elution volume instead of log[Z]M versus the elution volume according to the ‘‘classical’’ universal calibration [27]. Only if we apply the modified universal calibration for the denaturated proteins, considering these polymers as polymers presenting a draining effect, and taking F ¼1.23  1023 for the low molecular mass region and F ¼1.77  1023 for the high molecular mass region (these values are obtained from Eq. (2)) we have obtained an applicability of the modified universal calibration for these macromolecules [14]. Let us indicate that the different denaturated proteins behave, even in the GPC, as a homologous series, and this is also observed in the work of le Maire et al. [28]. Despite their draining effect, the denaturated proteins should still be considered as random coiled structures, because, as we have seen, they obey most of the proposed structure property relationships for synthetic polymers, as it has been also shown by Kohn et al. [29]. Millett et al. [8] have shown this to be true even when denaturated proteins contain helical segments. Moreover we have shown that even complexes SDS-proteins present a random coil behavior [30].

4. Conclusions Considering the denaturated proteins as macromolecules presenting a draining effect, when are found in solution in concentrated GuHCl, we have calculated their molecular dimensions. In these calculations, we have used values for the Flory’s parameter F obtained by an equation proposed for the synthetic polymers presenting a draining effect (Eq. (2)). This equation relates the F value to the exponent of the MHS equation obtained with the fractions of the polymer. Here, we consider the different denaturated proteins as fractions of a same polymer with compositions which are not enough different from one protein to the other. The obtained dimensions (D ¼92 in Eq. (9)) are in

A. Dondos / Physica B 405 (2010) 3572–3575

good agreement with the dimensions calculated by Flory and his collaborators [3] (D ¼90). With a statistical segment length of 24.4 A˚ (comparable to the statistical segment length of polystyrene), we could not consider the denaturated proteins as wormlike polymers, but we consider that the draining effect should be attributed to the fact that the solvent is water. In this solvent, draining effect has been also observed with the flexible poly(ethylene oxide) and some polyelectrolytes [15,16]. Finally, commenting on the difference in the statistical segment length between the Flory’s group and the other authors, we give here two sentences from Ref. [5]. An additional possibility is that the value of F which we have used is too large. A smaller value would bring our dimensions closer to the values of Flory and co-workers. References [1] [2] [3] [4] [5] [6]

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