Light scattering studies of proteins under compression

Biochimica et Biophysica Acta 1764 (2006) 405 – 413 http://www.elsevier.com/locate/bba

Review

Light scattering studies of proteins under compression Ewa Banachowicz ⁎ Faculty of Physics, Adam Mickiewicz University, Umultowska 85, PL-61-614 Poznañ, Poland Received 18 October 2005; received in revised form 23 December 2005; accepted 17 January 2006 Available online 9 February 2006

Abstract The scattering techniques are very convenient and effective in investigation of the shape, size and interactions of biological molecules close to their natural states in solution. However, it seems that from among a wide spectrum of scattering techniques, the light scattering ones have been relatively rarely used for the study of proteins under elevated hydrostatic pressure. This paper gives a brief description of the well developed possibilities of this technique for potential applications in the study of dissociation, aggregation and structural changes in proteins under compression. A short review of the already known applications is also given. Finally, the high-pressure dynamic light scattering results obtained by author on the lysozyme solution are shown and discussed. © 2006 Elsevier B.V. All rights reserved. Keywords: Light-scattering; Diffusion coefficient; Dissociation; High hydrostatic pressure; Protein

1. Introduction The scattering of electromagnetic waves by matter (bioparticles included) has been for many decades used as a source of information on the structure and dynamics of the scattering medium. The general theory of the process of electromagnetic wave scattering can be applied to describe this phenomenon in the complete range of electromagnetic radiation wavelengths, from microwave to X-ray, and even to scattering of neutrons (to whom a certain wavelength (λ) can be assigned on the basis of the de Broglie relation). The intensity of the scattered electromagnetic field depends on the product between the particle size (Rg—gyration radius) and the magnitude of the scattering wave vector q. The scattering vector q: q¼

4pno sinðh=2Þ k

ð1Þ

where λ is a radiation wavelength, θ is a scattering angle, no is a refractive index of the solvent. The shorter the wavelength the smaller size molecules are effective in the process of scattering. Each molecule is treated as an independent scattering centre. The intensity of light scattered by small size non-interacting particles is the same in each direction. Large molecules are ⁎ Tel.: +48 61 8295257; fax: +48 61 8257758. E-mail address: [email protected]. 1570-9639/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.bbapap.2006.01.014

assumed to be made up of small scattering elements. Each of these elements scatters independently of any other. However, the fact that the distance among the elements belonging to the same molecule is constant has a significant effect on the phase difference between the scattered waves. The phase difference depends on the distance between the scattering centres so on the size of the molecule, on the wave vector k = 2πno/λ and the angle of observation θ. The waves can superimpose producing constructive or destructive interference in certain directions. The difference in the intensity of light scattered at different angles is the greater the larger the molecule (or the shorter the wavelength). Changes in the scattered wave intensity as a function of θ of small size molecules do not exceed 2%, but if the size of the particles is larger than λ/20, it is possible to observe the angular dependence of the scattered wave intensity. Light scattering techniques (or laser light scattering— because laser is usually used as a source of light) use electromagnetic waves from the visible range of ∼400– 800 nm. These techniques have been widely applied in the study of mass, shape and aggregation behaviour of proteins in aqueous solutions. One of these techniques is known as static light scattering (SLS) and involves measurements of the timeaveraged intensity of scattered light as a function of the concentration of scattering particles and/or scattering angle. Another one, known as dynamic light scattering (DLS) or photon correlation spectroscopy (PCS), is applied to study the

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time-dependent scattered intensity following from the fluctuations of the local concentrations of the particles. In general, SLS is applied in the equilibrium conditions for determination of the mass and size of the protein molecules or in the study of intermolecular interactions in solutions versus, e.g., ionic strength [1]. In the applications of this type, both SLS and DLS methods provide a possibility of finding the optimum conditions of the crystal growth [2,3] and can be employed in the study of crystallization under high pressure. If the size or shape of the biomolecules changes in time, the method of static light scattering permits detection of these changes, provided that the time of measurement is shorter than that of the process observed. Direct observation of changes in the scattered light intensity is sufficient for monitoring of the changes in the state of the proteins studied, induced by a factor disturbing the equilibrium. Hence, measurements of the changes in the scattered light intensity in time, taking place in response to pressure changes, can provide information on the kinetics of dissociation of complexes and aggregates [4–6] and the degree of dissociation [7,8]. In the pressure-induced dissociation reported by Gebhardt et al. [8] the average light intensity was used as an indicator of the degree of dissociation, with the size distribution of scattered particles determined by the DLS method. Oligomeric proteins generally tend to dissociate under a relatively low pressure, while aggregation is often observed in the pressure-denatured state. Kinetics of association of the native protein in the lowpressure region has been also studied by SLS combined with DLS [9]. A possibility of monitoring the association and dissociation processes induced by pressure changes is of great significance for medical and biological applications. In some individual proteins or more complex macromolecules such as enzymes, chaperones, ribosomes or viruses, the dissociation or association can cause a loss of biological activity [10–13]. The DLS method is based on measurements of the intensity–intensity time correlation function G(τ). The shorttime fluctuations of the scattered light intensity are related to the motions of the scattering particles, e.g., those undergoing diffusive Brownian motion. The intensity of motion of the particles is inversely proportional to their size. It means that a single small protein molecule moves faster then a dimer or a larger one. Additionally, the proteins of the same molecular weight but of different shape are characterised by different diffusion coefficients, D. The globular aggregate made of a few monomers will perform different movements than a fibrillike or amyloid-like rod made of the same elements. Different diffusion coefficients of the same molecular weight particles and of the same hydration point to differences in the radii of the spheres being hydrodynamic equivalents of the molecules. As long as the scattering particles are relatively small, the dynamic light scattering permits only a detection of the translation of the centre of the mass, so the translational diffusion. The conclusions on the shape of the scattering molecule can be drawn on the basis of the hydrodynamic models. If the size of the objects studied is sufficiently large and their shape anisotropy is significant, also the rotational diffusion can be measured, Θ. A combination of the rotational and translational diffusion provides the data allowing unique

determination of the molecular shape. Measurements of Θ are based on the time correlation function of depolarised light (the so-called Depolarised Dynamic Light Scattering, DDLS). Successful interpretation of the DLS and DDLS results requires the application of appropriate hydrodynamic models taking into account the degree of hydration and partial specific volume of the protein. The usefulness of DLS for investigation of proteins under compression depends on the type and magnitude of the changes observed. Although DLS is very sensitive to changes in the shape and size of the particles observed, detection of the structural changes taking place in small globular proteins prior to denaturation is very difficult. On the other hand, the processes of aggregation or nucleation or transitions to the denatured form can be easily detected [14–16]. A detailed description of the SLS/DLS experiment and the relevant theory has been given for example by Chu, Berne, Pecora [17–21]. The laboratory equipment for SLS and DLS is commercially available (Brookhaven Instruments, ALV GmbH) but the cells for high-pressure experiments are usually laboratory-built [22–24]. A scheme of a typical experimental setup for SLS and DLS measurements is given in Fig. 1. 2. Static light scattering (SLS) The static light scattering is based on measurement of the angular dependence of the time-averaged intensity I in terms of the scattering wave vector q. The total scattered intensity is directly proportional to S(q)—the structure factor, describing the intermolecular interactions (particles distribution in solution) and P(q)—the form factor (or particle scattering factor) depending on the particle size and shape (the Fourier transform of segment distribution within one molecule). In order to relate the intensity of the scattered light to the properties of proteins taking part in the light scattering, the scattering by background sources (solvent, stray light, etc.) must be subtracted from the total intensity and the latter should be normalised by the incident light intensity. After the subtraction and normalisation, the contribution of proteins in the light scattering can be expressed by the excess Rayleigh ratio:
 I  I o no 2 RH ¼ Rst : ð2Þ Ist nst I, Io, Ist are scattered intensities of solution, solvent and standard (benzene or toluene), respectively, no, nst refractive indexes of solvent and standard, respectively. Hence, RH ¼ KMcPðqÞSðqÞ

ð3Þ

where M is the molecular weight of the proteins in the solution, c is the protein weight concentration and K is the system specific constant (optic constant) given by:
 4pn2o dn 2 K¼ ; NA k4 dc

ð4Þ

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Fig. 1. Schematic diagram of a standard setup for (High-Pressure) Laser Light Scattering.

with NA Avogadro's number, dn/dc the protein refractive index increment. The term S(q) can be neglected when the interparticle interactions are very small or when the mean concentration of the particles tends to zero (S(q)≈1). The form factor P(q) tends to 1 if the scattering molecules are very small size in comparison to the incident wavelength (i.e., Rg ≪ q−1 or equivalently qRg ≪ 1) otherwise its value changes as a function of q. For a few most common types of particle shape like rigid-rod, solid sphere (possible model for protein aggregates), Gaussian coil etc., mathematical expressions have been derived for P(q) [21]. Unfortunately, for the majority of proteins, this condition (qRg ≪ 1) is not satisfied in the range of visual radiation wavelengths. To be able to measure Rg of an aggregate by SLS, i.e., to see the difference in the angular dependence of the scattered light intensity (that originates from the phase difference between the scattered wavelets within a single molecule), its size will roughly have to be, over about 0.1 of the incident beam wavelength (e.g., for λ = 488 nm the protein size should not be smaller than 48 nm). For more detailed description please refer to the textbooks [17–21]. Under the above restrictions, the classical equation used in SLS for determination of the mass and the second virial coefficient becomes [25]: Kc 1 ¼ þ 2B22 c; RH M

ð5Þ

where B22 is the second virial coefficient, taking positive values for net repulsion, negative for net attraction and vanishing for non-interacting particles (ideal solution).

In the field of protein studies under compression, the SLS technique can be used for observation of at least two processes: changes in the interparticle interactions—by measurement of B22 as a function of pressure, and dissociation of oligomers—by measurements of time changes in the intensity of scattered light (changes in the mean molecular weight of the scattering particles). Knowledge of the changes in the 2nd virial coefficient as a function of pressure is important, not only in the aspect of the crystal growth. For instance, Loupiac and contributors [26] reported changes in B22 as a function of pressure measured for metmyoglobin solution (“pH” of deuterated solution: pD 6.6) by the small-angle neutron scattering. In their experiment B22 was 7.2 × 10 − 4 cm 3 mol/g 2 at 54 MPa and decreased to 5.6 × 10−4 cm3 mol/g2 at 302 MPa, which meant that the repulsive interactions decreased with increasing pressure. This interesting observation can explain the proteins tendency to form aggregates in the native conditions in the low-pressure regime [9]. The changes in the second virial coefficient as a function of pressure have been explained by variations in the gyration radius of the protein. According to Loupiac [26] the pressure dependence of B22 can follow from two reasons. 1. The magnitude of the charge on the surface of the protein particle changes with pressure as a result of the changes in the dissociation constant pK of the amino acids residues, which leads to weakening or enhancement of the interactions. 2. Changes in the particle size induced by alteration in the protein hydration state and not by structural changes, can take place. These two processes can occur simultaneously, as the presence of charge and changes in its density on the protein surface are strongly related to the degree of hydration and the properties of

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the hydration shell. Explanation of this phenomenon (changes in the second virial coefficient) for myoglobin and other proteins, for which the B22 changes with pressure show a different tendency, is still an open question. Another important application of the SLS technique is for monitoring of the oligomers dissociation. The intensity of scattered light is, according to Eq. (3), proportional to the product of the molecular weight and concentration of a given type protein. For polydispersive diluted solutions (S(q)→1, ideal solution) of small size proteins (P(q)→1) the Rayleigh ratio is the sum of the Rayleigh ratios for scattering of proteins of each possible molecular weight: X X RH ¼ K ci M i ¼ K ni Mi2 : ð6Þ i

i

ci and ni are the weight and number concentration of the component of the molecular weight Mi. The larger the particles (the oligomers), the stronger the scattering, in spite of a decreasing number concentration of the scattering particles. For example, the scattering by a solution of fully dimerised protein is twice stronger than that of a solution of monomers of a twice higher number concentration. This fact is the basis of most experiments on the stability of protein complexes. In such SLS experiments the effect of pressure on the constant K (changes in dn/dc, nst) and on possible changes in B22 is generally disregarded as the measurements of the quantitative changes in the particle state in solution have been very difficult. For investigation of the kinetics of the dissociation process it is thus sufficient to observe directly the time changes in the scattering light intensity. 3. Dynamic light scattering (DLS and DDLS) 3.1. Sample preparation To illustrate the potential possibilities of the high-pressure DLS, a few measurements of the translational diffusion coefficient of the Hen Egg White Lysozyme (HEWL) solution were made. Lysozyme (SIGMA) was dissolved in 50 mM sodium acetate buffer, 100 mM NaCl, pH4.6. Four protein concentrations were prepared: 5, 10, 25, and 50 mg/mL, The samples were filtered (Millipore filters, pore size 0.22 μm) into light scattering cuvettes and stored for 2 days at room temperature in the conditions favouring the formation of dimers [27]. The high-pressure cell suitable for light scattering experiments was built by Dr. Gerd Meier from the IFF-Institute of Research Centre Jülich (Germany). In this high-pressure cell the medium transmitting the pressure is gas nitrogen, the possible pressure range is between 1 and 1500 bar and the temperature range is between −20 and 150 °C. The cell allows measurements at three scattering angles, 45°, 90° and 135°. Because of the use of special glass windows of an extremely low Pockels coefficient, also measurements under polarisation control are possible. The high-pressure cell was mounted in a standard DLS setup (argon laser, λ = 514 nm, scattering angle 135°, T = 20 °C, ALV-correlator). The relaxation times were determined by standard methods: the cumulant analysis and the exponential fit.

3.2. Dynamic light scattering In the dynamic light scattering the intensity–intensity time correlation function G(2) (τ) is measured by means of a multichannel digital correlator:   Gð2Þ s ¼ A 1 þ bjgð1Þ ðsÞj2 ; ð7Þ where A is the background, b is a coherence factor, and g(1)(τ) is the normalised electric field autocorrelation function. For a monodisperse sample (a solution of proteins of the same molecular weight) the dependence g(1)(τ) takes the form:
 s gð1Þ ðsÞ ¼ exp  : ð8Þ so τo is the characteristic correlation time of the scatterer motion. The majority of lasers used as sources of light in such experiments generate the beam of vertical polarisation. The translational motion of the particles in the scattering medium does not cause its depolarisation so the correlation time of translational motion should be determined form the correlation function of the vertically polarised scattered light intensity (IVV). Taking into account small size of protein molecules, especially globular, this condition is in general neglected. Many advanced methods have been proposed to analyse the correlation function in order to elicit information about the characteristic time τo or its distribution in polydisperse solutions. The time τo provides information on the mobility of the scattering molecules: 1 ¼ Dapp q2 : so

ð9Þ

Dapp is the average apparent diffusion coefficient which depends on the protein concentration (Fig. 2): Dapp ðcÞ ¼ Do ð1 þ kd cÞ;

ð10Þ

kd is the parameter describing interparticle interactions, Do is the translational diffusion coefficient at a concentration c equal to 0 and is related to the hydrodynamic radius RH through the Stokes–Einstein relation: Do ¼ kT =6pgRH ; k—Boltzman constant, g—solvent viscosity.

ð11Þ

3.3. Depolarised dynamic light scattering The rotations of protein molecules are responsible for depolarisation of the light scattered, so in order to determine the rotational diffusion coefficient the intensity–intensity time correlation function of the depolarised component is measured (IVH), where V refers to the vertically polarised incident beam and H to the horizontal polarisation of the scattered light. The characteristic decay time of the correlation function can be expressed by two terms: 1 ¼ Dapp q2 þ 6H sVH

ð12Þ

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Fig. 2. Protein concentration dependence of the apparent lysozyme diffusion coefficient under atmospheric (square) and elevated pressure 100 MPa (circle).

so the rotational diffusion coefficient Θ can be obtained by a combination of DLS and DDLS. Rotational diffusion coefficient is related to the hydrodynamic radius RH through the Stokes–Einstein–Debye relation: H ¼ kT =8pgR3H :

ð13Þ

The coefficient Θ is much more sensitive to changes in the size of protein molecules than Dapp (Eq. (11)), but it is much more difficult to determine it for small size protein molecules. Small globular protein molecules are practically optically isotropic and the depolarised component of the scattered light has low intensity, so a measurement of the correlation function requires a long time of accumulation (therefore, in measurements of translational diffusion the effect of depolarisation is usually neglected). Besides, the time of rotational diffusion is usually by a few orders of magnitude shorter than that of translational diffusion and for the particles of molecular weight smaller than 30–50 kDa it may not be measurable by a standard correlator. For instance for a protein of the size of the lysozyme molecule (14.3 kDa, about 18 Å) the correlation time of translational diffusion is shorter than 20 microseconds, while that of the rotational diffusion is of an order of a few nanoseconds and is beyond the limit of measurable times of the majority of correlators. The rotational diffusion coefficient of small size molecules can be measured by a Fabry–Perot interferometer. The anisotropic large protein molecules that can be approximated by an ellipsoid or a rod (e.g., amyloids) are best suitable for DDLS method investigation. Knowing the two diffusion coefficients it is possible to find the parameters of the models, that is the ratio of the ellipsoid semi-axes (b/a) or the ratio of the length to the diameter (L/d) and their pressure changes.

409

kf is the coefficient of the linear term in the development of the friction factor, ν2 is the protein partial specific volume, M is the molecular weight, and B22 is the second virial coefficient. B22 is related to the sum of spherically symmetric potentials of mean force, W22. In the limit of the low salt concentration W22 is described by DLVO (Derjaguin– Landau–Verwey–Overbeek) theory: W22 = WHS + WC + WA, WHS is the hard sphere potential, WC is the screened Coulomb potential, WA is the van der Waals potential. B22 can be directly measured by SLS. The pressure dependence of kd observable by DLS is a resultant of changes in all potentials describing the interactions among the proteins and between the protein molecules and the solvent (Fig. 3). The value of kd depends on the electric charge of the protein molecule, pH and the ionic strength of the solution. Changes in pressure induce changes in the electric charge (protonation/ deprotonation), in the ion concentration and in pH. Moreover, as a result the dielectric constant of water, used as a solvent, changes (Fig. 4). Consequently, the changes in kd as a function of pressure are an interesting source of information on the mutual relations between protein molecules in the range of pressures below that of denaturation. 3.5. Hydrodynamic models The DLS method is sensitive to changes in the shape of the molecules of the protein studied. Hence, it seems particularly well suited for investigation of structural changes in biological molecules under compression. The method is very effective in studying the processes of denaturation and dissociation; however, it may not be able to detect subtle changes in the molecule volume or in the density of the hydration shell. The first approximation of the protein size is a sphere with the radii RH. The hydrodynamic radius RH is found directly from Eqs. (11) or (13) and contains the information on the size and the hydration/ solvation of the protein. If the protein molecule is approximated by a ball of the radius Ro, then the ratio RH/Ro is greater than unity because of the presence of the hydration shell. The radius of the “dry” protein molecule can be calculated

3.4. Protein–protein interactions The parameter kd from Eq. (10) describes interparticle interactions [19]: kd ¼ 2MB22  kf  2m2 :

ð14Þ

Fig. 3. Parameter kd decreases with increasing pressure. This kind of behaviour was also observed during the lysozyme crystal growth under pressure.

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Fig. 4. Changes in the water parameters (viscosity [28], refractive index [29], specific volume [30], dielectric constant [31]) with pressure increasing up to 120 MPa (1.2 kbar).

knowing its partial specific volume: Ro = [(3/4 π)(Mw/NA)v2]1/3. The volume of a hydrated protein molecule of the radius RH should also include the volume of the hydration shell δ1, ν1, where δ1 is the so-called molecular hydration and ν1 is a partial specific volume of bound water. Thus, from the following equation [21]: RH =Ro ¼ ½ðm2 þ d1 m1 Þ=m2 1=3

ð15Þ

the parameters of the hydration shell can be estimated for the experimentally determined hydrodynamic radius. Actually, even globular protein molecules are not ideal spheres so a special shape function is introduced into Eq. (15). If the protein molecule can be described by an ellipsoid of revolution, the shape function (in this case Perrin function) P, is a combination of the ellipsoid's semi-axes [32]. For simple models (oblate and prolate ellipsoids, thin rigid rod etc.) the analytical formulae for P are known. The shape function value is found for pairs of parameters, e.g., Do and Θ, or Do and Rg. Theoretically, starting from (RH/Ro) determined at ambient conditions, we are able to follow conformational changes with pressure just by measuring Do, however, in the pressure range up to 200 MPa, the changes in δ1, ν1 and ν2 are too small to be observed by the light scattering method. The effect of high pressure on the lysozyme structure has been studied by many authors by NMR up to 200 MPa (2 kbar) [33], by UV fluorescence up to 1.1 GPa (11 kbar) [34], by CD spectra up to 600 MPa (6 kbar) [35], by NMR up to 200 MPa (2 kbar) [36] and by X-ray diffraction up to 100 MPa (1 kbar) [37]. As follows from the above works, the pressure induced changes in the solution structure of hen egg-white lysozyme, in the pressure range from atmospheric pressure to 1.2 kbar, are

too small to be observed by DLS. For instance, in the pressure range 3–200 MPa (30–2000 bar) a change in the partial specific volume of this protein molecule does not exceed 2.4% [33]. Detection of such a subtle difference requires extremely precise measuring methods. The model most faithfully reproducing the shape of a protein molecule is the so-called “bead model”, based on the known crystallographic or NMR structure. The reproduction of the molecule shape is achieved by replacement of the domains, amino acids or individual atoms by beads of the radius adjusted to take into account the hydration effect. In this paper the model of “a bead per atom” proposed by the group of Prof. Garcia de la Torre, with the bead radius σ = 2.4 Å was used. The diffusion coefficient value calculated assuming this radius was equal to the experimental value. The hydrodynamic parameters were calculated with HydroPro software developed by Garcia de la Torre [38]. On the basis of the atomic coordinates of the lysozyme dimer deposited at the Protein Data Bank, PDB [39], the bead models of lysozyme monomer and dimer were made (Fig. 5) for which the diffusion coefficients monomer (PDB ID:

Fig. 5. Bead model of (A) lysozyme monomer molecule (B) lysozyme dimer based on the coordinates from 2lyz.

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2lyz) [40] and dimer under atmospheric pressure were calculated. Dimer of lysozyme was generated using the Protein Quaternary Structure file Server (PQS) based on the crystal symmetry matrices to generate symmetry-related copies of the chains in a PDB entry [41]. The diffusion coefficient for the lysozyme monomer was Do = 11.38 × 10−7 cm2/s (RH = 18.9 Å) and that for the lysozyme dimer was Do = 8.48 × 10−7 cm2/s (RH = 25.9 Å). If the data on the pressure changes in the atom coordinates are available, the effect of these changes on the hydrodynamic parameters measured can be estimated. The shape of the model of the protein molecule can be then modified and verified by comparing the experimental and calculated values of the hydrodynamic parameters. The ratio of the change in the translation diffusion coefficient obtained on the basis of the atomic coordinates for lysozyme monomer under atmospheric pressure relative to the diffusion coefficient determined on the basis of the atomic coordinates for lysozyme monomer under 200 MPa (PDB ID: 1gxv and 1gxx, respectively) [33] is only 1.2%. 3.6. Polydispersity The sample polydispersity can be a consequence of the presence of impurities, proteins of a different molecular weight [42], or the process of self-association [27]. Measurements of the diffusion coefficient of the non-identical scattering particles give the mean of the diffusion coefficients of all components: Do ¼

Rni Mi2 Di ; Rni M12

ð16Þ

where Di is the diffusion coefficient of the macromolecule of molecular weight Mi and at the numerical concentration of ni. A generally recommended method of the correlation function analysis permitting a determination of the distribution of the diffusion coefficient is the method of cumulants. The quantity used to specify the degree of polydispersity is

411

Fig. 7. Lysozyme diffusion coefficient Do (value extrapolated to 0 concentration and calculated taking into account the pressure changes in n, and η) versus pressure. Horizontal line corresponds to the diffusion coefficient calculated for the lysozyme monomer on the basis of the coordinates from 2lyz.

the polydispersity index, PD.I. For the protein in purely monomeric state PD.I should be lower than 0.01. For the protein monomer accompanied with lower order oligomers or a relatively low concentration of larger aggregates of other proteins PD.I may vary from 0.05 to 0.50. Monitoring the values of PD.I and the mean Do by the High Pressure DLS is an excellent method for determination of the hydrostatic pressure effect on the processes of dissociation/ association and their reversibility. The use of properly adjusted hydrodynamic models for the calculations of Di permits a quantitative determination of the contributions of particular components Eq. (16). We assume that the solution of lysozyme in a buffer of pH 4 is composed mainly of protein monomers and small amount of dimers [27]. Thus, the diffusion coefficient of the monomer found in experiment is D1 = 11.4 × 10−7 cm2/s, while the diffusion coefficient of the dimer (calculated for the bead-model 1hf4) is D2 = 8.48 × 10−7 cm2/s. For such values of Di (D1 for monomer and D2 for dimer) the relation between the mean diffusion coefficients on e.g. the monomer concentration can be obtained from Eq. (16), see Fig. 6. The value Do = 10.5 × 10−7 cm2 /s, obtained experimentally indicates dimerisation of about 18% of the molecules. For all samples at atmospheric pressure the PD.I of about 0.200–0.300 was observed. At elevated pressure (120 MPa) PD.I was significantly lower (about 0.07) and the diffusion coefficient reached a plateau at the Do of the monomer (Fig. 7). 4. Concluding remarks

Fig. 6. Mean value of the diffusion coefficient Do of lysozyme dependence on the percent proportion of the contribution of monomers and dimers derived from Eq. (16). The horizontal lines determine the interval of Do values published by different authors as obtained for lysozyme in similar conditions (salt concentration, pH and temperature) [43].

Light scattering studies in water solutions of proteins in an environment close to native should be performed taking into account the pressure dependence of the physical parameters of the buffer: no, η, ε, ν1 (Fig. 4). Monitoring changes in RH (or Do), due to conformational changes and/or aggregation processes provides the information on the size, shape or hydration changes with elevating pressure. The conformational changes and alterations taking place in the surrounding water in the pressure range below denaturation are

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rather too small to be detectable by this method to a standard accuracy. Taking into account the differences in the hydrodynamic radii of the lysozyme models based on the atomic structures determined under different pressure values, the effect of compression should be measurable by DLS, unless the change in RH caused by that in the partial specific volume ν2 is compensated by changes in the properties of the hydration shell δ1ν1. By observing the changes in kd due to the pressure dependence of dielectric permittivity, one can follow the changes in the protein–protein interaction (association and disassociation) with pressure. Dynamic light scattering has been shown to be an excellent method to investigate the protein–protein interactions in solution, determine the crystallisation conditions and detect changes in the size and shape of biomolecules under varying environmental conditions.

[13]

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[15]

[16]

[17] [18] [19] [20]

[21]

Acknowledgments [22]

The financial support of the SoftComp Network of Excellence (No. S080118) is gratefully acknowledged. I wish to thank Dr. G. Meier from the Institute of Solid State Research (IFF), Research Centre Juelich (Germany) for his kind assistance.

[23]

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