Theoretical and Experimental Investigation of the Translational

Article pubs.acs.org/JPCB

Theoretical and Experimental Investigation of the Translational Diffusion of Proteins in the Vicinity of Temperature-Induced Unfolding Transition Stanislav Molchanov,*,†,∥ Dzhigangir A. Faizullin,⊥ and Irina V. Nesmelova*,‡,§ †

Department of Mathematics and Statistics, ‡Department of Physics and Optical Sciences, and §Center for Biomedical Engineering, University of North Carolina, Charlotte, North Carolina 28223, United States ∥ National Research University “Higher School of Economics”, Moscow 101000, Russia ⊥ Kazan Institute of Biochemistry and Biophysics RAS, Kazan 420011, Russia S Supporting Information *

ABSTRACT: Translational diffusion is the most fundamental form of transport in chemical and biological systems. The diffusion coefficient is highly sensitive to changes in the size of the diffusing species; hence, it provides important information on the variety of macromolecular processes, such as self-assembly or folding−unfolding. Here, we investigate the behavior of the diffusion coefficient of a macromolecule in the vicinity of heat-induced transition from folded to unfolded state. We derive the equation that describes the diffusion coefficient of the macromolecule in the vicinity of the transition and use it to fit the experimental data from pulsed-field-gradient nuclear magnetic resonance (PFG NMR) experiments acquired for two globular proteins, lysozyme and RNase A, undergoing temperature-induced unfolding. A very good qualitative agreement between the theoretically derived diffusion coefficient and experimental data is observed.

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INTRODUCTION The theory of transition between folded (globular) and unfolded states is widely reviewed in the literature for synthetic polymers and biological macromolecules1−5 (and references therein). Temperature-induced transition from globular to unfolded state involves the change in the size of a macromolecule that leads to the decrease of its translational selfdiffusion (i.e., random Brownian motion, hereafter denoted simply as diffusion) coefficient. Thus, diffusion measurements can be used to analyze the conformational properties of synthetic polymers or non-native states of proteins. The measurements of the diffusion coefficient, D, have been used to estimate the change of size of proteins due to unfolding caused by heat or denaturing agents.6−15 All of these estimates are based on using a well-known Stokes−Einstein formula, D = kT/6πηr, for the diffusion coefficient of the Brownian sphere of radius r, which is used to approximate the protein molecule. Other variables in this formula include the temperature T, the viscosity of pure solvent η, and the Boltzmann constant k. While the Stokes−Einstein relation is applicable to two terminal states (i.e., folded and unfolded compact states) of proteins, it does not predict the temperature dependence of the protein diffusion coefficient in the vicinity of the transition when the size of the molecule gradually increases. In this work, we derive the theoretical relation describing the diffusion coefficient of a globular macromolecule in solution in the vicinity of the transition from folded to unfolded state, © 2016 American Chemical Society

induced by the increase in temperature. We consider the diffusion of a macromolecule when the interactions with other macromolecules are negligible; hence, the predictions are applicable to dilute solutions. The unfolded state of the macromolecule is treated as a continuous random process. The derived formula is used to fit the experimental data on the temperature dependence of the diffusion coefficient of two globular proteins. We show that there is a good qualitative agreement between theoretical predictions and experimental data. Limitations of this approach and directions for future development are further discussed.

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EXPERIMENTAL METHODS Materials. Hen egg lysozyme and bovine pancreatic ribonuclease A (RNase A) were purchased from Sigma. Both proteins were homogeneous by SDS-PAGE and used without further purification. D2O, DCl, and NaOD were purchased from Cambridge Isotope Laboratories. Lysozyme was dissolved in 0.1 M solution of NaCl in D2O at pH 1.6 at the concentration of protein 18 mg/mL as determined from UV−visible spectroscopy. The value of pH was checked after this procedure and adjusted to 1.6 by adding microliter quantities of DCl or NaOD to the sample. RNase A Received: June 9, 2016 Revised: September 11, 2016 Published: September 14, 2016 10192

DOI: 10.1021/acs.jpcb.6b05834 J. Phys. Chem. B 2016, 120, 10192−10198

Article

The Journal of Physical Chemistry B

Figure 1. Example of NMR spectra for diffusion coefficient determination. (A) Stack plot of 1D NMR spectra for diffusion coefficient measurement of 18 mg/mL lysozyme at 303 K. The bracket indicates the region of the spectrum used for diffusion coefficient determination. (B) 1D spectrum expansions for 18 mg/mL lysozyme at 303 and 333 K exemplifying spectral changes caused by temperature-induced protein unfolding. Spectral regions used for diffusion analysis are shown by the dashed line box.

mobility of the protein molecule, the convection effect still could be noticed in dilute solutions at high temperatures used in this study. Therefore, using the DSTE sequence was essential for obtaining accurate data at elevated temperatures. Additionally, the chemical exchange (in our case, the exchange between folded and unfolded protein states could be expected) affects the diffusion attenuation profile.22 The DSTE pulse sequence with bipolar gradients eliminates the distortion of signal intensity caused by chemical exchange and is a better choice for systems where the exchange is present.23 To record the diffusion attenuation of the spin−echo signal, a z-gradient was employed. The maximum value of the gradient was calibrated using Varian’s standard procedure and was found to be 56 G/ cm, which was consistent with the value obtained from analysis of diffusion data on water using its known diffusion constant.24 The x- and y-gradients were employed during z-storage periods to achieve proper magnetization selection. The diffusion coefficient D was calculated from the diffusion attenuation of the spin−echo signal A(g2) at the increasing strengths of the pulsed field gradient g according to the equation20

was dissolved in pure D2O at pH 2.5 at protein concentrations of 11 and 32 mg/mL. The choice of acidic pH values was based on the literature data on protein volume change due to the thermal unfolding of lysozyme and RNase A, as well as to avoid the aggregation of proteins upon unfolding.16−19 Furthermore, no intermediate states with extensive internal hydration have been detected in RNase A at this pH.18 Solutions of lysozyme and RNase A were heated for 1 h to 318 and 315 K (45 and 42 °C), respectively, to achieve the hydrogen−deuterium exchange required in our infrared spectroscopic experiments, cooled down, and then lyophilized. All solutions were prepared in amounts of 0.5 mL, of which 0.2 and 0.12 mL were used to run NMR and FTIR experiments, respectively. The experiments were repeated three times using freshly made samples. NMR Diffusion Measurements. Diffusion coefficients were measured in the temperature range from 293 to 348 K (20 to 75 °C) for lysozyme and from 278 to 358 K (5 to 85 °C) for RNase A. Changes in solution pH after the experiments were insignificant: 0.1 and 0.02 for lysozyme and RNase A, respectively. To avoid variations due to the speed of heating, diffusion measurements for all samples were done within 30 min after a 15 min exposure of the sample to each temperature. Experiments were performed at 600 MHz using a Varian Unity INOVA-600 instrument equipped with a 5 mm triple resonance probe with three-axis shielded magnetic resonance gradients. The temperature in the NMR probe was calibrated using methanol or ethylene glycol. The double-stimulated-echo (DSTE) pulse sequence with longitudinal eddy-current delay20 (LED) and bipolar gradients21 was used. The DSTE pulse sequence allowed eliminating the contribution of convection currents to the diffusion coefficient. We estimated that, for the probe used in this study, convection currents led to a significant error in the diffusion coefficient of low viscous liquids (e.g., water) measured at temperatures above 303 K (30 °C). When the sample volume was decreased to 0.2 mL by using a Shigemi NMR tube, this temperature was shifted to 328 K (55 °C). Although less severe in protein solutions due to a lower

⎡ ⎛ 4δ 3τ ⎞⎤ ⎟⎥ A(g 2) = A(0) exp⎢ −Dγ 2δ 2g 2⎜Δ + + ⎝ ⎣ 3 2 ⎠⎦

(1)

where Δ is the diffusion delay, γ is the gyromagnetic ratio for protons, δ is the duration of the field gradient pulses, τ is the delay between the edge of the gradient pulse and the consequent radio frequency pulse, and A(0) is the intensity of the echo signal in the absence of a gradient pulse. For these experiments, a series of 12−22 one-dimensional (1D) proton spectra were collected using a spectral width of 10 kHz, Δ of 62 ms, δ of 4 ms, and a delay τ of 200 μs. These parameters were kept the same for diffusion measurements at all temperatures. A representative stacked plot of 1D spectra for diffusion coefficient measurement of lysozyme at T = 303 K is shown in Figure 1A. The residual water signal is indicated by an arrow. The region of the lysozyme 1D spectrum used to determine the diffusion coefficient is indicated by a bracket. Figure 1B shows 10193


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⟨Vx̃ (t )⟩ = 0, ⟨Vx̃ (t ) ·Vx̃ (t + τ )⟩ = B(τ ) = σx 2 exp( −α|τ |)

expansions from 1D lysozyme spectra collected at 303 and 333 K exemplifying changes due to temperature-induced protein unfolding. The integrated area of the peak between 0.4 and 2.4 ppm, as shown by the dashed line box, was used to calculate the diffusion coefficient. Supporting Figure 1 shows temperatureinduced changes in RNase A. Note, only monoexponential diffusion decays were observed in the whole temperature range, indicating the fast exchange between folded and unfolded protein molecules throughout the transition. The standard error of diffusion coefficients, estimated by the fitting procedure, was below 2%. Fourier Transform Infrared Spectroscopy (FTIR). Infrared spectra were collected on a Tensor 27 (Bruker) FTIR spectrometer in the range 4000−1000 cm−1 at 4.0 cm−1 resolution. Protein solution was placed in a 100 μm CaF2 cell with a jacket through which water from a thermostatic bath was circulated. All reported spectra are the averages of 256 scans. The corresponding buffer baseline was subtracted from each spectrum, and a smoothing has been applied. Fourier selfdeconvolution (FSD) was applied to the amide I spectral region (1700−1600 cm−1) to increase the visual resolution. A bandwidth value of 2 cm−1 and a resolution enhancement factor of 1.5 were used. FSD spectra were decomposed into Gaussian components using the Fityk 0.9.8 software.25 The number and position of the components were determined from the second derivative of collected spectra.

(5) σ2 . 2α

where α = 6πrη/m and σx 2 = B(τ) is the correlation function of the stationary process Ṽ (t). The dependence of Ṽ x on temperature is given by the equipartition theorem of classical statistical physics: 1 ̃2 mVx 2

x(t ) − x(0) =

t

V ̃ (τ ) d τ

Var(x(t ) − x(0)) = ∼t

(7)

∫R′ B(τ) dτ =

∫0

t

dt1

∫0

t

B(t1 − t 2) dt 2

2tσ12 kT t = α 3πηr

(8) 26

which gives the Stokes−Einstein result for the diffusion coefficient D, where D is proportional to temperature T:

D=

kT 6πηr

(9)

The diffusion of a biopolymer (in particular, of a protein) has several important features, related to the transition from a compact globular state, where the random density fluctuations are small, to an unfolded (or diffusive) state, where the biopolymer becomes more flexible and extended; e.g., the density fluctuations become significant and their correlation length is of the same order as the dimensions of a biopolymer.2 For a biopolymer consisting of l monomers, the diffusive (unfolded) state can be considered as the continuous Brownian trajectory x(λ), λ ∈ [0, l], of length l (each monomer unit of the biopolymer chain has unit length). The transition from folded to unfolded state can be initiated by increasing the temperature. Then, at temperatures of T < Tcr, where Tcr is the critical temperature of the transition, and large l, the biopolymer molecule can be approximated by a sphere of some effective radius reff(T, l). At T ↑ Tcr, reff(T, l) increases, and according to eq 9, the diffusion coefficient of a biopolymer is expected to decrease. On the basis of statistical physics of phase transitions, one can expect the following asymptotic behavior of effective radius with temperature in the vicinity of the transition: reff(T, l) ∼ (Tcr − T)−ξ, ξ > 0, where ξ is some critical exponent.27 In the following section, we derive this dependence using a self-consistent theory approach by considering the model of the long homogeneous polymer chain in an attracting potential field.28 According to this model, the statistical weight of a particular trajectory representing a polymer chain is given by the Gibbs distribution

(2)

Random collisions of the solvent molecules with the particle produce a stochastic, rapidly fluctuating force FR(t) (3)

where σ̇(t) represents the white noise of intensity σ. Assuming that there is no external potential (“free” particle), the equation of motion of the particle (the Langevin equation) is written according to Newton’s second law as mV̇ (t ) = −6πrηV (t ) + σ (̇ t )

∫0

In equilibrium, for a random process, the average displacement is ⟨x(t) − x(0)⟩ = 0. However, the mean squared displacement of the particle from the starting point is not zero and is given by the following relation

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FR (t ) = σ (̇ t )

(6)

The displacement of the particle in the direction x, x(t), is then found from the relation between the position and velocity by integrating Ṽ with respect to time:

THEORETICAL CONSIDERATIONS Several simplifying assumptions are made in deriving the diffusion coefficient of the macromolecule: (1) the globular state of the macromolecule is represented by the Brownian particle; (2) the unfolded state of the macromolecule is modeled by the random Brownian trajectory of infinite length (l → ∞); and (3) a single macromolecule is considered, i.e., the interactions between different macromolecules are neglected (e.g., the results are applicable to dilute solutions). First, let us briefly summarize the derivation of the equation for the diffusion coefficient of the Brownian particle, representing the globular state of the macromolecule. Consider a large solid spherical particle of mass m and radius r immersed in an incompressible liquid with viscosity η at temperature T. The effect of solvent molecules on the particle can be represented by two forces. The motion of the particle is decelerated by a frictional (or Stokes) force FS(t). At time t, FS(t) is reversely proportional to the velocity vector of the particle V(t): FS(t ) = −6πrηV (t )

1 kT mσx 2 = 2 2

=

(4)

l

Independent of the initial condition, the solution V(t) of eq 4 converges to the stationary Gaussian process Ṽ (t) with independent coordinates, each satisfying the following relations

e β ∫0 U (x(λ))dλ Pβ(x) = , Ζ(β , l) 10194

β=

1 kT

(10)


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Figure 2. Temperature dependencies of diffusion coefficients and secondary structure content for lysozyme and RNase A. (A) Temperature dependence of the diffusion coefficient of lysozyme (squares) at 18 mg/mL and pH 1.6 and of RNase A at 11 mg/mL (triangles) and 32 mg/mL (inverted triangles) at pH 2.5. Solid lines connecting the data points are guides to the eyes. The dashed line shows the slope of the temperature dependence T/η, calculated from water viscosity. (B) Intensity vs temperature plots of bands at 1640 cm−1 (combined α-helical and β-sheet secondary structure) and 1659 cm−1 (β-turns) in lysozyme and at 1631 cm−1 (β-sheet structure), 1640 cm−1 (α-helical structure), and 1659 and 1668 cm−1 (β-turns) in RNase A reflect the disruption of the secondary structure of proteins. No intermolecular structure is formed, as indicated by the lack of temperature dependence of the bands at 1614 cm−1 (RNase A) and 1615 cm−1 (lysozyme).

where U(x) ≥ 0 is the rapidly decreasing potential around some point O ∈ R3, for example, the center of mass of the macromolecule, and Z(β, l) is the partition function. We assume that trajectories x(λ) are distributed according to the Wiener measure (which represents the limit of the physical Brownian motion measure when the correlation length of the underlying stationary Ornstein−Uhlenbeck process 1 m = 6πηr ≪ 1). Then, the partition function can be written as α

E0(β) ∼ c3(β − βcr )2 ,

Ζ 0(β , x) = βG E0U (x)Z0 ∼ c·e−

Ζ(β , l) = ⟨e

⟩,

x(0) = 0

(14)

E 0 |x |

c3 (β − βcr ) |x|

∼ c · e−

(15)

Here, G E0(x , y) = ( −Δ + E0)−1(x , y) =

e− E0 |x − y| 4π |x − y|

is the Green

function of the classical Laplacian. Note, Z0(β, x) does not contain singularity. The stationary distribution of the trajectory x(λ) is equal to Z02(β, x), i.e.,

l

β ∫0 U (x(λ))dλ

c3 = c3(U )

(11)

According to the Feynman−Kac formula, the function ⟨x 2(λ)⟩ =

l

Ζ(β , l ; x) = ⟨e β ∫0 U (x + x(λ))dλ⟩, where the trajectory starts from the arbitrary point x ∈ R3, satisfies the following equation: ∂Ζ = Δx Ζ + βU (x)Ζ = −H Ζ ∂l

= (12)

Ζ(β , 0; x) = 1

3

∞ 4 − c (β − β ) |x| cr x e 3 0 ∞ 2 − c (β − β ) |x| cr x e 3 0

∫

dx

∫

dx

Then, at T ↑ T cr , reff =

For β < βcr (high temperatures, T > Tcr), the distribution of trajectories representing a polymer is close to the Wiener distribution; i.e., polymer molecules exist in a “diffusive” state where x(λ) − x(0) ∼ l , and multiple conformations are equally possible. On the contrary, for β > βcr (low temperatures, T < Tcr), x(λ) is close to the stationary process, x(λ) ∼ const, 0 < λ < l. This fact is related to the existence of the ground state (globular state) Z0(β, x) > 0 with normalization ∫ 3 Ζ 0 2 dx = 1 R and the ground state energy E0(β) > 0, which tends to zero 1 1 when β = kT ↓ βcr = kT , so that

∼

12 c3(β − βcr )2

⟨x 2(λ)⟩ ∼

12 / c3 β − βcr

(16)

, and the

temperature dependence of the diffusion coefficient is described by the following relation: D∼

kT 6πη

(Tcr − T ) 12/c3 ·kT ·Tcr

=

1 (Tcr − T ) 6πη 12/c3 ·Tcr

(17)

Here, ξ = 1 and the constant c3 depends on the choice of U(x) (e.g., eq 14). Remark 1. The result given by eq 17 is derived under the assumption that the potential U(x) is bounded and supported in the sphere of finite radius. If the potential U(x) is more singular, the critical exponent ξ may have a different value. For example, Grosberg et al.1 consider a point potential and show that E 0(β) ∼ const·(β − βcr ), which would lead to

cr

H Ζ 0 = ΔΖ 0 + βU (x)Ζ 0 = E0(β)Z0

∫R |x|2 Z02(β , x) dx

(13)

For an infinitely smooth finite range potential U(x), Cranston et al.28 have derived the following asymptotic result for E0(β) and Z0(β, x) at β ↑ βcr (in three-dimensional space):

reff = 10195

⟨x 2(λ)⟩ ∼ (β − βcr )−1/2 and D ∼

kT 6πη

Tcr − T kT ·Tcr

, i.e., ξ =


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The Journal of Physical Chemistry B 1/2. Depending on the potential, E0(β) ∼ const·(β − βcr)ζ, with 1 ≤ ζ ≤ 2. Remark 2. In a theoretical model of the long homogeneous polymer chain in an attracting potential field,28 the length of the polymer is infinite, while the real biopolymer has a finite length. Consequently, the results of the model are applicable if (β − βcr)2l ≫ 1; i.e., the critical temperature of a real biopolymer is always lower than the theoretical Tcr of a polymer chain with l → ∞. The overall temperature dependence of the diffusion coefficient is then specified by the term Tcr − T , where the

temperature. No noticeable changes in the secondary structure of lysozyme or RNase A were observed up to about 313 K. In lysozyme, the intensity of peaks corresponding to combined αhelical and β-sheet structure and β-turns starts to decrease at 318 K, indicating the onset of temperature-induced unfolding. In RNase A, the temperature-induced unfolding starts at 315 K and a steep decrease in the β-sheet content is observed as the protein is heated. Temperatures corresponding to the onset of the secondary structure loss are consistent with the abnormal decrease of the diffusion coefficient of lysozyme and RNase A. At the same time, at all temperatures investigated, no significant changes in the intensity of bands at 1615 cm−1 (lysozyme) and 1614 cm−1 (RNase A) are observed, indicating that the aggregation of proteins does not occur at the chosen experimental conditions. Thus, the change in the size of lysozyme and RNase A (hence, the decrease of diffusion coefficient) is due to the temperature-induced secondary structure loss. The change of the size of lysozyme and RNase A can be estimated from the temperature dependence of their diffusion coefficients. Note that the slope of the temperature dependence of the diffusion coefficient before the unfolding transition is the same as the slope of the temperature dependence of water viscosity T/η (Figure 2A), as expected for the Brownian particle (eq 9). Thus, if the temperature dependence of the diffusion coefficient is extrapolated to higher temperatures (above the transition), as it would follow the temperature dependence of water viscosity, it would represent the diffusion coefficient of proteins with unperturbed structures (or not significantly perturbed, so that the hydrodynamic volume of the protein molecule remains the same). Consequently, the size change caused by the temperature-induced secondary structure loss can be estimated from the ratio of extrapolated diffusion coefficients above the unfolding transition to experimental values taken at the same temperature. According to these estimates, the volume of lysozyme and RNase A molecules changes approximately 1.8 times, in agreement with literature data for similar experimental conditions.16−19 Two observations important for the subsequent discussion should be noted. First, both proteins represent good experimental models to test the theoretical prediction for the temperature behavior of the protein diffusion coefficient in the vicinity of the unfolding transition given by eq 17, because at the chosen experimental conditions the change of size is due to the disruption of the secondary structure and no protein aggregation is observed. Second, a relatively small size change (1.8 times) and the fact that the slope of the temperature dependence of the diffusion coefficient above the transition remains approximately T/η indicates that after the transition both proteins remain in the compact state, hydrodynamically comparable to that described by eq 9. The Comparison of Theory and Experiment. We next sought to compare the theoretically predicted temperature dependence of the macromolecule diffusion coefficient in the vicinity of the transition from globular to unfolded state with experimental data obtained for two proteins, lysozyme and RNase A. Under simplifying assumptions of the theoretical model used, several specific features of the protein are not taken into account. For example, even in the globular state, proteins are inherently dynamic molecules and their shape deviates from spherical. Both of these factors do not alter the slope of the temperature dependence of the diffusion coefficient, because globular proteins diffuse as rigid Brownian

η

temperature dependence of the solvent viscosity is given by the

( RTE ) with the energy of activation E ,

Arrhenius law η = η0 exp

A

A

where R is the gas constant. Near the threshold temperature Tcr, the viscosity can be considered constant. Accordingly, when the molecule of a globular protein is heated to temperatures close to Tcr, the diffusion coefficient is expected to decrease rapidly due to the presence of the (Tcr − T) term.

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RESULTS AND DISCUSSION

Temperature Dependence of Protein Diffusion Coefficients. The diffusion coefficient of lysozyme at pH 1.6 and RNase A at pH 2.5 measured as a function of temperature at different protein concentrations is shown in Figure 2A. The diffusion coefficient increases as the temperature increases in the range from 293 to 318 K (20 to 45 °C) for lysozyme and from 278 to 315 K (5 to 42 °C) for RNase A, respectively, indicative of increasing protein mobility. The dashed line corresponds to the slope of the temperature dependence T/η and represents the temperature dependence of solvent viscosity (shifted along the y-axis for a better view). It is apparent that the viscosity of water defines the temperature dependence of the diffusion coefficient of lysozyme and RNase A up to 318 and 315 K, respectively. This behavior is expected for a globular molecule in dilute solution, where it can be approximated by the Brownian sphere (see eq 9). When the temperature rises above 318 K for lysozyme and 315 K for RNase A, the diffusion coefficient of both proteins demonstrates an abnormal behavior, i.e., significantly weaker temperature dependence as compared to that of water viscosity. The temperature range of this abnormal temperature dependence is relatively narrow, 5 K for lysozyme and 8 K for RNase A. As the temperature increases further, the diffusion coefficient of both proteins behaves normally again; i.e., it increases with increasing temperature, approximately following the temperature dependence of water viscosity T/η. Temperature-Induced Changes in Protein Structure. The decrease of the diffusion coefficient indicates the change of size of the diffusing species (see eq 9) that could be caused by protein unfolding and/or aggregation. Therefore, to detect structural changes and/or aggregation of lysozyme and RNase A, we recorded FTIR spectra on protein samples in the range of temperatures from 278 to 358 K. In lysozyme, the intensity of bands at 1640 cm−1 (combined α-helical and β-sheet secondary structure), 1659 cm−1 (β-turns), and 1615 cm−1 (intermolecular extended β-sheet structure due to aggregation) was monitored.29 In RNase A, the intensity of bands at 1631 cm−1 (β-sheet structure), 1640 cm−1 (α-helical structure), 1659 and 1668 cm−1 (β-turns), and 1614 cm−1 (intermolecular extended β-sheet structure due to aggregation) was monitored.19,30,31 These band intensities are shown in Figure 2B as a function of 10196


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parameters, are listed in Table 1. Details of the fitting procedure and other fitting parameters are provided in the Supporting

particles in the whole solubility range32,33 and taking the nonspherical shape of the protein molecule into account only changes the absolute value of the diffusion coefficient.34−36 In fact, our data on lysozyme and RNase A confirm that the slope of the temperature dependence of their diffusion coefficients outside the transition region is close to T/η of water (Figure 2A), demonstrating that their hydrodynamic properties are similar to rigid particles. Additionally, the hydrodynamic behavior of proteins may be affected by the hydration of charged amino acids on the surface and different dynamics of water molecules near the protein surface.37,38 Furthermore, the protein molecule has a finite length and is chemically heterogeneous, consisting of nonpolar, polar, and charged amino acids. When hydrophobic and hydrogen bond interactions that stabilize the folded protein state are disrupted, the unfolded chain may rearrange to compact conformations with favorable long-range electrostatic interactions.39 Lysozyme and RNase A remain compact above the transition, retaining residual secondary structure9,19,31 (Figure 2B). Thus, the unfolded state deviates from random as assumed by the Brownian trajectory and is “semi-diffusive”, where residual secondary structure elements can be viewed as “beads” joined by unfolded segments of the protein molecule. The configuration of beads and unfolded segments can be treated as the Brownian trajectory of length l /l1 , where l1 is the size of the bead. These effects are not taken into account by the theoretical model presented here, and the multiscale approach is required for a more accurate description of protein unfolding. Thus, given the simplifying assumptions of the model, we anticipate observing a qualitative agreement between theoretically predicted and experimental temperature dependences of protein diffusion coefficients. Figure 3 shows temperature dependences of lysozyme and RNase A diffusion coefficients plotted within 10 K below the

Table 1. Values of Activation Energies EA and critical temperatures Tcr, obtained by fitting experimental data to eq 17 protein

EA (kJ/mol)

Tcr (K)

lysozyme, 18 mg/mL RNase A, 11 mg/mL RNase A, 32 mg/mL

42.5 ± 2.3 42.0 ± 1.9 43.3 ± 1.8

342 ± 2 339 ± 1 339 ± 1

Information. The fits produce overestimated but reasonable activation energies EA in agreement with literature data.40,41 The value of Tcr yielded by fits corresponds to the temperature at the end of the transition to a new conformation with disrupted secondary structure (Figure 2B).

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CONCLUSIONS We present a simple theoretical model to describe the diffusion coefficient of the macromolecule in the vicinity of the transition from globular to unfolded state. This model is applied to describe the diffusion of two globular proteins, lysozyme and RNase A, undergoing the temperature-induced secondary structure disruption. A very good qualitative agreement between the proposed model and the experimental data is observed. Although we used NMR spectroscopy to measure protein diffusion coefficients, the proposed theoretical model is applicable to describe self-diffusion coefficients obtained by other techniques, e.g., dynamic light scattering.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b05834. 1D NMR spectra of Rnase A (PDF) Fitting of experimental data (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*Address: Department of Mathematics and Statistics, University of North Carolina, Fretwell 355G, 9201 University City Boulevard, Charlotte, NC 28223, USA. Phone: (+1) 704-6871379. Fax: (+1) 704-687-1392. E-mail: [email protected]. *Address: Department of Physics and Optical Science, University of North Carolina, Grigg Hall 306, 9201 University City Boulevard, Charlotte, NC 28223, USA. Phone: (+1) 704687-8145. Fax: (+1) 704-687-8197. E-mail: Irina.Nesmelova@ uncc.edu. Author Contributions

Figure 3. Theoretical fits of experimental temperature dependencies of lysozyme and RNase A diffusion coefficients. The experimental values (squares, lysozyme; triangles, RNase A at 11 mg/mL; inverted triangles, RNase A at 32 mg/mL) are shown within 10 K from the onset of the conformational transition due to the decrease in the secondary structure induced by the increasing temperature. Solid lines represent fits of the experimental data to eq 17.

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are thankful to Prof. V. D. Fedotov and Prof. V. D. Skirda for useful discussions. This work was supported by the Faculty Research Grant from the University of North Carolina to I.V.N. and by the National Science Foundation [1410547] to S.M.

transition. Theoretical fits of experimental data to eq 17 are shown by solid lines. Despite simplifying assumptions of the model, the diffusion coefficients predicted by eq 17 are in good agreement with the experimental values. The values of the activation energy EA and critical temperature Tcr, used as fitting 10197


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ABBREVIATIONS NMR, nuclear magnetic resonance; FTIR, Fourier transform infrared spectroscopy; DSTE, double-stimulated-echo; FSD, Fourier self-deconvolution

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Theoretical and Experimental Investigation of the Translational

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