Nano-rheology of the folded state

The experimental system consists of a layer of diameter gold nanoparticles (GNPs) tethered to a gold surface through the protein under study (Fig. 1). The surface is a gold film ( thick) evaporated on a glass slide, which serves both to anchor the proteins (through the SH group of specifically introduced Cysteins) and as a semi-transparent electrode. A thick flow cell is constructed with this slide and a similarly gold coated cover slip, in a parallel plates capacitor configuration. An AC voltage applied to these electrodes drives the GNPs through the electrophoretic force, the GNPs carrying a large negative charge due to surface bound charged polymers (ss DNA 32mers). The “vertical” (perpendicular to the surface) motion of the GNPs, averaged over a large ensemble of GNPs (), is detected by evanescent wave scattering [15], the signal being recovered at the forcing frequency in a phase locked loop. The combination of noise rejection due to the synchronous detection and averaging over many particles makes it possible to measure the ensemble average amplitude of oscillation of the GNPs with sub-Angstrom resolution (see Figure S1, Figure S2, Figure S3, and Figure S4).

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Figure 1. Experimental setup.

(a) Schematics of the experimental setup including the chamber, electric excitation, and optical readout. The inset shows the geometry of the protein (Guanylate Kinase, GK) attached by the 171 and 75 sites to a gold nanoparticle (GNP) and a gold film evaporated on the glass slide (not to scale). (b) Crystal structure of GK (PDB: 1S4Q) with the attachment sites 75 and 171 highlighted.

https://doi.org/10.1371/journal.pone.0028097.g001

The protein of this study is Guanylate Kinase (GK) from ; we [16], [17] and others [18] have been exploring its mechano-chemical properties with different methods over the past few years. GK is an essential enzyme which catalyzes the transfer of a phosphate group from ATP to GMP. The substrate binds in the cleft between the two lobes of the structure (Fig. 1b); GMP binding is of the induced fit type [18]–[21], the two lobes closing on the substrates through a conformational change. The specific molecule of this study is the 75/171 mutant of [16], where Cys have been substituted at positions 75 and 171 (Fig. 1b); through these Cys the enzyme is anchored to the gold surfaces. We know from previous studies [16], [21], [22] that a mechanical stress in the 75–171 direction couples to the enzymatic function, specifically the binding affinity for GMP can be modulated through such stresses. A beautifully detailed representation of the mechanics of this enzyme is given in the simulations of the Baaden group [18], [23].

For this system, we measured the response to a sinusoidal applied force in the frequency range 10 Hz–10 kHz [13], [14]. Below we summarize our previous results as they are relevant for what follows. For low amplitude of the force, the response (the amplitude of oscillation of the GNPs at the forcing frequency, averaged over many GNPs) is given by the squares in Fig. 2a. This is the response of a mass-less damped spring (continuous line in Fig. 2a), exhibiting a corner frequency where is the spring constant and the dissipation coefficient. The low frequency plateau () is due to the elastic constant of the protein (represented by ), the high frequency cutoff () is due to the hydrodynamic dissipation of the GNP (represented by ). We have shown through these measurements that when the enzyme binds the substrate GMP, it stiffens by , i.e. increases by with a corresponding increase in while remains the same [13]. Since the enzyme is able to selectively bind its substrates, it is presumably in the folded state. This is confirmed by measurements of the enzymatic activity of the surface-immobilized enzyme (see Figure S5).

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Figure 2. AC susceptibility of GK (from[14]).

(a) Amplitude of the protein deformation vs frequency for two different amplitudes of the force (corresponding to driving voltages 550 and 600 mV). (b) Magnitude of the complex modulus (arbitrary units) of the protein + GNP system; this is the same data as in (a) replotted as vs . For mV (squares) the behavior is elastic; for mV (circles) the behavior is viscoelastic. The error bars represent standard deviations of 5 measurements. The real “complex modulus” corresponding to in the graph is pN/nm. The points are experimental data while the lines are fits with the corresponding elastic or viscoelastic models.

https://doi.org/10.1371/journal.pone.0028097.g002

For larger amplitude of the force there is a transition to a qualitatively different response, given by the circles in Fig. 2a. The amplitude “diverges” at low frequencies. This is the response of a viscoelastic element (a spring and dashpot in series) attached to the GNP (dotted line in Fig. 2a), exhibiting two characteristic frequencies , where is the spring constant, the dissipation coefficient of the dashpot (representing internal dissipation in the protein), the hydrodynamic dissipation coefficient of the GNP [14]. The increase of the response amplitude for is the signature of viscoelasticity; the system “flows” at low frequencies.

Complex Modulus formalism and Maxwell model

We start with the generalized linear relation between force and displacement: [24], [25], (1)

where is the applied force, is the displacement, and is the “relaxation modulus”, implicitly satisfying the causality condition if . In the frequency domain, , , we have the generalized Stokes Einstein relation (GSER) [26]–[28], (2)

where (3)

Here is a “complex spring constant”, but we find this term awkward so we will borrow a term from rheology instead and call our the “complex modulus”. In the rheological literature the relations above are written for the stress and strain; the function (the complex modulus) has then dimensions of a force per unit area. Here we choose to use force and displacement, instead of stress and strain, to describe the system for the reason that force and displacement are operationally well-defined in our experiments. In contrast, stress and strain are properly defined over length scales which are large compared to the atomic structure, but small compared to the macroscopic volume of the material under consideration [24]. Thus it is questionable whether stress and strain are well defined quantities for a single protein molecule. On the other hand, the relation (1) just expresses a general linear relationship, which here we assume between force and displacement. From (1) and (2) has then dimensions of a force per unit length. In analogy with the rheological terminology, where the real and imaginary parts of the complex modulus are often called the storage and loss modulus, respectively, in the following we call the real and imaginary part of our the same.

Let us form a qualitative picture of what to expect for the measurements of the complex modulus of the protein, . In the elastic regime () the “complex modulus” is simply a real constant: . The eigenfrequencies of the protein lie in the range above Hz [29]; this is also easily estimated from a typical “spring constant” pN/nm [30] and the mass of the protein kDa which gives a fundamental mode Hz, or equivalently from a typical Young's modulus MPa, density g/cm, and size nm, giving Hz. Therefore in the frequency range of the experiments, Hz kHz, in the elastic regime one expects to see the low frequency response of a spring which corresponds to a constant complex modulus.

In the experiments, the protein is attached to a Gold nanoparticle (GNP). The equation of motion of the GNP is [13], [14]: (4)

where is the electrophoretic force applied to the GNP, the force on the protein, the hydrodynamic dissipation coefficient of the GNP (because of the proximity of the surface, , the Stokes drag coefficient, where is the viscosity of the fluid and the radius of the GNP). The inertial term has been neglected in eq. (4) because in the experiments the driving frequency is “small”: where is the mass of the GNP. Also, there is no Brownian motion term because the measurement method averages it out (this is the reason why with the present method one can measure displacement amplitudes of a fraction of 1 Å, see Figure S1, Figure S2, Figure S3 and Figure S4).

We wish to show that the measured response of the system is consistent with a viscoelastic transition of the protein [14]. The simplest model of viscoelasticity is the Maxwell element, which is a spring and dashpot in series. The corresponding equation of motion (5)

( is the stiffness of the spring, the dissipation coefficient of the dashpot, the applied force, the displacement) gives a complex modulus (6)

where Re is the storage modulus and Im is the loss modulus. From eqs. (5) and (4) we obtain [14]: (7)

With a sinusoidal force of amplitude : the response is an oscillation of amplitude : (with the usual convention of taking real parts to obtain the physical quantities); substituting in (7) we obtain the complex modulus of the Maxwell element + GNP system (the protein + GNP system if the protein behaves like a Maxwell element): (8)

where is given by (6). The following remarks will be useful in understanding the arguments below. First, a moment's reflection shows that the decomposition (8) is valid independently of the specific model assumed for , i.e. the contribution from the GNP to the complex modulus is purely imaginary, . Therefore the storage modulus of the protein is in fact exactly the same as that of the protein GNP system. Second, in the frequency response experiments, the amplitude of the driving force is kept constant at different frequencies, . Then (see eq. (8)), i.e., the magnitude of the complex modulus is inversely proportional to the amplitude of the protein deformation , which is the quantity measured in the experiments. Therefore we may plot the experimental measurements either as vs , as in Fig. 2a, or equivalently as vs , which is, apart from a multiplicative constant, the same as vs . This is the way the experimental data are plotted in Fig. 2b. In conclusion, Fig. 2b shows the magnitude of the complex modulus vs frequency measured for GK for two values of the forcing amplitude. These are the same data as in [14], presented in terms of . A transition between two different behaviors is apparent. For low forcing amplitude (squares), we get the response of a spring: the solid line is a fit using eq. (8) with , which is (9)

where . The increase of at high frequency ( rad/s) reflects the hydrodynamic dissipation of the GNP (i.e. ). At higher forcing amplitudes (circles) we get the response of a viscoelastic element: the solid line is a fit using eq. (8) with given by (6). This form is (10)

where , . The fits give the values: rad/s, rad/s, rad/s [14]. Since and are essentially the same, i.e. and in the viscoelastic regime can also be written as: (11)

The drop of at low frequency is the signature of viscoelasticity. This transition from elastic to viscoelastic behavior is reversible: turning up the driving force the sample jumps from the elastic to the viscoelastic behavior, and turning the driving force down to the original value the same sample reverts to the elastic behavior [14].

We can now show that the piecewise linear response of Fig. 3 is quantitatively consistent with the measurements in the frequency domain Fig. 2, in the framework of the Maxwell model (6). Namely, the formulas (9), (11) read for the response amplitude in the elastic and viscoelastic regimes: (12)

(13)

At fixed driving frequency ( for the data of Fig. 3) the amplitude of the driving force is proportional to the amplitude of the protein deformation in both regimes. But the proportionality constants, or the slopes, are different in the two regimes. In the elastic regime, the slope is ; while in the viscoelastic regime the slope is . The ratio of the slopes is: (14)

and using the value rad/s measured in the frequency response experiments (Fig. 2) and rad/s we obtain . On the other hand, from the linear fits in Fig. 3 we can measure directly the ratio of the slopes and obtain . This shows a remarkable consistency of the two measurements Fig. 3 and Fig. 2 in the framework of the simplest viscoelastic model. We may also say that the ratio of the two slopes measured from Fig. 3 provides another way to estimate the internal friction (and thus the internal viscosity) of the protein, by assuming (14) and using the spring constant for the protein reported previously [30], pN/nm. In this way we obtain the internal friction coefficient kg/s (or an internal viscosity of the protein Pas), which is close to the value we obtained in [13] from the characteristic frequencies , . In summary, the ratio of the slopes in Fig. 3, which is about 6, is consistent with the characteristic frequency at which the response departs from simple elasticity (Fig. 2).

Given the nonlinear (piecewise linear) response displayed in Fig. 4, the question arises whether such nonlinearity alone, in the absence of internal dissipation, can give rise to the frequency response of Fig. 2, or in other words, is this a viscoelastic system or a nonlinear, non-dissipative system. The question can be answered (in favor of viscoelasticity) by numerically computing the frequency response of a nonlinear spring such as the one of Fig. 4. We do this in Figure S6. The result is that the piecewise linear response displayed in Fig. 3 cannot by itself, in the absence of internal dissipation in the protein, give rise to the frequency response displayed in Figs. 2. Instead, the experimental measurements show that the force vs displacement response of the protein is piecewise linear, corresponding to a transition from elasticity (where no internal dissipation of the protein is seen) to viscoelasticity, where there is internal dissipation and the protein is mechanically softer. This transition happens, in this case ( Hz), for a yield deformation Å corresponding to a yield strain Å .

If the deformation is not too small it is possible to measure directly in the experiments the real and imaginary parts of the deformation at a fixed frequency. Then the storage modulus is (15)

and represents the storage modulus (real part of the complex modulus) of the protein only, with no contribution from the GNP (see eq. (8)). The measurements are shown in Fig. 4, vs amplitude of the applied force. We now show that the measurements of displayed in Fig. 4 are consistent with the measurements of of Fig. 3, if a viscoelastic model is assumed. Because the frequency of these measurements is so low (10 Hz), the complex modulus obtained from the data of Fig. 2 reports on the complex modulus of the protein, the contribution from the gold nanoparticles being negligible. In the viscoelastic regime, assuming the response is that of a Maxwell element, the real and imaginary parts of the complex modulus (16)(17)

can be written in terms of as: (18)(19)

where , , and . Similarly for the elastic regime: (20)(21)

where and (see Figure S7).

The relation (18) for the storage modulus is somewhat model independent, in the sense that if , which is true in our experiments [10], [14]. The triangles in Fig. 4 are computed from the measurements of of Fig. 2, using the relation (18) to obtain (normalized by a multiplicative constant). The signal over noise of the measurement obviously improves at larger forcing amplitudes, so particularly the direct measurements have larger error bars at low frequency (the error of the storage modulus is estimated through error propagation where and are the standard deviation of five measurements of and , respectively); nonetheless Fig. 4 shows that no systematic deviation is seen between the storage modulus measured directly and computed from the measured assuming the response of a Maxwell model. The inset in Fig. 4 shows the ratio between the two, the red line indicating a ratio of 1.

We mentioned that the transition from the elastic ( Å) to the viscoelastic ( Å) regime is reversible (the same piecewise linear curve can be repeated for the same sample as a function of the driving amplitude), but not if one exceeds a certain driving voltage ( V in these experiments). There is indeed a second, irreversible transition, which is more apparent if we plot the same data of Fig. 3 in terms of the complex modulus vs applied force , which is displayed in Fig. 5. There are three regimes: the elastic regime where is a constant (independent of applied force), the viscoelastic regime which also entails a reversible, progressive softening of the protein with increased applied force, and finally an irreversible transition to a regime where the complex modulus also decreases with increased applied force, but slower than in the reversible viscoelastic regime. This graph also shows that not all properties of the system can be described by a linear model such as the Maxwell model, since in the viscoelastic regime the experimental depends on the applied force. This is seen equivalently in Fig. 3, where the linear behavior in the viscoelastic regime does not extrapolate to the origin.

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Figure 5. Complex modulus from the dynamic stretching experiments.

Magnitude of the complex modulus of the protein as a function of the amplitude of the applied force. The error bars represent standard deviation of 5 measurements.

https://doi.org/10.1371/journal.pone.0028097.g005

Materials

Gold nanoparticles (GNP, 20 nm diameter) were from Nanocs (New York, NY); other chemicals from Sigma-Aldrich. Experiments were performed in saline-sodium citrate buffer (SSC; Invitrogen) diluted with deionized water to a final concentration of 50 mM sodium chloride and 5 mM trisodium citrate, pH 7.0 (SSC/3).

Sample Preparation

The protein under study (Guanylate kinase or GK, a 24 kDa, 4 nm sized globule) was prepared by mutagenesis with the internal Cys changed to Ser and the residues at the positions 171 and 75 substituted by Cys, as described in [22]. The introduced Cys are essential for the protein to tether gold nanoparticles to a gold thin film on a glass slide (Fig. 1). The glass slide was thoroughly cleaned before evaporating a 3 nm layer of Cr followed by 30 nm of gold using an e-Beam vacuum evaporation system. The purpose of the gold layer is to obtain a conducting electrode (see details below) and also take advantage of the affinity of the thiol groups (Cys on the protein surface) for gold surfaces [33], [34]. In practice, the Au-slide was immersed in GK solution (2 M in 1 M KHPO at pH 7.0) overnight and then washed with a large amount of deionized water. Gold nanoparticles (20 nm diameter) were then introduced and incubated at room temperature for 2 hours, followed by washing with water. In order to make the gold nanoparticles charged through surface modifications, and to remove nonspecific protein immobilization, the slide was then immersed in a solution of thiol-modified DNA (32 bases, 1 M in 1 M KHPO at pH 4.0) overnight and washed with water. The prepared slide forms the bottom of a chamber, which is constructed in a parallel capacitor configuration with a cover slip (also coated with a gold thin film) on top of the slide, separated by 200 m spacers. After external electrodes are attached to the gold films, the chamber is filled with SSC buffer diluted by 3 and ready to use.

Frequency Response Experiments

Frequency response experiments, which measure the AC susceptibility of the sample vs frequency, are described in detail in [13]. The experiment consists in mechanically forcing the gold nanoparticles using an AC electric field and detect their motion along the direction of the forcing by evanescent wave scattering in a phase locked loop (Fig. 1). The GNPs carry a large negative charge due to the DNA “brush” anchored to their surface, and thus can be driven by the electrophoretic force established by applying a potential difference across the chamber. The motion of the gold nanoparticles along the direction of the electric field, which is perpendicular to the gold surface, is monitored by evanescent wave scattering, where the displacement of the GNPs is proportional the change in the scattered light intensity , , where (64nm in our setup) is the penetration depth and is the total light intensity scattered from a collection of gold nanoparticles ( GNPs in a filed of view mm). Therefore we detect the average displacement of this collection of GNPs, which reports on the average deformation of the tethered proteins. In the frequency response experiments, AC voltages at frequencies, ( and rad/s), are applied to the chamber and the displacements are measured in a phase locked loop, using a lock-in amplifier. At each frequency, the amplitude of the GNP displacement, called the “response”, is averaged over 50 seconds. The resultant averaged response as a function of the driving frequency, referred to as the “frequency response”, is then used to calculate the mechanical properties of the sample.

Dynamic Stretching Experiments

Dynamic stretching experiments share the same setup (Fig. 1) with the frequency response experiments. The difference from the latter is that we now vary the amplitude of the driving AC voltage at a fixed driving frequency. In this study we chose 10 Hz as the driving frequency based on the observation that the difference in the response of the sample to driving forces between the elastic regime and the viscoelastic regime lies in the low frequency range [13] (and see “Results and Discussions” below). The amplitude of the AC voltage on the chamber is a linear series, ranging from 0.450 V to 1.025 V. At each amplitude, the response is measured and also averaged over 50 seconds, as in the frequency response experiments. By varying the amplitude of the applied AC voltage, a dynamic stretching experiment measures the relation between the amplitude of the driving force (proportional to the AC voltage) and the amplitude of the displacement, which we refer to as the “dynamic force-extension curve”.