Nanosecond Stokes Shift Dynamics, Dynamical Transition, and

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Nanosecond Stokes Shift Dynamics, Dynamical Transition, and Gigantic Reorganization Energy of Hydrated Heme Proteins Dmitry V. Matyushov* Center for Biological Physics, Arizona State University, P.O. Box 871504, Tempe, Arizona 85287-1504, United States

bS Supporting Information ABSTRACT: We report numerical simulations of three hydrated heme proteins, myoglobin, cytochrome c, and cytochrome B562. The properties of interest are the dynamics and statistics of the electric field and electrostatic potential at heme’s iron, as well as their separation into the protein and water components. We find that the electric field produced by both the protein and the hydration water relaxes on the time scale of 3
6 ns, and the relaxation time of the electrostatic potential is close to 1 ns. The slow dynamics of the electrostatic observables is accompanied by their large variances. For the electrostatic potential, a large amplitude of its fluctuations leads to a gigantic reorganization energy of a half redox reaction changing the redox state of the protein. Both a large magnitude and a slow relaxation time of the electric field fluctuations are required to explain the onset of large meansquare displacements of iron at the point of protein’s dynamical transition. These requirements are met by the simulations which are used to explain the temperature dependence of heme iron displacements measured by M€ossbauer spectroscopy. All three phenomena, (i) nanosecond dynamics, (ii) protein dynamical transition and a large high-temperature excess of atomic mean-square displacements, and (iii) the gigantic reorganization energy, are explained here by one physical mechanism. This mechanism involves two components: nanosecond motions of the protein surface residues and polarization of the interfacial water by the protein charges. Global nanosecond conformations of the protein move the surface water. Since water is polarized, these movements create large-amplitude electrostatic fluctuations, sufficient to modify displacements of groups inside the protein and yield reorganization energies of protein electron transfer far exceeding those found for small molecules. Water follows adiabatically the protein motions. Therefore, the relaxation times of the protein and its hydration layer are close, leading to matching temperatures of the dynamical transition for the two components.

’ INTRODUCTION Polar dynamics near solutes of molecular size is typically faster than the dielectric Debye relaxation of bulk polar solvent. The speedup comes from ballistic motions of the surface solvent molecules which can contribute up to 80% of the relaxation.1 The situation is often reversed when the solute grows to a nanometer scale. Instead of acceleration, slowing down of the water dynamics has been documented in recent theoretical2
6 and experimental7
20 studies. The question that these observations, many of them for proteins in solution, raise is whether slow dynamics can be attributed to the properties of the extended solvent interface or to solutes. The network of bulk hydrogen bonds is extensively broken at the interface with a nanometer solute, resulting in the formation of distinctly new density and orientational interfacial structures.21
23 Both the density and orientational distributions of interfacial waters become exceedingly sensitive to the properties of the solute surface.24,25 Density diminishes at a hard-wall surface, and water partially dewets for a weak solute
solvent attraction.26 On the contrary, when the solute
solvent attraction is increased by introducing charged and polar groups at the surface, the interfacial water density exceeds that of the bulk. This is the case typical of proteins27
29 for which the density of the interfacial layer exceeds bulk density by =10
15%. A similar r 2011 American Chemical Society

picture is valid for hydrated silica nanoparticles with dense surface charge.30 Sharper density peaks, with a possibility of water layering, are observed both for proteins29 and nanoparticles.30 However, in contrast to nanocrystals, the water
protein interface is highly chemically heterogeneous, and one might ask if this mosaic of attraction sites of different strengths slows the dynamics, surface diffusion in particular.31 Optical spectroscopy of probe dyes attached to the surface or intercalated inside the proteins3,16 has been extensively used to characterize the electrostatic interfacial response.32
34 The general observation, confirmed by the majority of studies, is the appearance of a slow component in the Stokes shift dynamics, ranging from tens to hundreds of picoseconds32
34 to several nanoseconds.7,8,10,11,13
16,20 One might suggest that slow Stokes shift dynamics reflects the orientational structure of the interfacial water. Indeed, in-plane preferential orientation of surface dipoles is universally found at water interfaces ranging from nanometers to macroscopic surfaces.21,22,35,36 Since many waters are involved in the electrostatic response, Stokes shift dynamics might report on correlated Received: January 14, 2011 Revised: August 4, 2011 Published: August 04, 2011 10715

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The Journal of Physical Chemistry B relaxation of large ensembles of interfacial waters. The universality of slow dynamics at extended water interfaces would then reflect collective relaxation of these orientationally correlated clusters. Although attractive, this proposal does not explain a significant amount of data showing very slow Stokes shift dynamics in the range of nanoseconds.7,8,10,11,13,14,16,20 Studies of rigid interfaces show that orientational relaxation times do not exceed tens of picoseconds5,24 and have never been found to extend to nanoseconds. Protein or a self-assembled structure with similar characteristics12 might be essential for reaching the nanosecond relaxation time. An alternative explanation proposed by Halle and co-workers37,38 offers a mechanism extending the dynamical response of the protein
water interface to much longer relaxation times. In this view, the slow relaxation component is attributed to slow conformational motions of a protein moving the dielectric protein
water interface and by that means affecting the electrostatic response of the protein
water solvent. Water, being a faster component of the system, follows adiabatically protein’s conformational motions, and the observed dynamics is purely protein’s. Therefore, there is nothing universal about slow dynamics found in many nanometer solutes except for the universal abundance of global conformational motions on the sub-to-nanosecond time scale, even though the corresponding relaxation times might be greatly influenced by hydration. The extensive body of work briefly overviewed here poses a question of whether the unusual properties found for protein solutions are significant for biological function. Electrostatics is obviously of great importance for the protein stability, but it also greatly affects enzymatic and redox functions of proteins39,40 and the rates of electron transport in respiratory and photosynthetic energy chains.41 We have recently suggested that slow dynamics of the electrostatic potential is critical for the energetics of biological electron transfer since it allows proteins to control activation barriers of electronic transitions by adjusting the reaction time scale, in addition to the commonly anticipated adjustments of the reaction Gibbs energy.42,43 The slow dynamics makes polar solvation kinetically arrested on short reaction times, thus preventing significant losses of energy to solvation when an electron is transferred in a series of hops between cofactors or protein active sites.43 Further, it was found that the breadth of the electrostatic fluctuations linked to slow dynamics greatly exceeds standard expectations.44 It is not only that some parts of the nuclear response can be kinetically arrested but also that the energies involved are very large. This observation brings in a new perspective that electric field fluctuations might be strong enough to alter and facilitate the flexibility of charged atoms and groups inside the protein.45 This notion then requires considering atomic motions of proteins typically recorded by neutron scattering and M€ossbauer spectroscopy.46
49 These three observables, dynamics of electrostatic response, statistics of electrostatic fluctuations, and protein’s atomic displacements affected by electrostatics, define the scope of this paper. We approach the problem by simplifying the typical observables, involving groups of atoms, and considering instead two physically well-defined properties: electric field and electrostatic potential at a single atom of the protein, heme’s iron. These properties are not directly reported by experimental techniques. Stokes shift spectroscopy reports on the statistics and dynamics of field fluctuations, but a chromophore only approximately can be considered as a point dipole probing the field. Likewise, a half

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redox reaction changes the charge of the active site, but the transferred electron spreads over several atoms coordinated to heme’s iron. Finally, M€ossbauer spectroscopy reports on a single atom, but extracting the effect of electrostatics on its meansquare displacement (msd) requires a theory not quite developed at the moment. All of these deficiencies of this presentation are recognized and can be improved upon in more accurate calculations.42,44,50
52 The goal of this paper is to formulate a conceptual framework to look at these phenomena from a perspective of a unifying physical mechanism. We indeed find here remarkably slow, on a time scale of =3
6 ns, dynamics of the electric field at heme’s iron; the long-time dynamics of the potential is somewhat faster, on a time scale of 1 ns. The amplitude of the electrostatic noise linked to these slow dynamics is also very large, producing the reorganization energy of a half redox reaction =25 eV in the case of cytochrome B562. The values of the electric field variance obtained from the simulations are also consistent with what is required to reproduce the temperature dependence of the iron msd from M€ossbauer measurements. We therefore confirm a long-proposed conjecture53 that M€ossbauer spectroscopy reports on the intensity of the electrostatic noise at heme’s iron, relevant to the energetics of protein electron transfer.

’ RESULTS The results presented here were obtained by numerical Molecular Dynamics (MD) simulations of three heme proteins: metmyoglobin (metMB, protein database entry 1YMB), reduced state of cytochrome B562 (cytB, 256B), and reduced state of bovine heart cytochrome c (cytC, 2B4Z). The details of the simulation protocols are given in the Supporting Information (SI). Briefly, the proteins were solvated by ∼33 000 TIP3P water molecules to approach millimolar protein concentrations typically used in solution experiments. Following NPT equilibration, NVE trajectories of the length 65
123 ns were produced using the NAMD integrator.54 Particle-mesh Ewald sums with tinfoil boundary conditions were used for both the trajectory production and the analysis. The proteins carry charge of
2 (metMB),
5 (cytB), and +6 (cytC) at physiological conditions. The simulation cell was neutralized by adding Na+ and Cl
ions to bring the ion concentration to 0.1 mol/L. We report here time correlation functions and statistical averages of the electrostatic potential and electric field produced by the protein
water solvent at the position of heme’s iron. Accordingly, the dynamics of the electric field is represented by the time autocorrelation function of the electric field E(t), which is further split into the protein (p) and water (w) contributions, E = Ep + Ew. The same separation is implemented for the electrostatic potential, ϕ = ϕp + ϕw. The iron atom carries the partial charge qFe, which depends on the protein’s oxidation state and is specified by a force field assigned to the heme (see the SI). The left panels in Figure 1 show the normalized electric field correlation functions for all three proteins studied here SX ðtÞ ¼ ½ÆðδXÞ2 æ
1 ÆδXðtÞ 3 δXð0Þæ

ð1Þ

where X stands for either the electrostatic potential, X = ϕ, or the electric field, X = E; δX(t) = X(t)
ÆXæ. The right panels in Figure 1 show the loss functions55 00

χX ðωÞ ¼ βωÆðδXÞ2 æSX ðωÞ=2 10716

ð2Þ

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Table 1. Variances of the Electrostatic Force F = qFeE and Potential ϕ at the Position of the Heme Irona,b protein metMB cytB cytC

Æ(δF)2æ 541

λvar.

Æ(δFw)2æ

1.8

6.5 69

λvar. w

109

25.5 6.7

5.7 4.2

Æ(δFp)2æ

7.9 21.5 3.3

291 20.7 99

λvar. p 9.9 5.9 6.9

a

The force variance is given in [kcal/(mol Å)]2, and the variance of the potential is expressed in terms of the reorganization energy (eV) λvar defined by eq 4. Water (w) and protein (p) components of the force variance and the reorganization energy are also listed. b Error estimates for the reported values of λvar. are: 0.01(1%) (metMB), 0.4(1%) (cytB), 0.3(4%) (cytC). Error estimates for the reported values of the force variance Æ(δF)2æ are: 4 (1%) (metMB), 0.1 (2%) (cytB), and 4 (6%) (cytC).

peaks’ heights and the corresponding relaxation times through the peaks’ positions. The loss functions are decomposed into the water and protein components  Æ|δXw,p(ω)|2æ and a cross term  ÆδXw(ω) 3 δXp(ω)æ Figure 1. Time correlation functions SE(t) (left panels) and the 00 corresponding loss functions βe2χE(ω) (right panels, in Å
2) for metmyoglobin (metMB), cytochrome B562 (cytB), and cytochrome c (cytC); e is the unit electric charge. The functions refer to the electric field from the entire protein
water system (P+W) and protein (P) and water (W) components separately.

00

Figure 2. Loss functions of the electrostatic potential βe2χϕ(ω) for metMB, cytB, and cytC. The functions refer to the electrostatic potential from the entire protein
water system (P+W) and protein (P) and water (W) components separately.

where SX(ω) is the time Fourier transform of SX(t) and β = 1/kBT is the inverse temperature. 00 The visual advantage of loss functions χX(ω) compared to SX(t) is that they show both the relative contributions of different components of the system to the overall variance through the

00

00

00

00

χX ðωÞ ¼ χX, p ðωÞ þ χX, w ðωÞ þ χX, p, w ðωÞ

ð3Þ

Only the self (water and protein) loss functions of the electric field are shown in Figure 1, and the loss functions of the electrostatic potential are presented in Figure 2. All three heme proteins show the same basic dynamics of the electric field at the heme iron. The correlation functions are biexponential, with 10
20% amplitude of a fast relaxation component, followed by an 80
90% amplitude of a long-time tail, with the relaxation time of τE = 3
6 ns (see the SI for the list of fitted relaxation times and amplitudes). This is clearly seen in the loss functions in the left panels in Figure 1 where the main Debye peak corresponds to the 00long relaxation time ωX = τ
1 X . The 00 protein, χE,p(ω), and water, χE,w(ω), loss functions have nearly equal peak frequencies ωE, testifying to the linked dynamics of the protein and hydration water. The relative contributions of water and protein to the total variance (heights of the peaks, eqs 2 and 3) vary significantly among the proteins. While water and protein contribute nearly equally to the field fluctuations for cytB, the protein is the main contributor for metMB and cytC. One might also notice that the protein and water contribute constructively to the overall electric field for metMB, but they add up destructively for the two cytochromes. Table 1 lists the variances of the electric force F = qFeE and electrostatic potential at heme’s iron. The force variance is higher for metMB than for the cytochromes due to a higher charge, qFe = 1.34, of Fe in the oxidized state compared to qFe = 0.24 in the reduced state (see the SI). The variance of the electric force determines the high-temperature softening of the iron’s msd that we discuss below. The higher value of the electric force variance projects itself into a higher msd. To characterize the strength of the electrostatic potential fluctuations, we use, instead of the potential variance, the reorganization energy λvar: ¼ βe2 ÆðδϕÞ2 æ=2

ð4Þ

where e is the unit electric charge. This parameter, although calculated for a single iron atom, is a fair estimate of the reorganization energy of an electrode half redox reaction involving the protein. The difference between the experimental reorganization energy and the one listed here is in the 10717

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slow, nanosecond-scale nuclear motions (Table 1 and Figure 2). The nanosecond peak, dominating the potential spectrum in the water and protein components separately, is almost entirely lost in the overall loss function of metMB. On the contrary, this nanosecond peak is most pronounced in cytB, which has the highest overall reorganization energy (Table 1). It appears that not allowing this compensation to occur is critical for the energetically efficient electron transport on the nanosecond and longer reaction time scale (see below), and that is what cytochromes, participating in electron transport chains, seem to be able to achieve. The relative weights of the slow and fast relaxation components differ between the two cytochromes: cytC displays a 00 noticeable weight of fast relaxation in 00 χE(ω), which becomes dominant in the potential loss function χϕ(ω) (Figure 2). It is not clear if these differences present any functional advantage. One might, however, notice that all proteins simulated here are taken at the ionization conditions of pH = 7.0. Their ionization state might be different at the pH of operation, and that might contribute to enhancing or depressing the fast component in the dynamics. Given a long range of electrostatic interactions and many particles involved, one might expect that the electrostatic observables follow the Gaussian distribution. In the case of electrostatic potential, one gets Figure 3. Distributions of the electrostatic potential (upper and middle panels) and electric field (lower panel) for metMB and cytC (see the SI for cytB). Components due to protein (P), water (W), and the overall response (P+W) are shown. The dashed lines in the upper and middle panels show Gaussian fits (eq 5) of the simulated distributions. The dashed line in the lower panel is the Maxwell
Boltzmann (MB) distribution (eq 6). The values of the observables are normalized to their corresponding variances, X/σX, σ2X = Æ(δX)2æ, for each distribution shown in the figure; X = E, ϕ. The inset in the lower panel shows the distribution of the angle θ between the protein and water components of the electric field at heme’s iron of cytC and metMB.

assumption that the entire charge e is transferred to iron in an electronic transition. Delocalization of the electronic density to atoms neighboring iron requires calculation of the potential at all atoms involved.44,56 However, given that the electrostatic potential is a slowly varying function in space, eq 4 gives a reasonable estimate of the electron-transfer reorganization energy, as is also confirmed by calculations including different extent of electronic delocalization.50 Within these limitations, which imply that we do not provide accurate calculations of reorganization energies measured in redox experiments, we will call λvar. the reorganization energy, also implying that the average calculated over the entire simulation trajectory is close to the corresponding equilibrium value, λvar. = λvar. eq . The dynamics of the electrostatic potential follows trends similar to that of the electric field. The dynamics is clearly biexponential, but the slow component tends to be somewhat faster than for the electric field, probably due to a longer range of the electrostatic potential (1/r) compared to the electric field (1/r3). The potential is thus more sensitive to faster bulk relaxation than the field (Figures 1 and 2 and the SI). The case of metMB is particularly interesting. This protein does not participate in electron transport chains. Correspondingly, its overall reorganization energy reaches the “normal” value as a result of a substantial compensation between the large protein and water reorganization energies originating from

2

Pϕ ðxÞ  e
ðx
ÆxæÞ =2

ð5Þ

where x = ϕ/σϕ and σϕ = (Æ(δϕ)2æ)1/2. The magnitude of the electric field will follow the Maxwell
Boltzmann (MB) distribution if its orientation averages out to zero PE ðxÞ  x2 e
x =2 2

ð6Þ

where x = E/σE and σE = (Æ(δE) æ) . These expectations are met with varying accuracy when tested against the simulation data. The upper and middle panels in Figure 3 show the distributions of the electrostatic potential Pϕ(ϕ), and the lower panel shows the distributions of the electric field PE(E). Electrostatic potential distributions fit reasonably well to Gaussians, although these fits are not very restrictive since the positions of the maxima are adjusted to their simulated values. On the other hand, the deviations from the MB distribution are significant for both the protein and water components of the electric field. In addition, all distributions are clearly bimodal. A part of the deviation of the field distribution from the MB form is a nonzero average field vector. As is shown by the inset in the lower panel of Figure 3, water is pinned by the protein field, maintaining the orientation of Ew almost entirely antiparallel to Ep in the case of cytC. The two fields do not have to be antiparallel, and they in fact show a positive average scalar product in the case of metMB (inset in Figure 3). However, in all cases studied here, the water’s field keeps a significant average projection on the protein’s field, which implies that there are large patches of surface water that are polarized by the surface protein residues on the entire simulation trajectory. The reorganization energies (eq 4) found in the present simulations are significantly larger than values =1 eV typically assigned to small redox molecules in laboratory and computer experiments. Recent numerical simulations have indicated a possibility of gigantic (compared to small molecules) reorganization energies λvar. for active sites of proteins (calculated with charges delocalized over the active site43,57). Indeed, the 2

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reorganization energies for metMB and cytC reported here are consistent with our previous results for nonheme redox proteins,43 but the value of =25 eV for cytB exceeds all previous reports (Table 1). The main conclusions here are that these new extensive simulations performed on heme proteins support the existence of gigantic reorganization energies previously found for nonheme redox proteins.43

’ MECHANISM The slow nanosecond dynamics of the water component of SX(t) and the corresponding gigantic reorganization energy have the same physical origin—polarization of the interfacial water by protein’s surface residues. We provide here arguments based on response functions, while a more traditional reasoning in terms of free energy surfaces of electron transfer reactions can be found in ref 44. We also note that we do not address here a separate issue of calculating the rates of electron transfer42 and the related problem of global free energy surfaces of electron transfer. These require umbrella sampling to be calculated,58
60 in particular in cases of highly nonparabolic free energy surfaces.61 The variance reorganization energy λvar. considered here defines the curvature of the free energy surfaces at their minima and is directly accessible from laboratory measurements by either electrochemistry62 or optical spectroscopy.63 It is therefore a welldefined physical parameter, which is still only approximately related to our main focus here—the distribution of electrostatic potential and electric field at one atom, iron of protein’s heme. The dipolar polarization of the heterogeneous protein
water solvent P(ω) at frequency ω can be given as a sum of the response due to the electric field of heme’s iron EFe and the polarization P0 created within the hydration layer by polar/ ionized surface residues of the protein PðωÞ ¼ χ Fe ðωÞ 3 EFe þ P0 ðωÞ

ð7Þ

Here, χFe(ω) is a linear response function (generally a tensor) of the hydration shell surrounding the protein and the protein itself to the electric field of the iron atom. Since we present here qualitative arguments only, the dependence of the response on the position is suppressed in all functions for brevity. The overall electrostatic response is given by integrating the dipolar polarization with the electric field EFe.50 The interfacial polarization P0(ω) in eq 7 is frequency dependent to indicate that it is modulated by conformational fluctuations of the interface altering the positions of the polar/ ionized residues P0 ðωÞ ¼ χ p ðωÞ 3 Ep

ð8Þ

Here, χp(ω) is the response functions projecting the electric field of the protein surface charges and dipoles Ep into the frequencydependent dipolar polarization P0(ω) of the hydration layer. In the case of a small molecular solute, there is essentially no density of vibrational states overlapping with the picosecond modes of the solvent, and one typically puts χp(ω)  δ(ω). The static polarization P0 then shifts the average electrostatic potential at the site of electron localization, which projects itself into a shift of the frequency of a corresponding charge-transfer band,64 but does not contribute to the reorganization energy. One then gets the traditional description of electronic transitions in molecular systems65 in which intramolecular vibrations, contributing to the internal reorganization energy, are decoupled from the solvent fluctuations contributing to λvar.. The situation is

Figure 4. Cartoon of the water (W) polarization induced by a surface charge inside the protein (P). The polarization of the water dipoles will produce an average potential at the position of Fe opposite in sign to the potential of the surface charge (Figure 3).

dramatically different for a hydrated protein which typically has the density of low-frequency vibrational states maximized (∼40 cm
1) in resonance with the vibrational density of states (VDOS) of bulk water.66
69 This critical distinction dramatically changes the polar response of the interface. If the electric field of the surface residues exceeds that of iron, one can neglect the first summand in the rhs of eq 7 with the result PðωÞ = χ p ðωÞ 3 Ep

ð9Þ

The polar response is then fully determined by the conformational modulation of the interface by the protein, and one expects a close match between the protein and water loss functions, as is indeed the case (Figure 1). Further, the relative contribution of the fast water response (the first summand in eq 7) will depend on how close the site is to the surface. For sites close to the surface, the fast linear response of polar interfacial water will dominate. The slow dynamics will only be observed for sites buried inside the protein, as is seen in the laboratory experiment.16 The two-exponential decay invariably seen in all studies of the Stokes shift dynamics in proteins16,32
34 roughly represents two terms in eq 7: a linear response to the field of an optical probe (fast) and surface polarization modulated by protein’s conformational motions (slow). The opposite signs of the average protein and water potentials4,37,38 naturally appear from this surface polarization picture, as is illustrated in Figure 4. A charge of a surface residue induces an equilibrium polarization of the water dipoles pointing their opposite-charge ends toward the protein charge. This polarization will yield the average potential opposite in sign to that of the surface charge itself, and one gets opposite signs for Æϕpæ and Æϕwæ (Figure 3). This observation is true for cytC and metMB, but both potential averages are positive for cytB (see the SI). A local distribution of charge should be considered in each individual case, while Figure 4 illustrates a probable common scenario. Further, if the interfacial polarization P0 is little affected by the change of the charge of a buried atom (either structurally or dynamically because the new conformational state takes too long 10719

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to achieve44), P0 will cancel in the difference of vertical transition energies64 ΔEi = ΔE0
eÆϕæi, where ΔE0 includes all nonelectrostatic components and i = 1 and 2 refer to two different redox states. The difference of vertical transition energies defines the observable optical Stokes shift when optical transitions can be recorded, |ΔE1
ΔE2| = eΔϕ = e|Æϕæ2
Æϕæ1|. On the other hand, the same property defines the reorganization energy λSt ¼ eΔϕ=2

ð10Þ

equal to λvar. in traditional theories.64,70,71 Since P0 cancels from eΔϕ, the reorganization energy λSt is defined by the linear response function χFe in eq 7 (and thus can be calculated by linear response theories50), while the variance and the reorganization energy λvar. are mostly defined by the response function χp. The result is that λSt and λvar. are given in terms of different response functions thus breaking down44,72 a fundamental relation of traditional (Marcus theory) models of electron transfer: λSt = λvar.. Further, because the modulation of the water polarization by the surface residues is more intense than by the buried iron, one gets the inequality λvar: . λSt

ð11Þ

and a gigantic value of the reorganization energy λvar. observed in simulations44,43 (Table 1). One needs to stress here that inequality (11) does not imply breaking of the linear response approximation for solvation of a charge, as it might appear. The linear response approximation connects the average electrostatic potential/field induced by a charge/dipole in the surrounding medium to the fluctuations of the medium in either the absence or presence of the multipole. Here the situation is quite different. While the medium responds linearly to the field of iron EFe, the electrostatic fluctuations at the iron position are not related, or little related, to this field. The electrostatic fluctuations are created by surface polarization modulated by low-frequency vibrations of the protein and are therefore distinct from the electrostatic perturbation induced by the iron charge. Both the polarization induced by the transferred charge and the electrostatic fluctuations of the interface can therefore in principle be described by the linear response approximation, but with different response functions. The reorganization energy λvar. reflects the electrostatic noise of the polarized protein
water interface, while λSt is the traditional Marcus reorganization energy. The total reorganization energy λvar. will be a sum of individual water and protein components and a cross-correlation term,  Æδϕwδϕpæ (as in eq 3). On the basis of the picture shown in Figure 3 (suggested by Halle et al.37,38), the cross-correlation terms are negative, often reducing the overall reorganization energy below its water and protein components72 (see, e.g., metMB in Table 1). One can simply visualize this compensation by imagining the positive and negative charges at the interface in Figure 4 close to each other and assuming them to shift in tandem by a surface conformational motion. Such a motion, observed from a distant active site, will not produce a sizable fluctuation of the electrostatic energy since positive and negative charges will effectively cancel each other. The actual magnitudes of these cross correlations will depend on the specific modulation of the surface charge distribution by low-frequency vibrations changing the protein shape. These depend on the protein sequence and fold and will vary among proteins (Table 1). Since the total reorganization energy is a

result of compensation between several large terms, one might anticipate a strong sensitivity of λvar. to the interfacial charge distribution which can be altered by either surface mutations or by the solution pH.

’ DISCUSSION The slow dynamics of electrostatic fluctuations at the protein
water interface affects several observables accessible to laboratory measurements. Time-resolved fluorescence measurements directly report on the dynamics of the electric field at the position of an optical dye. In addition, electronic transitions between proteins and protein electrochemistry are sensitive to the statistics of the electrostatic potential at the protein active site.73 We show below that the displacement of heme’s iron is also strongly affected by the electrostatic fluctuations of the interface, but the observed temperature dependence and the phenomenon of dynamical transition require arguments involving the instrumental observation window. These are relevant to electronic transitions in proteins as well, and we discuss them first. The recognition of the importance of at least two reorganization energies in protein electron transfer, λSt and λvar., significantly changes the outlook at the energetics of biology’s electron transport chains.40 These are realized as a sequence of an often large number of electron hops between electron cofactors in the membrane, carrying the redox energy from a site of photoexcitation (photosynthesis) or redox energy input (respiratory chains) to a catalytic site. Since at least some of these electron transport steps are activationless, energetic efficiency becomes an issue of fundamental importance. The problem is rooted in the fact that standard theories require the free energy ΔG =
λSt to be lost to heat in an activationless electron hop. Since λSt = 0.8
1.0 eV,73 most of the input energy would be lost in a single electron transfer event. The appearance of a gigantic reorganization energy λvar. . λSt resolves this conundrum since the energy lost to heat becomes much smaller,43 ΔG =
(λSt/λvar.)  λSt. The efficiency of electron transport chains involving proteins is thus much higher than with typical redox pairs of synthetic chemistry. Both λSt and λvar. are statistical parameters obtained from ensemble averages. These two parameters alone are insufficient to describe electronic transitions when their rates k become comparable to the electrostatic relaxation times. This is where the dynamic part of the picture proposed here comes into consideration. Given the slow dynamics of the electrostatic field and potential found here (Figures 1 and 2), our ability to use statistical averages is compromised for most subnanosecond transitions. The system Hamiltonian alone is insufficient, and one has to specify the fraction of the phase space available to the system. This implies that ergodicity and canonical ensemble break down, and one has to replace the equilibrium reorganization energy with its nonergodic analogue71 Z ∞ var: λðkÞ ¼ λ Sϕ ðωÞðdω=πÞ ð12Þ k

Several phenomena are related to the nonergodic cutoff introduced in eq 12. First, the excess of the reorganization energy λvar. over λSt is produced by protein motions on the nanosecond time scale. Once those are cut from the frequency integral (or cancel out, as is the case for metMB), the standard near equality of two reorganization energies, λvar. = λSt, is restored.44 This result implies that in the reaction window k > τ
1 ϕ electronic 10720

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transitions become energetically inefficient again, in agreement with the Marcus picture of electron transfer.65 If the rate of electronic transitions is further increased, more nuclear modes are getting cut from the fluctuation spectrum, and both λvar. and λSt are further reduced, to the value of about 0.3 eV for a 3 ps time scale of bacterial charge separation.42,51 The living systems are therefore forced to choose either ultrafast reactions in the time window 10 ns time window. The slow relaxation of the electrostatic potential is therefore directly related to the energetic efficiency of life and to the design choices made by living systems in constructing their energy chains.43 The same phenomenology as outlined here for electronic transitions applies to the temperature dependence of the iron displacement and the phenomenon of dynamical transition, 74,75 where electrostatic fluctuations of the protein
water interface can explain the observed increase of atomic displacements. Atomic displacements of proteins and other biomolecules show a dynamical transition at TD = 200
240 K at which the dependence of the atomic msd on temperature changes its slope.74
76 The low-temperature msd is linear in temperature, well described by the VDOS of the protein.77 On the contrary, the high-temperature msd is caused by some overdamped (dissipative) process originating from the protein
water interface.78
81 When, with lowering temperature, the relaxation time of this dissipative process crosses the instrumental resolution time τobs, the nuclear mode responsible for the hightemperature msd excess undergoes a kinetic arrest.82 We suggest that this nuclear process is realized as the coupling between the interfacial region and the buried iron through the electric field of the protein
water solvent acting on the iron’s atomic charge.52 This mechanism allows us to reconstruct the observed msd by using the variance and relaxation time of the electric field fluctuations reported here. The problem of reconstructing the high-temperature excess of the msd was recently approached by Frauenfelder and co-workers who constructed the ergodicity loss function, similar in spirit to eq 12, from the dielectric response of partially hydrated myoglobin trapped in a polymer matrix.74 This successful approach required a very slow relaxation time, in the nanosecond range at room temperature, to reach the instrumental time τobs = 140 ns near TD. Although this relaxation time is indeed recorded by dielectric spectroscopy, its physical origin has remained obscure. The conceptual problem arising from that analysis is that any process related to a single-molecule water relaxation near the surface of the protein is much faster, in the range of a few picoseconds.24,38 The observation of a very slow collective relaxation of the electrostatic response reported here resolves the puzzle. A quantitative description of the temperature variation of the msd incorporates the coupling of the electric field produced by the protein
water interface to the displacements of the charged heme iron.52 The resulting msd is enhanced by the softening factor ME compared to the vibrational portion Æ(δx)2æv determined by the protein VDOS79 ÆðδxÞ2 æ ¼ ÆðδxÞ2 æv =ME

ð13Þ

Here, x is the projection of the iron displacement on the direction of radiation wavevector. Further, the factor ME is affected by the instrumental resolution and is given by the

ergodicity loss function

Z

ME ¼ 1
ðβq2Fe =3ÞÆðδxÞ2 æv

∞

ωobs

00

χE ðωÞdω=ðπωÞ

ð14Þ

00

where ωobs = τ
1 obs and χE(ω) is defined by eqs 1 and 2. Further, the factor of 1/3 is introduced to account for the use of all three Cartesian components of the field in the corresponding correlation function. The observation frequency ωobs starts to affect the frequency integral when it becomes comparable with the relaxation frequency ωE(T) = τE(T)
1. In the common case of a twoexponential decay of SE(t), the ME factor becomes ME ¼ 1
β2 ÆðδxÞ2 æv

ÆðδFÞ2 æ ½1
ð2AE =πÞtan
1 ðωobs τE Þ 6 ð15Þ

where the loss of ergodicity by the fast relaxation component has been neglected and AE refers to the amplitude fraction of the slow component in SE(t). Figure 3 shows the results of applying eqs 13 and 15 to fit the temperature-dependent msd of three proteins studied by M€ossbauer spectroscopy: metmyoglobin (metMB), oxidized form of cytochrome c (cytOx), and reduced form of cytochrome c (cytRed).83,84 The long relaxation times τE(300) and AE were taken from our simulation data for metMB and cytC at 300 K (see the SI). The temperature dependence of the relaxation time was assumed in the Arrhenius form, with the activation energy Ea equal to 4000 K for the cytochromes and 3000 K for metMB. The variance of the electrostatic force acting on the iron was considered a fitting parameter. Polycrystalline samples used in M€ossbauer experiments hardly match the protein solution simulations performed here. This comparison therefore aims at showing that experimental results can be reproduced with a choice of the electrostatic field variance in the same range as reported by the simulations. The fitted values of the force variance (155, (cytRed), 80 (cytOx), and 85 (metMB) [kcal/ (mol Å)]2) are indeed in the range of the simulated values (Table 1). The main result of this analysis is that both features of the electrostatic response found in the present simulations, a large breadth of the electrostatic noise and a slow relaxation time, are essential for reproducing the temperature dependence of the iron msd.

’ SUMMARY AND CONCLUSIONS This study suggests that three phenomena characteristic of hydrated proteins, (i) nanosecond Stokes shift dynamics, (ii) dynamical transition of protein’s atomic displacements, and (iii) the gigantic reorganization energy of electronic transitions, are different signatures of the same physical mechanism—movements of the polarized hydration shell by protein’s conformational fluctuations. Dynamics of confined and interfacial water is a complex phenomenon, clearly involving dynamical slowing down. However, studies of model rigid interfaces5 show that this effect does not reach the time scale of nanoseconds, and protein motions are required.4 The present simulations clearly show that nearly 90% of the electric field response of a buried atom occurs as a single-exponential relaxation with the relaxation time of 3
6 ns at 300 K. The relaxation times of protein and water match closely each other (Figure 1) pointing to linked nuclear fluctuations of the two components of the protein
water interface. 10721

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The Journal of Physical Chemistry B

Figure 5. Msd of three proteins measured experimentally by M€ossbauer spectroscopy (points)83,84 and obtained by fitting the experimental data to eqs 13 and 15 (dashed lines). The long relaxation times at 300 K were taken from MD simulations (τE = 6 ns and AE = 0.93 for metMB; τE = 3 ns and AE = 0.89 for cytC), while their temperature dependence was calculated from an Arrhenius law with the activation energy of 4000 K for two cytochromes and 3000 K for metMB. The variance of the electrostatic force Æ(δF)2æ acting on iron has been adjusted in each case, and the results are: 155 (cytRed), 80 (cytOx), and 85 (metMB) [kcal/(mol Å)]2. The vibrational msd Æ(δx)2æv was linearly extrapolated from its low-temperature part with the slopes: 0.55 (cytOx), 0.31 (cytRed), and 0.61 (metMB) 10
4 Å2 K
1. The linear form for Æ(δx)2æv vs T and the slope agree well with independent calculations using the VDOS of the protein.77

The idea that the electrostatic noise is produced by relatively slow motions of the ionized/polar surface residues of the protein, pushing the polarized water domains attached to them, allows one to explain a number of observations in this paper and in the literature. These include the dependence of the dynamics on the distance between the optical probe and the interface16 and the compensating effect between the protein and water components in the overall electrostatic potential38 (Figure 3). The proposed mechanism decouples the average of the electrostatic potential from its variance, allowing gigantic values of the redox reorganization energy.44 The overall reorganization energy is strongly affected by negative cross-correlations between the protein and water components of the electrostatic potential. The extent of this compensation varies strongly among the proteins. It is therefore conceivable that this compensation provides a mechanism linking the protein function to the protein sequence. The energetic efficiency of electron transport in biology’s energy chains requires this compensation not to occur and thus to retain the gigantic value for the overall reorganization energy. The nanosecond dynamics of the electrostatic potential determines the design choices made by living systems demanding either very fast electron transfer in the picosecond time window, as is the case for natural photosynthesis, or much slower reactions, in tens of nanoseconds, when gigantic reorganization allows lowering of the amount of free energy dissipated into heat in an electron transfer event. An intense electrostatic noise coupled to the atomic partial charge explains large atomic displacements of proteins at high temperatures (Figure 5). Slow dynamics, with the relaxation time reaching nanoseconds at 300 K, is required to explain the temperature dependence of the atomic msd. The relaxation time grows with cooling and reaches the instrumental observation window at the temperature of dynamical transition at which electrostatic fluctuations freeze in74 and the slope of the msd vs temperature turns into its low-temperature branch.77

ARTICLE

Energy chains of biology use the same mechanism of freezing polar solvation, although on much shorter times of a few picoseconds in the case of photosynthetic charge separation. This dynamical control of the reaction energetics avoids solvation penalties for moving charge over large distances in a sequence of nearly activationless transitions.43 All these new phenomena become possible because of two signatures of proteins making them different from small molecules: (i) a significant spectral density of protein vibrations in the range of low frequencies67,69 and (ii) surface charges moved by these vibrations. The combination of the two is responsible for both the long relaxation times and the large amplitudes of the electrostatic fluctuations. The absence of the dynamical transition in dehydrated protein samples is often interpreted as evidence that either hydration shells or bulk water cause large high-temperature msd.74,85 The electrostatic mechanism advocated here suggests that the effect of hydration might be not in water itself but in the alteration it makes to the protein. Indeed, comparable contributions of water and protein to the electrostatic variances point to a major difference between dehydrated and hydrated samples in the presence of surface residues ionized by water. The mechanism of msd enhancement in partially hydrated protein powders, typically studied by scattering techniques, then consists of concerted motions of clusters of surface waters with the protein charges polarizing them. This mechanism naturally explains the near equality of the dynamical transition temperatures TD for the water and protein components.86,87

’ ASSOCIATED CONTENT

bS

Supporting Information. Description of the simulation protocol and multiexponential fits of time correlation functions. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This research was supported by the National Science Foundation (CHE-0910905). CPU time was provided by the National Science Foundation through TeraGrid resources (TG-MCB080116N). ’ REFERENCES (1) Jimenez, R.; Fleming, G. R.; Kumar, P. V.; Maroncelli, M. Nature 1994, 369, 471–473. (2) Makarov, V. A.; Andrews, B. K.; Smith, P. E.; Pettitt, B. M. Biophys. J. 2000, 79, 2966–2974. (3) Golosov, A. A.; Karplus, M. J. Phys. Chem. B 2007, 111, 1482–1490. (4) Li, T.; Hassanali, A. A.; Singer, S. J. J. Phys. Chem. B 2008, 112, 16121–16134. (5) Castrill on, S. R.-V.; Giovambattista, N.; Aksay, I. A.; Debenedetti, P. G. J. Phys. Chem. B 2009, 113, 7973–7976. (6) Laage, D.; Stirnemann, G.; Hynes, J. T. J. Phys. Chem. B 2009, 113, 2428–2435. (7) Bashkin, J. S.; McLendon, G; Mukamel, S.; Marohn, J. J. Phys. Chem. 1990, 94, 4757–4761. (8) Pierce, D. W.; Boxer, S. G. J. Phys. Chem. 1992, 96, 5560–5566. 10722

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