Ergodicity breaking of iron displacement in heme proteins
aa r X i v : . [ phy s i c s . b i o - ph ] A ug Ergodicity breaking of iron displacement in hemeproteins † Salman Seyedi a and Dmitry V. Matyushov ∗ b We present a model of the dynamical transition of atomic displacements in proteins. Increasedmean-square displacement at higher temperatures is caused by softening of the vibrational forceconstant by electrostatic and van der Waals forces from the protein-water thermal bath. Vibra-tional softening passes through a nonergodic dynamical transition when the relaxation time of theforce-force correlation function enters, with increasing temperature, the instrumental observationwindow. Two crossover temperatures are identified. The lower crossover, presently connected tothe glass transition, is related to the dynamical unfreezing of rotations of water molecules withinnanodomains polarized by charged surface residues of the protein. The higher crossover tem-perature, usually assigned to the dynamical transition, marks the onset of water translations. Allcrossovers are ergodicity breaking transitions depending on the corresponding observation win-dows. Allowing stretched exponential relaxation of the protein-water thermal bath significantlyimproves the theory-experiment agreement when applied to solid protein samples studied byMössbauer spectroscopy.
Atomic displacements in proteins are viewed as a gauge of theoverall flexibility of macromolecules. Displacements of the hy-drogen atoms are reported by neutron scattering, and mean-square displacements (B-factors) of all atoms are known fromX-ray crystallography. Neutron scattering reports ensemble av-erages of scattering from many hydrogen atoms of a single pro-tein. In contrast, Mössbauer spectroscopy often probes the dis-placements of a single atom in the protein, which is the hemeiron in this study focused on cytochrome c (Cyt-c) and myoglobinproteins.The temperature dependence of atomic displacements fromboth neutron scattering and Mössbauer spectroscopy shows anumber of crossovers. They are marked by changes in the slopeof atomic mean-square displacement vs temperature, deviatingfrom expectations from the fluctuation-dissipation theorem. This problem has attracted significant attention in the litera- a Department of Physics, Arizona State University, PO Box 871504, Tempe, Arizona85287. b Department of Physics and School of Molecular Sciences, Arizona State University, POBox 871504, Tempe, Arizona 85287; E-mail: [email protected] † Electronic Supplementary Information (ESI) available: [details of any supplemen-tary information available should be included here]. See DOI: 10.1039/b000000x/ ture.
The accumulation of the data over several decadesof studies, combined with their recent refinements through thecomparison of the results obtained on spectrometers with differ-ent resolution, have lead to a convergent phenomenologicalpicture.Two low-temperature crossovers are now identified (Fig. 1).The higher-temperature crossover T d , originally assigned to theprotein dynamical transition, depends on the observationwindow of the spectrometer and shifts to lower tempera-tures when the resolution is increased (a longer observation time τ r in Fig. 1). The lower crossover temperature, T g ≃ − K,is independent of the observation window (in the range of reso-lution windows available to spectroscopy) and is assigned to theglass transition of the protein hydration shell.
All motions, rotations and translations, in the hydration shell(except for cage rattling) terminate at the lower temperature T g .While this interpretation is consistent with the basic phenomenol-ogy of glass science, it does not address the question of how thestructure and dynamics of the hydration shell affect atoms insidethe protein, the heme iron for Mössbauer spectroscopy. The basicquestion here is whether the observations can be fully related tostiffening of the hydration shell at lower temperatures, thus re-ducing elastic deformations of the protein, or there are somelong-range forces acting on the heme, which are reduced in their uctuations when the hydration shell dynamically freezes. It ispossible that no simple answer to this question can be obtainedin the case of neutron scattering since there are several classesof motions of protein hydrogens: cage rattling, methyl rotations,and jumps between cages. To avoid these complications, we fo-cus here on a single heavy atom, heme iron, probed by Mössbauerspectroscopy on the resolution time τ r = ns.The question addressed here is what are the physical mecha-nisms propagating fluctuations of the protein-water interface toan internal atom within the protein. This question, also rele-vant to how enzymes work, was addressed by the electro-elasticmodel of the protein, where both the effect of the viscoelas-tic deformation and the effect of the long-range forces acting onthe heme iron were considered. The main conclusion of that the-oretical work was the recognition of the two-step nature of thecrossover in the mean-square fluctuation (MSF) of the heme iron.The low-temperature crossover, T g ≃ − K, was assignedto an enhancement of viscoelastic deformations above the glasstransition of the protein-water interface. The increment in theMSF at T g was, however, insignificant, as confirmed below basedon new molecular dynamics (MD) simulations. It was, therefore,concluded that altering elastic stiffening is not sufficient to de-scribe the rise of the MSF above T d and long-range forces need tobe involved.The iron MSF significantly increases when electrostatic forcesacting on the iron are included. The dynamical transition andthe corresponding enhancement of the MSF are promoted by er-godicity breaking when the longest relaxation time crosses theinstrumental time. The equation for the MSF resulting from thisperspective involves the MSF from local vibrations of the heme h δ x i vib and the global softening of the heme motions throughthe long-ranged forces acting on it. This second component en-ters the denominator of Eq. (1) through the variance of the forceacting on heme’s iron h δ F i r h δ x i r = h δ x i vib − β h δ F i r h δ x i vib (1)The subscript “r” in the angular brackets, h ... i r , indicatesthat the average is constrained by the observation window τ r .Correspondingly, the fluctuations of the long-range forces aremostly frozen at low temperatures when h δ F i r is low, yield-ing h δ x i ≃ h δ x i vib . Since the relaxation time of the long-rangeforces τ ( T ) depends on temperature according to the Arrheniuslaw, it shortens with increasing temperature, ultimately reach-ing the point τ r ≃ τ ( T d ) , at which the high-temperaturecrossover occurs. Fluctuations of the long-range forces becomedynamically unfrozen at this temperature, leading to an increaseof both h δ F i r and h δ x i r .In the present paper, we present new extensive simulations ofCyt-c in solution at different temperatures. The goal is to assert ! δ x " TT g T (1) d T (2) d τ (1) r τ (2) r τ (1) r > τ (2) r Fig. 1
Schematic representation of two crossovers in the temperaturedependence of the mean-square fluctuation (MSF) h δ x i . The lowercrossover, T g , is independent of the instrumental resolution window andcorresponds to the glass transition of the protein-water interface. Theupper crossover (dynamical transition), T d , does depend on theobservation window and is related to the entrance of the relaxation timeof the force acting on the coarse-grained unit (residue, cofactor, etc) intothe resolution window of the experiment. The temperature T d shifts tothe lower value when the observation time is increased. the role of long-range forces in achieving the vibrational soften-ing of atomic displacements at high temperatures (Eq. (1)). Weconsider the entire heme as a separate unit experiencing the forcefrom the surrounding thermal bath. This coarse graining allowsus to focus on the long-time relaxation of the force-force correla-tion function relevant for the long observation time, τ r = ns,of the Mössbauer experiment. We find that the longest relaxationtime τ ( T ) follows the Arrhenius law with the activation barriercharacteristic of a secondary relaxation process ( β -relaxation ofglass science ). We therefore support the proposal advancedby Frauenfelder and co-workers that the higher-temperaturecrossover is caused by ergodicity breaking when the relaxationtime of the secondary process characterizing the protein-waterinterface enters the experimental observation window. This re-laxation process effects the heme iron through the combinationof non-polar (van der Waals) and polar (electrostatic) forces.Our focus on the protein in solution has a limited applicabilityto experiments done with solid samples. Nevertheless, computersimulations produce results close to observations for the reducedstate of Cyt-c. The length of simulations is also insufficient to sam-ple the dynamics on the time-scale of τ r = ns. In addition,the solution setup does not reproduce highly stretched dynam-ics observed in protein powders. We find that the agree-ment between theory and experiment is much improved whenstretched exponential dynamics from dielectric spectroscopy are used in our model.Despite limitations of our simulations in application to exper-imental data, there is one significant advantage of the solutionsetup. Experiments done with solid samples cannot claim thatthe observed phenomenology directly applies to solutions. Thesimilarity between our simulations and such experiments gives redit to the idea that dynamical transition, caused by ergodicitybreaking, is a general phenomenon relevant to physiological con-ditions. From a more fundamental perspective, ergodicity break-ing is broadly applicable to enzymetic activity at physiologicalconditions and is described by a formalism carrying significantsimilarities with the problem of dynamical transition of atomicdisplacements. The standard definition adopted for the fraction of recoiless ab-sorption of the γ -photon in Mössbauer spectroscopy is throughthe average f ( k ) = (cid:12)(cid:12)(cid:12)D e ikx E r (cid:12)(cid:12)(cid:12) . (2)The average h ... i r is over the statistical configurations of the sys-tem accessible on a given time resolution of the experiment speci-fied through the observation (resolution) time τ r . Further, k is thewavevector aligned with the x -axis of the laboratory frame and x is the displacement of the heme iron.The average over the stochastic variable of iron displacement x can be represented by an ensemble average with the free energy F r ( x ) D e ikx E r = Z dxe ikx − β F r ( x ) , (3)where β = / ( k B T ) is the inverse temperature.The free energy F r ( x ) is distinct from the usual thermodynamicfree energy in two regards. First, it is a partial free energy cor-responding to the reversible work performed by all degrees free-dom of the system at a fixed displacement x . Therefore, F r ( x ) is analogous to the Landau functional of the thermodynamic or-der parameter. There is another distinction of F r ( x ) from thethermodynamic free energy specified by the subscript “r”. Thisfree energy is defined by sampling the constrained part of thephase space Γ r which can be accessed on the resolution time τ r .The definition of F r ( x ) should thus include two constraints: (i)a fixed value x and (ii) a restricted phase space available to thesystem. Both constraints are mathematically realized by the fol-lowing equation e − β F r ( x ) = Z Γ r d Γδ ( x − ˆx · q ) e − β H . (4)Here, ˆx is the unit vector along the x -axis and q is the iron’s dis-placement vector. The restriction of the phase space is realizedas a dynamical constraint on the frequencies over which the cor-relation functions appearing in the response functions are inte-grated. A simple cutoff, ω > ω r = τ − r , is used in the statisticalaverages below.We will next consider the displacement of the iron as composedof the displacement of the heme’s center of mass and the normal-mode vibrations relative to the center of mass. The Hamiltonianin Eq. (4) can therefore be separated into a linear term involving the external force F acting on the heme from the protein-waterthermal bath and the Hamiltonian H vib of intra-heme vibrations H ( q ) = H ( ) − q · F + H vib . (5)By expanding the iron’s displacement q in the normal-mode vi-brations Q α , we can re-write the free energy F r ( x ) in the form e − β F r ( x )+ β H ( ) = Z d q δ ( x − ˆx · q ) h e β q · F i B Z ∏ α d Q α δ (cid:18) q − ∑ α ˆe α Q α √ m (cid:19) e − β H vib , (6)where m is the mass of the iron atom. Further, the average h ... i B is over the fluctuations of the classical protein-water thermal bathwhich creates movements of the heme as a whole. It is reason-able to anticipate that these relatively large-scale fluctuations fol-low the Gaussian statistics with the force variance σ F = h ( δ F ) i , δ F = F − h F i . The average over such fluctuations in Eq. (6) thenbecomes h e β q · F i B = e ( β q σ F ) / . (7)In addition, the integral over the normal modes in Eq. (6) is aGaussian integral such that Z ∏ α d Q α δ (cid:18) q − √ m ∑ α ˆe α Q α (cid:19) e − β H vib = e − q / ( σ vib ) , (8)where the variance due to intramolecular vibrations is σ vib = ¯ h m ∑ α ˆ e α n α + ω α . (9)Here, ¯ n α is the average occupation number of the normal mode α with the frequency ω α . By substituting Eqs. (7) and (8) into Eq.(6), one obtains the harmonic free energy function β F r ( x ) = H ( ) + x σ (10)with the variance σ = σ vib − ( βσ F σ vib ) . (11)The basic result of this derivation is straightforward: addingGaussian fluctuations of the heme’s center of mass to intramolec-ular vibrations of the heme leads to the softening of the forceconstant of the harmonic free energy F ( x ) . Combining this re-sult with Eqs. (2) and (3), one obtains the Gaussian form for therecoiless fraction f ( k ) = e − k h δ x i r (12)with h δ x i r given by Eq. (1) in which h δ x i vib = σ vib .The subscript “r” in h δ F i r specifies that the average over thestochastic fluctuations of the force F acting on the heme from !" & ’( )$ * + , - . /"$0" )$ ,23*4 ) . Fig. 2
Long relaxation time of the force-force autocorrelation function ofthe total force acting on the heme vs / T . The results of MD simulationsfor the reduced (Red, filled circles) and oxidized (Ox, open squares) arefitted to Arrhenius linear functions with the slopes E Red / k B = E Ox / k B = K. the thermal bath is understood in the spirit of the dynamicallyrestricted average over a dynamically accessible subspace of thesystem Γ r , as specified in Eq. (4). In practical terms, this impliesthat only frequencies greater than ω r = τ − r can contribute to theobservables. The effective variance can therefore be calculatedas h δ F i r = Z ∞ω r ( d ω / π ) C F ( ω ) . (13)Here, C F ( ω ) is the Fourier transform of the time auto-correlationfunction C F ( t ) = h δ F ( t ) · δ F ( ) i , (14)where δ F ( t ) = F ( t ) − h F i . The force acting on the entire heme, F H , was calculated from MDsimulations. This procedure averages out the short-time fluctua-tion of the forces caused by internal vibrations and allows us tofocus on the long-time dynamics, produced by the bath, and itspotential effect on the observable displacement of the iron. Wefound that the force-force time correlation function calculated forthe iron atom is dominated by intramolecular vibration and is os-cillatory (see ESI † ). The long-time dynamics is hard to extractfrom that correlation function, which is the reason for our focuson the overall force acting on the heme. However, this overallforce needs rescaling when applied to the individual iron atom.Assuming that the heme moves as a rigid body, the re-scaling isgiven by the ratio of the iron mass m = g/mol and the mass ofthe heme M = g/mol F = mM F H . (15)This re-scaling, assuming the heme moving as a rigid body,can obviously apply only to the slowest dynamical components of the force. In contrast, the correlation function C F ( t ) calcu-lated from simulations shows a number of time-scales, from sub-picoseconds, to long-time dynamics on the time-scale of 6–25 ns( T ≃ K). While the slowest relaxation process usually consti-tutes about half of the amplitude of the time correlation function,the scaling in Eq. (15) does not discriminate between the slowand fast dynamics. It is therefore clear that our estimate of theoverall amplitude of the force acting on heme’s iron is good onlyup to some effective coefficient accounting for imperfect rigidityof the heme. Elastic deformations of the heme shifting its cen-ter of mass are effectively disregarded in the re-scaling assumingthe rigid-body motions. Given these uncertainties, we estimate h δ F i r in Eq. (1) from the following equation h δ F i r = f ne ( T )( m / M ) h δ F H i . (16)The nonergodicity parameter f ne ( T ) here comes from the dy-namic restriction imposed on the integral over the frequencies inEq. (13). Assuming that only the slowest component in the relax-ation of the force can potentially enter the observation window, τ r = ns, we can write f ne ( T ) in the form corresponding toexponential relaxation of C F ( t ) in Eq. (14) (see below the discus-sion of non-exponential, stretched dynamics) f ne ( T ) = ( / π ) cot − [ τ ( T ) / τ r ] . (17)In this equation, τ ( T ) is the relaxation time of the slowest com-ponent of C F ( t ) . A similar expression accounting for the finiteresolution of the spectrometer was used in the past for the inte-grated elastic intensity. It is clear from Eq. (17) that the nonergodicity parameter isequal to unity when τ ( T ) ≪ τ r and the fluctuations of the forceare ergodic. In the opposite limit of slow fluctuations, τ ( T ) ≫ τ r ,the force fluctuations are dynamically frozen on the observationtime and do not contribute to the softening of iron’s displacement, f ne → . This corresponds to low temperatures when intra-hemevibrations dominate. The crossover temperature T d is reached at τ r ≃ τ ( T d ) .The long-time relaxation times τ ( T ) are shown in Fig. 2. Theactivation barrier of this relaxation time, E a / k B ≃ K, is belowthe typical values for the α -relaxation of condensed materials,thus pointing to a localized (secondary) relaxation process of theprotein-water interface. This relaxation time is determinedin the range of temperatures ≤ T ≤ K, where our sim-ulations demonstrate sufficient convergence. The Arrhenius fitsof the simulation data (lines in Fig. 2) are then extrapolated tolower temperature where the experimental Mössbauer data areavailable. These extrapolated relaxation times are used in Eq.(17) to calculate the nonergodicity factor in Eq. (16).Calculations of displacements of the heme iron based on Eqs.(1), (16), and (17) are shown in two panels of Fig. 3. The exper- ! " ! % & ’ " ( ) * + " , "$ " -./ ! 0 ! " ! % & ’ " ( ) * + " , "$ " Fig. 3 h δ x i for reduced (Red, upper panel) and oxidized (Ox, lowerpanel) states of Cyt-c. The points are experimental data and the solidlines are calculations according to Eqs. (1) , (16) , and (17) . The dashedlines are low-temperature interpolations of the experimental data. Thedashed-dotted line in the lower panel is based on multiplying therelaxation time τ ( T ) for the Ox state with the constant coefficient equalto 2.65. imental results are reasonably reproduced by our calculationsin the Red state of the protein without any additional fitting. Theshift of the crossover temperature to a higher value in the Oxstate observed experimentally would imply, in our model, slowerdynamics of the force or a larger value of h δ F H i . While a largervalue of h δ F H i is indeed observed (Table 1), its overall result isinsufficient to explain the shift of the experimental crossover tem-perature. The experimental results are recovered by multiplying τ ( T ) from simulations by a factor of 2.65. While this factor is ob-viously arbitrary, the need for a correction might be related to ourinsufficient sampling of the long-time dynamics, extrapolation ofthe high-temperature relaxation times to lower temperatures, andthe assumption of exponential dynamics not supported by mea-surements with protein powders (see below). Table 1
Separation of h δ F H i (nN ) into the electrostatic (El.) andnon-polar (vdW) components and the splitting into the protein (Prot.)and water contributions ( T = K).
Redox State El. vdW Prot. Water TotalRed 8.86 23.17 9.60 3.12 14.52Ox 16.62 12.65 19.34 2.87 17.05
Despite some difficulties with the long-time dynamics in thecyt-Ox state, the short-time dynamics produced by simulationsare consistent with experiment. This is confirmed by the calcula-tion of the vibrational density of states D ( ω ) = N ∑ α = ˆ ( e α · ˆx ) δ ( ω − ω α ) , (18)where ˆe α are expansion coefficients for the linear transformationfrom the Cartesian displacement of the Fe atom to normal coor-dinates Q α in Eq. (6). The normalization of the density of statesadopted in producing the experimental data shown in Fig. 4 re-quires Z ∞ D ( ω ) d ω = (19)With this normalization, the density of states from simulationswas computed from the velocity-velocity autocorrelation function(see ESI † for more detail) and displayed in Fig. 4. There is an ex-cess of low-frequency modes relative to experiment, whichmight be related to the expansion of the protein at T = K, atwhich simulations were performed, compared to the experimen-tal temperature of T = K.Table 1 shows the splitting of the variance of the force actingon the heme into electrostatic and van der Waals (vdW) compo-nents and, additionally, into the components from the water andprotein parts of the thermal bath. Note that the components donot add to the total force variance because of cross-correlations.The splitting into components indicates that vdW interactions andelectrostatics contribute comparable magnitudes to the force vari-ance. The softening of iron vibrations cannot therefore be fully ! " ! % & ’ ( ) * + , - . / " - )234!)(5,! Fig. 4
Experimental (Exp., T = K ) and simulation (Sim., T = K) vibrational density of states for Cyt-Ox ( ¯ ν = ω / ( π c ) , c is the speed oflight). Simulations were done for 1 ns in the NVE ensemble withnon-rigid protons and 0.25 fs integration step (configurations savedevery 1 fs). attributed to electrostatics (dielectric effect ). It cannot beattributed to the hydration shell either and is in fact a com-bined effect of protein and water, with the dominant contributionfrom the protein. The water contribution can be further dimin-ished in solid samples used in neutron scattering or Mössbauerspectroscopy.The separation of the force variance between protein andwater allows us to comment on the idea of “slaving” of theprotein dynamics by water suggested by Frauenfelder and co-workers. The “slaving” phenomenology implies the equal-ity of the enthalpy of activation for a relaxation process in theprotein with the enthalpy of activation for the structural relax-ation of bulk water ( α -relaxation). When plotted in the Arrheniuscoordinates ( − ln [ τ ] vs / T ) the two plots are then parallel.The origin of this phenomenology is easy to appreciate withinthe framework of Kramers’ activated kinetics dominated by fric-tion with the thermal bath (Fig. 5). The rate constant of anactivated process ∝ ω R exp [ − β∆ F † ] is the product of an effec-tive frequency in the reactant well ω R with the Boltzmann fac-tor exp [ − β∆ F † ] involving the free energy of activation ∆ F † . Ifthe motions along the reaction coordinate are represented by anoverdamped harmonic oscillator with the frequency ω and thefriction coefficient ζ , the direct solution of the Langevin equationleads to the relaxation frequency ω R = ω / ζ . Therefore, “slav-ing” appears when most of energy dissipation occurs to the wa-ter part of the thermal bath (which has a higher heat capacitythan the protein ). In that case, the temperature dependenceof ζ ( T ) , and of the corresponding relaxation process in water,would determine the temperature dependence of the relaxationrate in the protein, which is only shifted to lower rates due to anadditional activation barrier ∆ F † (assuming ∆ F † is temperature-independent). This is the “slaving” scenario. Fig. 5
Activated Kinetics in the Kramers’ friction dominated limit. Thecharacteristic frequency of vibrations in the well is given by ω R = ω / ζ for an overdamped harmonic oscillator with the eigenfrequency ω andfriction with the medium ζ ; ∆ F † is the free energy of activation along thereaction coordinate q . !" ! & ’ ( ) * + + + + % + , + $ )-(./* -%0 -1-,, -1 .2 Fig. 6
Normalized force-force correlation function S F ( t ) = C F ( t ) / C F ( ) for the protein (p) and water (w) components at the temperaturesindicated in the plot. While there are reported instances when this picture is cor-rect, one can argue that energy dissipation for localized pro-cesses occurs to the protein hydration shell, which possesses itsown relaxation spectrum. Indeed, Frauenfelder and co-workers argued that localized processes in the protein have to be “slaved”to the relaxation of the hydration layer. Consistently with thatnotion, relaxation processes related to protein function are oftencharacterized by the activation barrier much lower than those for α -relaxation of bulk water (Fig. 2). For instance, the Stokes shiftdynamics directly related to the redox activity of Cyt-c show theactivation barrier of its relaxation time E a / k B ≃ K. This ismuch lower than ∼ K (increasing to ∼ K upon cooling)from diffusivity and viscosity of water ( α -relaxation). The ideaof “slaving” to β -relaxation of the hydration shell is less useful,and is harder to prove, since relaxation of the shell is mostly in-accessible experimentally. Our simulation results allow us sucha test since the dynamics of both the hydration layer and of theheme’s iron are available.In application to Mössbauer experiment, our data do not sup- !" & ’( )$ * + , - . /"!/"0/"1/"2/"$2"!2"0 )$ / ,34*5 ) . Fig. 7
Long relaxation time of the force-force autocorrelation function ofthe force acting on the heme vs / T (black circles) for reduced Cyt-c.Also shown are the relaxation times for the force on the heme producedby the protein (squares) and by water (triangles). Fits to Arrhenius linearfunctions are shown by the dashed lines. port “slaving”. Only ∼ % of the force variance acting on heme’siron comes from from hydration water (Table 1). This also im-plies that the dynamics should be biomolecule-specific. In thisscenario, “slaving” would be only possible if the protein dynamicsfollowed the dynamics of water. The results of simulations do notsupport this conjecture: the dynamics of S F ( t ) = C F ( t ) / C F ( ) aredistinctly different for the protein and its hydration water (Fig. 6).The dynamics of water is on average significantly faster (a largerdrop from the initial value S F ( ) = , not resolved in Fig. 6), andthe slow dynamics of the protein and water are not consistent ei-ther. Nevertheless, the temperature dependence of the relaxationtime of the force-force correlation function is consistent betweenthe protein and water components (Fig. 7). The enthalpies of ac-tivation for the protein and water relaxation are, therefore, closein magnitude, in a general accord with the “slaving” phenomenol-ogy. The origin of this effect can be traced to coupled fluctuationsof the protein and hydration water, without invoking a dom-inant role of water in the dynamics.Water is a faster subsystem producing a shorter relaxation timeof C F ( t ) . One therefore anticipates that the temperatures of er-godicity breaking should separate for the water and protein com-ponents of the thermal bath. This indeed happens, as is illus-trated in Fig. 8 for the reduced state of Cyt-c. The rise of h δ x i due to water occurs at ≃ K, while the transition temperaturefor the protein is ≃ K. The water’s onset is hard to disentan-gle because the force produced by water on the heme is relativelylow. One might expect that the water transition is better resolvedin neutron scattering experiments since a large number of pro-tons located close to the interface potentially contribute to thesignal. Overall, this calculation clearly points to a nonergodicorigin of the dynamical transition, as we stress again below whenconsidering the separation of rotational and translational motions ! " ! ! $ ! % & ’ " ( ) * + " , "$ " -)*., )/’0)-1234)5612789):3276 Fig. 8 h δ x i for the reduced state of Cyt-c. The points are experimentaldata and the solid lines are calculations according to Eqs. (1) , (16) ,and (17) . The calculations are done for the total force-force correlationfunction (black) and for its components from the protein (orange) andwater (blue). The dashed lines refer to the low-temperature linear fit ofthe experimental data and to the high-temperature linear fit of the irondisplacement produced by the protein. of water in the hydration shell. Difficulties with reproducing ergodicity breaking of Cyt-Ox (Fig.3 lower panel) might be related to a limited applicability of theresults obtained for solutions to dynamics in protein powders andcrystals studied experimentally. In addition to the obvious uncer-tainty of extrapolating the high-temperature simulation resultsto lower temperatures, the dynamics of hydration water can bequalitatively different in those environments compared to solu-tions. The relaxation of hydration water in powders was associ-ated by Ngai and co-workers with the general phenomenol-ogy of confined water in water-containing glass-formers. The ν -process characterizing such dynamics is highly stretched, with avery slow decay of the high frequency tail of the loss function: ε ′′ ∝ ω − γ for the dielectric loss and χ ′′ ( ω ) ∝ ω − γ for the neu-tron scattering loss. A low value of stretching exponent, γ ≃ . ,is observed in both cases.The ν -process observed in lysozyme and myoglobin powdersby dielectric spectroscopy was identified to cause the dynamicaltransition in neutron scattering. We can therefore use the corre-sponding relaxation time τ ( T ) reported from dielectric measure-ments to explain Mössbauer data for met-myoglobin (oxidizedform of myoglobin). Before we do that, we have to extend thenonergodicity parameter obtained in Eq. (17) for exponential re-laxation to stretched exponential relaxation. Cole-Cole functionwas used to fit the dielectric data. We therefore can re-write thenonergodicity parameter f ne ( T ) as follows f ne ( T ) = π Z ∞τ ( T ) / τ r d ωω Im h ( + ( i ω ) γ ) − i , (20) ! " ! $ % & ’ ( ) * + ’ , - ’. ’ /. / . )2&3!)24!)*’/, 567897:;< Fig. 9
MSF of heme iron in oxidized myoglobin. Points indicateexperimental results, solid line refers to the fit to Eq. (1) and (16) withthe nonergodicity factor f ne ( T ) determined from stretched dynamicsaccording to Eq. (20) . The nonergodic force variance is determinedaccording to Eq. (21) with the fitting constant A = . nN/Å (correspondsto h δ F i = . nN at T = K). where γ is the stretching exponent of the Cole-Cole function. At γ = , Eq. (20) transforms to Eq. (17). This nonergodicity factorcan be used in the following form for the force variance β h δ F i r = A f ne ( T ) (21)where, according to the standard prescription of the fluctuation-dissipation theorem, the amplitude A is held constant. The useof this form along with γ = . and the experimental τ ( T ) (seeFig. S4 in ESI † ) in Eq. (20) produce the MSF of myoglobin shownby the solid line in Fig. 9. The fit requires h δ F i ≃ . nN at T = K, which is roughly consistent with h δ F i ≃ . nN forCyt-Ox in Table 1 when Eq. (15) is applied. The quality of the fitis significantly reduced with γ = , which testifies to the need ofapplying stretched relaxation to describe ergodicity breaking inprotein powders. The lower crossover temperature T g of the protein MSF repre-sents the glass transition of the hydration shell. It was previouslyidentified with the onset of translational diffusion of the watermolecules in the shell. However, glass science requires one topay attention not only to translations, but also to molecular rota-tions. There are a number of reasons for that. First, the configura-tional entropy of fragile glass-formers is mostly rotational (e.g.,the heat capacities of supercooled ethanol and its plastic crystalare nearly identical ). Reducing the configurational entropy isrequired for reaching the glass transition and, therefore, the ro-tational configuration space has to be strongly constrained closeto T g . Second, the temperature dependence of the dielectric re-laxation time can be superimposed with the relaxation time fromviscosity and with the diffusion coefficient. Therefore, both ro- !"! ! $ %! !!"%!"!! &’()* ’+,’(-’.’/’0*’+,’(-’.’ Fig. 10
The dipolar susceptibility of the hydration shell water calculatedfrom MD simulations according to Eq. (22) for shells of thickness a around Cyt-Ox (open points) and Cyt-Red (filled points) at differenttemperatures (some Red and Ox points coincide on the scale of theplot). The dotted lines connect the points to guide the eye. tations and translations are expected to dynamically freeze near T g .The density of water in the hydration shell is enhanced com-pared to the bulk, and shell water, being heterogeneous andmore disordered than the bulk, is close in physical propertiesto a mixture of low-density and high-density amorphous ice. Nevertheless, the positional structure of the shell (pair distribu-tion function) does not change with cooling, and there is no struc-tural transition associated with crossing the temperature T d . Compared to the positional structure and diffusional dynamics, there is much less experimental and computational evidence onorientational correlations and rotational dynamics of water inthe hydration shell. The single-particle rotational dynamics areslowed down by a factor of 2–4, as is seen by NMR and com-puter simulations. Collective relaxation probed by Stokes shiftof optical dyes are much slower, in the range of sub- to nanonosec-onds, pointing to a significantly slower collective response ofwater dipoles compared to single-molecule rotations.The fact that the collective response of the shell dipole is quitedifferent from single-particle MSF is illustrated in Fig. 10, whichshows the dipole moment variance for hydration shells of Cyt-Ox and Cyt-Red with varying temperature and thickness of theshell. More specifically, we present the dimensionless varianceof the shell dipole moment defined analogously to the dielectricsusceptibility of bulk dielectrics χ ( a ) = [ k B T v w N w ( a )] − h δ M ( a ) i . (22)Here, v w is the volume of a single water molecule (effective di-ameter . Å ) and N w ( a ) is the number of water moleculesin the shell of thickness a measured from the van der Waals sur-face of the protein; M ( a ) is the total dipole moment of the watermolecules in the shell, δ M ( a ) = M ( a ) − h M ( a ) i . he main qualitative difference between the temperature de-pendence of the MSF and the shell dipole is that the latterclearly violates the fluctuation dissipation theorem, which pre-dicts h δ M ( a ) i ∝ T . The phenomenology of susceptibility decay-ing with temperature, in violation of the fluctuation-dissipationtheorem, is shared by most polar liquids. However, in con-trast to homogeneous liquids, the protein hydration shells areheterogeneous and highly frustrated. This is because polarizedinterfacial water has to follow a nearly uniform mosaic of posi-tively and negatively charged surface residues. Surface chargesorient water dipoles into polarized domains. These domains aremutually frustrated by altering sign of the charged residue, butstay in the fluid state with the fluctuations of the shell dipolesignificantly slowed down (hundreds of picoseconds to nanosec-onds ) compared to the bulk. This new physics, quite distinctfrom bulk polar liquids, connects hydration shells to relaxor fer-roelectrics, where mutual frustration of dipolar crystalline cellsbreaks the material into ferroelectric nanodomains at the glasstransition reached above the Curie point. The phenomenology of relaxor ferroelectrics suggests that thedipolar response of the shell is determined by reorienting thepolarized domains, instead of predominantly single-particle ro-tations found in bulk polar liquids. This interpretation is sup-ported by nanosecond time-scales characterizing the dynamic sus-ceptibility of the shell χ ( ω , a ) (see the ESI † ). This picture doesnot contradict to the dynamic (fluid) nature of the hydration shellin which water can diffuse along the surface visiting a residue per ≃ ps. Moving from a positive to a negative residue can beaccompanied with a dipole flip, still preserving the domain struc-ture, which requires much longer times to be altered. The dipoleflip of a water molecule moving to a neighboring residue will alsoproduce a short relaxation time for single-particle rotations.
A sharp drop of χ ( a ) at about ≃ K signals reaching the glasstransition on the time scale of MD simulations (Fig. 10). This T g issomewhat lower than experimental T g ≃ K from calorimetryof concentrated solutions of Cyt-c. The glass transition of thehydration shell prevents elastic motions of the protein, making ahydrated protein harder at low temperatures than the dry one. One wonders if rotations and translations of water molecules inthe shell terminate at the same temperature. Figure 11 showsthat this is not the case (see ESI † for the details of calculations).The glass transition for χ ( a ) coincides with freezing of water rota-tions. The onset temperature depends on the observation window(cf. filled to open squares in Fig. 11), consistent with ergodicitybreaking at the transition. On the contrary, the onset of watertranslations occurs at a higher temperature, ≃ K. A similarphenomenology was recently reported from neutron scattering ofprotein’s hydration shell, where the onset of water’s transla-tions also followed the onset of rotations. The temperature oftranslational onset is close to T d , as was noted in the past. !" ! % & ’ ( ) * + , ) /)*0, Fig. 11
Center of mass MSF (trans., circles) and the MSF due tomolecular rotations (rot., squares) of water molecules within thehydration shell 6 Å thick around the Ox Cyt-c. The center of masstranslations and molecular rotations are calculated within thetime-window of 100 ps (filled points) and 1 ns (open points). The MSFsfor center-of-mass translations are reduced by a factor of 40 to bringthem to the same scale with the results for rotations. The dashed linesare linear fits through subsets of points to illustrate differences in theonset temperatures ( T rot ( ) = K, T rot (
100 ps ) =
K, and T tr (
100 ps ) =
K. The dotted lines connecting the points are drawn toguide the eye.
A crude estimate of the “dielectric constant” of the shell mightbe relevant here. If, for the sake of an estimate, one adopts theconnection between the dielectric constant and the susceptibil-ity of bulk dielectrics, ε ( a ) = + πχ ( a ) , then the inspection ofFig. 10 suggests ε ( a ) ≃ at T = K and a = Å. This veryhigh dielectric constant is consistent with recent dielectric spec-troscopy of protein powders, reporting high dielectric incre-ments ∆ε ≃ − for the relaxation process reaching 1-10 µ sat the room temperature. Given the temperature dependence ofthis relaxation process, it appears likely that it is responsible forglass transition of hydrated protein samples. The drop of χ at T g seen in Fig. 10, and a similar behavior observed previouslyin simulations of lysozyme, suggests a possible connection be-tween high ∆ε and polarized domains formed in the hydrationshell. Equation (1) offers a natural explanation of the extended flexibil-ity of proteins at high temperatures in terms of the force constantassigned to a cofactor or residue in the folded protein. Accord-ing to Eq. (1), softening of the protein matrix due to collective ag-itation of the protein-water thermal bath reduces the vibrationalforce constant κ vib = ( β h δ x i vib ) − by the magnitude κ b = β h δ F i r , (23) !" ! $ % & ’ ( ) * + , - /&’0, &123456&728&92: ; Fig. 12
Force constant of the protein-water medium κ b = β h δ F i r calculated from κ ( T ) and κ vib ( T ) according to Eq. (24) . Points indicatethe experimental results for Cyt-c (Ox), myoglobin (Myo), and forlysozyme dissolved in 50:50 glycerol-D O solvent at h = . g D O/gLys. The results for lysozyme are multiplied by a factor of 10 to bringthem to the scale of the plot. The dotted lines connecting the points aredrawn to guide the eye. which reduces the total force constant κ = ( β h δ x i ) − κ = κ vib − κ b . (24)Using Eq. (24), Fig. 12 shows κ b ( T ) for Cyt-c (Ox) and myo-globin (Figs. 3 and 9). We have additionally included the resultsfrom neutron scattering of lysozyme (Lys) in 50:50 glycerol-D Osolution ( h = . g D O/g Lys), which display a crossover tem-perature at ≃ K. All these data point to a rise of κ b at T d to anearly constant value charactering the protein flexibility at GHzfrequencies. The Young’s moduli of the hydrated protein fall withincreasing temperature in a fashion consistent with κ b in Fig.12.The notion of protein dynamics as proxy for enzymatic activ-ity has been actively discussed in the recent literature. Onehas to clearly distinguish flexibility, i.e. the ability to samplea large number of conformations, from the actual dynamics, i.e.the time-scales involved in usually dissipative decay of correlationfunctions. Whether flexibility and activity must accompany eachother for slow (in milliseconds) enzymetic reactions remains to beseen, but there is one class of enzyme reactions where proteinconfigurational space has to be dynamically restricted for the re-action to occur. This is the process of protein electron transportessential to production of all energy in biology, either throughphotosynthesis or through mitochondrial respiration. The fluctuation-dissipation theorem connects fluctuations to re-sponse to an external perturbation. In this framework, high flex-ibility implies high solvation, or trapping, energy. Electrons inbiological energy chains have to perform many tunneling stepswithin a narrow energy window consistent with the energy inputfrom food or light. In order to accomplish vectorial electron trans- !" ’ ( ) * + , - . /*+0. *%*12*$*12*%&*12*3 Fig. 13 ∆ F † given by Eq. (26) vs T calculated from MD simulations( ∼ ns of simulations at each temperature ). The legend indicatethe reaction times τ r = k − R . Deviations from the thermodynamicbehavior, k R = , are determined by the nonergodic factor f ne ( T ) (Eq. (17) ) calculated from the observation window τ r = k − R and therelaxation time τ X ( T )( s ) = exp [ − . + / T ] . The reorganizationenergies from long simulation trajectories are approximated by linearfunctions of temperature: λ St ( T ) = . − . × T eV, λ ( T ) = . − . × T eV ( T is in K). port, energy chains have to avoid deep energy traps. Therefore,large conformational motions producing asymmetries in solvationenergies between the initial and final tunneling states have to bedynamically frozen on the reaction time. This phenomenology is consistent with what we have foundhere for the dynamical transition of atomic displacements. Therole of the force constant in Eq. (23) is played by the reorgani-zation energy λ determined through the variance of the donor-acceptor energy gap X used to gauge the progress of the reaction.The reorganization energy is determined through the variance of X by the equation inspired by the fluctuation-dissipation theorem(cf. to Eq. (23)) λ ( k R ) = β h δ X i r / (25)Here, h δ X i r = h δ X i f ne ( T ) depends on the observation windowthrough the nonergodicity factor f ne ( T ) (Eq. (17)) multiplyingthe thermodynamic ( τ r → ∞ ) variance h δ X i . The only differ-ence of this problem from our discussion of iron’s MSF is thatone has to replace the relaxation time of the force τ ( T ) withthe relaxation time τ X ( T ) of the Stokes-shift correlation function C X ( t ) = h δ X ( t ) δ X ( ) i . The role of the observation window is nowplayed by the reaction time τ r = k − R given in terms of the reactionrate constant k R .The reorganization energy λ ( k R ) quantifies the depth of thetrap created for a charge by the protein-water thermal bath. Theamount of energy to de-trap the electron and bring it back to thetunneling configuration specifies the activation barrier ∆ F † . Itis given in terms of two energy parameters: the difference offirst moments of X in the initial and final states, known as theStokes-shift reorganization energy λ St , and the second moment
10 | 1–14 f X specified by λ ( k R ) ∆ F † = ( λ St ) / [ λ ( k R )] . (26)The parameter λ St specifies the energy difference between twostates of the protein (Red and Ox in the case of Cyt-c). It doesnot reach its thermodynamic value because of the inability of theprotein to sample its entire phase space on the reaction time. Instead of reaching, through a conformational change, two ther-modynamic minima of stability (for Red and Ox states), the pro-tein gets trapped in intermediate local minima. The time sepa-ration k − R ≪ τ conf between the reaction time and the time of theconformational transition τ conf constrains the availvble configura-tion space allowing a relatively small value of λ St such that thecondition λ St ≪ λ ( k R ) keeps the reaction barrier in Eq. (26) rel-atively low. The reorganization energy λ ( k R ) in the denominatorin Eq. (26) is, however, directly affected by nonergodic freezingof a subset of degrees of freedom, which can lead to a significantincrease of the reaction barrier at low temperatures and to thetermination of the protein function.This perspective is illustrated in Fig. 13 showing the effect ofthe observation window on ∆ F † ( T ) . The input parameters to theresults shown in Fig. 13 are λ St ( T ) and λ ( T ) taken from longtrajectories ( k R → ) and the Stokes-shift relaxation time τ X ( T ) calculated for Cyt-c. As the temperature decreases, the relax-ation time τ X ( T ) leaves the observation window, τ r = k − R , and λ ( k R ) drops. The activation barrier grows at low temperatures(see Eq. (26)) and the reaction slows down due to ergodicitybreaking qualitatively consistent with the dynamical transition forthe atomic MSF (at faster rates, such as those involved in primaryevents of photosynthesis, λ St becomes affected by k R and the pic-ture changes again ). The overlap of the time-scales probed bythe neutron scattering and Mössbauer spectroscopy with the typ-ical reaction times of protein electron transfer suggests that thefluctuations of the protein-water thermal bath responsible for thehigh-temperature part of the displacement curve are the same asthose involved in activating redox activity of proteins. The present model assigns atomic displacements in the proteinto two factors: (i) high-frequency vibrations within the subunit(residue, cofactor, etc.) and (ii) fluctuations in the position of thesubunit caused by thermal fluctuations of the entire protein andits hydration shell. The second component enters the observableMSF in terms of the variance of the force applied to the centerof mass of the subunit (denominator in Eq. (1)). This equationcan be alternatively viewed as softening of a stiff vibrational forceconstant by the protein-water thermal bath (Eq. (24)). Since thevariance of the force depends on the observation window, soft-ening of vibrations is achieved at the temperature above T d al-lowing the long-time relaxation of the force autocorrelation func- tion to remain within the observation window. An experimentallink to this picture is provided by inelastic x-ray scattering recording softening of the protein phonon-like modes represent-ing global vibrations. In line with the common observations ofthe dynamical transition, softening of the protein phonon modesis strongly suppressed in dry samples. Similar phenomenologyis provided by the temperature dependence of the protein bosonpeak reflecting the density of protein collective vibrations onthe length-scale of a few nanometers and THz frequency.
Forinstance, the frequency of the boson peak for myoglobin falls from ∼ cm − to ∼ cm − when the temperature is raised from 170to 295 K. The forces produced by the protein-water thermal bath at inter-nal sites inside the protein are strongly affected by the structureand dynamics of the hydration shell.
Shell dipoles clusterin nanodomains pinned by charged surface residues. Dynamicalfreezing of these nanodomains occurs at the glass transition of thehydration shell corresponding to the lower crossover temperature T g (Fig. 1). Rotations of water molecules in the shell dynamicallyfreeze at this temperature. Translations dynamically freeze at ahigher temperature close to T d . Therefore, the existence of twocrossover temperatures in the dynamical transition of proteins re-flects two separate ergodicity breaking crossovers for rotationsand translations of hydration water (Fig. 11).The entrance of the relaxation time into the resolution win-dow, resulting in the dynamical transition of a specific relaxationmode, is often considered to be a “trivial” effect, in contrast toan anticipated true structural transition. However, this ergod-icity breaking allows protein-driven reactions to proceed withoutbeing trapped into deep solvation wells. The link between flex-ibility and solvation, and thus the ability to produce traps, hasbeen under-appreciated in the literature on enzymatic activity. Asan illuminating example, protein electron transfer occurs in dy-namically quenched proteins where ergodicity breaking preventsfrom developing deep solvation traps along the electron-transportchain.
Conflict of Interests . There are no conflicts of interest to de-clare
Acknowledgement . This research was supported by the NSF(CHE-1464810) and through XSEDE (TG-MCB080116N). We aregrateful to Antonio Benedetto for useful discussions and thepreprint (Ref. ) made available to us. We acknowledge help byDaniel Martin with the analysis of the simulation trajectories. References
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