Chemical Physics 418 (2013) 57–64

Contents lists available at SciVerse ScienceDirect

Chemical Physics

journal homepage: www.elsevier .com/locate /chemphys

Viscosity-dependent structural fluctuation of the M80-containingX-loop of horse ferrocytochrome c

0301-0104/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.chemphys.2013.04.009

⇑ Corresponding author. Tel.: +91 0175 239 3832; fax: +91 40 2301 2460.E-mail address: rajesh.kumar@thapar.edu (R. Kumar).

Rajesh Kumar ⇑, Rishu Jain, Rajesh KumarSchool of Chemistry and Biochemistry, Thapar University, Patiala 147 004, India

a r t i c l e i n f o

Article history:Received 2 July 2012In final form 13 April 2013Available online 24 April 2013

Keywords:Kramers modelInternal frictionStructural fluctuationSolvent viscosityMotional dynamics

a b s t r a c t

To determine the effect of solvent viscosity on low-frequency local motions that control the slow changesin structural dynamics of proteins, we have studied the effects of solvent viscosity on the structural fluc-tuation of presumably the M80-containing X-loop by measuring the rate of thermally-driven CO-disso-ciation from a natively-folded carbonmonoxycytochrome c (NCO-state) in the 0.65–92.5 cP range ofviscosity at pH 7.0. At low viscosities (68 cP), the rate coefficient, kdiss for dissociation of CO from theNCO-state varies inversely with the viscosity, but saturates at high viscosities, suggesting that CO-disso-ciation reaction involves sequential stages that depend differently on solvent friction, i.e., solvent coupledand nonsolvent-coupled stages of the process. In the low viscosity regime (0.65 6 gs 6 8.0 cP), the rate-viscosity data were fitted to modified Kramers model, kdiss = [A0/(r + gs)n]exp(�DG/RT), which producedinternal friction, r = 1.35 cP (±0.88), which suggests that the speed of CO-dissociation from NCO atgs 6 8.0 cP is controlled by internal friction.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Proteins are dynamic molecules which undergo structural fluc-tuations ranging from ultrafast (in the femtoseconds to picosec-onds range) to relatively slow (in the range of seconds) [1–5].Structural fluctuations that occur on the ultrafast timescales en-able a protein to sample a complex conformational energy land-scape. These rapid motions ultimately facilitate slower, largerscale protein rearrangements that are accountable for modulatingthe biological functions [6–8]. The stability and dynamics of a pro-tein in solution are strongly coupled to the dynamics of the solvent[8–12]. The comprehensive investigations of protein dynamics as afunction of solvent viscosity have provided insights into funda-mental biochemical reactions [12–14] and protein–solvent interac-tions [5,8–10,15–17].

A common approach for investigating the significance of proteindynamics in a chemical reaction is to study the effect of the solventviscosity on reaction rate. Kramers theory models the kinetics ofchemical reactions and asserts that the reaction rate, k for a heavilydamped process depends on both the height of the potential bar-rier, DG and a reaction friction parameter, c through k a c�1

exp(�DG/RT) [18,19]. According to Stokes law, one should expectcag, which finally gives kag�1. But this simple behavior is rarelyfound with protein reactions. Ansari et al. [13] suggested that the

total damping rate is a linear combination of the internal (ci) andthe solvent (cs) contributions, where ci is the friction coefficientfor motion in the protein, and cs is the friction coefficient for mo-tion in the solvent, which according to Stokes’ law is proportionalto its viscosity.

Useful extensions of the Kramers theory to intramolecular acti-vated processes in proteins have been provided by Gavish [20,21],Doster [22], and Schlitter [23]. Gavish [20] has suggested that vis-cosity plays a vital role in determining the dynamic state of theprotein through structural fluctuations. The viscosity-dependentexchange of energy between the protein and the solvent was pro-posed to be a major source of energy for the protein structural fluc-tuations in determining its function [20,24]. Since proteinstructural fluctuations are mainly driven by Brownian motion,therefore, the Kramers theory has been applied to describe the rateof entry and escape of small ligands [12,25] and enzymatic reac-tions [20,26]. Indeed, several investigators have considered thephenomenological Kramers model to examine the effect of solventviscosity on several types of protein reactions, including proteinfolding reactions [27–43], enzyme kinetics [44,45], and the bindingof ligands to heme proteins [12,13,21,26,39,46,47]. In particular,for enzyme reactions or protein–ligand interactions, where diffu-sive motions of protein segments through the solvent alone donot contribute to the observed rates, the rate constant has beenfound to be inversely proportional to the fractional power of theviscosity, k / gs

�n (0 < n < 1) [48,49]. The validity of Kramers the-ory has been tested experimentally for polymer dynamics [50]and is well supported by computer simulations [51,52].

58 R. Kumar et al. / Chemical Physics 418 (2013) 57–64

A large body of work on protein dynamics in viscous solventindicates that there are three possible origins for fractional viscos-ity dependence – (i) breakdown of Kramers theory, (ii) the devia-tion from Stokes law [12,22,34,47,49], and (iii) the reactioninvolves sequential stages that depends differently on solvent fric-tion, i.e., solvent coupled and nonsolvent-coupled stages of theprocess [22]. The meaning of fractional exponent, n suggests thatthe friction coefficient, c does not vary linearly with the solventviscosity, gs. This is essentially a violation of Stokes law (c / gs)and is not clear at present.

Although commendable progress has been made in the under-standing of the effect of solvent viscosity on the fast proteindynamics [53–56], very little has been done to record the effectof solvent viscosity on low-frequency local motion that controlthe relatively slow changes in structural dynamics of proteins. Inthis work, we have examined the influence of solvent viscosityon the structural fluctuation of presumably the M80-containingX-loop by measuring the rate of thermally-driven CO-dissociationfrom the natively folded carbonmonoxycytochrome c (NCO-state).Because of its smaller size and the content of the heme iron, car-bonmonoxycytochrome c is a model in experimental proteindynamics studies [41,57–63]. The CO-dissociation rate of NCO-state varies inversely with viscosity initially when the solvent vis-cosity is low (68.0 cP), but saturates at higher viscosity, indicatingthat the CO-dissociation from the NCO-state involves sequentialstages that depend differently on solvent friction, i.e., solvent cou-pled and nonsolvent-coupled stages of the process [22,47]. In thelow viscosity regime (0.65 6 gs 6 8.0 cP), the rate-viscosity datawere fitted to a modified Kramers model, kdiss = [A0/(r + gs)n]-exp(�DG/RT), which produced the internal friction, r = 1.35(±0.88) cP. This suggests that in the low viscosity regime, the speedof CO-dissociation from a native-like folded intermediate, NCO iscontrolled by internal friction.

2. Materials and methods

Horse cytochrome c (Type VI) was purchased from Sigma. Un-less otherwise indicated, all the experiments were performed un-der anaerobic conditions in 15 mM phosphate buffer (pH 7,25 �C), containing �2 mM freshly prepared sodium dithionite.The guanidine hydrochloride (GdnHCl) concentrations were deter-mined by refractive index measurements. The compositions ofglycerol, sucrose, and glucose are expressed in (w/w)%. We directlymeasured the kinematic viscosity of glycerol, sucrose, and glucosesolutions at 25 �C using a Brookfield digital viscometer (BrookfieldEngineering Labs., Inc., USA).

2.1. Measurement of refolding kinetics of unfoldedcarbonmonoxycytochrome c

Cyt c (�0.4 mM), initially unfolded in�6.5 M GdnHCl, pH 7, wasreduced under N2 by the addition of a concentrated solution of so-dium dithionite to a final concentration of 3.0 mM. The unfoldedferrocyt c was liganded with CO, and was mixed with the refoldingbuffer in the stopped-flow in a desired ratio to record refoldingkinetics. In each case, before mixing, an equilibration time of�5 min was given. Stopped-flow experiments used a BiologicSFM400 module. A constant temperature of �25 �C was regulatedby the use of an external water bath. The spectrometer was config-ured for fluorescence detection (ex: 280 nm; em: 340 nm). A two-syringe mixing (1:7; protein: buffer) at a total flow rate of 8 ml/swas employed. The instrument dead-time for a high-density mixerand a 0.8 mm flow cell was determined to be 3.0 (±0.1) ms. Typi-cally, 10–15 shots were averaged.

2.2. Preparation and identity of native carbonmonoxycytochrome c(i.e., NCO-state)

The NCO-state is prepared from chemically denatured cyt c bylowering the denaturant concentration in the presence of carbonmonoxide, which binds tightly to the heme iron and prevents for-mation of the native M80-heme (sulfur–iron) link [57,58,64].Briefly, cyt c is unfolded in the presence of �6 M GdnHCl. This un-folded protein was deaerated and reduced by adding sodiumdithionite. The heme iron of the unfolded protein (called U-state)was liganded with a CO molecule by adding the dry gas at 1 atmpressure to the unfolded protein solution. Unfolded CO-ligandedprotein (UCO) thus obtained is diluted out with a CO-free refoldingbuffer (15 mM phosphate, pH 7, 25 �C). The protein chain refoldstrapping the CO molecule which is still bonded to the heme iron.Cyt c with the trapped CO (called NCO-state) is structurally na-tive-like [59,60] and it represents a long-lived late kinetic interme-diate [61,65,66,67]. Folding is blocked as long as the NCO-statestays frozen in the kinetic trap. The folding of NCO to N-state re-quires the escape of CO molecule from the NCO due to thermal dis-sociation of the Fe2+�CO bond and simultaneously reorganizationof the misconfigured interactions. Actually, the reorganization ofthe misconfigured interactions involves the ligation of the side-chain of the amino acid residue M80, which is the intrinsic N-stateligand. The N and NCO conformations are not different from eachother to any considerable extent [68], and the motional dynamicsof the two protein states are not expected to be different either.

2.3. Thermal dissociation of CO from the natively foldedcarbonmonoxycytochrome c (NCO-state)

Within the limit of the stopped-flow resolution (�3 ms dead-time), the UCO refolds extremely fast in refolding buffer [64]. Thekinetics exemplified in Fig. 1c give an observed rate of �310 s�1

in the presence of �0.3 M GdnHCl. A minor refolding phase of�40 s�1 accounts for remaining 8% of the total observed amplitudearises most likely from a fraction of oxidized protein that does notbind CO. The refolding process UCO ? NCO proceed after�101 folddilution of the UCO solution with the CO-free folding buffer. Sincethe concentration of CO in the refolding medium is substantially re-duced, and because the affinity of native protein for CO is lower rel-ative to that for the intrinsic M80 ligand, the UCO ? NCO processleads immediately to the NCO ? N + CO conversion. TheNCO ? N + CO conversion is basically a unimolecular thermallyactivated reaction, where the NCO molecules accumulate energynecessary for the CO-dissociation by thermal fluctuation. Collisionsbetween different groups of atoms or structural elements affordedby internal dynamics of the protein act as an intramolecular sourceof energy. Thermal dissociation of CO from the NCO-state is essen-tially the Fe2+�CO + M80 ? Fe2+�M80 + CO replacement. The Fe2+-

�CO ? Fe2+�M80 conversion does not involve any majorconformational adjustment and occurs because of its instabilitydue to greater affinity of M80 sulfur for Fe2+ relative to that of CO.Fig. 1d typifies the kinetics of the NCO ? N + CO dissociation reac-tion, recorded after 101 fold dilution of the unfolded CO-ligandedprotein into a CO-free refolding buffer. The slow increase in absor-bance at the heme p ? p⁄ a-band (550 nm) in a single exponentialis due to the dissociation of CO (Fig. 1). The slowness of the reactionallows accurate determination of kdiss (s = 23 min with no addition,Fig. 1d) by conventional UV–visible spectrophotometer.

2.4. Equilibrium unfolding measurements of ferricytochrome c (i.e.,ferricyt c) in the presence of different composition of glycerol

The GdnHCl-induced unfolding transition curves of ferricyt c inthe presence of different composition of glycerol solutions were

Time (sec)101 102 103 104

OD

, 550

nm

0.06

0.08

0.10

0.12

0.14

λ, nm500 550 600

Abs

orba

nce

0.00

0.05

0.10

0.15(b)

(d)

λ, nm500 550 600

Abs

orba

nce

0.05

0.10

0.15

(a)

Time of folding (s)10-3 10-2 10-1 100

Fluo

resc

ence

0.0

0.4

0.8

1.2

Time of folding (s)0.00 0.01 0.10

Fluo

resc

ence

0.0

0.5

1.0UCO signal

(c)

Fig. 1. (a) Steady-state visible absorption spectra of NCO (dashed line) and N (solid line) states. The NCO ? N reaction was probed at 550 nm, the kmax of the N-statespectrum. The spectra were recorded in 15 mM phosphate buffer, pH 7, containing �0.3 M GdnHCl and 10% glycerol (mass percent) at 25 �C. (b) Steady-state visibleabsorption spectra of native ferrocytochrome c (N-state) at 0.0 (long dash line), 10 (solid line), 50 (dotted line), and 80 (short dash line)% glycerol (mass percent) at 25 �C. Theintensity of the absorption spectrum (a band (550 nm) and b band (520 nm)) of native ferrocytochrome c initially increased up to�10% glycerol (mass percent) and on furtherincrease of glycerol composition it remained almost constant. (c) Unfolded carbonmonoxy-cytochrome c (UCO) refolds extremely fast (UCO ? NCO; k = 310 s�1, �0.3 MGdnHCl, 25 �C). 8% of the observed amplitude is due to a slow minor phase (k = 40 s�1). The normalized fluorescence signals (inset) indicate occurrence of processes faster thanwhat is seen in the stopped-flow. (d) Kinetics of CO-dissociation from NCO at 25 �C, pH �7 as monitored by change in absorbance at 550 nm. The solid line show least-squarefit of the data to a single exponential function (s � 23 min (0.3 M GdnHCl and 2% glycerol (mass percent))).

R. Kumar et al. / Chemical Physics 418 (2013) 57–64 59

recorded by tryptophan fluorescence excited at 280 nm (slit-width2 nm) and emission at 358 nm (slit-width 5 nm) using a 1 cmsquare cuvette. Protein concentration was 10 lM, and all experi-ments were carried out in 15 mM sodium phosphate buffer.

3. Results and discussion

3.1. Binding of carbonmonoxide to ferrocyt c in the presence of glycerol

Fig. 1b shows the visible absorption spectra of NCO and Nstates (pH 7, 25 �C) which shows that the CO binding to ferrocytc causes both a- and b-bands (550 and 520 nm, respectively) tobroaden and red-shift with a large fall in the extinction coefficientof the a-band [64]. Fig. 1c shows the steady-state visibleabsorption spectra of native ferrocyt c (N-state) in the absenceand presence of glycerol at 25 �C. The intensity of the absorptionspectrum (a band (550 nm) and b band (520 nm)) of native ferr-ocyt c initially increased up to �10% glycerol (mass percent) andon further increase of glycerol composition it remained almostconstant.

3.2. Challenges in viscosity dependence of protein reactions

A major experimental challenge in the study of the viscositydependence of protein reactions such as protein folding using visc-ogens is the proper accounting for the accompanying changes inprotein stability [43]. In fact, the NCO ? N + CO conversion ratemay change if the protein gains stability with increments of visco-gen composition. The challenge is to solely vary the viscosity keep-ing the energy of activation between the initial state and the

transition state constant. Obtaining isostability conditions in thepresence of different concentrations of viscogens are not straightforward either. According to Kramers theory for barrier crossingreactions in the high friction limit [18,19], that correspond to mostreactions in solution and also to the NCO to the N-state conversion,the rate of a reaction depends on the reaction friction (c) as well asthe height of the activation free energy barrier, DG. Study of viscos-ity effects on the NCO to the N-state conversion must therefore ac-count for the influence of viscogens on the stability of the N-state,because similar shifts in DG may generate an extra perturbation tokdiss. The DG shifts for protein reaction can be compensated bymany methods such as adding denaturant simultaneously withthe viscogen [29] or choosing experimental conditions where theviscogens does not affect DG [31]. In the present case, Fig. 2a showsthat the rate of conversion of the NCO-state to the N-state showslittle sensitivity to the overall free energy of folding, DG. The veryweak effect of GdnHCl on kdiss (olnkdiss/o[GdnHCl] =(0.015 ± 0.01) M�1) thus indicates that any destabilization/stabil-ization of the N-state (such as from GdnHCl or other cosolutes)does not affect the activation free energy (DG) of the NCO to theN-state conversion. To further determine whether the addition ofviscogen affects the activation energy of the NCO to the N-stateconversion, we investigated the temperature dependence of therate of dissociation of CO from the NCO-state at several fixed glyc-erol compositions (0%, 35%, 65%, and 90% of glycerol (w/w)). Arrhe-nius plots of kdiss in 0.0% and 90% glycerol are shown in Fig. 2b. Thestraight lines were obtained by a least squares fit of the data.Fig. 2b shows that in the region 10–42 �C, the best-fit straight linesare nearly parallel and the activation energies deduced from theirslopes are �24.5 (±0.5) and 24.9 (±0.5) kcal/mol at about 0.0% and90% glycerol, respectively. Thus, the temperature dependence of

(1/T)x1000, (K-1)3.2 3.3 3.4 3.5

ln k

diss

(s-1

)

-10.5

-9.0

-7.5

-6.0

-4.5

0 35

Glycerol (w/w%)

65 90

η (cP)1 10 100

E a (k

cal m

ol-1

)

24

25

26

27107

(c)

(b)

[GdnHCl] (M)0 1 2 3 4

k diss

(s-1)

3x10-4

6x10-4

9x10-4

(a)

Fig. 2. (a) Denaturant dependence of kdiss of NCO at 25 �C, pH 7. Denaturantconcentrations up to �4.0 M GdnHCl do not affect CO-dissociation rate of NCO. (b)Arrhenius plots in the absence (.) and presence (r, 90% (w/w)) of glycerol. Theactivation energies (Ea) derived from the linear least-squares fitting of the data are�24.5 (±0.5) and �24.9 (±0.3) kcal mol�1 for 0.0% and 90% of glycerol, respectively.(c) The plot of Ea as a function of glycerol composition (w/w)% and solvent viscosity(cP).

Fig. 3. (a) GdnHCl-induced equilibrium unfolding curves of ferricyt c in thepresence of 0 (j), 30 (h), 40 (D), and 60 (N) (w/w%) glycerol solution at 25 �C, pH�7. Iterated fit values for a two-state equilibrium [87] are DG0 � 7.5 (±0.3), 8.54(±0.2), 8.6 (±0.4), and 8.5 (±0.4) kcal mol�1 and Cm � 2.6, 2.73, 2.81, and 2.84 MGdnHCl corresponding to 0%, 30%, 40%, and 60% (w/w) glycerol composition,respectively. (b) The plot of DG0 as a function of glycerol composition and solventviscosity (cP).

60 R. Kumar et al. / Chemical Physics 418 (2013) 57–64

the rate of dissociation of CO from the NCO-state does not suggestany significant change in the activation energy of the CO-dissocia-tion reaction in the 0–90% range of glycerol (Fig. 2c). Therefore, thedata do not require compensation for viscogen effects on DG.

3.3. Effect of glycerol on the stability of cyt c

To determine the extent to which glycerol affects the stability ofcyt c, we measured the GdnHCl-induced unfolding curves of ferri-cyt c in the presence of 0%, 30%, 40%, and 60% glycerol (w/w) at25 �C (Fig. 3a). The unfolding free energy (i.e., the difference of freeenergies of unfolded and native states (DG�)) of ferricyt c increasesby �1.0 (±0.4) kcal mol�1 when the protein is held at 40% glycerol,and remains constant at higher concentrations of glycerol (Fig. 3b).Thus, these results did not suggest any significant change in thestability of cyt c in the 0–60% range of glycerol (Fig. 3b). These datawere recorded by using ferricyt c which is �11 kcal mol�1 less sta-ble than the ferrocyt c [69]. The oxidized and reduced states of cyt care nearly identical both structurally and conformationally [64,70–73], therefore, the extent of stability offered by glycerol is likely tobe the same. Further, as discussed above, the NCO-state is structur-

ally native-like, although the value of Gibbs free energy of NCO-state is larger than the N-state by �2 kcal mol�1 [64], indicatingthat the glycerol-induced changes in the free energies of the NCOand N states would be parallel. Thus, glycerol would affect thereactant and the product to the same extent. For the present none-theless, the use of glycerol as a viscogen does not appear to influ-ence the NCO ? N + CO conversion rate considerably.

3.4. Solvent composition modulation of the NCO ? N + CO dissociationrate

The NCO ? N + CO process (i.e., Fe2+�CO + M80 ? Fe2+-

�M80 + CO) can be viewed as a bimolecular displacement reaction,where the two reacting sites diffuse together to form an encountercomplex in which the relevant sites collide with each other at somefrequency before either dissociating away or reacting to form theproducts. Under steady-state assumption, kdiss = Kk, where K isthe equilibrium constant for the formation and the breakdown ofthe encounter complex, and k is the rate coefficient for the bar-rier-activated reaction. Since the reacting sites forming the reac-tion volume are uncharged and have no special interactionotherwise, K equals the inverse of the encounter complex volume.The rate of the barrier-activated reaction depends on the frequencyof collisions during an encounter. For NCO ? N + CO reaction, theM80-resident segment of the polypeptide, which is linked to theheme iron through the Fe2+�M80 bond in N but is free in NCO, pro-vides a reacting site. The segment of the polypeptide between res-idues 70 and 85 (Fig. 4a) forms a small X-loop [74]. The collectivemotion of the X-loop is expected to be the leading determinant ofthe CO-dissociation process. This is because of the two main rea-sons – (i) the local mobility of the heme ring is suppressed bythe intrinsic size and rigidity of the ring system [75], and (ii) the

(w/w)% Viscogens 0 30 60 90 120

k diss

(s-1)

10-3

10-4

(b)

Structure of horse cytochrome c(PDB: 1 HRC)

(a)

Fig. 4. (a) A ribbon representation of horse cytochrome c; accession code PDB 1HRC[88]. The side-chain of M80 and some other key residues are shown explicitly in balland stick display. The segment of the polypeptide (residues 70–85) containing M80forms an X-loop. The figure has been drawn using Cerius2 Version 4.0 (MSI). (b) The[Viscogens] dependence of rate coefficients, kdiss for CO-dissociation of NCO(glycerol (s), sucrose (D), and glucose (h)) at 25 �C, pH �7. The error barsrepresent the standard deviations from the kdiss values.

ηs(cP)100 101 102

k diss

(s-1)

10-3

10-4

(a)

ηs(cP)10-1 100 101

k diss

(s-1)

10-3

10-4

(b)

Fig. 5. Non-Kramers scaling observed for CO-dissociation (NCO ? N + CO) rateconstant of NCO as a function of viscogen (glycerol (s), sucrose (D), and glucose(h)) viscosity in 15 mM phosphate buffer, 0.3 M GdnHCl, pH 7, 25 �C. The kdiss

values below viscosity 1.01 cP (Fig. 5a and b) were eventually measured at differenttemperatures (i.e., 30, 35, and 40 �C) in the absence of viscogen and by Arrheniusequation using activation energy of �24.5 kcal/mol these were corrected to 25 �C(Fig. 2b). The kdiss values below viscosity 1.01 cP were the temperature correctedvalues at 25 �C. The viscosities of water below 1.01 cP (i.e., gs = 0.65, 0.80, and0.89 cP are the viscosity of water at temperature �40, 30, and 25 �C, respectively)were taken from the literature [89]. Fits of the data according to Eq. (3) in Fig. 5ayields, n � 0.75 (±0.02), A � 6.5 � 1015 s�1, and G = 26.5 kcal mol�1. The fit to theexperimentally observed rates using Eq. (5) is shown in Fig. 5b with a value ofr = 1.35 (±0.88) cP.

R. Kumar et al. / Chemical Physics 418 (2013) 57–64 61

neighboring residues of M80 have significantly higher thermal fac-tors in the X-ray structure of cyt c [76]. The CO-dissociation rate ofNCO is found to decrease with increasing composition of the visco-gen (glycerol, sucrose, and glucose) (Fig. 4b). The solvent composi-tion modulation of kdiss (Fig. 4b) reveals the way the collectivemotion of the loop or of a part of it responds to solvent contentin the reaction medium. Since atomic fluctuations or high-fre-quency local motions involve only small spatial displacements,the thermal motion viewed here must be of collective characterin which groups of atoms in a part or in the entire X-loop movein a correlated manner or as a unit. The observation that it re-sponds to increments of solvent composition implies that it is a lo-cal motion, and hence is a low-frequency (s, millisecond or longer)large-amplitude mode (several Å) [77].

3.5. The effect of solvent viscosity on the motional dynamics of nativecarbonmonoxycytochrome c (i.e., the NCO-state)

Occurrence of a large variety of intramolecular thermal motionsand collision-mediated relaxations renders internal dynamics ofproteins a complex phenomenon. In terms of time scale and spatialdisplacement two broad classes of motions emerge: (i) atomic fluc-tuations or individual vibrations that are fast and random, andhave sub-angstrom coverage, and (ii) large-scale collective fluctu-ations in which a sizable set of atoms or several side chains or res-idues in a subglobal structure move in unison covering adisplacement of several angstroms [77,78]. Either type of motionsdepends in some way on solvent composition and viscosity. Thiswork investigates the effect of solvent viscosity on the low-fre-quency local motions such as thermal collisions due to collectivemotional mode of a subglobal structural unit of ferrocyt c. TheNCO ? N + CO conversion is a dynamical event associated withFe2+�CO + M80 ? Fe2+�M80 + CO reaction. The effect of the sol-

vent viscosity on the structural fluctuation of presumably theM80-containing X-loop has been deduced by measuring the ratecoefficient of M80 interaction with Fe2+ of NCO under conditionsof progressively increasing the viscogen composition. Fig. 5a showsthe plot of kdiss as a function of solvent viscosity of three differentviscogens (glycerol, sucrose, and glucose). The values of kdiss weredetermined at different viscosities of viscogens holding the GdnHClconcentration (�0.3 M) constant at 25 �C. The kdiss values belowviscosity 1.01 cP (Fig. 5a and b) were eventually measured at dif-ferent temperatures (i.e., 30, 35, and 40 �C) in the absence of visco-gen and finally corrected to 25 �C by Arrhenius equation usingactivation energy of �24.5 kcal/mol (Fig. 2b).

Each of the three different viscogens used here had a similarslowing effect on the CO-dissociation kinetics, causing the equaldecrease in kdiss with increasing solvent viscosity (Fig. 5a and b).The rate-viscosity data in Fig. 5a illustrates that spatial displace-ment of thermal fluctuations of a subglobal structural unit is re-duced at low viscosity (68 cP), but saturates at high viscosity.The finding that the rate of thermally-driven CO-dissociation fromthe NCO state decreases up to �8 cP imply that the motional re-sponse of the M80 containing X-loop at lower viscosities (68 cP)is different from what is seen at higher viscosities (>8 cP) (Fig. 5a).

3.6. Phenomenological description of viscosity dependence of theNCO ? N + CO conversion rate

Kramers theory for unimolecular reactions asserts that the ratek for a barrier-crossing process subject to strong damping frictionwill vary as [18,19]:

62 R. Kumar et al. / Chemical Physics 418 (2013) 57–64

k ¼ A0

c

� �exp

�DGRT

� �ð1Þ

where A0 is a constant (that depends on m, xA, xB etc., here, xA andxB denote the frequencies of motion in the starting well and on topof the barrier, respectively, and m is the effective molecular mass ofthe particle that is crossing the barrier) and DG is the barrier height.If the friction in the NCO ? N + CO conversion arises from gs (thedynamic viscosity of the solvent) then the CO-dissociation rateshould scale as kdiss a c�1 a gs

�1. Fig. 5a shows the plot of kdiss asa function of solvent viscosity, gs. The value of kdiss varies inverselywith viscosity when the solvent viscosity is low (68 cP), but satu-rates at higher viscosity, indicating a deviation from the k / gs

�1

relation. In general, for protein reactions, the functional dependenceof the reaction rate constant, k for the rate limiting step on solventviscosity, gs has the form [12,20],

k ¼ A0

gns

� �exp

�DGRT

� �ð2Þ

Several groups have tested solvent viscosity effects on protein reac-tions [12,20,25,26,46,79,80], and in most of the cases the observedrate constant has been found to be inversely proportional to thefractional power of the viscosity, k a gs

�n (0 < n < 1). A number of ef-forts have been devoted to explain the fractional viscosity depen-dence in protein reactions, but so far there is no generalagreement about the origin of the fractional n value [46]. In the con-sequence of protein dynamics, reactions with n close to 1 are tightlycoupled to surface motion while reactions with n close to zero arewell shielded by rigid protein structure [22]. It has been suggestedthat fractional exponent, n is the degree with which solvent viscos-ity is coupled with (frequency dependent friction) [22,23,52], orpenetrates into (position dependent friction) [21,81] the proteininterior. Any of these variants results in the modification of theStokes law. Yedgar et al. [46] revealed that the fractional index ofa power n is a function of cosolvent molecular weight. If one variesthe solvent viscosity through large cosolvent molecules with highmolecular weight that do not penetrate into protein then one canobtain, n ? 0, i.e., the reaction rate constant does not depend on sol-vent viscosity. With the decrease of cosolvent molecular weight, thevalue of n increases [46].

To describe the deviation at high viscosities for protein reac-tions, Beece et al. [12] added a viscosity-independent term, A toEq. (2),

kdiss ¼ Aþ A0

gns

� �exp

�DGRT

� �ð3Þ

Fits of the rate-viscosity data according to Eq. (3) finally yields, n�0.75 (±0.02) and A �6.5 � 1015 s�1. Equation (3) predicts thedecoupling of structural motion from the solvent at high viscosity[12]. In an early work, to explain why the molecular oxygen rebind-ing rates in myoglobin decreases at low viscosity but saturates athigh viscosity, Doster [22] derived a dynamic friction model assum-ing that the fluctuation spectrum at the reaction site involves twocomponents: solvent independent diffusion of local structural de-fects in the protein matrix and global fluctuations coupled to thesolvent [22]. In general, global fluctuations involve a large part ofthe protein and are therefore coupled to the solvent. Here in ourcase, the global fluctuations of a subglobal structural unit (theM80 containing X-loop) of the protein coupled to the solvent atlow solvent viscosity and thus the rate coefficient for CO-dissocia-tion from the NCO decreased significantly (Fig. 5).

The diffusion of local structural defects with a diffusion coeffi-cient results in the exchange of thermal energy between internaldegrees of freedom in a protein, and is independent of the solventviscosity [22]. When the protein is suspended in a solid surround-ing like poly vinyl alcohol, frozen solvents or dry films etc. the

diffusion of local structural defects can explain the reason of finitereaction rates at infinite external viscosities [12,82]. For proteinreactions, the propagation of the defects (i.e., a hole, or a volume,or a chain fluctuation) to the reaction site speeds up the necessarymotions of the relevant structural element. The physical nature ofdefects may differ from one system to another, and we suggest theoperative defect for the NCO ? N + CO dissociation step in cyt c isthe void created by the departing CO ligand by thermal fluctua-tions. Because of the availability of an increased volume in theproximity due to loss of CO, the movement of the void to the siteof the M80 side-chain can allow a rapid reconfiguration of the lat-ter. Since at high viscosity, the structural motions of reactive modeare faster than the global fluctuation [22,41], therefore the ratecoefficient for CO-dissociation varies only slightly (Fig. 5a). The de-fect diffusion dominates at high solvent viscosity, therefore, tofacilitate structural element motions, the distance migrated by adefect should depend on the diffusion coefficient. The actual valueof diffusion coefficient in a protein interior is independent of thesolvent viscosity and depends only on the internal friction. Themisorganized side-chain of M80 can add onto the internal fric-tional effect for the slow folding of NCO (NCO ? N + CO) by pro-ducing additional drag forces. Thus, at high solvent viscosity, thethermally-induced slow CO-dissociation process or slow foldingof a compact native-like intermediate will depend on the size,the configurational nature, and the collapsed state of the protein.

3.7. Internal friction and its role in protein folding and dynamics

For simple chemical reactions, solvent friction generally need tobe taken into account, however in proteins, where the amino acidresidues are only partially exposed to solvent, other dissipative,‘‘internal friction’’ mechanisms are possible and result in a slow-down of the conformational dynamics. Internal friction, which re-flects the ‘‘roughness’’ of the energy landscape, plays a crucial rolefor proteins through modulating the dynamics of their folding andother conformational changes. However, the experimental quanti-fication of internal friction and its role to protein folding anddynamics has remained elusive.

The microscopic barrier hopping due to long-range inter-resi-due interactions [83] and backbone rotations [81,84] generally re-sults into internal friction in polymers and proteins. Further, thegeometric frustrations of varying degrees in the late folding struc-tures can also contribute to the internal friction. Portman et al. [39]presented both theory and simulations regarding the effect ofinternal friction in barrier crossing by a folding protein. They ob-served that the higher frequency modes of the protein mainly de-pend on large-amplitude local reconfigurations of the backbone,and are controlled by internal friction from dihedral isomeriza-tions, whereas lower frequency modes involve large scale motionsthat are more influenced by solvent friction. They also found thatfor a partially-ordered molecule or a compact state, barrier cross-ing on a multidimensional free energy surface may depend sensi-tively on the higher frequency modes. In this case, even a smallcontribution of internal friction may have large number of conse-quences for the dynamics.

Eaton and coworkers [13,85] proposed an empirical equation todescribe the viscosity dependence of the conformational relaxationrate of myoglobin, by taking into consideration both the solventfriction (gs) and internal friction (r),

kdiss ¼A0

gs þ r

� �exp

�DGRT

� �ð4Þ

Ansari et al. [13] studied nanosecond conformational dynamics infolded myoglobin and suggested a positive value of r(r � 4.1(±1.3) cP) for native myoglobin. Pabit et al. [62] studied la-

R. Kumar et al. / Chemical Physics 418 (2013) 57–64 63

ser induced microsecond folding kinetics of a natively folded car-bonmonycytochrome c (i.e., NCO ? N + CO) and suggested a posi-tive value for r (r � 2.1(±0.3) cP). For the NCO ? N + CO folding,the data clearly extrapolate toward an x-intercept (�r) that is sig-nificantly different from zero, i.e., r � 2.1 (±0.3) cP [62]. Jacobet al. [31] studied millisecond folding kinetics of CspB and also sug-gested a positive value for r. Even for proteins that fold throughcompact transition states, the protein folding data from Eq. (2)has showed no clear evidence for r > 0. Plaxco et al. [29] studiedmillisecond folding kinetics of protein L and calculated a small neg-ative value for r (r = �0.1 (±0.2) cP) from Eq. (2), a result that sug-gests no internal friction in folding [29]. Cellmer et al. [86] studiedfolding/unfolding kinetics of an ultra-fast folding villin headpiecesubdomain and also observed a negative value for r (i.e., r < 0). Inthis very special case, the r < 0 is attributed to the increase of theeffective viscosity, which is probably associated with a shift of thetransition state along the reaction coordinate toward the nativestate [86]. Pradeep et al. [37] studied millisecond unfolding kineticsof barstar [37], and suggested a negative value for r (i.e., r < 0). Oneof the possible reasons for r < 0 is that the local viscosity at the pro-tein–solvent interface is lower than the bulk solvent viscosity [52].In the millisecond unfolding kinetic experiments of barstar, thechange in the folding barrier by viscogen is compensated by addi-tion of denaturant; therefore, the variation of folding rates is solelydefined by the viscosity. Based on this, it was also concluded thatthe violation of Kramers theory with r > 0 is mainly because ofinternal friction [29,37]. This nature of internal friction is mostlikely related to the fact that only small parts of a polypeptide chainare participating in the rate-limiting step of folding.

For the present work, activation free energy barrier for theNCO ? N + CO conversion is not greatly affected by viscogen,therefore, this study does not require any chemical denaturant toobtain isostability conditions in the presence of viscogen. For theNCO ? N + CO case considered here, for example, the misorganizedside-chain of M80 can add onto the internal frictional effect pro-ducing additional drag forces. If the rate-limiting step in a foldingreaction involves internal rearrangement, then the reaction rate isexpected to be strongly influenced by the internal friction as com-pared to the solvent friction [30].

Provided that both internal and solvent frictions are frequency-dependent, an empirical formula was proposed by Wallace et al.[69]:

kdiss ¼A0

ðgs þ rÞn� �

exp�DGRT

� �ð5Þ

The data in Fig. 5 indicate that for viscosities between 0.65 and8.0 cP, the CO-dissociation rate, kdiss is inversely proportional toan effective viscosity, (r + gs)n. Fits of the data according to Eq. (5)finally yields DG = 25.9 (±0.5) kcal mol�1 and r = 1.35 (±0.88) cP.Thus, in low viscosity regime, the speed of CO-dissociation from anative-like folded intermediate, NCO is controlled by internalfriction.

4. Conclusion

In summary, we find that the rate coefficient for the dissocia-tion of CO from the natively folded carbonmonoxycytochrome c(NCO) varies inversely with viscosity at low solvent viscosities(68 cP), but saturates at high viscosities. This result suggests twomajor results: (i) the motional response of the M80 containingX-loop at lower viscosities is different from that of the higher vis-cosities, and (ii) the CO-dissociation reaction involves sequentialstages that depend differently on solvent friction, i.e., solvent cou-pled and nonsolvent-coupled stages of the process. In the low vis-cosity regime (0.65 6 gs 6 8 cP), the rate-viscosity data were fitted

to an alternative modified Kramers model, kdiss = [A0/(r + gs)n]exp(�DG/RT). Fits of the data according to this model finally producedinternal friction, r = 1.35 (±0.88) cP, which suggests that in the lowviscosity regime, the speed of CO-dissociation from the NCO stateis controlled by internal friction.

Acknowledgments

We gratefully acknowledge many helpful suggestions and sup-port of Abani K. Bhuyan (UOH) and Deepak Sharma (IMTECH). Thiswork was supported by the Department of Science and TechnologySERC Fast Track Research Grant (to R.K., project No. SR/FT/CS-070/2009), a DST/INSPIRE Fellowship (to R.J., Fellow code-IF 110240)and Council of Scientific and Industrial Research Grant (to R.K.No. 37(148s)/1r/EMR -II), Government of India.

References

[1] S. Sottini, S. Abbruzzetti, C. Viappiani, S. Bettati, L. Ronda, A. Mozzarelli, J. Phys.Chem. B 109 (2005) 11411.

[2] C.A. Sawicki, M.A. Khaleque, Biophys. J. 44 (1983) 191.[3] M. Lim, T.A. Jackson, P.A. Anfinrud, J. Am. Chem. Soc. 126 (2004) 7946.[4] K.A. Merchant, W.G. Noid, R. Akiyama, I.J. Finkelstein, A. Goun, B.L. McClain,

R.F. Loring, M.D. Fayer, J. Am. Chem. Soc. 125 (2003) 13804.[5] K.D. Rector, J. Jiang, M. Berg, M.D. Fayer, J. Phys. Chem. B 105 (2001) 1081.[6] C. Viappiani, S. Bettati, S. Bruno, L. Ronda, S. Abbruzzetti, A. Mozzarelli, W.A.

Eaton, Proc. Natl. Acad. Sci. USA 101 (2004) 14414.[7] A. Ansari, M.J. Colleen, E.R. Henry, J. Hofrichter, W.A. Eaton, Biochemistry 33

(1994) 5128.[8] H. Frauenfelder, B.H. McMahon, R.H. Austin, K. Chu, J.T. Groves, Proc. Natl.

Acad. Sci. USA 98 (2001) 2370.[9] P.W. Fenimore, H. Frauenfelder, B.H. McMahon, R.D. Young, Proc. Natl. Acad.

Sci. USA 101 (2004) 14408.[10] P.W. Fenimore, H. Frauenfelder, B.H. McMahon, F.G. Parak, Proc. Natl. Acad. Sci.

USA 99 (2002) 16047.[11] F. Parakand, H. Frauenfelder, Phys. A 201 (1993) 332.[12] D. Beece, L. Eisenstein, H. Frauenfelder, D. Good, M.C. Marden, L. Reinisch, A.H.

Reynolds, L.B. Sorensen, K.T. Yue, Biochemistry 19 (1980) 5147.[13] A. Ansari, C.M. Jones, E.R. Henry, J. Hofrichter, W.A. Eaton, Science 256 (1992)

1796.[14] R.E. McKinnie, J.S. Olson, J. Biol. Chem. 256 (1981) 8928.[15] R. Walser, W.F. Gunsteren, Proteins 42 (2001) 414.[16] M. Tarek, D.J. Tobias, Phys. Rev. Lett. 88 (2002) 138101.[17] G. Caliskan, D. Mechtani, J.H. Roh, A. Kisliuk, A.P. Sokolov, S. Azzam, M.T.

Cicerone, S. Lin-Gibson, I. Peral, J. Chem. Phys. 121 (2004) 1978.[18] H.A. Kramers, Physica 7 (1940) 284.[19] P. Hanggi, P. Talkner, M. Borkovec, Rev. Mod. Phys. 62 (1990) 251.[20] B. Gavish, M.M. Werber, Biochemistry 18 (1979) 1269.[21] B. Gavish, Phys. Rev. Lett. 44 (1980) 1160.[22] W. Doster, Biophys. Chem. 17 (1983) 97.[23] J. Schlitter, Chem. Phys. 120 (1988) 187.[24] B. Gavish, Biophys. Struct. Mech. 4 (1978) 37.[25] D. Lavalette, C. Tetreau, Eur. J. Biochem. 177 (1988) 97.[26] K. Ng, A. Rosenberg, Biophys. Chem. 39 (1991) 57.[27] B.A. Chrunyk, C.R. Matthews, Biochemistry 29 (1990) 2149.[28] M. Jacob, T. Schindler, J. Balbach, F.X. Schmid, Proc. Natl. Acad. Sci. USA 94

(1997) 5622.[29] K.W. Plaxco, D. Baker, Proc. Natl. Acad. Sci. USA 95 (1998) 13591.[30] R.P. Bhattacharyya, T.R. Sosnick, Biochemistry 38 (1999) 2601.[31] M. Jacob, M. Geeves, G. Holtermann, F.X. Schmid, Nat. Struct. Biol. 6 (1999)

923.[32] A.G. Ladurner, A.R. Fersht, Nat. Struct. Biol. 6 (1999) 28.[33] M. Jacob, F.X. Schmid, Biochemistry 38 (1999) 13773.[34] G.S. Jas, W.A. Eaton, J. Hofrichter, J. Phys. Chem. B 105 (2001) 261.[35] M. Silow, M. Oliveberg, J. Mol. Biol. 326 (2003) 263.[36] O. Bilsel, C.R. Matthews, Adv. Protein Chem. 53 (2000) 153.[37] L. Pradeep, J.B. Udgaonkar, J. Mol. Biol. 366 (2007) 1016.[38] D.K. Klimov, D. Thirumalai, Phys. Rev. Lett. 79 (1997) 317.[39] J.J. Portman, S. Takada, P.G. Wolynes, J. Chem. Phys. 114 (2001) 5082.[40] R.B. Best, G. Hummer, Phys. Rev. Lett. 96 (2006) 228104.[41] R. Kumar, A.K. Bhuyan, J. Phys. Chem. B 112 (2008) 12549.[42] L. Qiu, S.J. Hagen, Chem. Phys. 307 (2004) 243.[43] S. Hagen, Curr. Protein Pept. Sci. 11 (2010) 385.[44] A.E. Sitnitsky, Chem. Phys. 369 (2010) 37.[45] A.A. Rauscher, S.Z. Szöllosi, G.J. Gráf L, I. Derenyi, A.M. Csizmadia, FASEB J. 25

(2011) 2804.[46] S. Yedgar, C. Tetreau, B. Gavish, D. Lavalette, Biophys. J. 68 (1995) 665.[47] T. Kleinert, W. Doster, H. Leyser, P. Winfried, V. Schwarz, M. Settles,

Biochemistry (1998) 717.[48] H. Oh-oka, M. Iwaki, S. Itoh, Biochemistry 36 (1997) 9267.

64 R. Kumar et al. / Chemical Physics 418 (2013) 57–64

[49] R.B. Gregory, in: R.B. Gregory (Ed.), Protein-solvent intereactions, MarcelDekker Inc., New York, 1995, p. 343.

[50] A.T. Bullock, G.C. Cameron, P.M. Smith, J. Chem. Soc. Farad. Trans. II (70) (1974)1202.

[51] J.A. Montgomery Jr., D. Chandler, B.J. Berne, J. Chem. Phys. 70 (1979) 4056.[52] C.L. Brooks, M. Karplus, J. Mol. Biol. 208 (1989) 159.[53] M. Settles, F. Post, D. Muller, A. Schulte, W. Doster, Biophys. Chem. 43 (1992)

107.[54] A.M. Massari, I.J. Finkelstein, M.D. Fayer, J. Am. Chem. Soc. 128 (2006) 3990.[55] J. Finkelstein, A.M. Massari, M.D. Fayer, Biophys. J. 92 (2007) 3652.[56] M.D. Fayer, Annu. Rev. Phys. Chem. 52 (2001) 315.[57] R.F. Latypov, K. Maki, H. Cheng, S.D. Luck, H. Roder, J. Mol. Biol. 383 (2008)

437.[58] C.M. Jones, E.R. Henry, Y. Hu, C.K. Chan, S.D. Luck, A. Bhuyan, H. Roder, J.

Hofrichter, W.A. Eaton, Proc. Nat. Acad. Sci. USA 90 (1993) 11860.[59] A.K. Bhuyan, Biochemistry 41 (2002) 13386.[60] R. Kumar, N.P. Prabhu, M. Yadaiah, A.K. Bhuyan, Biophys. J. 87 (2004) 2656.[61] M. Yadaiah, R. Kumar, A.K. Bhuyan, Biochemistry 46 (2007) 2545.[62] S.A. Pabit, H. Roder, S.J. Hagen, Biochemistry 43 (2004) 12532.[63] R. Kumar, N.P. Prabhu, A.K. Bhuyan, Biochemistry 44 (2005) 9359.[64] A.K. Bhuyan, R. Kumar, Biochemistry 41 (2002) 12821.[65] P.G. Wolynes, Q. Rev, Biophysics 38 (2005) 405.[66] J.D. Bryngelson, P.G. Wolynes, Proc. Natl. Acad. Sci. USA 84 (1987) 7524.[67] P.G. Wolynes, J.N. Onuchic, D. Thirumalai, Science 267 (1995) 1619.[68] P. Ascenzi, M. Coletta, R. Santucci, F. Polizio, A. Desideri, J. Inorg. Biochem. 53

(1994) 273.

[69] M.I. Wallace, L.M. Ying, S. Balasubramanian, D. Klenerman, Proc. Natl. Acad.Sci. USA 98 (2001) 5584.

[70] T. Takano, R.E. Dickerson, J. Mol. Biol. 153 (1981) 79.[71] T. Takano, R.E. Dickerson, J. Mol. Biol. 153 (1981) 95.[72] L. Banci, I. Bertini, H.B. Gray, C. Luchinat, T. Reddig, A. Rosato, P. Turano,

Biochemistry 36 (1997) 9867.[73] L. Banci, I. Bertini, J.G. Huber, G.A. Spyroulias, P. Turano, J. Biol. Inorg. Chem. 4

(1999) 21.[74] J.F. Leszczynski, G.D. Rose, Science 234 (1986) 849.[75] W.Y. Yang, M. Gruebele, Nature 423 (2003) 193.[76] M.C.R. Shastry, H. Roder, Nat. Struct. Biol. 5 (1998) 385.[77] G.A. Petsko, D. Ringe, Ann. Rev. Biophys. Bioeng. 13 (1984) 331.[78] M. Karplus, Methods Enzymol. 131 (1986) 283.[79] W. Doster, T. Kleinert, F. Post, M. Settles, in: R.B. Gregory (Ed.), Protein-solvent

intereactions, Marcel Dekker Inc., New York, 1995, p. 375.[80] A. Rosenberg, K. Ng, M. Punyiczki, J. Mol. Liq. 42 (1989) 31.[81] M. Fixman, Faraday Discuss. Chem. Soc. 83 (1987) 199.[82] R. Korenstein, B. Hess, Nature 270 (1977) 184.[83] P.G. de Gennes, Cornell University Press, New York, 1979.[84] M. Fixman, J. Chem. Phys. 89 (1988) 2442.[85] S.J. Hagen, J. Hofrichter, W.A. Eaton, Science 269 (1995) 959.[86] T. Cellmer, E.R. Henry, J. Hofricheter, W.A. Eaton, Proc. Natl. Acad. Sci. USA 105

(2008) 18320.[87] M.M. Santoro, D.W. Bolen, Biochemistry 27 (1988) 8063.[88] G.W. Bushnell, G.V. Louie, G.D. Brayer, J. Mol. Biol. 214 (1990) 585.[89] J. Kestin, M. Sokolov, W.A. Wakeham, J. Phys. Chem. Ref. Data 7 (1978) 941.