Selected Topics in Physical Chemistry
Selected Topics in Physical Chemistry CHEM 860
Popular in Course
Popular in Chemistry
verified elite notetaker
verified elite notetaker
verified elite notetaker
verified elite notetaker
verified elite notetaker
verified elite notetaker
This 17 page Class Notes was uploaded by Ms. Jerrell Lind on Thursday September 17, 2015. The Class Notes belongs to CHEM 860 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/205352/chem-860-university-of-wisconsin-madison in Chemistry at University of Wisconsin - Madison.
Reviews for Selected Topics in Physical Chemistry
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
Date Created: 09/17/15
J Phys Chem B 2000 104 1127 11 FEATURE ARTICLE Aspects of Protein Reaction Dynamics Deviations from Simple Behavior Martin Karplus Laboratoire de Chimie Biophysique ISIS Universite Louis Pasteur 67000 Strasbourg France and Department of Chemistry amp Chemical Biology Harvard University Cambridge Massachusetts 02138 Received October 6 1999 Proteins are complex systems so that their reactions may have characteristics not often observed in small molecules Two common aspects of small molecule reactions are an exponential time dependence and Arrheniuslike temperature dependence This review shows that corresponding behavior is not always observed in reactions involving proteins Bondm aking and bondbreaking reactions as well as conformational changes which do not involve chemical bonds are discussed As examples of bond making bond breaking reactions enzyme catalysis by triosephosphate isomerase and ligand binding by myoglobin are considered Several reactions involving conformational changes are described They include aromatic side chain rotations loop lid opening and closing motions quaternary structural changes in hemoglobin and protein folding In some of these cases deviations from simple small molecule behavior have been observed but it is evident that more information is needed for a complete understanding The protein folding reaction which is most complex because it involves the entire polypeptide chain is analyzed in some detail I Introduction Macromolecules of biological interest include proteins nucleic acids lipids and carbohydrates They form complex mesoscopic systems with many degrees of freedom and multidimensional potential energy surfaces Their motions cover a wide range of time scales from femtoseconds to seconds or even longer and length scales from 001 to 10 A or even larger A fundamental question concerns the relationship between the microscopic interactions which govern the potential energy surface energy landscape and the observed macro scopic behavior Of particular interest is whether and if so how the microscopic complexity manifests itself in the simplicity or complexity of macroscopic observables This review focuses on proteins with the expectation that a corresponding analysis is applicable to other macromolecular biological systems Proteins participate in a wide range of reactions all of which involve internal motions Some as in the binding and release of oxygen in myoglobin and hemoglobin and the catalysis of reactions by enzymes involve the making and breaking of chemical bonds Many other reactions of proteins involve conformational changes such as hingebending motions subunit reorientations and the folding and unfolding reactions of the polypeptide chain They are governed by relatively weak nonbonded interactions arising from the van der Waals and electrostatic terms in the potential energy Any or all of these reactions may have complex time and temperature dependences because of the wide range in the time and length scales of the underlying protein motions Proteins in their native state have been shown to have a large number of similar con gurations or substates This conclusion is based on a variety of experiments the earliest of these involved studies of the ligand rebinding reaction of myoglobin The nature of and the transitions between these substates have been analyzed by quenched molecular dynamics simulations2 101021jp993555t CCC 1900 Since these substates are likely to have functionally different properties their role in protein reactions has to be considered More generally there is the question of whether the distribution of protein con gurations plays a role in the kinetics of bio logically important processes at room temperature or at other temperatures What is the effect on protein reactions of the rate at which substates interconvert Are the effects of the con gurations of the protein that govern the rate of a given phenomenon localized or do they involve the protein as a whole Does nonexponential behavior arise from relaxation phenomena in a nonequilibriurn situation Questions of this type which are of interest for localized events eg ligand binding become even more important for global reactions such as protein folding for which much less is known about the nature and complexity of the energy surface To focus the present analysis it is useful to de ne rst what is meant by simplicity and complexity in a reaction Given this framework some examples are used to illustrate what has been found theoretically and experimentally in reactions involv ing proteins The examples include the catalysis of the conver sion of DHAP to GAP by the enzyme triosephosphate isomerase ligand rebinding a er photolysis of myoglobin and certain cases of localized conformational changes in native proteins Finally the protein folding reaction which involves the entire polypep tide chain is discussed The concluding section summarizes the present situation and the outlook for a more complete under standing of reactions involving proteins 11 Kinetic Background and Definitions Simple versus complex behavior for a reaction is de ned here in terms of two not unrelated criteria A reaction is considered to have simple behavior if a phenomenological expression involving rate constants describes the time dependence of the 2000 American Chemical Society Published on Web 12111999 12 J Phys Chem B Vol 104 No 1 2000 reaction and if the temperature dependence of the rate is Arrheniuslike We consider the unimolecular reaction A 2 B which is appropriate for many protein reactions including geminate rebinding in myoglobin an enzymatic reaction a er the Michae lis complex has formed conformational change and protein folding and unfolding The differential expressions for the rate of the reaction can be written Mr 7 0330 1a 7 kAAt kBBO 1b where kA and k3 are the rate coef cients for the forward and reverse reaction and At and Bt are the concentrations of species A and B respectively at time t39 low concentrations the limit of in nite dilution are assumed to avoid the need to consider interactions among molecules If the rate coef cients A an k5 are essentially independent of time ie kAt kA kBt k5 they are referred to as the rate constants for the reaction The integrated rate expressions are then Ma At 7 Aeq AA0 6 2a A30 31 7 Beq AB0 6 2b where Aeq and Beq are the concentrations of A and B at equilibrium and AAI and ABC and AA0 and AB0 are the deviations from equilibrium at time t and t 0 respec tively The constant kin eqs 2 is the sum of the rate constants k kA kg Equations 2 are the standard rate expressions for a unimo lecular reaction and their validity corresponds to the rst criterion for simplicity lfk is a function of time the reaction is not simple by our de nition For higher order reactions corresponding expression involving a rate constant can be written for the differential rate law though the integrated expression is more complex except under certain limiting conditions An insightful discussion of the treatment of cases where k depends on the time and the reaction rate is nonexpo nential has been given by Zwanzig124 He makes a use il separation between static disorder different molecules in the ensemble have different activation barriers and d3mamic disorder the value of kwithin a molecule varies with time The term strange kinetics has also been introduced to describe such systems125 I return to this point in section IVB The second criterion for a simplicity of a reaction is that the temperature dependence of the rate follows an Arrheniuslike equation This requires that the rate constant can be written to a good approximation in the form k AU am 3a where AEquot the activation energy is approximately independent of temperature andAT is the preexponential factor with only a weak temperature dependence Given eq 3a we have d ln k AEquot 017 T T 3b ie aplot of ln kversus lT approximates a straight line More generally we can write k AU 6W 4a Karplus and by the GibbsiHelmholtz relation d In k AHquot g 7 4b dlT R where AGquot and AHquot are the free energy and enthalpy of activation and A T is the appropriate preexponential factor which is signi cantly different from AT in many cases For elementary asphase reactions involving small mol ecules both conditions for simplicity are usually satis ed over a wide range of temperatures4 However even in these systems where the entropic contribution to the reaction is small it is important for quantitative interpretations to consider the tem perature dependence of the preexponential factor An example is provided by the H H2 exchange reaction The temperature dependence of the rate constant calculated by the quasiclassical trajectory method5 can be tted by the Arrhenius expression kT 43 gtlt lO13 exp77435RT cm3 mol 1 sf1 5a over the temperature range 300 to 1000 K A slightly better t is obtained by taking the temperature dependence of the pre exponential factor into account The result is kT 79 x 109T118exp7623Rr cm3 mol 1 s 1 5b Equations 5 make clear that even for this simple reaction with a high activation barrier some care has to be used in interpreting the numerical values obtained by tting Arrheniuslike equa tions By contrast for systems such as disordered crystals and glasses 8 nonexponential relaxation a er a perturbation and nonArrhenius temperature dependence for motional phenomena are commonly observed Although phenomenological models exist that ascribe such behavior to the multiplicity of time scales the details of the origins of the complexity are generally not fully understood at the atomic level Observing corresponding complexities in protein reactions would not be surprising given the multiplicity of time scales for their internal motions III Aspects of Simple Behavior IIIA Criterion for Existence of a Rate Constant There are three essential aspects of a reaction that are involved in the existence of a wellde ned rate constant see for example ref 3 We consider here a single elementary step in a reaction39 if the overall reaction has several steps as in many enzymes9 the discussion would apply separately to each one The rst requirement is that it is possible to de ne a coordinate the reaction coordinate or other progress variable for the transition from reactants to products the second requirement is that there exists a wellde ned barrier with a free energy several times kT separating the reactant and product states along the reaction coordinate and the third intimately related to the other two is that the relaxation time of all the degrees of freedom involved in the transformation other than the reaction coordinate is fast relative to motion along the reaction coordinate Given that these three requirements are satis ed a rate constant can be de ned and simple exponential behavior is expected It can be true that a rate constant simple rate expression exists at one temperature but not at another because the relaxation times of different modes of the system depend differently on the temperature To make clearer what is involved in the three requirements it is useful to review the activated dynamics method that is widely used for calculating the rates of chemical reactions The method was originally proposed for gasphase reactionsw 12 and Feature Article has been developed and applied in its present form to reactions in solution 14 and in proteins1 Following the formulation of Chandler3 we can write the rate coef cient in the form wimw wu 6 f mod where Z is the reaction coordinate and the double dagger 1 de nes the value of the coordinate at the transition state The symbol p represents the probability density for the system as a function of E which corresponds to the free energy potential of mean force W de ned by WZ RTln 0ZC 7 with appropriate normalization constant C The activation free energy in eq 4a can now be written m 11 MC 1 where the integral 1 is over the values of the reaction coordinate associated with the reactant valley Such a potential of mean force is meaning il if and only if for each value of the reaction coordinate Z the system can be assumed to be in equilibrium with respect to all of the other degrees of freedom ie the latter must relax rapidly relative to motion along the reaction coordinate That the barrier be suf ciently high and well de ned means that the essential step determining the rate of the reaction is the passage over the barrier which is a rare event If the barrier were very low or nonexistent one might expect to observe a diffusive rate for a reaction in solution that would not generally satisfy the conditions for a simple reaction The quantity m is the equilibrium average value of the time derivative of Z 7 dZdt evaluated with Z restrained to the transition state i This can be determined computationally by restricting the system to Z f and doing simulations that are long enough to obtain the average velocity39 again this procedure is meaningful only if there is a separation of time scales Finally K is the transmission coef cient which in gas phase reactions usually has a value close to unity ess surface crossing or very high collision energies are involved However in solution or in proteins K often is signi cantly less than unity The time dependent transmission coef cient Kt can be calculated from the reaction ux expression3v13 AGquot iRT ln 8 K0 Nlti09ZI ill 9 where 0 t f is a step function that equals 1 when the reaction coordinate is in the product state and is zero otherwise N is a normalization constant such that K0 is equal to unity Equation 7 corresponds to the transition state theory TST expression if Kt equals unity39 that is it is assumed in TST that there are no recrossings which would reduce the rate constant so that the TST value is an upper limit to the reaction rate The function Kt is in fact equal to unity at very short times corresponding to the fact that no recrossings have occurred For a wellde ned rate constant there is then a short relaxation period in which systems may recross the transition state one or more times until they settle into the reactant well no contribution to reaction rate or into the product well a contribution to reaction rate The requirement for a high activation barrier relative to kT corresponds to the fact that J Phys Chem B Vol 104 No 1 2000 13 particularly for a unimolecular reaction collisions remove kinetic energy from the reaction coordinate as soon as the system has le the transition state region so the system is quenched in the reactant or product region This is true in a suf ciently dense system like a macromolecule or aliquid15 If a separation of time scales were not present Kt would not reach a plateau value and the rate coef cient would have a signi cant time dependence39 ie the reaction rate would not show the simple exponential behavior corresponding to eqs 2 Concomitant with this is the possibility that the de nition of a potential of mean force barrier as in eqs 68 is not appropriate for the system39 that is the coupling of different degrees of freedom may be such that a reaction coordinate which varies slowly relative to other degrees of freedom does not exist In proteins where there are many different types of degrees of freedom with different time scales of relaxation deviations from simple behavior is not unexpected There are two additional points The rst is that we have used the description of the activated dynamics method primarily as a way of indicating the factors involved in the existence of a rate constant It is of course not necessary for the existence of simple exponential behavior that we can determine the reac tion coordinate or appropriate progress variable It is required only that there exists such a coordinate with the properties we have described This is clearly relevant to protein folding reac tions where as we discuss below the de nition of a reaction coordinate is a major problem More generally a dif culty in applying the activated dynamics method to complicated many body systems is the determination of an optimal reaction coor dinate and the transition state as a function of that coordinate This problem is rooted in the conceptual and computational complexity associated with nding the minimal number of atomic coordinates which adequately specify a dividing surface between a reactan con guration and a product con gura tion The second point is that the use of eq 6 does not require knowing the optimal transition state which is essential only for the application of transition state theory With eq 6 a poor choice of reaction coordinate or transition state dividing surface does not invalidate the method in principle The result is that the transmission coef cient will have a reduced value However the ef ciency of the transition state sampling by activated dynamics decreases rapidly as the choice of reaction coordinate and transition state become less than optimal To illustrate the possible complexities of determining the reaction coordinate we show two potential energy surfaces in Figure l The rst Figure la is a calculated collinear surface for the H H2 reaction5 Its shape is such that the reaction can be followed by the coordinate REC for example with RAB expressed as a mction of REC Alternatively the reaction coordinate can be expressed as a uniquel de ned linear combination of RBC and RAB4 Figure lb represents the free energy surface calculated with a lZSmer bead model for the protein folding reaction39102 a free energy surface is shown because the entropy plays an important role in the reaction The surface is drawn as a function of two coordinates that are adequate for describing the complexity of the reaction102 The variable Qc is the number of core contacts and Q5 is the number of surface contacts details are given in the original paper The gure shows two ominant average paths that make clear that a single reaction coordinate cannot be used For the slow path on the left Q is not a monotonic function of Qc because an intermediate 1 is formed and the Q5 values decrease as the reaction proceeds from 139 that is some contacts must be broken for the reaction to go forward from the intermediate 14 J Phys Chem B Vol 104 No I 2000 Figure 1 Simple and complex reaction surfaces In both cases the reaction proceeds from the upper right to the lower left a Potential energy surface for the reaction HAB Hc a HA HBc The surface is calculated for a linear collision as a function of RAB the HA HB distance and BBC the HB Hc distance The welldefined reaction coordinate is shown as the heavy white line in the gure adapted from refs 5 and 93 b Free energy surface for a 125mer heteropoly mer protein model The two coordinates are a minimal set required to describe the free energy surface and the possibility of multiple pathways on that surface QC represents the fraction of native core contacts and Q5 the fraction of native surface contacts Two average paths are shown one white arrows represents fast track folding without an intermediate and the other black arrows corresponds to the most common slow track with an intermediate adapted from ref 102 which presents the details of the model and its analysis By contrast for the fast track it appears that QC could be used as the reaction coordinate that is QS could be expressed a function of QC to describe the reaction in this region However this is not true because the path shown in the figure is an average over an ensemble of paths each with different values of QS as a function of QC IIIB Validity of Arrhenius Equation The second criterion for simplicity is concemed with the variation of the rate constant if it exists with temperature If the reaction has an Arrhenius like temperature as defined in eqs 3 and 4 it is regarded as simple from this point of View Gasphase reactions of small molecules with welldefined barriers at least several times kT tend to have rate constants that obey the Arrhenius law a plot Karplus of ln k versus lT approximates a straight line with an increase in ln k as T increases eqs 4b with AH constant Deviations from such a simple Arrhenius temperature de pendence in complex systems can be of several types At high temperature significant curvature in the Arrhenius plot may appear with the possibility of a decrease in the rate with increasing temperature This has been observed in the protein folding reactions and is discussed below At low temperatures there are several types of deviation from Arrhenius behavior One of these corresponds to a decrease in the temperature dependence ie the rate constant falls off more slowly than predicted by the Arrhenius law that can arise from the contribution of tunneling to the reaction rate The other deviation corresponds to a higher order temperature dependence that is the reaction rate decreases more rapidly at low temperatures than expected from the Arrhenius law Such behavior is often referred to as a superArrhenius temperature dependence We consider both types of deviations from Arrhenius behavior in the relevant sections IV Examples of Protein Reactions IVA Dynamics of Enzymatic Reactions The enzyme triosephosphate isomerase TIM catalyzes the interconversion of dihydroxyacetone phosphate DHAP and glyceraldehyde 3phosphate GAP in a central step of the glycolytic cycle9 TIM is of particular interest because it has been characterized as a perfect enzyme that is the chemical steps in the reaction are accelerated such that the overall rate is limited by the diffusioncontrolled binding of the substrate and release of the product17 The reaction DHAP lt GAP has several steps The first of these which we consider in what follows consists of the transfer of a proton from DHAP to the residue Glu 165 of TIM which acts as a base to form an enediolate intermediate see Figure 2 The rate enhancement of this step is about 5 x 105 relative to the reaction catalyzed by a general base in solution18 As has been shown19gt20 the reduction of the activation barrier is due to interactions of the reactants and the transition state with specific residues of the enzyme Displacements of atoms in the enzyme by l A or so can have a large effect on the activation barrier21gt22 Such a tight coupling of structure and reactivity raises the question of its possible effect on the dynamics of the reaction that is do the structural requirements for transition state stabilization result in a significant reduction in the effective rate of crossing the barrier Is the reaction simple in terms of the present definition To begin to answer these questions the activated dynamics method summarized in section IIIA was applied to the first step of the TIM reaction23 that is a reaction coordinate was selected the transition state was determined by calculating the potential of mean force along the reaction coordinate and the transmission coefficient was evalu ated by use of the reaction ux method This is the first time the transmission coefficient has been determined for an enzy matic reaction Warshel and coworkers have made important studies of the dynamics of reactions in solution and enzymes24gt25 but their conclusions were based on more approximate treat ments that did not involve calculations of the transmission coefficient Although the activation energy had been studied by a semi empirical molecular orbital QMMIVI method a simpler repre sentation of the potential energy surface was needed to calculate the large number of trajectories that are required for evaluating the transmission coefficient The representation used describes the surface for the reaction in terms of the coupling of two states one corresponding to the reactants and the other to the products Feature Article OH O H cf vVWVVVC O Glu1 l O CHK P04 DHAP J Phys Chem B Vol 104 No 1 2000 15 OH OH 01 gt Iwwwc c o Glu15 l 0 CH2 PO4 Endiolate Figure 2 Initial proton transfer step in the catalytic mechanism of triosephosphate isomerase A proton is transferred from the Cl carbon of t DHAP to one of the carboxyl oxygens of GLU165 yielding an endiolate interme a e 050 39l O1O O30 Veff Kcalmol the transmission coef cient The potential energy was calculated with all degrees of freedom other than 010 qas A Figure 3 Potential of mean force near the transition state and the potential energy for three initial con gurations used in the calculation of the reaction coordinate xed The zero of energy is at q and AV is the height of the instantaneous barrier that trajectories need to surmount In each state the potential surface is represented by a standard molecular mechanics MM force eld The possibility of reaction is introduced by coupling the two electronic states257 The potential function VR is chosen to have the EVB forrn2 gt27 VR gnu V102 50mm VPR2 4sz 10 where R is the set of all coordinates VRR and VpR are the MM potential energies in the reactants and products states respectively and VRp is approximated by a constant coupling assumed to be independent of R for simplicity An appropriate reaction coordinate for the proton transfer from carbon C1 of DHAP to the O of Glu 165 see Figure 2 is the asymmetric stretch de ned by mr imr ll qas mcmo CCH OOH where rm and rOH are the distances of the proton from the donor carbon and the acceptor oxygen respectively and mo and mo are their masses for a collinear arrangemth gas has a reduced mass of 097 amu In the reaction the proton moves from the vicinity of the donor carbon atom an ll A to the vicinity of the accepting oxygen qas 03 A To determine the location of the transition state the potential of mean force PMF28 as a function of qas was determined by use of umbrella sampling29 As already described the PMF is an effective potential for the reaction coordinate that is based on an equilibrium average over all other degrees of freedom The resulting free energy pro le for the reaction is shown in Figure 3 As a starting point for the transition state trajectory calcula tions equilibrium con gurations in the transition state were sample y running a 400 ps trajectory qa5t was constrained to q with a SHAKElike algorithm30 and 40 con gurations at 10 ps intervals were saved For each of these con gurations 100 trajectories were initiated with different values of mm sampled from a Maxwell distribution The function Kt calculated from these trajectories is shown in Figure 4 It decays rapidly and has almost reached a plateau value by 10 fs The plateau value is K 043 j 008 a signi cant reduction from the TST value of unity However the reduction by a factor of 2 is a small effect relative to the speci c interactions in the protein that lower the potential of mean force barrier for the reaction so as to increase the rate by 5 orders of magnitude The rapid decay of Kt to the plateau value makes it likely that most degrees of freedom coupled to the reaction coordinate do not change signi cantly during the time required to trap the traj ectory in the reactant or product wells De ning the substrate 16 J Phys Chem B Vol 104 No 1 2000 Karplus K0 30 4O 50 t fs Figure 4 The timedependent transmission coef cient Kt when all atoms are allowed to move i and in a rigid environment see text and the GLUI65 atoms as the intramolecular part and the rest of the system as the environment the calculation of the transmission coef cient was repeated from the same initial con gurations but with only the intramolecular atoms allowed to move and the environment atoms xed in their initial 1 0 positions The behavior of Kt is very similar to that obtained when all atoms are allowed to move Figure 4 This demon strates that the dynamics of the environment atoms is unim portant for the barrier crossing per se and suggests that the structure at t 0 determines whether a reaction takes place If all degrees of freedom other then the reaction coordinate are frozen frozen bath assumption and there is a barrier AV in addition to the potential of mean force barrier for a particular set of bath coordinates a con guration chosen from the 400 ps trajectory the transition state trajectory has to surmount that barrier to complete the reaction the range in AV observed in the calculations is about 05 kcalmol see Figure 3 This reduces the rate from the TST value which is based on the potential of mean force barrier With such a frozen bath assumption the transmission coef cient is given by31 Kfroz e 7A Wag r 12 where k5 is Boltzmann constant T the is temperature and the average is over all initial transition state con gurations Figure 3 The average over the 40 transitionstate con gurations gives Kfmz 040 in good agreement with the activated dynamics simulations This con rms that the frozen bath approximation is valid that is the surrounding system is xed within the short time scale of the barrier crossin wo mechanisms have been proposed for the in uence of the environment on the instantaneous barrier The rst found in a study of a symmetric SN reaction in water31 corresponds to a nonequilibrium solvation mechanism In that reaction the instantaneous barrier arises from different solvation states of the intramolecular subsystem for example a uctuation in the environment that results in better solvation of the reactant relative to the products introduces an instantaneous barrier located between the transition state and the products state and vice versa In an alternative mechanism for the environmental in uence the instantaneous barrier is determined by the con guration of the intramolecular subsystem but the intramo lecular dynamics is coupled to lowfrequency uctuations of the environment for example intramolecular motions that are required for the reaction are hindered by the environment which appears rigid on the time scale of the barrier crossing N10 fs but uctuates on a picosecond time scale The two mechanisms were tested for the proton transfer in the TIM reaction and it was demonstrated that the second mechanism is dominant in the barrier modulation and in the reduction of the rate from the transition state limit Which mechanism operates for the proton transfer in solution from DHAP to acetate for example and the magnitude of the effect ie the value of Kt have not been determined Cui and Karplus work in progress It is possible also that in enzymes or in solution a single reaction coordinate is not suf cient to describe the reaction as discussed by reference to Figure I This appears not to be the case for TIM unlike ligand rebinding to myoglobin after photodissociation see section IVB From the above analysis there is a wellde ned rate constant for the reactions ie the reaction is simple according to the rst criterion and eqs 2 are applicable As the calculations have shown there are three time scales for the reaction There is the very fast time scale of relaxation of the highenergy transition state trajectories to a thermal distribution N10 fs there is the time scale of the motions associated with the enzyme environ ment that determines the height of the transition state barrier NI ps The dominant modulations of the barrier are of a low frequency character a lt 300 cm l and originate from the interaction between the intramolecular subsystem and the rest of the enzyme Finally there is the time scale associated with the overall reaction rate NI ms Since the third is much longer than the other two simple behavior as exempli ed by the well de ned plateau value for the transmission coef cient is expected and was obtained in the calculations However it Feature Article should be noted that the trajectories covered a time scale 400 ps many orders of magnitude shorter than that of the overall reaction rate ms Consequently it is not excluded that there exist very slow structural uctuations on the same time scale or longer than the reaction that alter the barrier This would lead to nonexponential behavior but it appears very unlikely at room temperature for such a slow reaction at low temperatures nonex onential kinetics might occur see section IVb below Interestingly single molecule studies of alkaline phosphatase have shown signi cantly different activities for individual molecules32 However it is likely that in this case there was a difference in composition eg glycosylation that was the source of the variation in reaction rates In the future single molecule techniques are likely to provide important new information in this area Concerning the other aspect of simplicity the Arrheniuslike temperature dependence data for the reaction catalyzed by TIM are not avai a e IVB Rebinding of Ligands after Photodissociation of Myoglobin A welldocumented case of complex behavior in a protein reaction is the geminate rebinding of ligands to myoglobin after ash photolysislvmv 4 This reaction has been studied experimentally over a wide range of temperatures and time scales and numerous theoretical analyses of the results have been made Moreover the highresolution Xray structures of liganded and unliganded myoglobin do not reveal any path by which ligands can move between the heme binding site and the outside of the protein3535 Since uctuations must therefore be involved in the overall ligand binding reaction1gt2v37 39 myoglobin has become a model system for studying the relation of motion to function in proteins The phenomenological description of the kinetics of photo dissociation and rebinding of a ligand X to myoglobin Mb can be written as MbX Mb X pocket M3 X protein Mb X solution 13 where the subscripts refers to the location of the ligand X Each of the designated species may involve several different states or substates on a microscopic level At low temperatures below 160 K in mixed glycerolwater it is found that the geminate rebinding of the CO ligand Mb Xpocket m MbX is nonexponential As the temperature is raised the geminate recombination of CO becomes essentially exponential Thus the rst criterion for a simple reaction eqs 1 and 2 is satis ed at room temperature but violated at low temperature For NO in contrast to CO the geminate rebinding at room temperature is nonexponential40 the temperature dependence of NO rebind ing has not been studied Thus the difference between the CO and NO behavior can aid in elucidating the source of complexity in this reaction The very long times over which the CO rebinding is nonexponential and timedelay ash experiments at low tem peratures1 suggest that an inhomogeneous model is involved Thus a description corresponding to the frozen bath model ie protein molecules are in different substate with different activation barriers see section IVA with the equilibration between bath con gurations slow compared to the overall reaction appears to be appropriate This is in accord with equilibrium molecular dynamics simulations at 80 K41 and incoherent neutron scattering experiments 43 They indicate that protein uctuations at such temperatures are relatively harmonic and correspond to oscillations in a single well or substate This J Phys Chem B Vol 104 No 1 2000 17 means that a distribution of barriers for a population of photolyzed myoglobin molecules is required for describing the observed reaction rather than a single potential of mean force surface The number of unliganded Mb hemes Nt as a function of time after photolysis can then be written as NO f dEgOE CXPI MEW 14 where gE is the distribution of barrier heights and kE is the rate constant for a barrier of energy E Equation 14 results in nonexponential relaxation that is the rate coef cient is time dependent and its form is determined by the choice of gE and kE Both power law or stretched exponential expressions for Nt can be tted to the experimental data The temperature dependence of the rebinding was obtained by assuming that kE in eq 14 can be expressed in Arrhenius form eqs 3 with a temperatureindependent expression for gE39 that is the distri bution of substates with different barrier heights was taken to be frozen in below 160 K within the precision of the experiments Independent measurements of other markers of internal motion have demonstrated that certain relaxation phenomena in myo globin v45 are nonexponential in time and have a superArrhe nius temperature dependence that can be expressed in the form kA exp7ERT2 15 over a wide temperature range These results suggest that most of the rebinding below 160 K is a relatively local phenomenon in which protein relaxation does not play an important role This is in accord with a dissociation simulation at 10 K K Kuczera and M Karplus unpublished which demonstrated that the initial motion of the iron out ofthe heme plane still occurs in 40 fs as it does at room temperature and that at least some of the undoming required for rebinding can take place without protein relaxation For CO at room temperature because of the large barrier to ligand binding one does not even begin to observe recombina tion until about 100 ns after photodissociation33 Thus the transitions between the substates are fast relative to the overall reaction and the protein is expected to have relaxed nearly completely to the unliganded con guration by the time signi cant CO rebinding occurs A description of the type outlined in section IIIA is expected to be applicable with Kt having a wellde ned plateau value This implies that a rate constant should exist and exponential rebinding should occur in agree ment with experiment At some intermediate temperatures near 160 K the rate of protein relaxation and rebinding will be on the same time scale and complex behavior is expected It is in fact observed that the rate decreases with increasing temperature in this range45v45 This is a clear manifestation of nonArrhenius temperature dependence Because of the difference in the electronic structure of the two ligands there is a much lower barrier for rebinding of NO than CO This is exempli ed by the fact that in myoglobin at room temperature 80 of the N0 molecules have rebound a er 300 ps40 An analysis of the Mb potential energy surface for the native state2 based on a room temperature 300 ps molecular dynamics trajectory47 demonstrated that the interconversion rates between substates range from 01 ps for those that are very similar to at least the length of the simulations 300 ps for those with larger structural differences experiments 49 indicate that there are relaxation processes that extend from femtoseconds into the microsecond range These results show that the substate transitions are on the same time scale as protein relaxation after photolysis in myoglobin as would be expected since they cover 18 J Phys Chem B Vol 104 No I 2000 the same conformational space In fact the rms difference between MbCO and Mb in crystal structures is 03 A for the main chain and 07 A for all nonhydrogen atoms36 somewhat smaller than the conformational space about 2 A explored by the single Mb simulation at room temperature2 From these results it is clear that the rigid bath model used to explain nonexponential rebinding of CO at low temperatures is not applicable to NO at room temperature Instead the similarity of the time scales of the protein motions and the rebinding rate suggests that there is a coupling between the two This is likely to arise from the difference of the position of the heme iron relative to the heme plane in the liganded and unliganded structures39 that is the heme is essentially planar with the sixcoordinate iron in the heme plane in the former and the heme is domed with the iron out of the heme plane in the latter The barrier to rebinding is expected to be lowest when the fivecoordinate iron is in the heme plane and to increase as the heme moves out of the plane and the protein relaxes to the unliganded equilibrium structure To examine the possibility that the nonexponential rebinding of NO can be produced by a time dependent evolution of the barrier we describe a simple model39 simulations that support this model have been made Becker and Karplus to be published The disappearance of unliganded protein Nt is assumed to follow the modified firstorder decay law dNdt ktN 16 where kt is a timedependent rate coefficient which is assumed to obey the modified Arrhenius equation kt A exp EtRT 17 with E0 Eeq eX1 kbart Eeq Here kbar is the rate constant that determines the decay of the initial barrier height E0 to its equilibrium value Eeq Reasonable parameters ie E0 0027 kcalmol Eeq 12 kcalmol kbar 15 x 1010 s71 give a good fit to NO rebinding for myoglobin at room temperature39 the limiting rate constants are 42 ps at t 0 and 300 ps at t 0040 The model just described requires that there be a slow component to the iron motion outofthe heme plane The existence of such a component has been demonstrated in simulations by Kuczera et al3950 interestingly simulations with somewhat different protein models51gt52 did not find a nonex ponential component The simulations of Kuczera et al50 showed that after photodissociation there is an ultrafast relax ation process with an average time constant of about 40 fs39 the range of relaxation times varies from 30 to 70 fs in different trajectories It consists of an iron outofplane displacement coupled to local adjustments of the heme The ultrafast behavior is very similar to that obtained following the dissociation of a heme histidine CO complex in the absence of the protein The remaining slower processes correspond to structural and energetic relaxation involving the protein as well as the heme This is due to the fact that the doming of the heme group exerts a force on the surrounding protein atoms The iron outofplane displacement averaged over several trajectories zt which was used as the reaction coordinate for the structural transition leading from the liganded to the unliganded state shows nonexponential behavior Over the time scale of the simulations 100 ps the results can be fitted to a power law or a stretched exponential Figure 5 shows the calculated relaxation behavior39 Karplus 20 4O 60 80 100 Time in ps Figure 5 Myoglobin photodissociation showing the outofplane motion of iron zt in angstroms as a function of time in picoseconds The solid line is the average of four simulations smoothed with 2 ps windows a fit with a power law two slightly different fits are shown b superposition of the simulated zt for the outofplane motion of the iron and experimental Ayt for the spectral shift of band 111 measurements by Lim et al48 The experimental data lled circles were subjected to an overall linear scaling see text Adapted from Kuczera et al50 the fluctuations in zt correspond to highfrequency local oscillations superposed on the global relaxation Also shown is the measured frequency shift Avt of band 11148 scaled in magnitude to correspond to the amplitude of zt Band III is a heme chargetransfer band whose frequency is related to the iron outofplane distance The nonexponential time dependence of Avt and zt are in excellent agreement over the range of the simulation This supports the simulation results and indicates as well that the band HI shift is a linear function of the iron outofplane displacement Steinbach et al45 have suggested that there is also a linear relationship between the barrier height for rebinding and the position of band 111 Further the temporal evolution of 20 from the simulations is in agreement with time scales found in the experimental studies of NO ligand rebinding40 The position of band 111 is a marker for conformational relaxation of myoglobin after photodissociation It has been shown as already mentioned that the shift in band 111 is nonexponential and takes place over a time scale from femto seconds to microseconds48gt49 Hagen and Eaton53 have developed a model for the nonexponential decay to equilibrium of a protein after a perturbation such as the photodissociation reaction The model is based on the assumption of two relaxation time scales one faster being the transition from the liganded to a certain portion of the unliganded manifold of states and the other slower the redistribution of states to the equilibrium population in the unliganded manifold To obtain a description that can be Feature Article used for simple calculations Hagen and Eaton assumed a Gaussian random energy model for the distribution of energies in the unliganded and liganded manifold of states and introduced a single transition state54 that governs the rates of all transitions within each manifold and between manifolds With reasonable parameters they are able to t the model to the stretched exponential type behavior of the measured shift of band III The details of the relaxation behavior depend on the assumption of the connectivity between states ie whether any state can make a transition to any other state as in the protein folding model of Bryngelson and Wolynes55 see also ref 56 or whether the allowed transitions are restricted to states with similar energies A major question in this type of model concerns the relation between energy and structure that is what is the structure of the postulated single transition state if one exists and is it possible to assume that states of similar energy have similar structures since any physical transition will most likely involve two similar structures Hints concerning the complex nature of the behavior that can be expected are given by the topological and kinetic analysis57 of the potential surface calculated for an alanine tetrapeptide by Czerminsky and Elber58v59 as described in section V The present analysis suggests that there are two limiting possibilities for protein reactions in general In one limit many protein substates with different barrier heights exist at equilib rium and it is the substate interconversion rate versus the reaction rate that determines whether exponential behavior is observed In the other limit the barrier and binding rate are altered by protein relaxation from the liganded to the unliganded structure after dissociation so that the relative rate of relaxation and rebinding determines whether the latter shows nonexponential kinetics The two possible sources of nonexponential behavior are not mutually exclusive and are likely to play a greater or lesser role depending on the nature of the ligand the protein and the temperature as well as other conditions specifying the system Single molecule measurements would be use Jl to distinguish between these possibilities53v50 Another protein reaction for which nonexponential kinetics has been observed is the fast primary electrontransfer step in the bacterial photosynthetic reaction center The origin of this nonsimple behavior has been analyzed by use of molecular dynamics simulations127 I have gone into considerable detail in reviewing the myoglobin rebinding after photolysis because it is the protein reaction for which the most detailed experimental and theoretical studies exist Nevertheless many questions concerning this well studied reaction remain unanswered IVC Specific Studies of Conformational Change There are many reactions in proteins involving conformational change It is likely that one of the reasons for having molecules as large as proteins carrying out important functions in living systems is that this makes it possible to have the wellde ned confor mational transitions needed for control and signaling A con formational change in the context of this section is a reaction in which the initial and nal states have wellde ned structures unlike the protein folding or unfolding reactions where one of the states is a superposition of a very large number of different conformations this very interesting case is considered in section V The most common examples of conformational changes of which there are many involve an alteration in the relative stability of two structures due to the binding of a ligand or a substrate Such conformational changes include lid opening and closing motions eg triosephosphate isomerase51 loop reori entation eg as in rasp2152 hingebending motions eg as J Phys Chem B Vol 104 No 1 2000 19 in lysozyme i3 and in GroEL subunits54 and the relative motion of subunits e g as in the quaternary transition of hemoglobin 55 For many of these cases a reaction coordinate would be relatively easy to de ne This contrasts such transitions with those concerned with globally distributed conformational changes that involve motions of all or most of the atoms of the protein as in the transition of myoglobin from the liganded to the unliganded state discussed above section IVB and the protein folding transition considered below section V IVCi Ring Flips in Bovine Pancreatic Trypsin Inhibitor BPTI This conformational change whose dynamics has been simulated in detail is a degenerate one in that the initial and nal states are the same It consists of aromatic ring isomer ization ring ips in the BPTI1S55 58 Room temperature activated dynamics simulations very similar to those described for triosephosphate isomerase section IVA were performed for the tyrosine ring Tyr 35 which has the slowest ipping rate of the four tyrosines in BPTI Even for this apparently simple case the obvious reaction coordinate corresponding to the ring dihedral angle is not satisfactory39 instead a more complex reaction coordinate that takes account of essential coupling between the ring and the local protein backbone was introduced On the basis of this reaction coordinate the transmission coef cient was calculated eq 9 and a wellde ned plateau value equal to 022 was reached in 01 ps Thus the calculations indicate that the conformational change shows simple behavior at least at room temperature This is as expected because the calculated potential of mean force barrier for the ring ip is on the order of 10 kcalmol and the barrier crossing rate is much slower than the collisional relaxation of the activated ring kinetic energy15 It is not unlikely that at a lower temperature nonexponential behavior would be observed in analogy to that found for ligand rebinding in myoglobin However such lowtemperature behavior has not been studied 39 J 39 e 1 39 data on the ring ip from nuclear magnetic resonance spectroscopy have been used with the assumption of simple kinetic behavior of the rst and second type to obtain the reaction rate57 The measured values of the activation enthalpy and entropy above 300 K are 37 kcalmol and 68 eu respectively This contrasts with calculated values of 10 kcalmol for the enthalpy and approximately zero for the entropy Such a large disagreement suggested that the experimental separation into enthalpy and entropy of activation based on the assumption that both are temperature independent may be in error69 see also section VB In fact the measured rates as a function of temperature showed signi cant curvature suggesting that complexity of the second type is present Otting et al70 have remeasured the Tyr 35 ring ips as part of a study of the equilibration of two conformers of the Cys 14Cys 38 disul de bond of BPTI In the range 299323 K similar to that used by Wagner et al57 the measurements 33 kcalmol and 53 eu con rmed the activation parameters found in the earlier work39 as before a kTh preexponential factor was used For the same temperature range the disul de conformational exchange had similar activation parameters values are not given in the paper What is interesting is that at lower temperatures 277293 K the disul de activation parameters were found to be very different39 the activation energy was on the order of 10 kcalmol and the activation entropy was small and negative on the order of 15 eu T us there is a striking difference between the low and hightemperature disul de conformational transition with the former having activation parameters much closer to those calculated for the ring ip39 unfortlmately the Tyr 35 ring ip 20 J Phys Chem B Vol 104 No 1 2000 rate could not be measured at the lower temperature Otting et al70 suggested that at high temperatures the disul de transition and ring ip are concerted to explain the increased activation parameters This could indicate that the discrepancy between the earlier experiments and the calculations for the ring ip arises from the fact that the measurements represent the high temperature process while the calculations correspond to the lowtemperature process indeed no disul de transition was observed in the ring ip calculation It is likely therefore that the ring ip transition and the disul de reorientation are not simple by the second criterion ie in this case the reaction appears to follow a different path at low and high temperatures and the activation parameters have a complex temperature dependence even though the native structure of the protein is essentially independent of temperature IVCii Triosephosphate Isomerase TIAI LidLoop Transi tion For most other cases where the occurrence of a confor mational change is demonstrated by the observation of two different crystal structures see for example Gerstein et al71 for a list and classi cation of many conformational changes in proteins there are no measurements of the dynamics One case which has been studied experimentally and theoretically is the conformational change of a loop region that protects the active site of TIM from solvent during the reaction We brie y describe the results because they demonstrate what is known and what is not known about a rather localized conformational chan e in a protein An analysis based on the open and closed crystal structures of the enzyme suggested that the peptide loop motion in going from one structure unliganded open to the other liganded closed is best described as arigid lid motion ie the internal conformation of the looplid does not change in the transition and certain residues at the two ends of the lid can be identi ed as hinges by use of pseudodihedral angles based on the Cot atom of each residue51v7l Hightemperature molecular dynamics simulations were required to go from a closed to an open structure analogous to protein unfolding see section V in a reasonable simulation time It was found that the loop opens and closes in a jumplike fashion within approximately 20 ps at 1000 K This suggests that there is a barrier to opening even in the absence of substrate or product Wade et al73 simulated the loop motion by Brownian dynamics in the presence of the electrostatic eld due to the rest of the rigid protein They used a simpli ed representation of the loop residues analogous to that employed by McCammon et al74 in Brownian dynamics simulations of 0Lhelix unwinding Use of such a model permitted simulations as long as 100 ns which would have been impossible with a more detailed description The simulations showed that the opening and closing motion occurred on a 1 ns time scale Moreover many dihedral transitions within the loop lid were observed in contrast to the analysis of Joseph et al51 A recent simulation of the lid motion75 with an allatom model using stochastic dynamics supports the rigidlid description and suggests a sizable activation barrier for the transition There have been two experimental studies aimed at the dynamics of the looplid motion One of these uses line shape analysis of the quadrupole powder pattem determined by solid state deuterium NMR75 to probe the motion of the perdeuterated indole side chain of T 168 This residue is near the end of the loop 166176 in fact one of the hinges involves motion about the pseudodihedral angles 166167 and 167168 though the displacement of the former is larger than that of the latter The experimental data were interpreted by assuming displace ments of the Trp 168 side chain that correspond to the difference Karplus observed in the crystal structures with this assumption a transition rate of about 104 s 1 was obtained This is much slower than that observed in the Brownian dynamics simulation Using a preexponential factor of 1012 s l which may not be appropriate for what is likely to be a diffusive transition an activation energy of 12 kcalmol was estimated39 no temperature dependence measurements were made to con rm this Some what surprisingly the same rate was observed for the free enzyme and that with the bound inhibitors Also the same ratio of 10 to 1 between two forms assumed open to closed for the free enzyme and closed to open for the inhibited enzyme was obtained by tting the spectra An obvious complication in the interpretation is that there is no direct evidence that the two conformations used to t the data are actually the open and closed form of the loop rather than two unrelated slowly interconverting conformations of Trp 168 such as have been found to exist in studies of tryptophane uorescence in other proteins With a barrier of 12 kcalmol the 20 ps time for the transition found in the molecular dynamics simulation at 1000 K yields a rate of about 104 s 1 at room temperature close to that estimated from the NMR experiment One other study examined the temperature dependence of a TIM transition by using 311 NMR of Glu 165 modi ed covalently with the substrate analogue 3chloroacetol phosphate77 Signals corre sponding to two conformations were observed again there is no direct evidence that they represent the open and closed lid and the temperature dependence of the transition rate between them was tted to an Arrhenius plot39 it yielded a barrier of 34 kcalmol The actual data indicate signi cant curvature though it is dif cult to determine whether this is meaningful since only three temperature points were measured n summary it is evident from the differences in the results obtained from the various experiments and simulations that the present understanding of this transition is incomplete and that more work is needed to determine the details of its behavior including whether the reaction can be described as simple or complex IVCiii Quaternary Transition in Hemoglobin A confor mational transition that has been studied at various levels of detail over many years is the quaternary transition in hemoglo bin Many measurements of the kinetics of the transition have been made because of its important role in hemoglobin function as exempli ed by the MonodWymanChangeaux model78 From the structural data of Perutz and coworkers55 it is known that there are two quatemary structures often referred to as T and R with the former more stable in the unliganded state no 02 bound and the latter more stable in the fully liganded state one O bound to each heme Since both tertiary and quaternary transitions play a role in the different states of hemoglobin a full analysis of the kinetics is rather complicated7980 However in the present report we are not concerned with these aspects To a good approximation the quaternary transition of the hemoglobin tetramer can be described in terms of two 06 dimers 0131 and 0232 that undergo a relative rotation of 150 with respect to each other39 there is also a small relative translation coupled to the rotation81 The most complete kinetic data are available for component I of trout hemoglobin whose quaternary transition is similar to that of human hemoglobin79 The experiments were done by photolysis of hemoglobin fully liganded with CO which provides a trigger analogous to that used in the myoglobin experiments described in section lVB It was possible to isolate spectrally the R to T transition of the fully photodissociated tetramers ie those with no CO bound A simple rate expression was assumed in the analysis of the Feature Article data39 there is not enough information to determine a nonexpo nential time dependence for the reaction even if it were to occur W Eaton private communication Measurements of the temperature dependence over the range 275338 K showed simple Arrhenius behavior Thus within the limits of the available experimental data the quaternary transition appears simple The measured activation energy was 8 kcalmol Use of this value a gasphase pre exponential factor of kTh and a transmission coef cient of unity yielded an activation entropy equal to 117 cal mol 1 K l However a simple diffusive model is probably better for obtaining the pre exponential factor The Kramers equation in the high friction limit 83 and a simpli ed sphere representation of the ot dimer with a 20 A radius yield a time oleO9 s 1 for the 150 motion involved in the R to T transition in the absence of an activation barrier This suggests that the factor KkTh is signi cantly smaller than the value used by Hofrichter et al79 in agreement with a suggestion in their paper Use of eq 4a with this preexponential factor yields a positive entropy of activation This seems more reasonable since simulations suggest that there is a loosening of the tetramer in the transition Fischer et al unpublished V Protein Folding Protein folding is the protein reaction of greatest complexity The entire molecule is directly involved in the reaction unlike some of the cases considered above Moreover the nonbonded van der Waals and electrostatic interactions that determine the potential energy surface for protein folding and lead to the stability of the native state are individually weak between 01 and 23 kcalmol If one is considering effective energy surfaces or potentials of mean force surfaces these energy terms correspond to the effective interaction energy between pairs of atoms in the presence of a canonically averaged solvent and so depend on temperature84 There are a very large number of these contribution even for a small protein like the wellstudied C terminal fragment of C12 with 64 residues there are on the order of 105 potential energy terms if they are represented as atomi atom pair interactions Nevertheless the free energy difference between the native and denatured state is only on the order of tens of kcalmol at ambient temperatures e g 7 kcalmol for a rather stable protein like C12 at 298 K87 which corresponds to about 04 kcal per residue and 004 kcalatom the enthalpy of unfolding is signi cantly larger39 it is about 30 kcalmol for C12 at 298 K87 Clearly a delicate balance of interactions is involved in the effective energy and free energy surface Another element in the complexity of the protein folding reaction is the large number of conformations accessible to the polypeptide chain This has two aspects One of course is the search problem embodied in the socalled Levinthal paradox35v85 and the other closely related is the possibility that the con gurational entropy has an important role in determining the activation barrier Despite the microscopic complexity most measurements of folding kinetics87 89 indicate that the rate of formation of the native state obeys the simple unimolecular rate law given in eqs 1 and 2 More speci cally the observed rates of folding and unfolding can be represented either by a simple rate law or by a kinetic scheme consisting of a small number of intermediates in addition to the denatured and native state and each step of the overall reaction can be described by a simple rate law9v90 Thus the kinetics of the protein folding reaction appears to be simple in terms of the rst criterion ie an exponential time dependence for the folding rate is observed within the accuracy of available experiments However the J Phys Chem B Vol 104 No 1 2000 21 second criterion that the temperature dependence obeys an Arrhenius law is not satis ed by the protein folding reactions whose temperature dependence has been studied91 It appears to be Arrheniuslike at physiological temperatures but very clearly deviates from Arrhenius behavior at higher temperatures Thus essential questions concerning protein folding are why the reaction is simple in terms of the rst criterion despite the complexity of the energy surface and the many degrees of freedom involved and why the protein folding reaction is complex in terms of the second criterion VA Exponential Time Dependence of Protein Folding From the discussion of reaction rate theory see section lllA and its applications to reactions involving proteins a number of points are clear A simple kinetic rate law is applicable if and only ifthere is a separation of time scales that is the rate of the reaction measured as ux passing over a barrier is slow compared to the elementary collisional events that lead to equilibration of the other degrees of freedom of the system This requirement is expected to be satis ed if there exists a single signi cant free energy barrier several times kT for the reaction Experiments show that at room temperature proteins have activation free energy barriers for the folding reaction that are large by this criterion for C12 the value of AGquot based on the measured rate and an assumed preexponential factor of kTh is about 15 kcalmol at 298 K but see below91 so that the observed simple exponential behavior is in accord with this qualitative criterion However it raises the question of why there is a dominant free energy barrier if there is one a question which does not have a simple answer There is a biologically relevant reason for unfolding to be endothermic so as to satisfy the thermodynamic criterion for a unique native structure that is stable under physiological conditions However this does not provide areason for having alarge several kT activation barrier for folding In previous sections sections 1 and IV we described the multirninimurn surface of the native state and its effects on reactions in native proteins It is very likely that the polypeptide energy surface for nonnative states is also a complex multi rninimurn surface However no detailed calculations of effective potential energy surfaces for proteins including a satisfactory sampling of the nonnative conformations are available One of the few studies of a complete potential surface deals with the alanine tetrapeptide 58v59 Even this ultra small system with only seven so dihedral angles has a number of points of interest for the protein folding problem The surface has 139 minima and 502 transition states A diagram ofa part of the surface is shown in Figure 657v92 The minima can be organized into local basins by an energy criterion Each basin contains several minima but the barriers between them are such that there is generally rapid equilibration within a basin relative to the passage from one basin to another A kinetic analysis of the approach of the system to the native state from an initial distribution over the denatured states shows approximately exponential behavior at 300 K If such results can be extrapo lated to the effective energy surface for protein folding they suggest that despite the existence of many minima there is a dominant barrier the barrier to folding and that transition rates connecting different basins are fast relative to the passage along the reaction coordinate progress variable over the rate determining barrier The overall folding kinetics should then obey a simple rate law possibly complicated by intermediates and the global description in terms of simple kinetic schemes is a useful one even if the species involved may represent populations with a wide range of structures 3 22 J Phys Chem B Vol 104 No I 2000 Energy kcalmol m A 0 O 60 1st dimension deg v V m l 60 60 Karplus quot39 o it 339 0 I lquot3 i O i 2nd dimension deg Figure 6 The alanine tetrapeptide a simpli ed diagram of the energy surface of the main basin which includes the minimum energy conformer projected on to two principal components a reduced representation relative to the seven soft dihedral angles specifying the conformation It is interesting to note that the accessible region is rather broad down to an energy of about 4 kcal mol l when the different basins separate Adapted from Becker92 The most direct support for this view of protein folding were originally provided by simulations of a random heteropolymer 27bead model on a cubic lattice9495 closely related studies have been made by Socci et al96 and Chan and Dillg7 have used a similar model on twodimensional square lattices Although oversimplified relative to actual proteins such models have important proteinlike properties that is they have a large number of conformations e g 1016 for a 27mer and a unique ground state the unique ground state in this model encom passes the large number of substates with very similar structures within 2 A rms that in reality contribute to the native state of proteins and have a role in the complex kinetic behavior exhibited in the ligand rebinding to myoglobin for example see section IVB Unlike more detailed protein models calculations of the folding kinetics and thermodynamics are sufficiently fast that they can be explored in detail for a large number of sequences at different temperatures Figure 7 shows the calculated kinetics for a 27mer sequence86gt93gt98 For all the temperatures studied it is found that the rate of folding is exponential that is the heteropolymer folding reaction is simple by the first criterion This is true even though the time dependence of the decay of the correlation function for the structural overlap between coordinate sets is best fitted by a stretched exponential with the exponent of the time near 05 corresponding to an approximately diffusive time dependence Sali and Karplus unpublished For some designed sequences ie sequences that are optimized by selecting the interactions to speed up folding more complex folding behavior has been found99 at high temperatures folding can be described by a single exponential but at lower temperature there is both a fast and slow exponential phase This was explained in analogy to interpretations of experiments on protein folding by the existence of an offpathway intermediate The effective free energy surfaces at two different temper atures drawn as a function of the progress variable Q which is the fractional overlap number of native contacts divided by the total number of native contacts with the native state are in 00 T11 10 2 O I 3 T212 t 20 E 1 o T1 O T19 30 DT15 T17 O 20 4O 60 MC STEP 106 Figure 7 Distribution of the mean first passage folding times for a 27mer sequence cN represents the native fraction The distribution of the mean first passage times is shown at the temperatures indicated on the plot The lines are linear leastsquares fits to the points To obtain each of the points 100 independent folding trials were done Adapted from Karplus et al98 accord with the calculated simple rate behavior see parts b and d of Figure 8 For both cases the free energy surface has a significant barrier near the native state that dominates the folding kinetics In the high and lowtemperature limits shown for the more general temperature dependence of the surfaces see Sali et al95 the free energy surface appears relatively smooth and has a single important barrier as required for the validity of a phenomenological rate equation This is true even at the lower temperature for which the effective energy surface is quite rough see Figure 8a due to a partial cancellation between the roughness in the energy and entropy93gt94 In Figure 8 we have used Q the fractional overlap with the native state as the progress variable to describe the free energy of the reaction Because of the complexity of the protein folding reaction and the many degrees of freedom involved it is much less simple to define a reaction coordinate than for small molecule reactions In fact it is likely that a reaction coordinate Feature Article a 55 so 65 u 70 75 00 02 04 06 00 10 c 40 60 H1 80 100 00 02 04 O 06 08 10 J Phys Chem B Vol 104 No 1 2000 23 b 5 u 70 75 Q d 90 95 u 400 W 105 00 02 04 06 08 10 Figure 8 Folding effective energy and free energy surfaces of a 27bead heteropolymer on a cubic lattice a The average effective energy E as a function of Q the fraction of native contacts at a low temperature T 07 there are 28 contacts in the native state which is a 3 x 3 x 3 cube b The average effective free energy F as a mction of Q for T 07 c and d are the same as a and b but at ahigh temperature T 2 Results adapted from Figure 4 of Sali et al95 Details of the de nition and methods are given in that paper corresponding to that for small molecule reactions does not exist tis there is no wellde ned linear combination of coordinates that describes the motion from the reactant to the product state This follows if the folding trajectories of individual polypeptide chains are very different and cover a wide range of possibilities Clearly if the fractional number of native contacts Q is used to specify the progress of the reaction many different sets of contacts correspond to the same value of the progress variable except for the unique native state For the use of Q as the progress variable in a protein unfolding simulation with an atomic model see Lazaridis amp Karplus1 It is not selfevident that motion along Q is always an appropriate description of the protein folding reaction There is the question of whether motion along Q corresponds to the slow step so that the effective free energy as a function of its value can be used to analyze the reaction e g to show that there is a single dominant free energy barrier For the random 27mers studied by Sali et al94v95 consistent results are obtained with Q as the progress variable oreover the transition states as de ned by the free energy barriers see parts b and d of Figure 8 correspond to the values of Q where the probability of reaching the native state increases rapidly from essentially 0 to about 05 gali and Karplus unpublished The choice of Q for the reaction coordinate of short chains as in the 27mers has been supported by the work of Socci et al95 and Pande et al101 For longer chains more than one progress variable may be required to describe the folding process see section III and Figure lb93v102 Alternative progress variables for protein folding have been used by Du et al103 and by Chan and Dill104 For the 27mers Q satis es the conditions for a useful progress variable or transition coordi nate suggested by Du et al103 Chan and Dill104 found cases in lattice folding simulations where the condition of a separation of time scale for the reaction coordinate and the other coordinates of the system appears not to be satis ed it would be interesting to know whether the folding rate is nonexponential in that case In this regard it is important to point out again that for the validity of the simple phenomenological rate expression it is not necessary to know the appropriate progress variable it is necessary only that such a coordinate exists and that motion along it is slow relative to the relaxation of other degrees of freedom of the system3 However to do meaningf1 calculations or interpret the results at the atomic level a knowledge of the appropriate slow variable that describes the reaction is necess Given the simulations which demonstrate simple kinetic behavior of the rst type eqs 1 and 2 a number of models have been introduced to justify the experimental and theoretical results The essential idea97v1 1v1 5v1 5 Neria and Karplus unpublished is that the native state is coupled to the very large number of con gurations making up the denatured state by a much smaller number of doorway states The analysis of the 27mer lattice simulations at low temperature suggests on the order of 1000 such doorway states the transition state ensemble If the doorway states have similar free energies and the equilibration among the denatured states is fast relative to the motion that leads to the doorway states a transitionstate like expression for the ux through the doorway states can be derived and an expression for the rate constant can be obtained Several of the models 105 Neria and Karplus unpublished have used a master equation approach to calculate the kinetics A somewhat different modelm has elements in common with that of Hagen and Eaton53 for the conformational relaxation in myoglobin The two essential assumptions of the model of Gutin et al105 are that the mean energy of the system relaxes rapidly fast time scale to the equilibrium value given by the random energy model below a critical temperature the equilibrium energy is essentially that of the native state On a slower time scale the system nds one of the small number of transition states from which it folds rapidly to the native state It is assumed that the transition states all have essentially the same energy in line with the kinetic models of Kopfer and Hilhorst54 Hagen and Eaton53 and Savens et al55 The model differs from 24 J Phys Chem B Vol 104 No I 2000 that of Hagen and Eaton53 in that the equilibration of degrees of freedom other than the reaction coordinate is rapid so that exponential kinetics is obtained It is important to consider brie y the preexponential factor in eq 4a since given the activation enthalpy of the reaction from a measurement of the rate as a function of temperature eq 4b the activation entropy and activation free energy depend on its value Although the gasphase Eyring value kTh which equals 6 X 1012 at 300 K is often used it is not generally valid for solution reactions as has been mentioned already in section IV For the motions involved in the folding reaction the system is likely to be in the diffusive regime because of size of the particles residues or larger groups However the microscopic complexity is such that a range of different structural entities may be involved during various portions of the folding reaction This is clearly one of the problems that still needs to be resolved In fact it may be very difficult to determine the appropriate average value of the preexponential factor for use in eq 4a Values such as 1010 s 1 89 and 6 x 1012 s 1 91 have been used but there is little justification for these or any other values Since the elementary step in residuebased treatments of protein folding e g Monte Carlo lattice simulations is likely to be a dihedral or pseudodihedral angle transition a value in the neighborhood of 109 s 1 appears to be suitable for that case74 However if larger entities are involved e g as in the diffusion collision modelm values closer to 107 108 s 1 may be appropriate Hagen et al108 have suggested a value in the neighborhood of 106 s 1 for the fastest forming loops in proteins based on measurements of cytochrome c and a theoretical model for the formation of a contact by collapse of a random chain This value seems somewhat too small not for the process specifically considered but for use generally in the protein folding reaction Some proteins fold on the microsecond time scale and interestingly an Arc repressor mutant has a bimolecular rate constant for folding of 3 x 108 M 1 1109 There is also the question of whether the diffusive rate slows down signifi cantly as the transition state for the folding reaction is ap proached see Socci et al96 for a diffusive Kramerslike82 protein folding model It is not clear whether this is a major factor since both the effective diffusion constant for the entities involved and the range over which diffusion is required are expected to decrease simultaneously VB Temperature Dependence of Folding Rate One of the striking differences between small molecule kinetics and protein folding is the temperature dependence of the rate constant Although the unfolding rate is often approximately Arrheniuslike ie a plot of ln k vs lT is essentially a straight line110 the folding rate is not Parts a and b of Figure 9 show the results obtained respectively for folding with the 27mer cube lattice model that we have already discussed94gt95gt98 and for folding of C12 from the experiments of Oliveberg et al91 The interpretation of the lattice model results is straightforward and provides an increased understanding of the folding dynam ics At low temperatures lT large in Figure 9a there is Arrheniuslike behavior associated with a positive activation energy or enthalpy as in eqs 3 and 4 This means the barrier to folding is dominated by the apparent activation energy in correspondence with the effective energy surface in Figure 8a Actually the barrier in Figure 8a is smaller than that estimated from Figure 9a suggesting that activated diffusion may also contribute to the barrier As the temperature increases lT small in Figure 9a the Arrhenius plot turns over and the activation energy becomes negative In the present model this arises from temperatureindependent interactions and an effective funnel Karplus 4160 1 65 39 IH H4 470 g x C 4757 18039 18539 i 490 04 015 0607 08 09 10 1T 7050 macn obs 111ka m 20 I l I I J 2 10027 00029 0003 00031 00032 00033 00034 lT Figure 9 a Calculated Arrhenius plot of the folding reaction rate constant as a function of temperature for the sequence used in Figure 7 results adapted from Karplus et a198 b Arrhenius temperature dependence of the measured refolding rate constant for C12 adapted from Oliveberg et al91 like energy surface in which there is a monotonic decrease in the energy The corresponding surface is shown in Figure 8c which yields a negative energy of activation as confirmed by analyzing the results in Figure 9a with eq 4b However the free energy of activation AG is still positive due to the activation entropy TAS whose effect increases with in creasing temperature The activation entropy is associated with the decrease in the number of configurations accessible with increasing Q39 ie there is an entropic bottleneck associated with the transition state and this bottleneck becomes more important at higher temperatures because of the larger number of states accessible in the denatured region At very low temperatures there is no indication in the lattice calculations Figure 9a and Gutin et al106 or in the available experiments Figure 9b that there is a deviation from the Arrhenius law that is no superArrhenius behavior analogous to that described by eq 15 for example has been observed in protein folding The experimental Arrhenius plot for the folding of C12 shown in Figure 9b is very similar in form to that found in the lattice simulations Figure 9a Consequently the formal interpretation in terms of activation enthalpy AH and activation entropy AS is exactly the same As the temperature increases the activation enthalpy becomes negative while the activation free energy remains positive One explanation of these results is that the effective enthalpy surface of protein C12 and other proteins corresponds in fact to those shown in parts a and c of Figure 8 at low and high temperatures respectively and that the positive activation free energy at high temperatures arises from the configurational entropy term as in the lattice models If T 16 the maximum in the rate of the lattice simulations Figure 9a is set equal to 325 K a factor of 10 decrease in the rate at higher temperatures corresponds to about 400 K somewhat Feature Article larger than the experimental value of 360 K for C12 this is in accord with the fact that the entropic contribution in C12 which has 64 residues is expected to be larger than for the 27mer The effective energy as a function of the number of native contacts obtained in unfolding simulations of C12 at high temperature100 does show a monotornically decreasing energy with increasing Q in accord with this model This is a very interesting result100 Lazaridis and Karplus to be published because it suggests that the experimental temperature depen dence of the protein folding reaction is in accord with funnel like effective energy surfaces at high temperatures but not at room temperature Since for many proteins 91 7110 the temper ature dependence of the folding rate is similar in form to that of C12 these results suggest that under physiological conditions the effective energy surface is not funnellike that is it is more like that shown in Figure 8a than Figure 8c It has been suggestedggvlmv110 that the temperature dependence of the interactions themselves due to the hydrophobic effect is involved in the nonArrhenius behavior The hydrophobic effect increases with increasing temperature over the range of interest 275330 K as based on small molecule studies and their application to interpret protein thermodynamics111 The dominant contribution of the hydrophobic effect is on the denatured state which is enthalpically less stabilized as the temperature increases 1f the transition state is less unfolded than in the denatured state for an analysis of C12 see ref 100 it would have a proportionally smaller contribution from the hydrophobic effect and the magnitude of AHt would be expected to decrease with increasing temperature This contribution to the curvature in the Arrhenius plots contrasts with the lattice results in which the interactions are independent of temperature and the change in activation enthalpy is due to the difference in the energies of the con gurations sampled as a function of temperature Both types of contributions may play a role in protein folding VI Summarizing Discussion This review provides a brief background in chemical kinetics to serve as a basis for comparing the simple behavior of small molecule reactions with the more complex reactions of proteins Simple behavior was de ned by two related aspects of reactions The rst is that a simple phenomenological rate law with an exponential time dependence for the rate applies and the second is that the temperature dependence of the rate follows the Arrhenius equation We have seen that although simple behavior is found in some protein reactions signi cant devia tions from both types of simplicity have been documented and interpreted theoretically Ligand rebinding after photolysis in myoglobin was discussed in considerable detail because this reaction has been studied more than any other and has been shown to deviate from both requirements for simplicity under certain conditions For protein folding the situation is somewhat different Although deviations from simplicity might have been expected there is considerable evidence that a simple rate law with an exponential overall folding rate applies to most proteins that have been studied so far there are exceptions intermediates but they are trivial from the present viewpoint By contrast the second criterion for simplicity is not satis ed The temper ature dependence of the folding rate of all proteins that have been studied show signi cant deviations from Arrhenius behavior We have given a determination of the results based on lattice simulations but the isolation of the actual mechanism for the nonArrhenius type behavior in proteins will require additional studies J Phys Chem B Vol 104 No 1 2000 25 From the history of the experimental and theoretical inves tigations of ligand rebinding after photolysis in myoglobin it seems likely that the applicability of a simple phenomenological rate law to the overall protein folding reaction hides an underlying complexity One set of experiments that will help to reveal this complexity involves studies with a short time resolution for systems under nonequilibrium conditions A range of methods for triggering the folding and unfolding reaction have been developed and these processes can be studied in principle on time scales from nanoseconds to seconds or longer 112 However techniques for monitoring the structural details as a function of time still need to be improved Techniques that are likely to make important contributions are timeresolved X ray crystallography113 real time NMR93 as well as infrared spectroscopy114 and vibrational Raman optical activity115 More generally for protein dynamics infrared photon echo measurements115v117 and optical line broadening stud iesngv119 are expected to yield signi cant new results Another type of approach that is likely to be of great importance for many reactions involving proteins including enzyme catalysis ligand binding and protein folding is the monitoring of single molec111es3212 122 This provides a direct way of determining the diversity that exists for the various reactions and will make it possible to distinguish in principle whether the macroscopic complexity is evident in individual molecules or whether it is a consequence of averaging over a population of molecules It is interesting to mention in this regard that single molecule behavior has been studied for a long time for the case of membrane channel conductance in which the molecules involved undergo a signi cant structural change between open and closed states For most channel systems the structural data required for atombased simulations are not available However the highresolution structure of a potassium channel determined very recently123 opens the way for detailed theoretical studies on this very interesting system39 many simu lations will certainly be made Finally the more global variation of experimental conditions eg folding at low temperatures will aid in determining the difference in the temperature or other dependence of different relaxation processes The future holds great promise for this exciting eld It is likely that as measurements and simulations become more precise and cover a wider range of conditions complex behavior of the type discussed in this manuscript will be found for an increasing number of reactions in proteins Note Added in Proof As stated in the Summarizing Discussion it was to be expected that the tmderlying complex ity of the protein folding reaction would be revealed by studies with a short time resolution under nonequilibrium conditions In a recent study of the folding of phosphoglycerate kinase and an ubiquitin mutant Sabelko et al128 have observed nonexpo nential folding induced by a temperature jump The origin of the nonexponential behavior and its relation to that discussed in section IV is not clear Further experiments will be needed to determine the validity of the interesting interpretation proposed by Sabelko et al Acknowledgment 1 thank William Eaton and Hans Frauen felder for helpful comments and Aaron Dinner and Andrej Sali for preparing Figure l 1 thank Stefan Fischer for calling my attention to reference 70 Partial support for the research reported here was provided by the National Science Foundation and the National Institutes of Health A preliminary version of the material reported here is being published in the Dahlem Workshop Report on Simplicity and Complexity in Protein and 26 J Phys Chem B Vol 104 No 1 2000 Nucleic Acids H Frauenfelder J Deisenhofer and P G Wolynes Editors Freie Universitat Berlin 1999 References and Notes 1 Austin R H Beeson K W Eisenstein L Frauenfelder H Gunsalus I C Biochemistry 1975 14 53555373 2 Elber R Karplus M Science 1987 235 318321 3 Chandler D J Chem Phys 1978 68 29592970 4 Weston R E Schwarz H A Chemical Kinetics Prentice Hall New York 1972 Karplus M Porter R N Shanna R D J Chem Phys 1965 43 32593287 6 Bal Y Fayer M D Phys Rev 1989 B39 1106611084 7 Greer J Fan J Angell C AJ Phys Chem 199498 13780 13790 8 Angell C A Science 1995 267 19241927 9 Fersht A A Guide to Catalysis and Protein Folding W H Freeman amp Co New York 1999 10 Keck J CAdv Chem Phys 1967 13 85121 11 Anderson J B J Chem Phys 1973 58 46844692 12 Wigner E P Trans Faraday Soc 1938 34 2941 13 Grote R F Hynes J T J Chem Phys 1980 73 27152732 14 Montgomery J A Chandler D Beme B JJ Chem Phys 1979 70 40564066 15 Northrup S H Pear M R Lee CY McCarnmon J A Karplus M Proc Natl Acad Sci USA 1982 79 40354039 16 Skinner J L Wolynes P G J Chem Phys 1980 72 4913 4 17 Knowles J R Philos Trans R Soc LondonB 1991 332 115 121 18 Richard J P J Am Chem Soc 1984 106 49264936 19 Bash P A Field M J Davenport R C Petsko G A Ringe D Karplus M Biochemistry 1991 30 58265832 20 Aqvist J Fothergill M Biol Chem 1996 271 1001010016 21 Knowles J R Nature 1991 350 121124 22 JosephMcCarthy D Petsko G A Karplus M Protein Eng 1995 811031115 23 Neria E Karplus M Chem Phys Lett 1997 267 2330 24 Hwang J K King G Creighton S Warshel A J Am Chem Soc 1978110 52975311 25 Warshel A Sussman F Hwang J K J Mol Biol 1988 201 139 9 26 Brumer P Karplus M Faraday Soc Discuss 1973 55 8091 27 Aqvist J Warshel A Chem Rev 1993 93 25232544 28 McQuarrieD A Statistical Mechanics Harper ampRow New York 1976 29 Valleau J P Torrie G M In StatisticalMechanics Berne B J Ed Plenum Press New York 1977 Part A p 169194 30 Ryckaert JP Ciccotti G Berendsen H JC J Comput Phys 1977 23 327341 31 Bergsrna J P Gartner B J Wilson K R Hynes J T J Chem Phys 1987 86 13561376 32 Craig D B Arriaga E A Wong J C Y Lu H Douichi N J J Am Chem Soc 1996 118 52455283 33 Henry E R Sommer J H Hofrichter J Eaton W A J Mol Biol 1983 166 443451 34 Ansari A Berendzen J Bowne S F Frauenfelder H Iben I E T Sauke T B Shyamsunder E Young R D Proc Natl Acad Sci USA 1985 82 50005004 35 Perutz M Matthews B T J Mol Biol 1965 21 199202 36 Takano T J Mol Biol 1977 110 537568 37 Case D A K lus M J Mol Biol 1979 132 343368 38 Frauenfelder H Petsko G A Tsemoglu D Nature 1979 280 558563 39 Debrunner P G Frauenfelder HAnnu Rev Phys Chem 1982 33 283299 40 Petrich J W Lamb JC Kuczera K Karplus M Poyart C Martin JL Biochemistry 1 91 30 39753987 41 Kuczera K Kuriyan J Karplus MJ Mol Biol 1990 213 351 42 Doster W Cusack S Petry W Nature 1989 337 754756 43 Smith J Kuczera K Karplus M Proc Natl Acad Sci USA 199087 16011605 44 Iben I Braunstein D Doster W Frauenfelder H Hong M K39 Johnson J B Luc S Ormos P Sc ulte A Steinbach P J 39 A H oung R D Phys Rev Lett 1989 62 19161919 45 Steinbach P Arnsari A Berenndzen J Braunstein D Chu K Cowen B EhrensteinDFrauenfelder H Johnson J B Lamb D Luck S Mourant J Nienhaus G Ormos P Philipp Xie A Young R D Biochemistry 1991 30 39884001 46 Agmon M Hop eld J J J Chem Phys 1983 79 20422053 Karplus 47 Levy R M Sheridan R P Keepers J W Dubey G S Swaminathan S Karplus M Biophys J 1985 48 509518 48 Lim M Jackson T A An nrud P A Proc Natl Acad Sci USA 1993 90 58015804 49 Jackson T A Lim M An nrud P A Chem Phys 1994 180 131140 50 Kuczera K Lambry JC Martin JL Karplus M Proc Natl Acad Sci USA 1993 90 58055807 51 Li H Elber R Straub J E J Biol Chem 1993 268 17908 17916 52 Schaad O Zhan H X Szabo A Eaton W A Handy E R Proc Natl Acad Sci USA 1993 90 95479551 53 Hagen S J Eaton W AJ Chem Phys 1996 104 33953398 54 Kopfer G Hilhorst H Europhys Lett 1987 3 12131217 55 Bryngelson J D Wolynes P G J Phys Chem 198993 6902 6915 56 Saven J G Wang J Wolynes P G J Chem Phys 1994101 1103711043 57 Becker 0 M Karplus M J Chem Phys 199710614951517 58 Czerminski R Elber R Proc Nat Acad Sci 1989 86 6963 6967 59 Czerminsky R Elber R J Chem Phys 1990 92 55805601 60 Wang J Wolynes P G Phys Rev Lett 1994 74 43174320 61 Joseph D Petsko G A Karplus M Science 1990 249 1425 1428 62 Ma J Karplus M Proc Natl Acad Sci USA 1997 94 11905 11 10 63 McCarnmon J A Gelin B R Karplus M Wolynes P G Nature 1976 262 325326 64 Ma J Karplus M Proc Natl Acad Sci USA 1998 95 8502 07 65 Perutz M Nature 1971 232 408413 66 Snyder G H Rowan R Karplus S Sykes B D Biochemistry 1975 14 3765 67 Wagner G DeMarco A W thrich K Biophys Struct Mech 19762139158 68 McCarnmon J A Karplus MBiopolymers1980 19 13751405 69 Brooks C L III Karplus M Pettitt M Proteins Theoretical Perspective of Dynamics Structure amp Thermodynamics Adv Chem Phys XXI John Wiley amp Sons New York 1988 70 Otting G Liepinsh E W thrich K Biochemistry 1993 32 35713582 71 Gerstein M Lesk A M Chothia C Biochemistry 1994 33 67396749 72 Gerstein M Chothia C J Mol Biol 1991 220 133149 73 Wade R C Davis M E Luty B A Madura J D McCarnmon J A Biophys Soc 64 915 74 McCarnmon J A Northrup S H Karplus M Levy R M Biopolymers 1980 19 20332045 75 Derreumaux Schlick T Biophys J 1998 74 7281 76 Williams J C McDermott A E Biochemistry 1995 34 8309 19 83 77 Y ksel K U Sun AQ Gracy R W Schnackerz K D J Biol Chem 1994 269 50055008 78 Monod J Wyman J Changeux J P J Mol Biol 1965 12 88 18 79 Hofrichter J Henry E R Szabo A Murray L P Ansari A J n s C M Coletta M Falcioni G Bru ori M Eaton W A Biochemistry 1991 30 65836598 Henry E R Jones C M Hofrichter J Eaton W A Biochem istry 1997 36 65116528 81 Baldwin J Chothia C J Mol Biol 1979 129 175200 82 Krarners H A Physica 1940 7 284293 83 McCarnmon J A Karp us M Nature 1977 268 765766 84 KarplusM Shakhnovich EProtein Folding Theoretical Studies of Thermodynamics and Dynamics In Protein Folding Creighton T Ed W H Freeman amp Sons New York 1992 pp 127195 85 Levinthal C How to fold graciously In Mossbauer Spectroscopy in Biological Systems Proceedings of a Meeting held at Allerton House Monticello IL Debrunner P 39bris J C M Munck E Eds University of Illinois Press Urbana 1969 p 86 Karplus M Folding Des 1997 2 569576 87 Jackson S E Fersht A R Biochemistry 1991 30 1042810435 88 Schindler T39 Schmid FX Biochemistry 1996 35 1683316842 89 Scalley M L Baker D Proc Natl Acad Sci USA 1997 94 1063610640 90 Fersht A R Itzhaki L S ElMasry N F Manhews J M Otzen D E Proc Natl Acad Sci U 1994 91 1042610429 91 Oliveberg M Tan Y Fersht A R Proc Natl Acad Sci USA 1995 92 89268929 92 Becker OMJ Mol Struct THEOCHEllI 1997 398399 507 516 Feature Article 93 Dobson C M Sali A Karplus MAngew Chem Int Ed Engl 1998 37V 869893 94 Sali A Shakhnovich E Karplus M J Mol Biol 1994 235 16141636 95 Sali A Shakhnovich E Karplus M Nature 1994 369 248 96 Socci N D Onuchic J N Wolynes P GJ Chem Phys 1996 104 58605868 97 Chan H S Dill K A JVChem Phys 1994 100 92389257 98 Kasplus M Ca isch A Sali A Shakhnovich E In Modelling of Biomolecular Structures and Mechanisms Pullman A et al Eds Kluwer Academic Publishers NorWell MA 1995 pp 69784 99 Abkevich V 1 Gutin A M Shakhnovich E I J Chem Phys 1994 101 60526062 l 0 Lazaridis T Karplus M Science 1997 278 19281931 101 Pande V S Grosberg A Y Tanaka T Folding Des 1997 2 109114 102 Dinner A Karplus M J Phys Chem B 1999 103 7976 7994 103 Du R Pande V J Grasberg A Y Tanaka T Shakhnovich E J Chem Phys 1998 108 334350 104 Chan H S Dill K A Proteins 1998 30 233 105 ZWanZi R Proc Natl Acad Sci USA 1997 94 148150 106 Gutin A all A Abkevich V Kasplus M Shakhnovich E J Chem Phys 1998 108 64666483 107 Karplus M Weaver D L Nature 1976 260 404406 108 Hagen S J Hofrichter J Szabo A Eaton W A Proc Natl Acad Sci USA 1996931161511617 109 Waldburger C D Jonsson T Sauer R T Proc Natl Acad Sci USA 1996 93 26292634 110 Segawa J J Sugihara M Biopolymers 1984 23 247372488 111 Makhatadze G I Privalov P LAdv Protein Chem 1995 47 25 J Phys Chem B Vol 104 No 1 2000 27 112 Eaton W A Thomson D A Chan CK Hagen S J Hofrichter J Structure 1996 4 113119 113 Srajer V Teng T Ursby T Pradervand C Ren Z Adachi S Schildkarnp W Bourgeois D Wulff M Moffat K Science 1996 274 17261729 114 Phillips C M Mizutarni Y Hochstrasser R M Proc Natl Acad Sci USA 1995 92 72927296 115 Wilson G Hecht L Barron L D J Mol Biol 1996 261 341 116 Tokrnakoff A Fayer M D Acc Chem Res 1995 28 4377 445 117 Rector K D Rella C W Hill J R KWok A S Sligar S G Chien E Y T Dlott D D Fayer M D J Phys Chem 1997 101 14681475 118 Thorn Leeson D Wiersrna D A Nature Struct Biol 1995 2 848851 119 Thorn Leeson D Wiersrna D A Fritch K Friedrich J J Phys Chem B 1997 101 63316340 120 Askin A Proc Natl Acad Sci USA 1997 94 48534860 21 Lu H Xie S Nature 1997 385 143146 122 Zocchi G Proc Natl Acad Sci USA 199794 1064710651 Weiss S Science 1999 283 16761683 123 Doyle D A Cabral J M Pfuetzner R A Kuo A Gulbis J M Cohen S L Chait B T MacKinnon R Science 1998 280 69777 124 ZWanZig R Acc Chem Res 199023 148152 125 Shlesinger M F Zaslavsky G M Kla er J Nature 1993 363 31 126 Wang Z Pearlstein J M Jia Y Fleming G R Norris J R Chem Phys 1993 176 421 i425 127 Gehlen J N Marchi M Chandler D Science 1994 263 499 128 Sabelko J Ervin J Gruebele M Proc Natl Acad Sci USA 199996 60316036
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'