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2009, 113, 2225–2230 Published on Web 02/05/2009
Fluctuating Enzyme and Its Biological Functions: Positive Cooperativity without Multiple States Hong Qian* and Pei-Zhe Shi Departments of Applied Mathematics, UniVersity of Washington, Seattle, Washington 98195 ReceiVed: December 3, 2008
Positive, sigmoidal cooperativity is known to occur to a monomeric single-site enzyme with slow conformational fluctuations in its unbound (E) states when the enzyme undergoes steady-state catalytic turnover (mnemonic enzymes, hysteretic enzyme). We show that positive cooperativity occurs even when the E state is a single thermodynamic state, provided that the fluctuation amplitude of the state is sufficiently greater than that of the enzyme-substrate complex ES and the fluctuation times being comparable. This can occur even without mean structural change between E and ES. Slow conformational fluctuations are widely observed in enzymes. Our result suggests that enzymes with substrate association that reduces conformational fluctuations while maintaining fluctuation temporality can exhibit sigmoidal binding inside of living cells. Implications of this result on drug-target interactions are discussed. 1. Introduction Many enzymes involved in cellular metabolism speed up biochemical reaction rates;1 other enzymes involved in cellular signal transduction regulate information processing. In the latter scenario, the sigmoidal response of a biochemical binding as a function of substrate concentration, often termed a “molecular switch”, is widely considered to be a key functional component of a molecular circuitry.2 Two mechanisms leading to a sigmoidal response are widely known, (1) allosteric cooperativity of protein complexes with multiple subunits or binding sites following equilibrium thermodynamic linkage3 and (2) temporal cooperativity of triprotein reaction networks, such as the phosphorylation-dephosphorylation cycle and GTPase, with zeroth-order kinetics in a nonequilibrium steady state (NESS).4,5 What is less well-known is the mechanism of dynamic cooperativity, also known as mnemonic enzymes, which arises in monomeric enzymes with a single binding site.6 See ref 7 for a recent review. Two conditions are necessary for an single-site monomeric enzyme to exhibit positive cooperativity. One is intrinsic to the enzyme molecule, and one is the condition under which the enzyme operates. The intrinsic requirement is that the enzyme has a slow conformational fluctuation in its unbound, E state.6 Recent single-molecule measurements on enzyme fluctuations have definitively shown that many enzymes have such a slow dynamic disorder. See ref 8 for an earlier review and ref 9 for an extensive discussion of the recent literature. The second requirement is that the enzyme has to be sustained under a NESS by a continuous turnover of substrate to product. While a NESS is difficult to set up in a test tube without a regenerating system, inside of a living cell, most enzymes are operating naturally under such a condition.10 While the existence of conformational fluctuations is firmly established,9,11,12 further details remain to be elucidated. In * To whom correspondence should be addressed.
10.1021/jp810657j CCC: $40.75
particular, there will be a conceptual difference between whether the conformational fluctuations are within one thermodynamic state or multiple states. In several signaling proteins, fluctuations among multiple thermodynamic states (free-energy wells) have been shown. Phosphorylation of the enzymes shifts the population distribution from an inactive state to an active state.13-15 A thorough statistical thermodynamic analysis, in the context of protein folding, that compares and contrasts conformational transitions in a two-state protein and in a single-state protein can be found in the classic paper, ref 16.17 In this Letter, we show that single-state conformational fluctuations of E and ES can be sufficient to yield sigmoidal positive cooperativity, provided that the magnitudes of the fluctuations in state E are much greater than that in state ES and their fluctuation times are comparable. Therefore, substrate binding that reduces enzyme dynamic disorder is possible to exhibit positive cooperative binding.
2. Conformational Diffusive Michaelis-Menten (CDMM) Theory The question we ask, namely, whether a positive cooperative behavior is possible for a fluctuating enzyme with a single thermodynamic state E and a single thermodynamic state ES, can be naturally addressed by the conformational diffusive Michaelis-Menten theory which has a very long history. Briefly, let x denote the conformational fluctuations of a protein, with free-energy functions u1(x) and u2(x) for states E and ES, respectively. The fluctuations are characterized by diffusion in the energy landscapes, and for simplicity, we assume constant diffusion coefficients D1 and D2, which are related to the frictional coefficients ηi ) kBT/Di, (i ) 1, 2) 2009 American Chemical Society
2226 J. Phys. Chem. B, Vol. 113, No. 8, 2009
(
Letters
)
∂pE(x, t) kBT ∂2pE 1 ∂ du1(x) + ) p - ko1(x)[S]pE + ∂t η1 ∂x2 η1 ∂x dx E (k-1(x) + k2(x))pES (1a)
(
Ka[S] )
)
- (k-1(x) + k2(x))pES (1b)
This coupled diffusion equation has been used to study motor proteins, known as the Brownian ratchet model.18,19 M. V. Volkenstein and his Russian school have used such a model since the 1970s in the theory of electronic conformational interaction (ECI).20 More recently, general CDMM theory in terms of a two-dimensional energy landscape can be found in ref 21. The CDMM theory, in various special cases, has been applied to enzyme kinetics. For example, if one is only interested in the transient kinetics of dissociation reaction ES f E + S with k- 1(x), then the eq 1b is reduced to the celebrated Agmon-Hopfield theory for an enzyme fluctuating perpendicular to the reaction coordinate.22 Agmon23 has further developed a model in which continuous fluctuations of the enzyme conformation in both E and ES states are considered but with k1o(x) and k-1(x) * 0 only at x ) 0 and k2(x) * 0 only at x ) 1. Mathematical treatment of the k(x)’s thus became boundary conditions that couple two diffusion equations in eq 1a. On the other hand, if one considers only a simple reversible, catalyzed unimolecular reaction E + S h E + P with a single fluctuating enzyme, then a diffusion equation for p(x, nS, t), where nS is the number of substrate molecules, has been developed and analyzed.24 There is an implicit assumption in the CDMM formalism. It is assumed that the conformational changes in x and the catalytic reaction, represented by the k2, are not coordinated. Reactions always occur without conformational change. For example, a reaction such as ES(x) f E(x ) 0) + P is excluded from the CDMM formalism. In motor protein theory, such reactions are considered in the chemical models but not in the ratchet models.25 In the framework of eq 1a, the researcher in ref 26 carried out an elegant analysis showing that the enzyme kinetics exhibiting Michaelis-Menten behavior. The various regimes are9,26,27 (1) (eq 1a) the quasi-static E fluctuation, (eq 1b) the quasi-static ES fluctuation, (2) the quasi-equilibrium between E and ES, and (3) ES being at conformational equilibrium. When conformational fluctuations in both E and ES are slow, a nonMM behavior arises.27
ko1(x) k1(x) ) Ka
3. Kinetics of Fluctuating Enzymes with Slow Dynamic Disorder In this Letter, an analytical result for slow conformational fluctuations in E and ES is fully developed, but first, the detailed balance condition for the substrate-enzyme binding is expressed as
where [S]eq is related to the association constant Ka28
o
)
∫ e-u (x)/(k T)dx ∫ e-u (x)/(k T)dx
2
B
1
B
(3)
(2)
2
B
1
B
and
ko1(x)[S] (4) s) k1(x)
Then, the terms k1o(x)[S] in eq 1 become k1(x)s. In the absence of k2(x), the steady-state solution to eq 1 is the equilibrium Boltzmann distribution
pE(x) ) Ae-u1(x)/(kBT)
pES(x) ) Ae-u2(x)/(kBT)
(5)
where A is a normalization factor such that
∫-∞+∞ (pE(x) + pES(x))dx ) 1
(6)
Now, if k2(x) is not zero, there is a substrate-driven catalytic cycle E f ES f E. If we assume slow conformational fluctuation and fast enzyme catalysis, then we can applying the method of “rapid biochemical cycling” in motor protein ratchet theory19 and obtain
∂p(x) ∂2a(x)p(x) ∂ + (b(x)p(x)) ) kBT 2 ∂t ∂x ∂x
(7)
where p(x) ) pE(x) + pES(x), and a(x) and b(x) are given in the Methods Section, eqs 17 and 18. The steady-state solution to eq 7 is
(
B 1 exp a(x) kBT
p(x) )
)
b(z) dz ∫0x a(z)
(8)
where B is a normalization factor. Working out the mathematics (see the Methods Section), we obtain the probability of the enzyme in the ES complex
pES )
∫-∞+∞ pES(x)dx ) s ∫ e-u (x)/(k T)e-φ(x,s)dx ∫ (se-u (x)/(k T) + (1 + σ(x))e-u (x)/(k T))e-φ(x,s)dx 2
2
ko1(x)[S]eq ) e-(u2(x)-u1(x))/(kBT) k-1(x)
e-GE /(kBT)
∫ e-u (x)/(k T)dx ∫ e-u (x)/(k T)dx
To simplify the notations, we shall denote
∂pES(x, t) kBT ∂2pES 1 ∂ du2(x) + + ) p 2 ∂t η2 ∂x η2 ∂x dx ES ko1(x)[S]pE
e-GES /(kBT) o
eq
B
B
1
(9)
B
in which σ(x) ) k2(x)/k-1(x) and
φ(x, s) )
dzσ′(z)/η1
∫0x se-(u (z)-u (z))/(k T)/η 2
1
B
2
+ (1 + σ(z))/η1
(10)
where we denote derivative dσ(z)/dz by σ′(z). We shall call φ(x, s) the non-MM factor since its independence of s implies that the pES obeys a hyperbolic binding curve. The following scenarios were discussed in ref 26: (1) If η1/η2 ) 0 or ∞, that is, the conformational fluctuations of ES or E are quasi-static, then φ(x, s) is either independent of s or zero;
Letters
J. Phys. Chem. B, Vol. 113, No. 8, 2009 2227
(2) if σ(x) ≈ 0, that is, the E and ES are in quasi-equilibrium, then φ(x, s) ) 0; (3) if σ(x) is a constant, that is, ES being at conformational equilibrium, then again, φ(x, s) ) 0. In all of these cases, the enzyme catalysis follows MM kinetics. To find out when pES exhibits positive cooperativity, we consider small s and express pES following eq 26 in the Methods Section
pES )
s(c1 + c2s + ...) c3 + c4s + ...
(11a)
exp(-u2(x)/(kBT)) dx 1 + σ(x)
(11b)
where
c1 )
c2 )
∫-∞+∞
∫-∞
c3 )
+∞
φ1(x) exp(-u2(x)/(kBT)) dx 1 + σ(x)
∫-∞+∞ exp(-u1(x)/(kBT))dx
c4 ) c1 + c5 ) c1 +
(11c)
(11d)
∫-∞+∞ φ1(x) exp(-u1(x)/(kBT))dx (11e)
φ1(x) )
η1 η2
Figure 1. (A) Probability distributions for the conformational fluctuations of E and ES states. More precisely, the µ1(x) and µ2(x) in eqs 14a and 14b, with x1 ) x2 ) 0 and σ1 > σ2. (B) If σ(x) is symmetrically distributed around the most probable conformation (x ) 0), then σ′(x) is an odd function and φ1(x) is an even function, nonpositive, with φ1(0) ) 0. c2/c1 ) ∫ φ1(x)µ2(x)dx and c5/c3 ) ∫ φ1(x)µ1(x)dx. Hence, the figure shows that c2/c1 > c5/c3.
What if the conformational fluctuations in E and ES are both symmetric with respect to the most probable conformations but with different magnitudes, as shown in Figure 1A? Let us assume that one can represent
-(u2(z)-u1(z))/(kBT)
∫0x σ′(z)e(1 + σ(z))2
dz
(11f)
µ1(x) )
Positive cooperativity means pES, as a function of s, has positive curvature at s ) 0. That means
(
)
c2 c1c4 c1 c2 c5 c1 - 2 ) >0 c3 c3 c1 c3 c3 c3
≈
(12) µ2(x) )
4. Conditions for Fluctuating Enzymes Exhibiting Positive Cooperativity We shall now use the criterion that we established in eq 12 to see what minimal deviations from a simple enzyme are necessary for an enzyme to exhibit positive cooperativity. By a simple enzyme, we mean it has well-defined E and ES states and the associated rate constants k1o, k-1, and k2, as traditionally assumed in the Michaelis-Menten theory. Let us first consider that the conformational fluctuations in states E and ES are comparable, u1(x) ) u2(x). Then, φ1(x) ) -(η1)/(η2)[(1/(1 + σ(x))) - (1/(1 + σ(0)))]. Substituting this into eqs 11 and 12, we have
〉
c2 c5 η1 1 1 )e0 c1 c3 η2 1 + σ 〈1 + σ〉
(〈
)
(13)
≈
exp(-u1(x)/(kBT))
∫-∞ exp(-u1(x)/(kBT))dx +∞
1
√2πσ21
(
exp -
(x - x1)2 2σ21
)
(14a)
exp(-u2(x)/(kBT))/(1 + σ(x))
∫-∞ exp(-u2(x)/(kBT))/(1 + σ(x))dx +∞
1
√2πσ22
(
exp -
(x - x2)2 2σ22
)
(14b)
then
φ1(x) )
c1 η1σ1 c3 η2σ2
∫0x 1 +σ′(z)σ(z) e-(z-x ) /2σ e(z-x ) /2σ dz 2
2
2 2
1
2
2 1
(14c)
and
c2 ) c1
∫-∞+∞ φ1(x)µ2(x)dx
and
c5 ) c3
∫-∞+∞ φ1(x)µ1(x)dx (14d)
where the bracket
∫ · · ·e ∫e
-u1(x)/(kBT)
〈· · ·〉 )
-u1(x)/(kBT)
/(1 + σ(x))dx
/(1 + σ(x))dx
and we used the Cauchy-Schwarz inequality 〈f -1〉〈f 〉 g 〈(f -1)1/2(f)1/2〉2 ) 1 for f > 0. Therefore, the curvature in eq 12 is negative, and there will be no positive cooperativity.
Let
φ01(x) )
η1σ1 η2σ2
∫0x 1 +σ′(z)σ(z) e-(z-x ) /2σ e(z-x ) /2σ dz
then φ1(x) ) (c1/c3)φ10(x), and
2
2
2 2
1
2
2 1
(14e)
2228 J. Phys. Chem. B, Vol. 113, No. 8, 2009
c1 c2 ) c1 c3
c5 c1 ) c3 c3
∫-∞+∞ φ01(x)µ2(x)dx
Letters
∫-∞+∞ φ01(x)µ1(x)dx (14f)
The curvature at s ) 0 is
(
) ()∫
c5 c1 c1 c2 c1 ) [ c3 c1 c3 c3 c3 2
have recently shown that dynamic cooperativity is direcetly related to substrate specificity and not limited by the equilibrium affinity.7 Our current result suggests that in a living cell, the efficacy of a ligand interacting with a fluctuating eznyme needs not to be correlated solely with the affinity in general. 6. Methods Section
+∞
-∞
φ01(x)µ2(x)dx-
∫-∞+∞ φ01(x)µ1(x)dx - 1]
(14g)
If µ1(x), µ2(x), and σ(x) are all symmetrically distributed around x ) 0, then as shown in Figure 1B, σ′(x) is an odd function, and φ10(x) in eq 14e is an even, nonpositive function, with φ10(0) ) 0. Combining the Gaussian distributions in eqs 14a and 14b with σ2 < σ1, that is, the fluctuations in the E are much greater than those in ES, we have
∫-∞+∞ φ01(x)µ2(x)dx - ∫-∞+∞ φ01(x)µ1(x)dx > 0
With the assumption of slow conformational fluctuations and fast enzyme catalysis, we have k1(x)spE(x) ) (k-1(x) + k2(x))pES(x) for every x. Therefore
(k-1(x) + k2(x))p(x) k1(x)s + k-1(x) + k2(x) k1(x)sp(x) pES(x) ) k1(x)s + k-1(x) + k2(x)
pE(x) )
(16)
in which p(x) ) pE(x) + pES(x) satisfies a Fokker-Planck equation (eq 7) with
(15) a(x) )
With a sufficiently large magnitude of φ10(x), the curvature in eq 12 can be positive. Figure 2 shows two examples.
)
k1(x)s/η2 + (k-1(x) + k2(x))/η1 k1(x)s + k-1(x) + k2(x) se-u2(x)/(kBT) /η2 + (1 + σ(x))e-u1(x)/(kBT) /η1
(17)
se-u2(x)/(kBT) + (1 + σ(x))e-u1(x)/(kBT)
5. Concluding Remarks In recent years, molecular mechanisms of enzyme functions have been increasingly understood in terms of their dynamics in addition to their atomic structures.29 Internal motions of a protein have long been suggested as a source of enzyme catalytic ability and specificity,30,31 and the possible functions of slow conformational dynamics in metabolic regulations were noted.32 While it has been known that a monomeric enzyme with a single site can exhibit sigmoidal, positive cooperative binding of its substrate, the kinetic model in the past assumed multiple conformational states for unbound enzyme E.6,33 It is not known whether conformational fluctuation within a single thermodynamic state is possible to do the same. Slow conformational fluctuations in enzymes have been firmly established in recent years.9 In this Letter, on the basis of a conformational diffusive Michaelis-Menten theory, we show that large fluctuations of unbound E in a single state (single energy well) are capable of exhibiting positive cooperativity in its substrate binding, provided that the substrate binding reduces the range of the dynamic disorder while maintaining its rate. The conformational fluctuations in ES are much smaller than those in E. It is important to note that a conventional threedimensional structure of an enzyme provides only the average structure; it precisely lacks informations on the “range” and “rate” of the structural fluctuations. The slow dynamic disorder, however, is only one of the necessary conditions. For an enzyme to exhibit positive dynamic cooperativity, it has to operate under a sustained nonequilibrium steady state, with a constant turnover of substrate to product.10 In other words, a monomeric single-site enzyme exhibits that positive cooperativity has an energy cost. Proper information processing inside of living cells cannot be carried out without energy dissipation.5 Will such dynamic cooperativity be relevant in living cells and in medicine? Curiously, a recent examination of a large set of drugs shows that drug efficacy is more correlated with the dissociation rate rather than equilibrium affinity, the latter being widely accepted as standard in drug-target interaction.34 We
b(x) ) )
k1(x)s(u2 (x)/η2) + (k-1(x) + k2(x))(u1 (x)/η1) k1(x)s + k-1(x) + k2(x)
se-u2(x)/(kBT)(u2 (x)/η2) + (1 + σ(x))e-u1(x)/(kBT)(u1 (x)/η1) se-u2(x)/(kBT) + (1 + σ(x))e-u1(x)/(kBT)
(18) where σ(x) ) k2(x)/k-1(x). We note that (u2 (x)/η2) -u2(x)/(kBT) se /η2
-u2(x)/(kBT)
se b(x) ) a(x)
+ (1 + σ(x))e-u1(x)/(kBT)(u1 (x)/η1)
+ (1 + σ(x))e-u1(x)/(kBT) /η1 s d 1 + σ(x) -u1(x)/(kBT) )-kBT ln e-u2(x)/(kBT) + e dx η2 η1
(
)
-u1(x)/(kBT)
+
kBTσ′(x)e -u2(x)/(kBT)
se
/η1
/η2 + (1 + σ(x))e-u1(x)/(kBT) /η1
(19) Solving the steady state of eq 7 therefore yields Bse-u2(x)/(kBT) × se /η2 + (1 + σ(x))e-u1(x)/(kBT) /η1 x b(z) 1 exp dz kBT 0 a(z)
pES(x) )
-u2(x)/(kBT)
(
)Bse
)
∫
(20)
-u2(x)/(kBT) -φ(x,s)
e
pE(x) ) B(1 + σ(x))e-u1(x)/(kBT)e-φ(x,s)
(21)
with
φ(x, s) )
dzσ′(z)/η1
∫0x se-(u (z)-u (z))/(k T)/η 2
1
B
2
+ (1 + σ(z))/η1
(22)
Letters
J. Phys. Chem. B, Vol. 113, No. 8, 2009 2229
∞ Figure 2. Examples of dynamic cooperativity: The probability of the enzyme-substrate complex is pES ) ∫-∞ pES(x)dx as a function of the substrate concentration s ) [S]/[S]eq. (A) Dynamic cooperativity with Gaussian σ(x). The parameters used in the computation are σ(x) ) 1000 exp(-x2/ 2), u1(x) ) x2/1000 + 10, u2(x) ) x2/2, η1/η2 ) 1, and kBT ) 1. The solid line is according to eq 20, which is valid in the limit of large η1, 2, that is, slow conformational fluctuations. The symbols are from numerical solutions to eq 1, where filled squares are for η1 ) η2 ) 106 and open squares are for η1 ) η2 ) 103. This shows that the asymptotic solution eq 20 is a good approximation when the conformational fluctuation is slow but not accurate when the conformational fluctuation is not slow enough. (B) Dynamic cooperativity with a boxcar function, showing a sharper sigmoidal curve. The parameters are σ(x) ) 100[tanh(38/x4 - x4) + 1], u1(x) ) x2/200 + 10, u2(x) ) x2/2 + 1, η1 ) 106, η2 ) 105, and kBT ) 1. Numerical solutions (filled squares) fit well with the asymptotic analytical solution. (C) The graph of σ(x) used in panel B, which is given by σ(x) ) 100[tanh(38/ x4 - x4) + 1]. (D) Graph of the functions µ1(x) and µ2(x) in eqs 14a and 14b in the case of panel B, which are approximately Gaussian functions. The magnitude of conformational fluctuation of E (corresponding to µ1) is much greater than that of ES.
For small s
e-φ(x,s) ) exp(-φ0(x) + φ1(x)s + ...) )
1 + φ1(x)s + ... 1 + σ(x)
(23)
where
φ0(x) )
∫0x (1 +σ′(z)σ(z)) dz ) ln(1 + σ(x))
φ1(x) )
η1 η2
(24)
-(u2(z)-u1(z))/(kBT)
∫0x σ′(z)e(1 + σ(z))2
(25)
dz
Combining eqs 20-25, we have s pES )
∫
∫e
-u2(x)/(kBT)
(se
-u2(x)/(kBT)
(1 + sφ1(x))/(1 + σ(x))dx -u1(x)/(kBT)
+ (1 + σ(x))e
)(1 + sφ1(x))/(1 + σ(x))dx
+ o(s2)
(26)
2230 J. Phys. Chem. B, Vol. 113, No. 8, 2009 Acknowledgment. We thank Wei Min and Sunney Xie for sharing their unpublished manuscripts,35 Henri Buc, Ken Dill, Peter Luo, Wei Min, Bill Parson, Attila Szabo, Sunney Xie, and Jianhua Xing for helpful discussions. References and Notes (1) (a) Hammes, G. G. Enzyme Catalysis and Regulation, Student ed.; Academic Press: New York, 1982. (b) Fersht, A. Enzyme Structure and Mechanism, 2nd ed.; W. H. Freeman: New York, 1985. (c) Segel, I. H. Enzyme Kinetics: BehaVior and Analysis of Rapid Equilibrium and SteadyState Enzyme Systems; Wiley: New York, 1993. (d) Cornish-Bowden, A. Fundamentals of Enzyme Kinetics; Portland Press: London, 2004. (2) Beard, D. A.; Qian, H. Chemical Biophysics: QuantitatiVe Analysis of Cellular Systems, Cambridge Texts in Biomedical Engineering; Cambridge University Press: New York, 2008. (3) Wyman, J.; Gill, S. J. Binding and Linkage: Functional Chemistry of Biological Macromolecules; University Science Books: Mill Vally, CA, 1990. (4) Goldbeter, A.; Koshland, D. E. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 6840–6844. (5) (a) Qian, H.; Cooper, J. A. Biochemistry 2008, 47, 2211–2220. (b) Qian, H. Biophys. Chem. 2003, 105, 585–593. (c) Ge, H.; Qian, M. J. Chem. Phys. B 2008, 129, 015104. (6) (a) Cornish-Bowden, A.; Ca´rdenas, A. L. J. Theor. Biol. 1987, 124, 1–23. (b) Ricard, J.; Cornish-Bowden, A Eur. J. Biochem. 1987, 166, 255– 272. (c) Ricard, J. In Allosteric Enzymes; Herve´, G., Ed.; CRC Press: Boca Raton, FL, 1989; pp 1-25. (7) Qian, H Biophys. J. 2008, 95, 10–17. (8) Frieden, C. Annu. ReV. Biochem. 1979, 48, 471–489. (9) (a) English, B. P.; Min, W.; van Oijen, A. M.; Lee, K. T.; Luo, G.; Sun, H.; Cherayil, B. J.; Kou, S. C.; Xie, S. X. Nat. Chem. Biol. 2006, 2, 87–94. (b) Kou, S. C.; Cherayil, B. J.; Min, W.; English, B. P.; Xie, X. S. J. Phys. Chem. B 2005, 109, 19068–19081. (10) (a) Qian, H. Annu. ReV. Phys. Chem. 2007, 58, 113–142. (b) Qian, H. J. Phys. Chem. B 2006, 110, 15063–15074. (11) Eisenmesser, E. Z.; Millet, O.; Labeikovsky, W.; Korzhnev, D. M.; Wolf-Watz, M.; Bosco, D. A.; Skalicky, J. J.; Kay, L. E.; Kern, D. Nature 2005, 438, 117–121. (12) (a) The Fluctuating Enzyme; Welch, G. R., Ed.; John Wiley & Sons: New York, 1986. (b) Welch, G. R.; Somogyi, B.; Damjanovich, S. Prog. Biophys. Mol. Biol. 1982, 39, 109–146. (c) Whitehead, E. Prog. Biophys. Mol. Biol. 1970, 21, 321–97. (13) Cooper, J. A.; Howell, B. Cell 1993, 73, 1051–1054. (14) Volkman, B. F.; Lipson, D.; Wemmer, D. E.; Kern, D. Science 2001, 291, 2429–2433. (15) Conformational changes, well studied at the time scale of milliseconds due to ligand bindings and/or chemical modifications, and conformational dynamics, as demonstrated by amide proton hydrogen exchange and molecular dynamic calculations occurring on the picosecond time scale, are often lumped together in most biochemical discussions. A
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