Download Protein Degrees of Freedom & Nonrecrossing Hypothesis in Enzymatic Catalysis and more Summaries Biochemistry in PDF only on Docsity!
Quantifying the limits of transition state theory in
enzymatic catalysis
Kirill Zinovjev a^ and Iñaki Tuñóna,
aDepartament de Química Física, Universitat de València, 46100 Burjassot, Spain
Edited by Donald G. Truhlar, University of Minnesota, Minneapolis, MN, and approved October 3, 2017 (received for review June 15, 2017)
While being one of the most popular reaction rate theories, the applicability of transition state theory to the study of enzymatic reactions has been often challenged. The complex dynamic nature of the protein environment raised the question about the validity of the nonrecrossing hypothesis, a cornerstone in this theory. We present a computational strategy to quantify the error associated to transition state theory from the number of recrossings observed at the equicommittor, which is the best possible dividing surface. Application of a direct multidimensional transition state optimiza- tion to the hydride transfer step in human dihydrofolate reductase shows that both the participation of the protein degrees of free- dom in the reaction coordinate and the error associated to the nonrecrossing hypothesis are small. Thus, the use of transition state theory, even with simplified reaction coordinates, provides a good theoretical framework for the study of enzymatic catalysis.
enzymatic catalysis | transition state theory | dynamic effects |
dihydrofolate reductase | transmission coefficient
T
he origin of the enormous catalytic power of enzymes has been the subject of intensive research during recent decades.
determine the reaction rate. The basic assumption in TST is that there is a hypersurface in the configurational space—the tran- sition state (TS)—separating reactant and product basins that is never recrossed by trajectories coming from the reactants side. Then, the discussion about the validity of TST in enzymatic ca- talysis can be recast as a question about the validity of this nonrecrossing hypothesis and quantified by the transmission coefficient (κ), the ratio between the true rate constant and that obtained from the application of TST,
κ =
ktrue kTST
, [1]
where κ is smaller or equal to unity. When κ = 1, there are no recrossings of the TS and the TST rate is exact. The nonrecross- ing assumption also implies that the distribution formed at the TS by the reactive flux is the equilibrium one (28). According to this, the rate constant can be evaluated from the free energy difference between the reactants and the TS. Quantification of the inherent error of TST is not a straight- forward task because the TST rate constant and then the transmission coefficient depend on the choice of the dividing surface. This choice is typically made by selecting a function—a reaction coordinate (RC)—that describes the progress of the reaction and assigning the TS to some value of this function. In other words, the transmission coefficient is not an intrinsic property of the system. If barrier crossing involves changes in other degrees of freedom not included in a putative RC, there will be a friction (mean force) acting against the RC, causing recrossings and thus decreasing κ (24, 29). For example, enzy- matic compression along the donor–acceptor distance could as- sist a transfer reaction (30). As a result, some of those trajectories
Significance
Transition state theory (TST) is the most popular theory to cal- culate the rates of enzymatic reactions. However, in some cases TST could fail due to the violation of the nonrecrossing hypoth- esis at the transition state. In the present work we show that even for one of the most controversial enzymatic reactions—the hydride transfer catalyzed by dihydrofolate reductase—the er- ror associated to TST represents only a minor correction to the reaction rate. Moreover, this error is actually larger for the re- action in solution than in the enzymatic active site. Based on this finding and on previous studies we propose an “enzymatic shielding” hypothesis which encompasses various aspects of the catalytic process.
Many successful studies, including both interpretation and pre- dictions, have been carried out in the framework of transition state theory (TST) (1–7). However, because of the complex dy- namic nature of the enzymatic environment (8–10), it has been continuously questioned whether statistical rate theories, such as TST, can capture the whole impact of protein motions in the rate of the chemical reaction. The so-called “dynamic effect” hy- pothesis stresses the role of these motions in the interpretation of several phenomena. Strong temperature dependence of ki- netic isotope effects (KIEs) in H-transfer reactions was inter- preted as the result of enzymatic promoting vibrations that compress the donor–acceptor distance (11, 12). Correlation be- tween the frequency of conformational changes and reactive events was seen as a signal of the existence of protein motions essential for the reaction (9, 13). The decrease of the catalytic efficiency observed after the introduction of mutation of residues placed far from the active site was explained as a consequence of the perturbation of some essential protein vibrations that expand beyond the active site (14, 15). Simulations were also performed for various enzymes to identify protein vibrational modes that would increase the probability of barrier crossing (16–18), al- though the methods used for this identification have been the subject of debate (19, 20). According to some models, the role of these promotion motions could be relevant in early and late stages of the reaction and not during barrier crossing (21, 22). In any case, the participation of protein motions in the chemical event could be the origin of practical or fundamental limitations for the application of TST to enzymatic catalysis. However, the dynamic effects hypothesis bears a serious shortcoming: There is no consensus in the field regarding the definition of what these effects are and how to quantify them. As a consequence, all of the interpretations based on these effects remain somewhat speculative (3–5, 23 – 25). The term dynamic effect itself can be misleading (24, 26, 27), because TST is in fact a dynamical theory that accounts for the reactive flux, using an equilibrium description. There is nothing in TST saying that the dynamics of the system are unimportant to
PNAS Early Edition | 1 of 6
reactions using a simple antisymmetric combination of breaking and forming bonds are typically around 0.5 or lower (39, 40), while higher values can be obtained with more collective RCs based in an empirical valence bond description (41). Many geo- metric degrees of freedom are coupled to the hydride transfer because the reaction involves changes in the aromaticity of the two reacting fragments and then it should not be surprising that a simple antisymmetric RC was not able to provide a good dividing surface. Using transition path sampling techniques on this enzyme, Masterson and Schwartz (18) recently proposed that fast protein motions might be dynamically coupled to the RC, increasing the likelihood of barrier crossing. However, an increase of the mass of hsDHFR by isotopic substitution had no detectable effects on the KIEs for the hydride transfer, which led Kohen and coworkers (42) to the conclusion that protein motions promoting the hydride transfer are hardly affected by the mass change. Therefore, the hydride transfer in hsDHFR is a representative and challenging system for the study of the limits of TST applied to enzymatic catalysis. Our results demonstrate that the intrinsic error associ- ated to TST is small and that the participation of the environment in the definition of the best possible RC is reduced in the catalyzed reaction with respect to the counterpart process in solution.
Results We have analyzed the hydride transfer reaction catalyzed by hsDHFR, using the GHTS method (38) to optimize the TS on multidimensional free energy surface described by quantum mechanics/molecular mechanics (QM/MM) Hamiltonian both in solution and in the enzyme (Methods). Three different approaches to the RC were employed (Fig. 2). We first used as RC the anti- symmetric combination of hydride–donor and hydride–acceptor distances. We then optimized the RC, using a basis set of 62 collective variables (CVs) that basically includes all of the relevant degrees of freedom of the substrate and the cofactor (distances and hybridization coordinates) (Methods and RC Optimization and Analysis for the Enzymatic Reaction). The resulting RC is called here “the chemical RC” because it is assumed to include all of the degrees of freedom of the chemical system (substrate and cofactor) that might contribute to the RC. Finally, the ensemble generated in this way was used to characterize the properties of the equi- committor surface, using the reweighting procedure (Methods). The equicommittor as a coordinate involves all of the degrees of freedom of the system (chemical subsystem and environment) and provides the best possible RC, minimizing the error associ- ated to TST.
Transmission Coefficient at the Equicommittor Surface. The trans- mission coefficients and committor histograms obtained for the dividing surfaces defined by the three different RCs (antisym- metric, chemical, and equicommittor) are given in Table 1 and Fig. 3, respectively. The transmission coefficients obtained with the antisymmetric coordinate in solution and in hsDHFR are similar to those obtained in previous works using the same co- ordinate (40, 43, 44). The committor histograms clearly show
N
H O
NH (^2)
N H
H N
N
NH
O
NH (^2)
N H
N
O
NH (^2)
N H
H N
N
NH
O
NH (^2)
N H
Fig. 1. Hydride transfer in DHFR.
that cross the TS surface defined along a simplified RC (e.g., the antisymmetric stretch in a transfer reaction) eventually fall back to reactants, violating the nonrecrossing assumption of TST. Then, a better definition of the dividing surface (with higher κ) can be obtained by modifying the initial RC to include protein degrees of freedom. This is the motivation for the variational TST (VTST), the application of the variational principle in the space of RCs to maximize κ (31–34). However, there is no guarantee that an ideal RC, providing a value for κ equal to unity, exists, even if all en- vironmental degrees of freedom are considered in its definition. That would result in a limit for the exactness of TST. Thus, there are two different issues to be quantified regarding the applicability of TST in the context of enzymatic catalysis:
(i) How important is the inclusion of protein degrees of freedom in the definition of the RC? (ii) How important is the error as- sociated to the nonrecrossing hypothesis for the best possible RC (one that includes all of the relevant degrees of freedom)? While the first question addresses a practical problem during the ap- plication of TST to realistic systems, the second one refers to a more basic issue in TST: Does a nonrecrossing surface exist for enzymatic reactions? And, if not, does this have important consequences in the evaluation of the rate constant? To give an unambiguous answer to these two questions one has to look for intrinsic characteristics of the system, independent of arbitrary choices. If for a particular system a nonrecrossing surface exists, it must coincide with the 0.5 isosurface of the committor
function pB—the probability for a random trajectory initiated at some point on the dividing surface to end up in the products basin. If all of the trajectories that cross a given surface are reactive, exactly half of the equilibrium flux (the one coming from the re- actants side) will end up in products, while another half (the one coming from the products side) will commit to the reactants basin. The 0.5 isocommittor surface (the equicommittor) can be recrossed only either by trajectories that recross the equicommittor an even number of times (reactive recrossings) or by the so-called recrossing pairs: a reactants → reactants and a products → products trajectory pair that share a common configuration (35). If recross- ing pairs exist, then the distribution formed by reactive flux at the dividing surface is different from the equilibrium one (36). This nonequilibrium effect is an intrinsic property of the system: There is no surface that can avoid the recrossing pairs if they exist, so they cannot be eliminated by RC optimization and thus the equicommittor provides an upper limit for κ. Therefore, the value of the transmission coefficient at the equicommittor answers the two questions raised in the previous paragraph. The first question, “How important is the inclusion of protein degrees of freedom in the definition of the RC?”, is answered by measuring the importance of these degrees of freedom in the improvement of the transmission coefficient up to the value reached at the equicommittor. The answer to the second one, “How important is the error associated to the nonrecrossing hypothesis for the best possible RC, one that includes all of the relevant degrees of freedom?”, is provided by the deviation of κ from unity at the equicommittor. Analytic expressions of the committor are not available for realistic systems and thus cannot be directly used in combination with en- hanced sampling methods to characterize the TS and to obtain the transmission coefficient. However, it can be calculated indirectly
using a reweighting scheme (37) (Methods), provided that a good- overlapping ensemble can be adequately sampled. Such ensemble can be obtained using the generalized hyperplanar TS (GHTS) optimization technique, recently developed in our group, which provides the dividing surface with highest κ, using a linear com- bination of a set of collective variables (38). We here applied this methodology to analyze a hydride transfer reaction catalyzed by the human dihydrofolate reductase (hsDHFR). hsDHFR catalyzes the reduction of dihydrofolate to tetrahydrofolate by NADPH (Fig. 1). The κ values obtained for enzymatic hydride transfer
2 of (^6) Zinovjev and Tuñón
not be abused to make claims about the TS structure without checking the value of κ.
Discussion The lack of practical approaches to unambiguously quantify the error of TST has been one of the reasons for ongoing debates regarding the importance of this factor, frequently attributed to so-called dynamic effects, in enzymatic catalysis. Here we have shown how a combination of RC optimization and ensemble re- weighting can solve this issue. Using this methodological com- bination, we have been able to address two important concerns often mired in this debate: the environmental participation in the RC and the inherent error associated to the use of statistical rate theories. The last one can be quantified from the transmission coefficient at the equicommittor surface. The results for the hy- dride transfer catalyzed by hsDHFR clearly show that these two factors have a negligible effect on the evaluation of the enzymatic reaction rate. Therefore, despite existing claims (18), it is hardly credible that such small effects could have practical consequences, for example in the improvement of the catalytic efficiency of en- zymes during natural or directed evolution. Moreover, deviation from the equilibrium assumption in the enzyme is smaller than in aqueous solution. We have introduced the “enzymatic shielding” concept that summarizes this and other differences between en- zymatic reaction and reaction in solution, resulting in an increased reaction rate: less reorganization of the environment, smaller participation of environmental degrees of freedom during the barrier crossing, and finally reduced nonequilibrium effects. Al- though further research may be required to generalize these conclusions, some of the results presented here have been al- ready observed in systems involving larger charge redistributions. In particular, the methyl transfer reaction catalyzed by catechol O-methyl transferase (COMT) proceeds from charged reactants (catecholate and S-adenosyl methionine) to neutral products. In that case it was found that the transmission coefficient for the enzymatic TS defined using a simple antisymmetric transfer co- ordinate was quite high (52). In addition, the quality of the RC, determined from the committor histogram, was not noticeably improved when optimized, including also a collective environ- mental coordinate. Instead, the quality of the RC was noticeably improved when the same procedure was followed for the coun- terpart process in solution. The analysis of the TS geometry reveals an interesting and useful feature: For the studied reaction the antisymmetric RC provides an average TS geometry that is only slightly different from the one obtained with optimized and even ideal (committor function) RCs. While the quality of a RC should always be checked by computing the associated transmission coefficient before ex- tracting any conclusion, it seems that for most practical tasks, such as the calculation of activation free energies or the design of TS mimics, expensive RC optimization is usually not required. How- ever, RC optimization might be necessary for reactions with poorly understood mechanisms, where dividing surfaces with large values of κ are challenging to obtain (53). The results presented here also offer some guidelines for future research work. First, since TST is able to calculate successfully the rate of enzymatic reactions, the main effort should be fo- cused on the improvement of free energy calculation techniques. This includes sampling efficiency, the quality of Hamiltonians,
Table 2. Fraction of flux formed by trajectories with different number of recrossings at the equicommittor System No recrossings Reactive recrossings Recrossing pairs
Enzyme 0.94 (±0.02) 0.03 (±0.01) 0.03 (±0.01) Water 0.88 (±0.03) 0.08 (±0.02) 0.05 (±0.01)
0 0.2 0.4 0.6 0.8 1
p (^) B
P
(pB
κ = 0.
A
0 0.2 0.4 0.6 0.8 1
pB
κ = 0.
B
0 0.2 0.4 0.6 0.8 1
p (^) B
P
(p
)B
κ = 0.
C
0 0.2 0.4 0.6 0.8 1
pB
κ = 0.
D
Fig. 3. The committor histograms for the transfer and chemical dividing surfaces for enzymatic and aqueous solution reactions. (A) Enzyme, anti- symmetric RC; (B) enzyme, chemical RC; (C) water, antisymmetric RC; (D) water, chemical RC.
equicommittor. The total flux through the equicommittor surface can be then trivially split into contributions from trajectories with different numbers of surface crossings (Table 2). The results show that, if only recrossing pairs are considered, nonequilibrium ef- fects account for only 3% and 5% of the total flux at the equi- committor, for the reaction in enzyme and water, respectively.
Geometry of the TS. While in free energy terms the difference observed between the rate constants obtained with a naive and an optimized RC can be small, there is no guarantee that a lower-quality RC provides an adequate geometric description of the TS, which can be essential for the design of TS analogs or biocatalysts or the analysis of KIEs (49). To our knowledge, only one study explicitly addressed this question before for an enzy- matic reaction (50). We then used simulations and the reweighting procedure to calculate the distributions of essential geometric properties at the three dividing surfaces analyzed above (Table 3 and Table S1). The results are strikingly similar for the three TSs. The dif- ferences between the mean values in all cases are on the order of 0.01 Å and are barely statistically significant. It was shown above that the transmission coefficient is significantly different for the three TSs. So, it seems that κ is extremely sensitive even to tiny changes in the geometry of the dividing surface. Turning this finding upside down, we reach an important conclusion: The TS geometry has low sensitivity to the quality of the RCs tested. As a result, when one is interested only in the geometry of the TS, a simple coordinate (such as the transfer coordinate) can provide a reasonable result. This finding is in line with the tremendous success of free energy-based computational methods in enzymology: Although most of the studies used simple geometric RCs, the results are generally in very good agreement with experiments (2–4, 6, 7, 51). If the nature of the TS were very sensitive to the choice of RC, that probably would not be the case. However, we stress that this result is obtained by comparing the best possible dividing surface to others with already reasonable quality: For the worst RC analyzed here (the antisymmetric transfer coordinate) the transmission coefficient is 0.41, which in free energy terms means an underestimation of the activation free energy <1 kBT (at room temperature). Probably, in cases where the RC (and thus the dividing surface) deviates strongly from the correct one (κ << 1) this conclusion does not apply and, therefore, should
4 of (^6) Zinovjev and Tuñón
and the RC definition. Second, our analysis is based on a classical description of nuclear motion and the consequences of a quantum description of some degrees of freedom are unclear. A rigorous theoretical background and practical approaches with affordable computational cost are needed to address this issue, which could be important in reactions with strong tunneling contribution. We hope that the present work will stimulate advances in these directions.
Methods The simulation system was prepared based on the structure 4M6K (54) from Protein Data Bank (www.rcsb.org/). The enzyme was embedded in a TIP3P (55) water box of size 58 × 55 × 67 Å^3. The QM/MM scheme was used with Amber ff12SB (56) forcefield and AM1 (57) semiempirical Hamiltonian with specific AM1 parameters for DHFR catalyzed reaction taken from Doron et al. (58) for the MM and QM regions, respectively (Fig. S1). Langevin dy- namics at 300 K were used in all cases with periodic boundary conditions, a time step of 0.5 fs, and particle mesh Ewald (59) to treat long-range elec- trostatic interactions. The cutoff to limit the direct space sum was set to 8 Å. For the committor test and κ calculation the velocities for each trajectory were sampled from the Maxwell–Boltzmann distribution at 300 K and the dynamics were propagated deterministically both forward and backward in time. For the reaction in water NADPH and DHF were solvated in a water box of size 36 × 48 × 41 Å 3 and were kept close to their orientation in the Michaelis complex by applying weak harmonic restraints (K = 2 kcal·mol−^1 ·Å−^2 ) to the positions of four heavy atoms, one on the adenine base of NADPH and three on DHF (Fig. S2). Five sodium atoms were added to neutralize the system. The rest of the simulation protocol, including the definition of the QM part and the Hamiltonian, was identical to that of the enzymatic system. For the GHTS method (38) we used an active space defined by 62 CVs involving only degrees of freedom of the chemical system to localize the optimum dividing surface (chemical RC). We stress that while a large number of CVs were used to optimize the dividing surface, the RC itself is one di- mensional and, therefore, provides a valid 3N-1 dimensional TS. Essentially all of the bond distances and hybridization coordinates of the two conju- gated systems were considered (Fig. S2). The hybridization coordinate was defined as a point-plane distance from the central atom (carbon or nitrogen) to the plane formed by the three substituents in the sp^2 state (60). However, no protein degrees of freedom were included. It was done on purpose— since the active space covers all of the relevant degrees of freedom of the substrate, the difference between the true TS (equicommittor) and the hy- perplane should roughly correspond to the degree of participation of
environment in the TS crossing. The GHTS optimization protocol and an analysis of the resulting chemical RC are given in RC Optimization and Analysis for the Enzymatic Reaction. The equilibrium properties (including κ) on the equicommittor surface were calculated using the ensemble reweighting (37):
ÆXæpB =0.5 = ÆXe^
βVb (^) j∇p (^) B jδðpB − 0.5Þæ b Æe βVb^ j∇p (^) B jδðpB − 0.5Þæb
. [2]
Here the averaging is done over structures obtained using a harmonic bias that maintains the system close to the hyperplane (“b” subindex) and having pB = 0.5 ± 0.01 at the same time (Dirac’s delta function). The reweighting factor eβVb^ compensates for the biasing potential and j∇pB j recovers the surface integral over p (^) B = 0.5 from one defined using Dirac’s delta function. The latter is approximated as
j∇p (^) B j ≈ ∇p^ B^ ·^ ∇q j∇qj^2
· ∇q = ∇~q pB , [3]
where ∇q p (^) B is the directional derivative of pB along ∇q, which was evaluated numerically. For the reweighting procedure to converge, one needs the two ensembles to be as similar as possible. If not, the reweighting factors become too large, increasing dramatically the statistical error. The optimized chemical RC did not perform so well for the reaction in water; then an extended space of trial coordinates including environmental CVs was used to obtain a better RC with κ = 0.65 (RC Optimization and Analysis for the Reaction in Solution). A total of 20,000 structures were generated for surfaces in protein and water. For each structure, 100 trajectories were integrated forward and backward in time, resulting in 2· 106 trajectories for each system. For both systems, among the 20,000 structures sampled ∼500 structures had the value of committor func- tion = 0.5 ± 0.01. The importance analysis was performed as in ref. 38 and details are given in RC Optimization and Analysis for the Enzymatic Reaction. All of the error bars provided here represent 95% confidence intervals and were calculated by statistical bootstrapping with 10,000 bootstrap samples.
ACKNOWLEDGMENTS. The authors acknowledge computational facilities of the Servei d’Informàtica de la Universitat de València in the “Tirant” supercomputer. The authors gratefully acknowledge financial support from Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional funds (Project CTQ2015-66223-C2-2-P). K.Z. acknowl- edges a Formación Profesorado Universitario fellowship from Ministerio de Educación.
- Åqvist J, Warshel A (1993) Simulation of enzyme reactions using valence bond force fields and other hybrid quantum/classical approaches. Chem Rev 93: 2523 – 2544.
- Martí S, et al. (2004) Theoretical insights in enzyme catalysis. Chem Soc Rev 33:98–107.
- Garcia-Viloca M, Gao J, Karplus M, Truhlar DG (2004) How enzymes work: Analysis by modern rate theory and computer simulations. Science 303:186–195.
- Gao J, et al. (2006) Mechanisms and free energies of enzymatic reactions. Chem Rev 106:3188–3209.
- Kamerlin SCL, Warshel A (2010) At the dawn of the 21st century: Is dynamics the missing link for understanding enzyme catalysis? Proteins 78:1339–1375.
- Glowacki DR, Harvey JN, Mulholland AJ (2012) Taking Ockham’s razor to enzyme dynamics and catalysis. Nat Chem 4:169–176.
- Hammes-Schiffer S (2013) Catalytic efficiency of enzymes: A theoretical analysis. Biochemistry 52:2012–2020.
- Henzler-Wildman KA, et al. (2007) A hierarchy of timescales in protein dynamics is linked to enzyme catalysis. Nature 450:913–916.
- Henzler-Wildman KA, et al. (2007) Intrinsic motions along an enzymatic reaction trajectory. Nature 450:838–844. 10. Nashine VC, Hammes-Schiffer S, Benkovic SJ (2010) Coupled motions in enzyme ca- talysis. Curr Opin Chem Biol 14:644–651. 11. Wang L, Goodey NM, Benkovic SJ, Kohen A (2006) Coordinated effects of distal mutations on environmentally coupled tunneling in dihydrofolate reductase. Proc Natl Acad Sci USA 103:15753–15758. 12. Hay S, Scrutton NS (2012) Good vibrations in enzyme-catalysed reactions. Nat Chem 4:161–168. 13. Bhabha G, et al. (2011) A dynamic knockout reveals that conformational fluctuations influence the chemical step of enzyme catalysis. Science 332:234–238. 14. Watney JB, Agarwal PK, Hammes-Schiffer S (2003) Effect of mutation on enzyme motion in dihydrofolate reductase. J Am Chem Soc 125:3745–3750. 15. Benkovic SJ, Hammes-Schiffer S (2003) A perspective on enzyme catalysis. Science 301: 1196 – 1202. 16. Schwartz SD, Schramm VL (2009) Enzymatic transition states and dynamic motion in barrier crossing. Nat Chem Biol 5:551–558. 17. Schwartz SD (2013) Protein dynamics and the enzymatic reaction coordinate. Dynamics in Enzyme Catalysis, eds Klinman J, Hammes-Schiffer S (Springer, Berlin), pp 189–208. 18. Masterson JE, Schwartz SD (2015) Evolution alters the enzymatic reaction coordinate of dihydrofolate reductase. J Phys Chem B 119:989–996.
Table 3. Distribution of the donor–acceptor distance for the studied dividing surfaces Enzyme Water
Transition state μ σ μ σ
Antisymmetric 2.645 (±0.006) 0.059 (±0.005) 2.675 (±0.006) 0.067 (±0.005) Chemical 2.652 (±0.005) 0.062 (±0.004) 2.663 (±0.006) 0.064 (±0.004) Equicommittor 2.654 (±0.012) 0.058 (±0.005) 2.702 (±0.037) 0.066 (±0.008) Errors for the mean values and the SDs correspond to 95% confidence intervals. All values are given in angstroms.
Zinovjev and Tuñón PNAS Early Edition | 5 of 6
