UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI Date: I,, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair:

2 Nuclear Magnetic Resonance Studies of Side-Chain Motions in Calbindin D 9k : The Role of Conformational Dynamics in Protein Stability and Calcium Binding A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Ph.D.) in the Department of Molecular Genetics, Biochemistry and Microbiology of the College of Medicine 2005 by Eric Johnson B.S. Xavier University, 1998 Committee chair: Mark Rance, Ph.D.

3 ABSTRACT An accurate understanding of the role of conformational dynamics in proteins requires data at multiple timescales and sites within the protein of interest. Considerable progress has been achieved in characterizing the picosecond-to-nanosecond (ps-ns) dynamics of the protein backbone via NMR relaxation measurements of the 15 N nucleus. More recent developments in the measurement of 2 H quadrupolar relaxation rates are enabling an extensive characterization of the dynamics in methyl-containing side-chains as well. The aim of the present study is to characterize the effects of Ca 2+ binding on the side-chain dynamics of the protein calbindin D 9k. Calbindin is a small (~8.7 kd), single domain protein of the EF-hand family. It contains two Ca 2+ binding sites that exhibit high positive cooperativity. Longitudinal, transverse, quadrupolar order, transverse antiphase and double quantum relaxation rates are reported for both the apo (Ca 2+ -free) and Ca 2+ - loaded states of the protein at two magnetic field strengths. The relatively large size of the data set allows for a detailed analysis of the underlying conformational dynamics by spectral density mapping and model-free fitting procedures. The results indicate that a methyl group s distance from the Ca 2+ binding sites is a significant determinant of its conformational dynamics. Several methyl groups segregate into two limiting classes, one proximal and the other distal to the binding sites. Methyl groups in these two classes respond differently to Ca 2+ binding, both in terms of the timescale and amplitude of their fluctuations. Ca 2+ binding elicits a partial immobilization among methyl groups in the proximal class, which is consistent with previous studies of calbindin s backbone dynamics. The distal class, however, exhibits a trend that could not be inferred from the backbone data in that its mobility actually increases with Ca 2+ binding. We have i

4 introduced the term polar dynamics to describe this type of organization across the molecule. The trend may represent an important mechanism by which calbindin achieves high affinity binding while minimizing the corresponding conformational entropy loss. ii

5 iii

6 ACKNOWLEDGEMENTS Many people have contributed to my education as a scientist. I am particularly grateful for the instruction and guidance provided by my advisor Mark Rance. Under Mark s direction, I have received both valuable one-on-one instruction as well as the freedom to explore new ideas independently. I am lucky to have found such a fine mentor. I am also thankful for the care provided by my wife Tricia Johnson. She has seen me through the highs and lows of my education and has tolerated my clumsy attempts to navigate back-and-forth between a solitary world of ideas and our shared life at home. Together, we have recently discovered the joys of parenthood with our son Ben who is my constant source of happiness and amusement. iv

7 TABLE OF CONTENTS Abstract Acknowledgements i iv Table of Contents 1 List of Tables 3 List of Figures 3 CHAPTER 1: Literature Review General importance of protein dynamics 4 The dynamic interpretation of NMR relaxation data 4 Modeling the spectral density function 7 Special considerations for protein side chain dynamics 12 Motional dynamics in calbindin 18 CHAPTER 2: Effects of Calcium Binding on the Side-chain Methyl Dynamics of Calbindin D 9k : A 2 H NMR Relaxation Study Abstract 28 Introduction 28 Theory 32 Results Relaxation Rate Measurements 37 Spectral Density Mapping 39 LS2 Model-Free Analysis 44 LS3 Model-Free Analysis 48 Discussion 51 1

8 Conclusions 58 Materials and Methods Sample Preparation 59 Chemical Shift Assignments 60 Methyl 2 H Relaxation Rate Measurements 61 Data Analysis 62 Acknowledgements 64 CHAPTER 3: Future Directions 65 Appendix 1: Methyl chemical shift assignments 69 Appendix 2: Representative spectra 72 Appendix 3: Relaxation rates 74 Appendix 4: Model-free parameters 94 Appendix 5: Calbindin D 9k NMR sample preparation (step-by-step instructions) 100 Appendix 6: Tips to aid in future data analyses 120 Appendix 7: Side-chain amide 2 H relaxation rates 124 Literature Cited 129 2

9 List of Tables Table 1: Defining features of polar dynamics 56 List of Figures Figure 1: Ribbon diagram of calbindin D 9k 31 Figure 2: Relaxation decay curves 38 Figure 3: Spectral densities as a function of residue number 40 Figure 4: Correlation between experimental and back-calculated rates 43 Figure 5: Order parameters as a function of residue number 45 Figure 6: The order parameter difference as a function of residue number 47 Figure 7: Effective correlation times as a function of residue number 49 Figure 8: LS2 versus LS3 spectral density curves 50 Figure 9: Structural context of the dynamics parameters 52 Figure 10: Correlation between the dynamics parameters and the distance to the binding sites 54 Figure 11: Representative apo spectrum 72 Figure 12: Representative Ca 2+ -loaded spectrum 73 Figure 13: R(D z N z ) relaxation rates 125 Figure 14: R(D x N z ) relaxation rates 126 3

10 CHAPTER 1: Literature Review General importance of protein dynamics The study of protein structure is motivated by numerous factors. It is driven largely by the principle that a protein s function is determined by its structure. Thus, a protein s structure may provide direct evidence of its binding partners or the chemical reactions in which it may participate. Similarly, one may want to alter a protein s function pharmacologically or engineer a protein with a modified function. Its structure is again likely to provide valuable evidence of how these goals may be best accomplished. A protein s structure is not static, however. At equilibrium, a molecule s conformation fluctuates with time. These conformational fluctuations have important implications for protein function, which typically involves some structural reorganization or excursion from the ground state. 1 This realization was made early on in the study of protein structure. For example, when the crystal structure of myoglobin was first solved in 1958, it revealed no obvious diffusion pathway for oxygen to travel from the solvent to its heme binding site in the protein. 2 It is also important to note that a protein s structure is not uniquely defined within a large collection of independent molecules, referred to as a molecular ensemble. 3 Individual members of the ensemble each represent slightly different conformations of the protein at any given point in time. This conformational heterogeneity is sometimes referred to as the residual entropy of folding and is thought to act as a significant stabilizing factor for proteins. 4, 5 These functional and thermodynamic considerations provide much of the motivation for protein dynamics studies and act as conceptual starting points for this thesis work. The dynamic interpretation of NMR relaxation data 4

11 NMR spectroscopy is currently used to study numerous aspects of molecular motion at a high degree of spatial and temporal resolution. NMR applications exist to study the rotational diffusion of entire molecules as well as internal fluctuations of individual bond vectors in the protein backbone and side chains. These motions extend across multiple timescales. Rotational diffusion typically occurs on the nanosecond (ns) timescale, whereas internal motions have been characterized across much wider timescales, extending from picoseconds (ps) to seconds (s). NMR relaxation rates are an observable quantity that report on these motions. Relaxation is the process by which a system returns to equilibrium following some small perturbation. 6 In the case of NMR, nuclear spins within the molecule of interest are placed in a large, static magnetic field, resulting in a Boltzmann distribution across the spin energy states. 7 This scenario describes the system at equilibrium. Perturbation of the system occurs when radiofrequency (rf) pulses, consisting of small, oscillating magnetic fields, are applied in a direction perpendicular to the static field. These rf pulses modify the spin state populations and create observable coherence among the spins in the form of a net transverse spin polarization. Eventually, the nuclear spins return to equilibrium. Populations return to their Boltzmann distribution, and any coherences that were created by the rf pulses decay to zero. Spin relaxation results from the fact that molecular motions modulate the magnetic field experienced locally by individual nuclei. In terms of the underlying quantum mechanics, these fluctuations introduce a stochastic, time-dependent component to the Hamiltonian operator. Fortunately, weak coupling between the spin and non-spin degrees of freedom allows relaxation to occur slowly enough to be observed experimentally (over 5

12 the course of milliseconds (ms) to seconds in solution). If this were not the case, the NMR experiment would not be feasible. The formal theory of NMR relaxation was originally developed by Bloch, Wangsness, and Redfield (BWR) 8, 9 10, 11, 12, and has since then been extensively reviewed. 13, 14, 15 In the BWR formalism, relaxation rates are dependent upon the time-correlation function related to the reorientation of certain spin interactions. As an example, consider a representative 15 N- 1 H spin system along the protein backbone. Relaxation of the 15 N nucleus occurs as molecular motions reorient the dipole-dipole and chemical shift anisotropy interactions present in the spin system. Reorientation of the magnetic fields associated with these spin interactions can be expressed as a time-correlation function 7 C(t) = Y 20 (0)Y 20 (t) (1) where C(t) is the time-correlation function evaluated for a delay value t, and Y 20 (0) and Y 20 (t) are second-order spherical harmonic functions evaluated at time 0 and some later time t. The angular brackets indicate that C(t) is an ensemble-averaged quantity. The spherical harmonic functions play a significant role in the theory of angular momentum. Their use in eq. 1 suggests that rotational motion is specifically responsible for spin relaxation. Cosine Fourier transformation of the correlation function results in a spectral density function: J(ω) = C(t)cos(ωt)dt (2) where J(ω) is the spectral density evaluated at frequency ω. As a first approximation, C(t) is often modeled as mono-exponential decay, in which case, the functional form of J(ω) is a Lorentzian distribution centered about ω = 0. This topic will be addressed in greater detail in the following section. Physically, the distribution represented by J(ω) indicates 6

13 that molecular motions occur over a continuous range of frequencies. The intensity of the motion at a particular frequency is found by evaluating the spectral density function at that frequency. BWR theory establishes a quantitative link between the spectral density function and the observed NMR relaxation rates. It predicts that spin relaxation rates depend upon a linear combination of spectral densities: 16 R = a i, j J(ω i ) (3) j i where R refers to the various relaxation rates that would be measured experimentally. Common examples include the longitudinal relaxation rate R 1 and the transverse relaxation rate R 2. The terms that are represented in eq. 3 differ for each of these processes. The summation over j refers to the various mechanisms that contribute to the relaxation process. In the case of the 15 N nucleus, for example, relaxation is primarily due to fluctuations in the dipole-dipole interaction and, to a lesser extent, fluctuations in its chemical shift anisotropy. The summation over i specifies the frequencies relevant to the relaxation mechanism. In general, these frequencies correspond to the Larmor frequencies of the spins, as well as sums and differences of the Larmor frequencies. 17 In some instances, such as the transverse relaxation rate R 2, the spectral density at ω = 0 is also included in the summation. Measurement of multiple relaxation rates provides a system of linear equations, from which one can calculate the spectral density at a limited number of frequencies. This result from the BWR theory provides a quantitative description of the dependence of NMR relaxation rates on molecular motions. Modeling the spectral density function 7

14 Although the BWR theory establishes a necessary link between NMR-observable quantities and molecular dynamics, it provides a limited description of the motions that are present among an ensemble of protein molecules in solution. It is possible to increase the number of spectral densities sampled by making use of the fact that the Larmor frequencies are proportional to the magnitude of the static magnetic field. Acquisition of relaxation data at multiple field strengths, therefore, allows additional points along J(ω) to be sampled. A complete description of the molecular dynamics, however, would include additional features. For example, the ability to measure relaxation rates for multiple spins within the protein affords a high degree of spatial resolution, but additional modeling is required to separate contributions from internal motion and overall tumbling. A complete description of the internal motion would also include not only its frequency, but also a measure of its amplitude. There are multiple approaches that might be pursued in order to extract these additional features from the relaxation data. One general approach would be to assume an idealized model for the internal motion. Some of the more commonly employed models are: harmonic motion along some internal coordinate, free diffusion within a cone, and/or jumps among a discrete set of rotameric states. Each of these motional models contains one or more adjustable parameters that can be fitted to the experimental relaxation data. In the case of harmonic motion, that parameter would be a force constant; for free diffusion in a cone, it would be a cone angle; and for rotameric jumps, it would be rotamer populations and exchange rates. Many of these analytic motional models were first developed for the analysis of relaxation data by Woessner 18. Motional models continue to be employed today, 19 especially in thermodynamic interpretations of NMR 8

15 data. 20, 21 Most practitioners, however, do not appear to favor their use in routine applications. The chief limitation of this approach is the fact that multiple models often fit the relaxation data equally well, and there is no a priori reason to prefer one model over another. Molecular dynamics (MD) computer simulations may provide some justification for a particular model, 22 but there is no guarantee that a simulation represents all the relevant features of the molecular motion. The comparison between experiment and computation may also be turned around. For example, NMR measurements may act as benchmarks for evaluating the accuracy of simulations. 6 Fundamental obstacles limit the utility of such comparisons, however. Both experimental and computational methods employ numerous parameters, a scenario that typically precludes a unique solution to any discrepancies between computation and experiment. An alternative approach, generally referred to (somewhat misleadingly) as the model-free approach, was formulated by Lipari and Szabo. 23 In the Lipari-Szabo (LS) method, the molecular motion is again modeled by an assumed mathematical expression, but the exact geometry of the internal motion is not specified, nor is the potential that drives the motion. A key assumption in the LS method (as well as in most analytic motional models) is that overall tumbling and internal motions are uncorrelated. This assumption is considered valid in cases where the amplitude of the internal motion is small and its timescale fast relative to overall tumbling. In this scenario, the internal motion is not expected to affect the global shape of the molecule, nor its consequent rotational diffusion. These conditions may not pertain to proteins that are either intrinsically disordered or unfolded, in which case the lack of a well-defined globular structure prevents the separation of internal and overall motions. 24 There has been recent 9

16 progress in the development of methods to identify cases in which this separability condition is not fulfilled, 25 as well as theoretical modifications of the LS formalism to accommodate such cases. 26 The assumption that overall and internal motions are uncorrelated simplifies the expression for the correlation function: C T (t) = C O (t)c I (t) (4) where C T (t) is the total correlation function associated with NMR relaxation-active motion (i.e. rotation). It is expressed as a product of the correlation functions for overall tumbling C O (t) and internal motion C I (t) at some site in the molecule. In its simplest derivation, the LS formalism assumes that overall and internal motions are both monoexponential processes. In the case of overall tumbling, the correlation function may initially be modeled as isotropic rotation: C O (t) = 1 5 exp( t τ R ) (5) where τ R is the correlation time associated with overall molecular rotation. It is related to the rotational diffusion rate constant D iso = τ R /6. (Note that the prefactors in some of these expressions differ among various sources.) As the rate of the overall rotation increases, the correlation time τ R decreases, in which case C O (t) decays more rapidly. Although the overall rotation may be nearly isotropic in certain cases, the internal rotation is expected to be anisotropic due to intramolecular interactions. The anisotropy of the internal rotation necessitates an additional parameter S in its correlation function: C I (t) = S 2 + (1 S 2 )exp( t τ e ) (6) 10

17 S is a generalized order parameter and τ e is the effective correlation time associated with the internal rotation. S 2 is the limiting value of C I (t) at long times: The following inequality applies to S 2 : S 2 = lim t C I (t) (7) 0 S 2 1 (8) Equations 6 and 8 indicate that S 2 = 0 for isotropic internal rotation and S 2 = 1 in the absence of internal rotation. The order parameter, therefore, is associated with the amplitude of the internal rotation at a given site. Large amplitude motions that are less restricted generate S 2 values near zero, while small amplitude motions that are more restricted give S 2 values closer to one. The effective correlation time τ e describes the frequency of the internal rotation. As the rotational frequency increases, the correlation function for the internal motion decays more rapidly and the value of τ e decreases. with Insertion of eq. 5 and 6 into 4 gives: C T (t) = 1 5 [S 2 exp( t ) + (1 S 2 )exp( t )] (9) τ R τ τ 1 = τ R 1 + τ e 1 (10) Cosine Fourier transformation of eq. 9 then results in the following expression for the spectral density function J(ω): J(ω) = 2 5 [S τ 2 R 1+ (ωτ R ) + (1 S τ 2 ) 2 1+ (ωτ) ] (11) 2 which is a sum of two Lorentzian functions, the first term corresponding to overall rotation and the second term representing a combination of internal and overall rotation. 11

18 Recall from the previous section that NMR relaxation rates are functions of the spectral density function evaluated at certain characteristic frequencies. This relationship is described in a most general way in eq. 3. Lipari and Szabo s formulation of the spectral density function is very useful because it allows various relaxation rate expressions to be recast in terms of a small number of parameters (S 2, τ e, and τ R ) whose physical significance is readily grasped. It is for this reason that the vast majority of protein dynamics studies by NMR rely upon calculation of these model-free parameters. Special considerations for protein side-chain dynamics NMR studies of protein dynamics have focused mainly on backbone motions, which are typically probed by 15 N relaxation. Analysis of 15 N relaxation data is greatly facilitated by the fact that the relaxation mechanisms are clearly defined and are few in number. For a two-spin 15 N- 1 H group, the 15 N nucleus relaxes under the 15 N- 1 H dipolar interaction and, to a lesser extent, the 15 N chemical shift anisotropy. Progress in this area is also due, in large part, to the fact that recombinant proteins are now routinely expressed in high quantities and at high levels of isotopic enrichment for 15 N. NMR studies of side chain dynamics, on the other hand, are far less common. The majority of side chain dynamics studies have utilized methods developed by Kay s group to measure the relaxation rate of deuterons in 13 CH 2 D methyl groups. 27,28 (In the above notation, H represents a proton, 1 H, and D represents a deuteron, 2 H.) The proteins that are necessary for this work are generally expressed in minimal media containing 13 C 6 - glucose as the sole carbon source and ~50% D 2 O:50%H 2 O, resulting in uniform 13 C- labeling and fractional deuteration. The pulse sequences are triple resonance experiments 12

19 in which rf pulses are applied to 1 H, 2 H, and 13 C nuclei. The flow of magnetization during the course of the experiment can be represented by: 29 1 H 13 C 2 H(t) 13 C(t 1 ) 1 H(t 2 ) Diagrams of this kind identify the nuclei involved in a series of magnetization transfer steps without specifying the particular form of the populations and coherences that are created in the process. The diagram indicates that magnetization originating from a methyl proton is transferred to the attached carbon and then to a deuteron bound to the same carbon. Phase cycling of one of the pulses within this part of the sequence selects for methyl groups with a singly attached deuteron. Following transfer of magnetization to the deuteron, relaxation is allowed to occur for a variable period t. Magnetization is subsequently returned to the carbon, whose chemical shift is recorded during the indirect period t 1, and finally back to the originating proton, whose chemical shift is recorded during the detection period t 2. The result is a two-dimensional 13 C- 1 H correlation spectrum in which the intensity I of a methyl cross-peak is a function of the duration t of the relaxation period. The value of t is incremented across a series of spectra, and the cross-peak intensities from these spectra are fit to a decay curve to yield the relaxation rate R for a given resonance: I = I 0 exp( Rt) (12) where I 0 is the intensity at t = 0. In order to calculate the dynamics parameters S 2 and τ e, it is necessary to measure multiple relaxation rates for a specific nucleus. For example, a minimum of two relaxation rates, R 1 and R 1ρ, would typically be measured in the side chain experiments under consideration. The rates that are measured represent the decay of particular 13

20 populations or coherences that are created by the pulse sequence. These populations and coherences can be specified by the product operator notation, a shorthand that is commonly used in the design and analysis of NMR experiments. In order to measure R 1 or R 1ρ, magnetization proportional to the operators D z or D y, respectively, is produced prior to the relaxation period. In reality, the situation is more complicated, however, in that one first measures the relaxation rate for the operators I z C z D z and I z C z D y. An additional pulse sequence element is then implemented in order to subtract the relaxation rate contribution from the operator I z C z. This approach allows the decay of pure 2 H magnetization to be approximated as: R 1 (D z ) R(I z C z D z ) R(I z C z ) (13) R 1ρ (D y ) R(I z C z D y ) R(I z C z ) (14) The approximations in eq. 13 and 14 are good to ~2% under typical conditions. 27 Kay s method is successful in large part because the deuteron is a spin I = 1 particle whose relaxation is dominated by the quadrupolar interaction. This scenario greatly facilitates analysis of the relaxation data. The method also benefits from the high resolution typically observed in the methyl region of a 13 C- 1 H correlation spectra. Kay s group later expanded the method s capabilities by increasing the number of accessible relaxation rates from two to five. 28 These five relaxation rates correspond to the decay of three coherences: D +, D 2 +, and D + D z +D z D + and two populations: D z and 3D 2 z -2. Within the context of the BWR theory, these five relaxation rates are collectively a function of the spectral density J(ω) at just three frequencies: 0, ω D, and 2ω D, where ω D is the deuteron s Larmor frequency. This expanded data set also allows for more complex 14

21 modeling of the spectral density function. 30, 31 As a first step in the analysis of side-chain relaxation data, it is customary to attempt a two-parameter fit, referred to as the LS2 model. This type of analysis corresponds to the standard method of Lipari and Szabo. However, side-chain dynamics are less likely than are backbone dynamics to be adequately described by just two parameters, S 2 and τ e. For cases in which the LS2 model proves inadequate, it is necessary to investigate various sources of error. One possible complication is the presence of anisotropic overall rotation, which is most easily evaluated by noting the range of τ R values obtained from 15 N backbone relaxation measurements. It is also necessary to consider the whole range of internal motions that may contribute to the relaxation of a deuteron in a side chain methyl group. The LS2 model accommodates internal motions that are much faster than the rate of overall tumbling. For a side chain methyl group, these motions most likely include: spinning about the methyl symmetry axis, torsional and bond angle fluctuations, and fast rotameric transitions. These motions are thought to occur on a similar timescale of ~ ps and are considered fairly ubiquitous. Side chain dynamics, however, may also include slower motions that approach the rate of overall tumbling. Motions that are characteristic of this intermediate timescale would include: slow rotameric transitions and concerted movements among multiple chemical groups whose dynamics are correlated in some manner. In cases where one of these slower processes is superimposed on the typical fast side chain motions, the total correlation function is no longer a simple product of two exponential functions, one corresponding to overall tumbling and the other corresponding to an internal fluctuation. Instead, the tail the total correlation function is itself comprised 15

22 of at least two exponentials with similar correlation times. Therefore, the LS2 model is unable to adequately fit the relaxation data. One solution to the problem is to use an approach first proposed by Clore and coworkers in the analysis of backbone dynamics. 32 They expressed the spectral density function in terms of four parameters: S f 2, τ f, S s 2, and τ s, corresponding to square order parameters and correlation times for fast and slow internal motion, respectively. This approach is referred to as the LS4 model. Although the approach is sound in principle, it is often difficult to implement, particularly in cases where the slow internal correlation time τ s is very similar in magnitude to the overall correlation time τ R. Much of the difficulty arises from the fact that, when two decay processes occur at similar rates, even a small amount of noise will prevent one from being able to separate their relative contributions. Kay's group offered an alternative expression for the spectral density function, the LS3 model, that provides less information, but generally fits the side-chain relaxation data in cases where the LS2 model fails. For a side-chain methyl group analyzed by the LS3 model, the total correlation and spectral density functions are given by: with C T (t) = [ 1 9 S 2 f + (1 1 9 S 2 f )exp( t )]exp( t τ f τ ) (18) eff c J(ω) = 1 eff 9 S 2 τ c f 1+ (ωτ eff c ) + ( S f 2 ) 1 τ = 1 τ c eff + 1 τ f (20) τ 1+ (ωτ) (19) 2 16

23 Unlike the LS4 model, which explicitly includes the parameters S 2 s and τ s, the LS3 model introduces the parameter τ eff c, which represents the combined effects of slow internal dynamics and overall tumbling. The term in eq. 18 that contains τ eff c is analogous to eq. 5, the correlation function for overall tumbling, but it is now site-specific in order to include contributions from internal motions that occur on a similar timescale. Equation 18 also contains a contribution from fast local motions, which is identical in form to eq. 6, with the exception that eq. 18 includes a factor of (1/9) to account for fast spinning about the methyl symmetry axis. The work performed by Kay's group represents a powerful approach to the study of side chain dynamics by NMR. This area of research is still very much in its early stages, however, and other methods have also proven useful. Whereas Kay's method utilizes 2 H relaxation, alternative approaches generally measure relaxation of the 13 C nucleus. Progress in this direction is complicated by the fact that multiple spin interactions often contribute to 13 C relaxation in protein side-chains. For example, uniform, high level isotopic enrichment for 13 C introduces scalar and dipolar couplings between directly bonded 13 C nuclei. As a result, the relaxation rates that are measured for these nuclei do not reflect isolated motion of the 1 H- 13 C bond vector. Another complicating factor is the potential for 1 H- 13 C dipolar cross-correlation in methyl and methylene groups containing more than one 1 H- 13 C spin pair. Various labeling strategies have been used to produce isolated 1 H- 13 C spin pairs whose relaxation behavior is subject to a more limited number of spin interactions. LeMaster and Kushlan developed a procedure which 13 C labels protein side-chains at alternating carbon positions, thereby removing the effects of 13 C- 13 C scalar and dipolar couplings. 33 Their technique requires a 17

24 specially modified strain of bacteria and uses either [2-13 C]glycerol or [1,3-13 C]glycerol as the carbon sources. Another approach, recently proposed by Brüschweiler s group, relies upon a combination of radiofrequency pulses and pulsed field gradients during the relaxation delay in order to suppress the effects of dipole-dipole cross-correlated relaxation. 34 This method may prove useful in characterizing the motions in methylene groups. Motional dynamics in calbindin Calbindin D 9k is a calcium-binding protein of the EF-hand family. It is found predominantly in tissues involved in the uptake and transport of calcium, such as cells of the intestinal brush border membrane. 35, 36 The identifier D 9k refers to the fact that calbindin s synthesis is regulated by the activated form of vitamin D. The subscript 9k reflects the protein s molecular weight, which is around 8700 daltons. The EF-hand is characterized by a helix-loop-helix motif that binds calcium with ligands provided by the loop segment. In most cases, the basic functional unit in these proteins is a pair of EFhands. Calbindin D 9k (hereafter referred to simply as calbindin) contains of a single pair of EF-hands in which the N-terminal EF-hand is separated from the C-terminal EF-hand by a short, 10-residue linker. This protein has been the subject of numerous NMR studies, in large part due to its small size, high solubility, and good long-term stability. The main focus of these NMR studies has been to resolve the structural and dynamic features that allow calbindin and other molecules in the EF-hand family to bind calcium. In the case of calbindin, there are two EF-hands and hence two calcium binding sites. Such a scenario implies four possible binding states for the molecule: the apo state, 18

25 two half-saturated states, and a fully-loaded state. Transitions between these states are described by the site-specific (microscopic) binding constants: K I, K II, K I,II, and K II,I. K calbindin I (Ca 2+ ) I 1 calbindin (Ca 2+ ) 2 calbindin K calbindin II (Ca 2+ ) II 1 calbindin (Ca 2+ ) 2 calbindin K II, I K I, II It is also useful to define the stoichiometric (macroscopic) binding constants K 1 and K 2 associated with this less detailed description of the binding process: K calbindin 1 (Ca 2+ K ) 1 calbindin 2 (Ca 2+ ) 2 calbindin Note that the macroscopic binding constants do not specify the order in which the sites are filled. Binding constants for calbindin were provided in a series of studies by Linse and coworkers. 37,38 They found that the binding constants K I and K II are nearly equal, but the protein is, under most conditions, an asymmetric system in which 1 < K II /K I < 3. Their findings also indicate that log K 1 = 6.3 and log K 2 = 6.5 at 298 K and 150 mm KCl. These measurements allowed the free energy of interaction between the two sites, G, to be calculated. This quantity is defined as: G = RT ln( K I,II K I ) = RT ln( K II,I K II ) (21) The following relations provide a link between the microscopic and macroscopic binding constants: K 1 = K I + K II (22) K 1 K 2 = K I K II,I = K II K I,II (23) Substituting these two relations into eq. 21 allows G to be expressed as a function of K 1, K 2, and η = K II /K I : 19

26 G = RT ln[ K 2 2 (η +1) ] (24) K 1 η Using this approach, Linse and coworkers found that G amounts to around -5 kj mol -1, signifying positive cooperativity between the two sites. Positive cooperativity implies higher affinity for an ion to site II in the presence of an ion in site I than when site I is unoccupied ( G II,I - G II < 0), and vice versa ( G I,II - G I < 0). An additional aspect of cooperativity supported by the data is that the affinity for the second ion is higher than that for the first ion along either of the two stepwise binding pathways ( G II,I - G I < 0 and G I,II - G II < 0). The structural and dynamic basis for these binding events had been investigated at a high level of detail by Walter Chazin s group whose past members have included (among others) Mikael Akke, Nicholas Skelton, and Johan Kördel. A key feature of the positive cooperativity exhibited by calbindin is a depletion of the half-saturated states at equilibrium. As a result, experimental studies of the two half-saturated forms of the protein have relied upon model systems developed by Chazin s group. In developing these models, Chazin and coworkers have relied upon fundamental differences between the two binding sites in calbindin. Calbindin is a member of the S100 subfamily of EFhand proteins that is distinguished by a nonconsensus (or pseudo-) N-terminal EF-hand and a consensus C-terminal EF-hand. The consensus and nonconsensus EF-hands are known to have very different loop structures. 39 In the consensus EF-hand, the loop is 12 residues long. The nonconsensus sequence, on the other hand, contains two additional residues. Although both structures coordinate calcium with pentagonal bipyramidal geometry, they provide the ion with different ligands. The consensus loop of the C- 20

27 terminus coordinates calcium primarily with side-chain ligands: five side-chain carboxylate oxygens, one backbone carbonyl oxygen, and one water oxygen. In contrast, the N-terminal, nonconsensus loop coordinates calcium with four backbone carbonyl oxygens, two carboxylate oxygens from a bidentate Glu side-chain, and a water oxygen. It is also common among consensus loops to have a Gly at position 6, which results in a 90 change of direction. This Gly is absent in the nonconsensus loop of calbindin. In fact, postion 7 of the nonconsensus loop is a Pro, which is expected to be much more conformationally restrictive. As a result of these differences, the N-terminal EF-hand in calbindin has been described structurally as a consensus loop turned inside out. The fact that the majority of its ligands are provided by backbone carbonyls has the effect of orienting the nonconsensus side-chains away from the ion and out towards the solvent. The functional consequences of these differences are that the consensus loop has slightly higher affinity for calcium than does the nonconsensus loop, while the nonconsensus loop is considerably more selective than the consensus loop for calcium against other divalent cations. This latter observation has been used in the design of a model for the (Ca 2+ II ) 1 half-saturated state. 40 Cadmium binds calbindin sequentially, first entering site II. Whereas the nonconsensus loop at site I adopts a more rigid structure with backbone carbonyl oxygens specifically arranged to coordinate calcium, site II allows for a greater degree of flexibility among its side-chain coordinating oxygens and is thus better able to accommodate a cadmium ion. Cadmium will, however, bind to site I at higher [Cd 2+ ]. In fact, the fully Cd 2+ -loaded state is very similar in structure to the fully Ca 2+ -loaded 21

28 state. 40 The cadmium model also retains the features of positive cooperativity described above. 41 An alternative approach has been taken in the development of a model for the (Ca 2+ ) I 1 half-saturated state. In the site I model, a substitution at site II has been utilized to render the binding sequential. The N56A mutant form of calbindin is deficient in one of its coordinating ligands at site II, which consequently favors binding first to site I. 42 It is worthwhile to note that another calbindin mutant, E65Q, was used before the development of the N56A system. 43 It was found, however, that the E65Q mutant dramatically reduces the site II binding constant, K II. This effect is due in large part to the fact that Glu 65 is a bidentate ligand in the WT structure. Glu 65 also plays an important structural role that is disrupted by its mutation. In contrast, Asn 56 has few interactions with the rest of the protein, and its coordinating oxygen atom can be removed without changing the net charge of the protein. Therefore, the N56A mutation is currently favored as a model of the (Ca 2+ ) I 1 half-saturated state. 42 Structures of the apo and fully-loaded states of calbindin exhibit relatively few differences. 44, 45, 46 Even in the apo state, the coordinating ligands of site I appear poised and ready to bind the ion. The N-terminal EF-hand, therefore, requires few adjustments in order to bind calcium. Rearrangements in the C-terminal EF-hand are slightly larger in amplitude, but are still relatively minor. Upon binding calcium, Helix IV of the C- terminal EF-hand extends from a purely α-helical conformation to a mixture of and α-helix, and the C-terminal end of helix III and the N-terminal end of helix IV move closer together. Such effects are considered small, especially when compared to observations in other EF-hand proteins. Calmodulin and troponin C, for example, 22

29 undergo substantial rearrangements upon binding calcium. 47 Conformational transitions in these proteins are considered important in facilitating binding to their various targets. It is important to note that, although the structural rearrangements observed in the fullyloaded state of calbindin are small, they are present in both the half-saturated states, (Ca 2+ ) I 1 -N56A and (Cd 2+ ) II 1. In this respect, the half-saturated states more closely resemble the fully-loaded state than they do the apo state. Such an effect is expected to favor binding of the second ion. Despite the fact that the average structure of calbindin is affected relatively little by calcium binding, more significant dynamic changes are observed in the binding process. Not only do motions in the molecule change, but they appear to contribute to the cooperativity between the two sites. A major advance in the characterization of backbone dynamics by NMR occurred in the late 1980 s with the increased availability of 15 N- labeled proteins. 48 Calbindin was actually one of the first molecules to be characterized by the methods made possible with uniform 15 N-labeling. 49, 50 These initial backbone dynamics studies of calbindin are particularly significant in terms of the high degree of statistical rigor that was used in the analysis. It was revealed in this work that the fully-loaded and apo states share several dynamic features. Their hydrodynamic properties are similar, with overall correlation times, τ R, of 4.25 and 4.10 ns for the fully-loaded and apo states, respectively. The two states also exhibit similar internal dynamics in the helical and linker regions. In the helices, the backbone order parameters are uniformly high, averaging 0.84 for the fullyloaded state and 0.85 for the apo state. The order parameters are much lower in the flexible linker between the two EF-hands, reaching minimum values around 0.5 at the 23

30 middle of the segment. This trend, in which the helices are relatively rigid and the linker is flexible, is shared, not only between the apo and fully-loaded states, but with the halfsaturated states as well. Although the order parameters are low in the linker region for all cases reported thus far, the exact values vary among different calbindin mutants. In the wild-type (WT) sequence, a Pro at position 43 undergoes cis-trans isomerization at the Gly 42-Pro 43 peptide bond, resulting in duplication of resonances for several nuclei. 51 This problem was initially circumvented by use of a P43G mutant, which is the standard background mutation in most of the calbindin studies cited here. 52 Later a P43M mutant was adopted for structural studies because it allows CNBr cleavage of calbindin into its two EFhands. 53 In both cases, the effects of the mutation are almost entirely local, but the P43G mutant exhibits increased flexibility throughout the linker relative to P43M. 39 The most significant differences between the fully-loaded and apo states occur at site II. Order parameters at site II increase with calcium binding from an average value of 0.72 in the apo state to 0.83 in the fully-loaded state. This finding suggests that, in the absence of calcium, the backbone amide groups at site II exhibit large amplitude fluctuations that are significantly attenuated by ion binding to that site. Site I, however, does not mimic this trend. Site I, which contains the nonconsensus binding loop, is relatively rigid regardless of the protein s binding state. The backbone order parameters for site I are 0.82 and 0.80 in the fully-loaded and apo states, respectively. It is important to note, however, that order parameters are sensitive only to ps-ns timescale motions. While the dynamics at this faster timescale appear relatively limited at site I, additional evidence suggests that slower µs-ms timescale dynamics play an important role there. 24

31 In cases where motions are present at the µs-ms timescale, the transverse relaxation rate R 2 contains an exchange contribution R ex. The exchange contribution is caused by processes that dephase the transverse magnetization during the CPMG pulse sequence that is used in the R 2 measurements. 50 In order for an exchange contribution to be observed, the rate of exchange must be on the order of the inverse of the time between the refocusing pulse and the formation of the spin-echo within the CPMG sequence. For example, if the delay between refocusing pulses is set to 1 ms, then the exchange rate needs to be comparable to 2/( s) = 2000 s -1. One indication of exchange is an increase in the resonance linewidths for certain regions of the protein. More subtle exchange effects can be detected during the model-free analyses, in which case R ex terms are sometimes necessary in order to adequately fit the backbone relaxation data. In the case of apo calbindin, R ex terms were obtained for five of the fourteen site I residues. The two most likely sources of these R ex terms are conformational fluctuations at the relevant timescale and/or chemical exchange between the apo state and a smaller equilibrium population of a calcium-bound state. Although the details will not be presented here, Chazin and his colleagues provide a convincing argument that conformational rather than chemical exchange is the source of the R ex terms at site I. 50 A key component of their argument is a demonstration that chemical exchange in the calbindin system occurs at too slow a rate to affect the transverse relaxation. It was noted above that the two half-saturated states more closely resemble the fully-loaded state in terms of their average structures. Such a generalization, however, is not possible with regard to the dynamic properties of the half-saturated states. For example, the (Cd 2+ ) II 1 state resembles the apo state near site I where substantial 25

32 conformational exchange is again observed. 50 R ex terms were obtained in the (Cd 2+ II ) 1 state for six of the fourteen residues at site I. An additional four of these fourteen residues are not even observed in the (Cd 2+ ) II 1 state due to severe exchange broadening. (The linewidths are large enough that the resonances do not emerge above the noise of the spectrum.) Nevertheless, the order parameters at site I are high (~0.82) for those resonances that are detected in the (Cd 2+ ) II 1 state. Despite a resemblance to the apo state in its site I dynamics, motions near site II of (Cd 2+ ) II 1 more closely resemble those of the fully-loaded state. Recall that order parameters at site II are low in the apo state (~0.72), but are high in fully-loaded calbindin (~0.83). Backbone order parameters for these residues are even higher in the (Cd 2+ ) II 1 state, averaging These observations provide evidence that ion ligation by the protein requires attenuation of the fast timescale dynamics near the binding site. When the backbone dynamics in calbindin were first investigated, 49, 50 the model for site I binding, (Ca 2+ ) I 1 -N56A, had not yet been developed. A more recent study has provided this missing piece of the puzzle. 54 Its results are significant because they offer the most convincing evidence to date that backbone dynamics contribute to calbindin s positive cooperativity. Calcium binding to site I in the N56A mutant results in a modest increase in the order parameters at site I. In order to position the protein s ligands in the optimal coordination geometry, it is clear that certain motions at the binding site are dampened. What is perhaps more surprising is that the order parameters also increase at site II, despite the fact that an ion is not bound at the second site in the N56A model. Increases in order parameters at either site can be interpreted qualitatively as a local decrease in conformational entropy. This change represents the entropic cost 26

33 associated with calcium binding by the protein. An estimate of the magnitude of this entropic effect is provided by the method of Akke, Brüschweiler and Palmer. 20 This method establishes a quantitative link between changes in order parameters and conformational entropy. Using these methods, Chazin and coworkers estimated that the reduction in conformational entropy that occurs at site II as a result of binding to site I in (Ca 2+ ) I 1 -N56A is 1.5 kcal mol The second binding event costs only an additional 0.4 kcal mol -1. Consequently, much of the entropic penalty incurred at site II occurs prior to the actual binding event at that site. This finding provides evidence of site-site communication in calbindin and suggests that dynamics are partly responsible for cooperativity in the system. The story is not complete, however. The current thermodynamic interpretation of backbone order parameters is subject to several limitations: 1) It assumes that bond vector motions are independent of one another, 2) The method is insensitive to motions that are slower than overall tumbling, 3) The choice of motional model is not indicated by the data, and 4) Only a small subset of bond vectors are typically included in the analysis. Some of these limitations may be difficult to avoid, although extensive efforts are underway to account for all possible sources of error. 55 The first three limitations are especially problematic and will undoubtedly require novel insights and techniques. 56 The fourth limitation, however, is subject to improvement with currently existing methodologies. In the next chapter, we present the results of an extensive 2 H relaxation study in which calbindin s conformational dynamics are characterized at multiple sidechain positions. 27

34 CHAPTER 2: Effects of Calcium Binding on the Side-chain Methyl Dynamics of Abstract Calbindin D 9k : A 2 H NMR Relaxation Study The effects of Ca 2+ -binding on the side-chain methyl dynamics of calbindin D 9k have been characterized by 2 H NMR relaxation rate measurements. Longitudinal, transverse, quadrupolar order, transverse antiphase and double quantum relaxation rates are reported for both the apo and Ca 2+ -loaded states of the protein at two magnetic field strengths. The relatively large size of the data set allows for a detailed analysis of the underlying conformational dynamics by spectral density mapping and model-free fitting procedures. The results reveal a correlation between a methyl group s distance from the Ca 2+ binding sites and its conformational dynamics. Several methyl groups segregate into two limiting classes, one proximal and the other distal to the binding sites. Methyl groups in these two classes respond differently to Ca 2+ binding, both in terms of the timescale and amplitude of their fluctuations. Ca 2+ binding elicits a partial immobilization among methyl groups in the proximal class, which is consistent with previous studies of calbindin s backbone dynamics. The distal class, however, exhibits a trend that could not be inferred from the backbone data in that its mobility actually increases with Ca 2+ binding. We have introduced the term polar dynamics to describe this type of organization across the molecule. The trend may represent an important mechanism by which calbindin D 9k achieves high affinity binding while minimizing the corresponding loss of conformational entropy. Introduction 28

35 An accurate understanding of the role of conformational dynamics in proteins requires data at multiple timescales and sites within the protein of interest. Considerable progress has been achieved in characterizing the picosecond-to-nanosecond (ps-ns) dynamics of the protein backbone via NMR relaxation measurements of the 15 N nucleus. 57 More recent developments in the measurement of 2 H quadrupolar relaxation rates are enabling an extensive characterization of the dynamics in methyl-containing side-chains as well. 27,28 Methyl groups are particularly useful reporters of protein dynamics because they are widely distributed throughout the protein sequence. They are commonly located in the hydrophobic core of the molecule where the packing density is generally high, but varying degrees of internal mobility are permitted. 58,59 A useful analogy suggests that side-chain packing is more like the packing of nuts and bolts in a jar than the pairwise matching of jigsaw puzzle pieces. 60 These features allow many interesting aspects of ligand binding to be addressed by the study of methyl dynamics, in particular by examining the manner in which the dynamics are changed upon binding. The aim of the present study is to characterize the effects of Ca 2+ binding on the side-chain dynamics of the protein calbindin D 9k. Metal ions such as Ca 2+ are often essential components of a protein s structure. In many cases, they also serve to modulate the structure in order to accomplish important regulatory functions. 61 Calbindin is a small (~8.7 kd) single domain protein of the EF-hand family. It is found predominantly in tissues involved in the uptake and transport of calcium, such as cells of the intestinal brush border membrane. 62,63 The EF-hand is a helix-loop-helix motif that binds calcium with the ligands provided by the loop residues and helical residues immediately adjacent to the loops. 64 Calbindin consists of a single pair of EF-hands connected by a 10-residue 29

36 linker (Figure 1). The two binding sites exhibit high positive cooperativity and are coupled by a short β-type interaction formed by two backbone-backbone hydrogen bonds between L23 and V61. The N-terminal binding site (site I) has an S100-type EF-hand sequence containing a 14-residue binding site in which the Ca 2+ -coordinating ligands are mainly backbone carbonyl oxygens. In contrast, the C-terminal site (II) conforms to the consensus EF-hand sequence of 12 residues with the Ca 2+ ligands contributed primarily by side-chain carboxylate oxygens. The structure of apo (Ca 2+ -free) calbindin is remarkably similar to that of the Ca 2+ -loaded state. 45,46 Subtle changes have been noted primarily in the C-terminal EFhand, involving rearrangement of helices C and D. The backbone dynamics are also affected relatively little by Ca 2+ binding. 49,50 The backbone N-H order parameters are uniformly high in the helices and low in the linker for both the apo and Ca 2+ -loaded states. The most significant change occurs at site II where the N-H order parameters increase from a mean value of 0.72 in the apo state to 0.83 in the Ca 2+ -loaded state. This finding suggests that the backbone amide groups at site II undergo large amplitude fluctuations in the absence of Ca 2+ that are attenuated by ion binding. Site I, on the other hand, is relatively rigid in both states. The N-H order parameters for site I are 0.80 and 0.82 in the apo and Ca 2+ -loaded states, respectively. It is clear from investigations of other proteins, however, that backbone N-H order parameters are often insensitive to important conformational fluctuations involving the protein s side-chains. Consider, for example, the protein calmodulin, which is representative of a large subfamily of EF-hand proteins. Calmodulin has two domains 30

37 Figure 1: Ribbon diagram of calbindin D 9k (PDB entry 2BCA). 45 The structural elements include: helix A (residues 3-15), loop I (16-21), β-sheet (22-24), helix B (25-35), linker (36-45), helix C (46-53), loop II (54-59), β-sheet (60-62) and helix D (63-74). The loop and β-sheet residues are colored black. The molecular graphics in this paper were produced using the UCSF Chimera package (supported by NIH P41 RR-01081)

38 that each contains a pair of EF-hands. The binding of four Ca 2+ ions to calmodulin results in the formation of two discontinuous binding surfaces for several different target proteins. Wand and co-workers have studied the methyl dynamics of calmodulin in both the Ca 2+ -loaded state as well as a Ca 2+ -loaded state bound to a target peptide. 66 (To our knowledge, no studies of the side-chain dynamics in apo calmodulin have yet been reported.) Peptide binding to calmodulin is similar to ion binding to calbindin in that little change is observed in the backbone dynamics. More significant changes, however, have been reported among calmodulin s methyl groups. 66 For a majority of the protein s methyl groups, the order parameters increase with complex formation, indicating a partial immobilization and a loss of conformational entropy. For a few side-chains, however, the order parameters actually decrease, indicating increased mobility and a local gain of conformational entropy. This trend is not limited to peptide binding, but has been noted in protein-rna interactions as well. 67 These factors represent important determinants of binding and merit additional investigation in well-characterized systems such as calbindin D 9k. Theory NMR relaxation rates depend on the time-correlation function for the reorientation of particular spin interactions. 7 Consider, for example, an isolated deuteron, such as that found in a 13 CH 2 D methyl group. There are five independent spin operators associated with an isolated deuteron: longitudinal magnetization D z, single quantum inphase coherence D +, quadrupolar order magnetization 3D 2 z -2, single quantum antiphase coherence D + D z +D z D + and double quantum coherence D 2 +. Relaxation rates for each of these five operators can be measured using the methods developed by Kay and co- 32

39 workers. 27,28 In the case of the deuteron, the relaxation process is dominated by the interaction between the nuclear quadrupole moment and the electric field gradient associated with the surrounding electrons. Reorientation of this interaction occurs with rotation of the C-D bond in the 13 CH 2 D methyl group. These fluctuations are characterized by a time-correlation function whose cosine Fourier transform gives the corresponding spectral density function, J(ω). The spectral density function, in turn, represents the distribution of rotational frequencies for the fluctuations. The seminal work of Bloch, Wangsness and Redfield related NMR relaxation rates to weighted sums of the spectral density function at certain characteristic frequencies. 8,9 The spectral density function, therefore, represents the most direct source of dynamics information contained in the experimental relaxation rates. 68,69,70 In certain favorable cases, one can directly solve for the spectral densities at the characteristic frequencies by inverting the relaxation rate expressions. 30,68 For example, the five relaxation rates associated with the 13 CH 2 D methyl group exhibit the following dependence on the spectral density function 28 : R(D z ) = 3 40 (e2 qq h )[J(ω D) + 4J(2ω D )] (1a) R(D + ) = 1 80 (e2 qq h )[9J(0) +15J(ω D) + 6J(2ω D )] (1b) R(3D z 2 2) = 3 40 (e2 qq h )[3J(ω D)] (1c) R(D + D z + D z D + ) = 1 80 (e2 qq h )[9J(0) + 3J(ω D) + 6J(2ω D )] (1d) R(D + 2 ) = 3 40 (e2 qq h )[J(ω D) + 2J(2ω D )] (1e) 33

40 where (e 2 qq/h) and ω D are the quadrupolar coupling constant and deuteron Larmor frequency, respectively. The quadrupolar coupling constant varies little across the different methyl groups in a protein and is well-approximated by an average value of 167 khz. 71 Equation 1 indicates that the five rates are collectively a function of the spectral densities at three frequencies: 0, ω D and 2ω D. This set represents an overdetermined system of equations consisting of five knowns and three unknowns that can be solved by singular value decomposition. This method, which is referred to as spectral density mapping, provides a way to sample the spectral density function at three discrete values without the need for additional modeling. The procedure is especially informative in the present case because it requires no assumptions about the underlying form of the spectral density function. As a point of contrast, the standard set of 15 N backbone relaxation rates does not provide an overdetermined system of equations, therefore requiring the use of additional assumptions. 72 The generality of the spectral density mapping procedure may be regarded as both an advantage and a disadvantage. In order to extract more detailed information from the relaxation data, the spectral density function is often modeled by the method of Lipari and Szabo. 23 Several different dynamic processes, all of which are reflected in the spectral density function, contribute to the reorientation of a side-chain methyl group in solution. The so-called model-free analysis of Lipari and Szabo provides a means of characterizing these individual contributions. In its simplest form, the Lipari-Szabo spectral density function is expressed as a sum of two Lorentzian functions: 34

41 J(ω) = 1 9 S 2 τ R f 1+ (ωτ R ) + ( S f 1 τ = τ R τ f 2 ) τ 1+ (ωτ) 2 (2) The overall motion is parametrized by the overall correlation time τ R, while the internal motion is parametrized by the order parameter S 2 f and the internal correlation time τ f. This model will be referred to hereafter as the LS2 model, following the example of Skrynnikov et al. 30 The Lipari-Szabo formalism assumes that the overall and internal motions are uncorrelated, which is generally the case for a well-structured, globular protein like calbindin. This assumption is less likely to be valid for intrinsically disordered proteins where large amplitude internal motions affect the overall shape (and hence tumbling) of the molecule. 25 Equation 2 also assumes that the overall tumbling is isotropic. This approximation is also considered reasonable for calbindin whose shape is roughly spherical. A combined 13 C/ 15 N relaxation study by Lee et al. found that calbindin gives a D /D ratio of For an axially symmetric diffusion tensor, D = D zz and D = D xx = D yy are the two unique diffusion coefficients. Unlike the overall tumbling of the molecule, the internal motions are expected to be anisotropic, requiring the use of additional parameters. The factor of 1/9 that appears in eq. 2 accounts for the effect of spinning about the methyl symmetry axis. It is obtained by evaluating the square of a second order Legendre polynomial: [(3 cos 2 θ - 1)/2] 2, where θ is the angle between the methyl symmetry axis and the C-D bond vector. For a tetrahedral methyl group, θ = Equation 2 assumes that this process occurs uniformly across all the methyl groups in the molecule. It also assumes that the process is 35

42 uncorrelated with reorientation of the methyl symmetry axis itself, which provides an additional contribution to the spectral density function through the order parameter S 2 f. The order parameter is a measure of the amplitude of the internal motion. It has limiting values of 0 and 1, where 0 indicates that the methyl symmetry axis reorients isotropically and 1 indicates that the axis is fixed relative to the molecular reference frame. At least two different processes are thought to reorient the methyl symmetry axis. 30,74 One contribution arises from local fluctuations of the axis within a particular side-chain rotameric state, while a second contribution involves transitions between rotamers. Although both of these processes may affect the value of S 2 f, additional methods are required to separate their relative contributions. 74,75 Finally, eq. 2 includes an internal correlation time, τ f, which represents the timescale of the fluctuation. Methyl groups that reorient quickly exhibit low τ f values, while methyl groups that reorient more slowly exhibit high τ f values. The LS2 model is sensitive to internal motions that occur on a timescale faster than that of overall rotation. In some instances, however, reorientation of a methyl group includes a significant contribution from slower timescales. Skrynnikov et al. have proposed the LS3 model for such cases. 30 The LS3 spectral density function is expressed as: J(ω) = 1 eff 9 S 2 τ c f 1+ (ωτ eff c ) + ( S f 1 τ = 1 τ c eff + 1 τ f 2 ) τ 1+ (ωτ) 2 (3) where τ c eff is a correlation time that accounts for the combined effects of overall and slow internal rotation. These two processes are assumed to occur on a similar timescale (~ns) 36

43 and thus are inseparable. The parameters S 2 f and τ f retain their original meaning from the LS2 model. Note that the LS3 model does not include an order parameter to characterize the amplitude of the slow timescale internal motion. By definition, eq. 3 assumes that this quantity is zero. Results Relaxation Rate Measurements Relaxation rates for each of the five operators in eq. 1 were measured using the methods developed by Kay and co-workers. 27,28 Experiments were performed at 500 and 600 MHz ( 1 H Larmor frequency), providing a total of ten possible rates for each methyl group in both the apo and Ca 2+ -loaded states. The rates were measured from a series of 13 C- 1 H correlation spectra that were generally well-resolved and displayed high sensitivity. Note, however, that the two resonances associated with the residue L23 are weak, particularly in the apo state. L23 is located near binding site I and participates in a short β-type interaction between sites I and II. This interaction is stabilized by two crossstrand hydrogen bonds involving L23H N -V61O and V61H N -L23O. The decreased peak intensities for the L23 resonances are likely the result of conformational exchange during periods of 13 C chemical shift evolution in the pulse sequences. Interestingly, the resonances associated with the methyl groups in residue V61 show no signs of line broadening. This observation provides initial evidence that conformational dynamics in the two sites are fundamentally different. Representative decay curves are provided in Figure 2. The D + and D + D z +D z D + operators relax most rapidly due to a dependence on J(0) that does not affect the other three operators. In a molecule whose overall tumbling is nearly isotropic, the relaxation 37

44 Figure 2: Representative decay curves from the 600 MHz data for the five spin operators: (a) I9δ1, apo; (b) V61γ1, apo; (c) L39δ1, Ca 2+ -loaded and (d) T45γ2, Ca 2+ -loaded. In all plots, the points correspond to ( ) R(D z ), ( ) R(D + ), ( ) R(3D 2 z -2), ( ) R(D + D z +D z D + ) and ( ) R(D 2 + ). Each curve has been normalized to have an intensity of one at time zero. The plots in this paper were produced using the Grace software. 38

45 rate contribution from overall rotation is the same for all methyl groups. One may conclude, therefore, that rate differences among the methyl groups, such as those observed between Figures 2a and b or between Figures 2c and d, result from differences in internal dynamics. Spectral Density Mapping The entire set of ten rates that were measured in this study is a function of five spectral density values. (The zero frequency value is sampled at both fields, while the non-zero frequency values are field-dependent.) Among the five spectral densities obtained in this study, the value at zero frequency, J(0), is especially informative. 69,70 The significance of J(0) arises from the fact that the area under the spectral density curve is a constant. This condition provides a convenient interpretation of the experimental data. If the value of J(0) is high for a particular site, then the curve is sharply peaked near ω = 0. If the value of J(0) at a different site is low, however, then the curve is distributed over a much larger frequency range in order to maintain a constant area relative to the previous site. Although this reasoning provides a readily accessible interpretation of J(0), one may still question the significance of a fluctuation that occurs with zero frequency. It is important to note that J(ω) represents the combined effects of both overall and internal rotation. As a result, the corresponding time-correlation function is expected to decay to zero at long enough timescales. J(0) is then a limiting value due to fluctuations that occur on timescales longer than the time required for the complete decay of the correlation function. The J(0) values are plotted as a function of residue number in Figures 3a and b. A clear trend is observed for the Ca 2+ -loaded state in which the value of J(0) approaches a 39

46 Figure 3: Plots of J(0) as a function of residue number for the apo (a) and Ca 2+ -loaded (b) states. Plots of J(2ω D,600 ) as a function of residue number for the apo (c) and Ca 2+ - loaded (d) states. Error bars are less than the size of the symbols in most cases. 40

47 maximum near the two Ca 2+ binding sites. In contrast, the J(0) values approach a minimum near the termini and linker connecting the two EF-hands. When plotted as a function of residue number, as in Figure 3b, this trend is approximately M-shaped. These observations suggest that fluctuations near the binding sites are distributed across a narrow range of low frequencies, while fluctuations near the termini and linker are more broadly distributed across higher frequencies. A similar, albeit less noticeable, trend is observed for the apo state. In the apo state, the J(0) values again tend to increase near the binding sites and decrease near the termini and linker, but the trend does not occur as smoothly or dramatically as that in the Ca 2+ -loaded state. Calcium binding appears to elicit a stronger dependence between the J(0) value and the methyl group s position within the protein s structure. In Figures 3c and d, the spectral density value at 2ω D,600 is plotted as a function of residue number. The 600 in the subscript refers to the corresponding 1 H Larmor frequency (in MHz). This plot is representative of the trend observed in the spectral densities at the other non-zero frequencies. The most significant finding in Figures 3c and d is an increase in J(2ω D,600 ) near binding site I. This increase is most clearly observed for the methyl groups L23 δ1 and δ2 in the Ca 2+ -loaded state. Recall that residue 23 participates in the β-type interaction between the two binding sites. A similar increase is observed in Figure 3c, but large uncertainties are associated with the parameters for L23 in the apo state. It was previously mentioned that the resonances associated with L23 δ1 and δ2 exhibit significant line broadening in the apo state. Although the line broadening suggests the presence of potentially interesting dynamic events, it limits the certainty of quantitative statements pertaining to the site. Another interesting observation from 41

48 Figures 3c and d is that a corresponding increase in J(2ω D,600 ) is not observed in either state near binding site II. For example, the J(2ω D,600 ) values are unremarkable for V61, the β-sheet residue near site II. It is clear from this and several other observations that the structure and dynamics of the two binding sites are very different. 54 In addition to its role in identifying general trends among the protein s dynamics, spectral density mapping serves another useful purpose. It provides an assessment of the quality and consistency of the data. This assessment is accomplished by substituting the spectral density values into the relaxation rate expressions in order to back-calculate the rates. 30 If the experimental data are self-consistent, then the back-calculated and experimental rates should exhibit a strong correlation. Figure 4 shows the correlation between the experimental rates at 500 MHz and the back-calculated (fitted) rates when spectral density mapping is performed with the full data set, consisting of all ten rates. The systematic bias δ was calculated as the mean (R exp -R fit )/R exp ratio across all the methyl groups. The quality of the data is high by this measure, providing strong evidence that the method of Millet et al. is robust and reliable. 28 Among the ten rates measured in the Ca 2+ -loaded state, δ < 1% for seven of the rates, while δ > 2% for only one of the rates. The quality of the data is still high, although not quite as good, in the apo state. Among the ten rates measured in the apo state, δ < 1% for five of the rates, but for another four of the rates 2% < δ < 3%. The Ca 2+ -loaded and apo data sets were acquired with identical pulse sequences and spectral parameters. The fact that the correlations within the apo data are less consistent is likely due to sample factors. One complicating factor is that the protein is less stable in the apo state. 46 In the present study, a second set of peaks began to emerge 42

49 Figure 4: Correlations between the relaxation rates observed experimentally at 500 MHz, R exp, and the rates back-calculated from the five best-fit spectral density values, R fit : (a) R(D z ), apo; (b) R(D z ), Ca 2+ -loaded; (c) R(D + ), apo; (d) R(D + ), Ca 2+ -loaded; (e) R(3D 2 z -2), apo; (f) R(3D 2 z -2), Ca 2+ -loaded; (g) R(D + D z +D z D + ), apo; (h) R(D + D z +D z D + ), Ca 2+ -loaded; (i) R(D 2 + ), apo; (j) R(D 2 + ), Ca 2+ -loaded. The bias δ was calculated as the mean (R exp -R fit )/ R exp ratio across all the methyl groups. 43

50 after the original apo sample had been in solution for approximately one month. This problem necessitated the use of two different apo samples, one for the data acquired at 500 MHz and another for the data acquired at 600 MHz. The Ca 2+ -loaded state, on the other hand, appears to be stable in solution for many months, allowing the full data set to be obtained with a single sample. It should be noted, however, that all three samples (the Ca 2+ -loaded and two apo samples) were products of the same sample prep. Individual aliquots for each of the three samples were not taken until the purification of the protein was complete. In a couple of instances, there exist one or two data points that deviate more significantly than the rest (eg. R(D z ) and R(D + ) at 600 MHz). These points are associated with the residue L23, in which line broadening is again problematic in the apo state. LS2 Model-Free Analysis The best-fit squared order parameters are plotted as a function of residue number for both the apo and Ca 2+ -loaded states in Figures 5a and b. In order to facilitate comparisons between the different methyl group types, Figures 5c and d also present the normalized order parameters, (S 2 f ) norm. 76 This quantity is calculated as: (S 2 f ) norm = S 2 f µ methyl (4) σ methyl where µ methyl and σ methyl are the mean and standard deviation, respectively, in the order parameters for a particular methyl group type. A database of eight proteins was used to obtain these values. Mittermaier et al. reported that the value of µ methyl decreases with the number of dihedral angles separating the methyl group from the backbone. 77 With the 44

51 exception of methionine, there is also an increase in the value of σ methyl with increasing separation from the backbone. For the Ca 2+ -loaded state, the order parameters in Figure 5: Plots of S 2 f as a function of residue number for the apo (a) and Ca 2+ -loaded (b) states. Plots of (S 2 f ) norm as a function of residue number for the apo (c) and Ca 2+ -loaded (d) states. Note that the order parameters reported here correspond to the LS3 model for the limited number of groups assigned to that model (see text). 45

52 Figures 5b and d are closely correlated with the J(0) values described previously in Figure 3b. The trend is again roughly M-shaped, exhibiting maxima near the binding sites, where small amplitude fluctuations predominate, and minima near the termini and linker, where larger amplitude fluctuations are permitted. In contrast, the apo state exhibits a much weaker trend as a function of residue number. The order parameters do not reach a maximum at either of the binding sites, and the minima are not as clearly defined. Not only are the order parameters low near the termini and linker, where high flexibility was observed in the Ca 2+ -loaded state, but also throughout much of helix C. This structural element exhibited distinguishing characteristics in several previous studies. It is shorter in length than the other three helices, and its overall orientation relative to the rest of the protein is less well-defined. 46 Helix C also exhibits high H N exchange rates in the apo state as well as lower average H-N order parameters. 78,50 These observations are consistent with those made in other S100 proteins wherein the reorientation of helix C is the most significant change to accompany Ca 2+ binding In Figure 6, the order parameter difference between the two states, S f = S f,ca - S 2 f,apo, is plotted as a function of residue number. Interestingly, the data continue to 2 exhibit the familiar M-shaped profile noted in Figures 3b and 5d. Large positive S f values are concentrated among the methyl groups that are closest to the binding sites, while methyl groups with large negative S 2 f values tend to be located near the termini and linker. Qualitatively, these findings indicate that Ca 2+ binding elicits a decrease in the amplitude of fluctuations among methyl groups that are closest to the binding sites. This trend is consistent with observations made in previous backbone dynamics studies. 50,54 The methyl data, however, suggest an additional mechanism that could not be inferred 46

53 Figure 6: Plot of S f 2 as a function of residue number. 47

54 from the backbone data. For several distant methyl groups, the fluctuations actually appear to increase in amplitude with Ca 2+ binding. Whereas the change that was observed among the backbone order parameters was unidirectional, the side-chain order parameters exhibit a bidirectional reorganization with binding. The implications of these findings will be discussed below. In Figures 7a and b, the internal correlation time τ f is plotted as a function of residue number. The most prominent feature, which is observed in both the apo and Ca 2+ - loaded plots, is the large increase in τ f for L23 δ1 and δ2 at binding site I. It is also interesting to note that a corresponding increase is not observed at binding site II. This trend is consistent with that observed among the non-zero spectral densities in Figures 3c and d. LS3 Model-Free Analysis Most methyl groups in both the apo and Ca 2+ -loaded states of calbindin are eff adequately fit by the LS2 model. A few groups, however, require the addition of the τ c parameter to account for slower timescale internal motion. In Figure 8, the spectral density mapping results are plotted along with the best-fit LS2 and LS3 spectral density curves, which are represented in red and blue, respectively. Four groups from the apo state (L40δ2, L46δ2, L53δ2 and L69δ1) and three groups from the Ca 2+ -loaded state (L6δ2, L30δ1 and L30δ2) were assigned to the LS3 model. Note the fact that all of these groups belong to Leu side-chains. A similar trend was observed by Skrynnikov et al. who proposed that longer side-chains, such as Leu, are more likely to exhibit slow internal motions. 30 The mean τ eff c value across the four LS3 methyl groups in the apo state is 3.11 ns, which is slightly lower than the overall correlation time of 4.11 ns used in the LS2 48

55 Figure 7: Plots of τ f as a function of residue number for the apo (a) and Ca 2+ -loaded (b) states. 49

56 Figure 8: These four groups are representative of the total set of seven groups assigned to the LS3 model. The black circles correspond to the results of the spectral density mapping procedure, while the best-fit LS-2 and LS-3 spectral density curves are represented in red and blue, respectively. 50

57 model. Similarly, the meanτ eff c value across the three LS3 methyl groups in the Ca 2+ - loaded state is 2.76 ns, which is lower than the LS2 overall correlation time of 4.04 ns. This trend was also observed by Skrynnikov et al. 30 Additional experimental studies are needed in order to determine the prevalence of slow timescale internal motion among methyl groups in proteins. Molecular dynamic studies may also prove useful in interpreting these analyses. 79 Discussion For several different parameters, an M-shaped trend has been noted as a function of residue number. These parameters include the J(0) and (S 2 f ) norm values for the Ca 2+ - loaded state in Figures 3b and 5d, respectively, as well as the S 2 f values in Figure 6. In order to provide a higher order structural context for these findings, the values of each of these three parameters were mapped on to the experimental NMR structure of Ca 2+ - loaded calbindin (PDB entry 2BCA). The result is depicted in Figures 9b, c and d. The methyl carbons are colored along a gradient in which the largest value for a given parameter is represented in red, the mean value is represented in white, and the smallest value is represented in blue. Figure 9a also includes an illustration of the distance from each methyl carbon to the two Ca 2+ ions, represented as a gradient from red for the carbon closest to the Ca 2+ ions to blue for the carbon most distant to the Ca 2+ ions. (Due to the fact that there are two Ca 2+ ions, the value assigned to each methyl carbon is actually the average of the two distance measurements.) Two limiting classes of methyl groups are evident in these illustrations. The first is a proximal class located close to the binding sites. The proximal class exhibits the following characteristics: 1) high J(0) values in the Ca 2+ -loaded state, indicative of a narrow distribution of rotational 51

58 Figure 9: Structural context of the parameters obtained from the relaxation analysis. (a) Mean distance of each methyl carbon to the two Ca 2+ ions (red: short distance from the Ca 2+ ions, blue: large distance from the Ca 2+ ions). (b) J(0) values in the Ca 2+ -loaded state (red: high J(0) values, blue: low J(0) values). (c) (S 2 f ) norm values in the Ca 2+ -loaded state (red: high (S 2 f ) norm values, blue: low (S 2 f ) norm values). (d) S 2 f values (red: large positive S 2 f values, blue: large negative S 2 f values). 52

59 frequencies, 2) high (S 2 f ) norm values in the Ca 2+ -loaded state, consistent with small amplitude motions and 3) large positive S 2 f values, suggesting that a partial immobilization accompanies Ca 2+ binding. The proximal class includes residues from the cross-strand β-type interaction between the two binding loops. The second class of methyl groups is the distal class located in positions far from the binding sites. These methyl groups display a trend opposite that of the proximal class: 1) low J(0) values in the Ca 2+ -loaded state (a broad distribution of rotational frequencies), 2) low (S 2 f ) norm values in the Ca loaded state (large amplitude motions) and 3) large negative S f values (increased conformational freedom with Ca 2+ binding). Residues in the distal class belong to different structural elements than the proximal class. For instance, L6 and I73 are located in helices A and D, respectively, while L39 is part of the linker. The trend identified from these two limiting classes is not absolute, however. In Figure 10, the three characteristics listed above are plotted as a function of distance from the Ca 2+ ions. The distal class, in particular, exhibits a few notable exceptions. By the measure used here, the methyl group most distant to the Ca 2+ ions is T45γ2, which is located at the C-terminal end of the linker region. One might predict that T45γ2 would have a low (S 2 f ) norm value in the Ca 2+ -loaded state and a large negative S 2 f value, similar to residues L6, L39 and I73. In reality, the (S 2 f ) norm value for T45γ2 is close to zero, and the S 2 f value is positive, not negative. One possible explanation for the discrepancy is the role of T45 in stabilizing the N-terminus of helix C through an N-cap hydrogen bond between T45O γ and E48H N. 45 V70γ1 is another exception. V70γ1 is located in the same general area as the distal class residues, but both its (S 2 f ) norm value in the Ca 2+ -loaded state and its S 2 f value are near zero, suggesting that it is highly solvent-exposed in both states. 53

60 Figure 10: Correlation plots of the J(0) values in the Ca 2+ -loaded state (a), the (S f 2 ) norm values in the Ca 2+ -loaded state (b) and the S f 2 values (c) versus the mean distance of the corresponding methyl carbon to the two Ca 2+ ions. 54

61 Despite these and other more minor inconsistencies, the correlations in Figure 10 are relatively high. The linear correlation coefficients are r = -0.61, and in Figures 10a, b and c, respectively. The corresponding p-values are p = , and Several residues from both the proximal and distal classes exhibited distinguishing characteristics in previous biophysical studies. Kragelund et al. examined the effects of various mutations on Ca 2+ binding. 80 They found that mutants involving proximal class residues, such as L23A and V61A, have a much lower Ca 2+ affinity. This effect was shown to be the result of an increased Ca 2+ dissociation rate among the mutants, presumably due to an increase in binding site dynamics relative to the WT protein. In contrast, mutations in distal class residues, such as L6V and I73V, have little effect on Ca 2+ affinity. Some mutants actually showed a slight increase in affinity as well as a slight decrease in dissociation rate relative to WT. What is most significant relative to our findings here is a correlation between the mutational effect on Ca 2+ affinity and the distance of the mutated side-chain to the nearest Ca 2+ coordinating atom (r = 0.77). In a recent study by Malmendal et al., Ca 2+ binding was found to lead to formation of a hydrophobic patch involving the distal class residues L39 and I73, as well as F Malmendal et al. demonstrated by NMR chemical shift changes that the patch interacts with a detergent dodecyl phosphocholine (DPC) molecule. The increase in dynamics correlates well with exposure of this patch. Analysis of our results collectively reveal a trend that we refer to as polar dynamics. A summary of its relevant features is provided in Table 1. Like any other chemical phenomenon, polar dynamics result from statistical thermodynamic factors. 3 55

62 Defining Features of Polar Dynamics Proximal class Distal class Prime examples: L23, V61 Prime examples: L6, L39 and I73 Close to the Ca 2+ binding sites Distant from the Ca 2+ binding sites Narrow distribution of rotational frequencies Broad distribution of rotational frequencies in the Ca 2+ -loaded state (high J(0)) in Ca 2+ -loaded state (low J(0)) Small amplitude motions in the Ca 2+ -loaded Large amplitude motions in the Ca 2+ - state (high (S 2 f ) norm ) loaded state (low (S 2 f ) norm ) Decreased conformational freedom with Ca 2+ Increased conformational freedom with binding (positive S 2 f ) Ca 2+ binding (negative S 2 f ) Mutants exhibit a decrease in Ca 2+ affinity Mutants exhibit little change in Ca 2+ and an increase in the Ca 2+ dissociation rate. affinity or in the Ca 2+ dissociation rate. Residues directly involved in the H-bonding Residues directly involved in a calbindinphospholipid network between the Ca 2+ binding sites interaction 56

63 However, as is typically the case in experimental systems, we have limited access to the full set of thermodynamic variables. One way in which the data can be interpreted is within the framework of the conformational entropy change that accompanies Ca 2+ binding. 20,21,56 For example, the sum of the S 2 f values across all the methyl groups is The fact that the total S 2 f is positive indicates that Ca 2+ binding elicits a conformational entropy loss among the methyl groups. The methyl groups as a whole sample fewer conformations in the presence of Ca 2+. Although this contribution disfavors binding, there are many compensating factors that are not accounted for by our data. These factors are both enthalpic and entropic and arise from several sources including the Ca 2+ ions, the solvent and the other chemical groups in the protein. Despite the net entropy loss indicated by the methyl data, there appears to be a significant degree of compensation among the methyl groups themselves. Although the total S 2 f is positive, the mean value is near zero (0.05), and the standard deviation in S 2 f across all the methyl groups is large (0.14). A defining feature of the polar dynamics observed in Figure 9d is a segregation of the S 2 f values into two limiting classes at near opposite poles of the molecule. This trend is largely justified by the related structural constraints. Ca 2+ binding requires a specific metal-coordination geometry supported by a network of non-covalent interactions. These interactions require a relatively fixed conformation among an entire region of the molecule located near the binding sites. In order to partially compensate for this entropic cost, the opposite pole of the molecule appears to sample a larger number of conformations. Our results suggest that this conformational sampling occurs over a broad range of frequencies. 57

64 The observation of polar dynamics also provides some justification for the results from site-directed mutagenesis studies on calbindin D 9k. For example, Ca 2+ binding has been described as a process that requires specific packing interactions among methyl groups in the proximal class. These interactions are readily disrupted by mutation. The distal class, in comparison, exhibits a much higher tolerance to mutation. One might expect that regions exhibiting a higher level of conformational freedom would be more tolerant of mutation. It is possible, and perhaps even likely, that the large amplitude fluctuations present among these methyl groups in the Ca 2+ -loaded WT protein persist in the corresponding mutants. Finally, we return to the binding of DPC and reconsider its effects on the methyl dynamics. The formation of a DPC binding surface in the presence of Ca 2+ introduces an additional free energy cost due to the exposure of hydrophobic residues. However, the aforementioned increase in conformational entropy associated with these residues may act to partially compensate for this cost. Binding of a phospholipid molecule is expected to bury much of the exposed surface area, providing for more optimized protein-solvent interactions. Future studies might aim to determine whether DPC binding also attenuates conformational fluctuations in the region. Conclusions In the case of calbindin D 9k, a methyl group s distance from the Ca 2+ binding sites is a significant predictor of its conformational dynamics. As a result, several methyl groups segregate into two limiting classes, one proximal and the other distal to the binding sites. We have introduced the term polar dynamics to describe this type of organization across the molecule. The description is only approximate. The two limiting 58

65 classes are not located at exact opposite poles of a perfectly spherical molecule, and exceptions have been noted. Nevertheless, the term accounts for several defining features of calbindin s side-chain dynamics, related both to its timescale (through J(0)) and its amplitude (through (S 2 f ) norm and S 2 f ). The organization of polar dynamics in calbindin is likely promoted by several factors, the most obvious being that it is a single domain protein. Interdomain interactions present in multidomain proteins introduce an additional layer of complexity not found in calbindin. Second, the presence of two EF-hands incorporates some degree of symmetry into the molecule. This symmetry may allow global trends to emerge that are less common in more asymmetric molecules. Third, calbindin is a metal-binding protein. This point is significant because protein-metal bonds are highly directional, requiring a specific coordination geometry and electrostatic distribution across a large network of functional groups. Finally, calbindin has two binding sites that are situated close to one another, near the surface of the molecule. These sites bind with positive cooperativity, indicating a significant degree of structural and dynamic coupling across the region. An interesting question for future studies is whether or not polar dynamics persist in the absence of this cooperativity. We note that well-characterized model systems have been developed for both the half-saturated states of calbindin. 40,41,42,43 These model systems should provide an initial test of the generality of our current findings. Materials and Methods Sample Preparation Bovine calbindin D 9k P43M was expressed from a pet1120 plasmid in Escherichia coli strain BL21(DE3)-Star cells. Use of the P43M mutant avoids cis-trans 59

66 isomerism around the P43-S44 bond. 51,52 Samples for side-chain chemical shift assignments were expressed in M9 minimal media containing 15 NH 4 Cl and 13 C 6 -Dglucose as the sole nitrogen and carbon sources, respectively. Stereospecific methyl assignments also required that samples be prepared in a 1:9 mixture of 13 C 6 - and 12 C 6 -Dglucose. 82 Samples for the 2 H relaxation experiments were expressed in M9 minimal media containing 15 NH 4 Cl and 13 C 6 -D-glucose as the sole nitrogen and carbon sources, respectively, as well as 50% D 2 O. The protein was purified by heat extraction, followed by anion-exchange and size-exclusion chromatographies. 83 All NMR samples contained ~4 mm protein, 10 mm imidazole-d 4, 10 µm NaN 3, 1 mm DSS and 90% H 2 O/10% D 2 O. The Ca 2+ -loaded samples also contained 10 mm CaCl 2. In order to prepare the apo state, a 20 molar excess of EGTA was added relative to the total amount of protein and Ca 2+ present in the sample. The Ca 2+ -loaded and apo states are readily differentiated from one another by the appearance of the most upfield region of a 1D 1 H NMR spectrum. In the case of the Ca 2+ -loaded state, the most upfield resonance is a triplet corresponding to Ile 73, while the most upfield resonance in the apo state is a doublet corresponding to Val 70. Once the apo state was confirmed as the sole population in the sample, the sample was dialyzed against decreasing concentrations of EGTA (500, 50, and 5 µm) and finally against Chelex-treated H 2 O. The protein was lyophilized and then dissolved in Chelextreated sample buffer. The ph of all samples was adjusted to 7.0 (not corrected for isotope effects). Chemical Shift Assignments Experiments were performed at 300 K on a Varian Inova 500 MHz spectrometer using in-house pulse programs. Backbone 15 N and 1 H assignments were available for 60

67 both the Ca 2+ -loaded and apo states under similar sample conditions 84,85,86 and were confirmed using 2D [ 15 N, 1 H]-HSQC 87, 3D HNCA 88,89 and HN(CO)CA 90,89 spectra. 12 Side-chain aliphatic 13 C and 1 H assignments were obtained from 3D CBCA(CO)NH, (H)C(CC)(CO)NH, HBHA(CBCACO)NH and H(CC)(CO)NH spectra. 89 A 2D TOCSYrelayed [ 13 C, 1 H]-HSQC spectrum was also acquired with a 13 C natural abundance sample in order to verify the Val and Leu methyl assignments. 82 Stereospecific methyl assignments were obtained by the method of Neri and co-workers using the 10% 13 C- labeled sample. 82 Spectra were processed by NMRPipe 91 and analyzed within Sparky. 92 Methyl 2 H Relaxation Rate Measurements Experiments were performed at 300 K on Varian Inova spectrometers at 500 and 600 MHz. Pulse programs were written in-house following the methods described in references 2 and 3. Pulse scheme C was selected from the two options proposed by Millet et al. to measure the R(D + D z +D z D + ) rate. The rates at 500 MHz were acquired as follows: 1) R(D z ): 16 scans; complex points; 0.05 ( 2), 4.5, 9.5, 15, 21, 28, 36 ( 2), 45 and 58 ( 2) ms relaxation delays, 2) R(D + ): 16 scans; complex points; 0.2 ( 2), 1.3, 2.8, 4.4, 6.2 ( 2), 8.4, 10.9 and 15.5 ( 2) ms relaxation delays, 3) R(3D 2 z -2): 32 scans; complex points; 1.4 ( 2), 3.6, 7.6, 11.8, 16.5, 21.6 ( 2), 27.4, 33.8, 41.3 and 50 ( 2) ms relaxation delays, 4) R(D + D z +D z D + ): 32 scans; complex points; 0.48 ( 2), 1.5, 3, 4.7, 6.6, 8.7, 10.9 ( 2), 13.5, 16.5 ( 2) and 20 ms relaxation delays, 5) R(D 2 + ): 32 scans; complex points; 1.4 ( 2), 3.6, 7.6, 11.8, 16.5, 21.6 ( 2), 27.4, 33.8, 41.3 and 50 ( 2) ms relaxation delays. The relaxation delays at 600 MHz were optimized from the rates measured at 500 MHz so that the intensity of the last time point was on average 10% of the intensity of the first time point. The rates at 600 MHz were 61

68 acquired as: 1) R(D z ): 16 scans; complex points; 0.05 ( 2), 4.5, 9.5, 15.5, 22.5 ( 2), 31.5, 42.5, 58 and 85 ( 2) ms relaxation delays, 2) R(D + ): 16 scans; complex points; 0.2 ( 2), 1.9, 4, 6.4, 9.3 ( 2), 12.8, 17.4, 23.8 and 34.7 ( 2) ms relaxation delays, 3) R(3D 2 z -2): 32 scans; complex points; 1.4 ( 2), 5.4, 11.6, 18.7, 27.2 ( 2), 37.5, 50.8, 69.5 and ( 2) ms relaxation delays, 4) R(D + D z +D z D + ): 32 scans; complex points; 0.48 ( 2), 2.4, 5.1, 8.3, 12 ( 2), 16.6, 22.5, 30.8 and 45 ( 2) ms relaxation delays, 5) R(D 2 + ): 32 scans; complex points; 1.4 ( 2), 6.8, 14.6, 23.6, 34.2 ( 2), 47.2, 63.9, 87.5 and ( 2) ms relaxation delays. Relaxation rates were obtained by fitting the cross-peak intensities to a single exponential function using the program CurveFit. 93 Duplicate time points provided a measure of the random error in the peak intensities. Monte Carlo simulations were subsequently performed in order to estimate the uncertainty in the rates from the measured peak intensity errors. Data Analysis An analogue of eq. 1 from Skrynnikov et al. 30 was solved by singular value decomposition, yielding the five available spectral density values. Model-free parameters were obtained by least-squares minimization over the entire set of ten rates. 94,95 Errors in the spectral densities and model-free parameters were estimated by Monte Carlo analysis. 96 The LS2 model-free calculations also require the overall correlation time, which was obtained from backbone 15 N relaxation measurements. R 1, R 2 and { 1 H}- 15 N NOE measurements were performed using standard parameters 97 and 15 N relaxation methods. 98 Analysis of the backbone data was performed with the r2r1_diffusion software, 99,100 which gave overall correlation times of 4.11 and 4.04 ns for the apo and 62

69 Ca 2+ -loaded states, respectively. Inclusion of the 15 N label allowed the overall correlation time to be measured on the same samples used for the 2 H relaxation measurements. Although the 15 N relaxation rates obtained from these samples contain contributions from dipolar interactions between the 15 N nucleus and the adjacent 13 C and 13 C α nuclei, a recent study by Xu et al. has demonstrated that such effects are essentially inconsequential. 101 The data analysis may be complicated by observations made by Millet et al. regarding the measurement of the R(D 2 + ) relaxation rate. 28 In addition to the dominant quadrupolar interaction, it is necessary to consider other minor contributions as well. One contribution arises from dipolar interactions within the methyl group (i.e. between the deuteron and carbon and between the deuteron and two protons). These dipolar interactions contribute to the R(D 2 + ) rate primarily through the J(0) term. Equation 1 from Millet et al. indicates how to incorporate these contributions into the spectral density calculations. There exist two other contributions, however, that are perhaps more difficult to approximate. One of these contributions arises from the J CD coupling that occurs during the relaxation delay in the experiment. Simulations performed by Millet et al. indicate that evolution under the J CD coupling causes a systematic decrease of 2-3% in the measured rate. Therefore, they recommend that the measured values be multiplied by a factor of prior to the analysis. Another difficult contribution to approximate arises from dipolar interactions between the methyl deuteron and protons that are close in space, but not directly bonded to the methyl carbon. Although the magnitude of this contribution is difficult to ascertain, it is expected to cause a systematic increase in the measured rates. A procedure was proposed by Millet et al. to estimate the magnitude of this contribution 63

70 on a per-site basis. Their procedure relies upon the discrepancy observed in the experimental and calculated values of an additional rate, R(I z C z ). When the R(D 2 + ) values are corrected for both the J CD coupling and the dipolar contributions from external protons, the R(D 2 + ) rates increase on average by 1.7% for the Ca 2+ -loaded state at both 500 and 600 MHz. Similarly, the R(D 2 + ) rates increase on average by 1.9 and 2.1% for the apo state at 500 and 600 MHz, respectively. The spectral density mapping was repeated with the corrected R(D 2 + ) values and the rates were back-calculated as before (data not shown). However, the bias in this second round of calculations is slightly higher than the bias encountered with the uncorrected rate values. For example, the bias in the R(D 2 + ) value increases from δ = 2.1 to 3.8% for the apo state at 600 MHz. As one final check of the data, the R(D 2 + ) rates were excluded from the analysis, and the spectral density mapping was performed with the remaining set of eight rates. The results are essentially unchanged. The J(0) values, for instance, change by less than 0.5%. We elected, therefore, to include the uncorrected R(D 2 + ) values in the model-free calculations. Acknowledgements We thank Carol Caperelli and Susan Meyn for the instruction that they provided in protein expression and purification. We also thank Patricia Johnson for help in preparing the figures. Eric Johnson was supported by Training Grant HL07382 and Walter Chazin by Operating Grant GM40120 from the National Institutes of Health. 64

71 CHAPTER 3: Future Directions The work described in the previous chapter is best understood as part of an ongoing research program. We now present several directions for future research. We currently have available: 1) a protocol for the expression and purification of calbindin, 2) pulse sequences to implement the 2 H relaxation measurements and 3) several data analysis programs. These resources should greatly facilitate investigations of the sidechain methyl dynamics in model systems of the two half-saturated states. In order to perform these studies, it will be necessary to first demonstrate that the half-saturated states can be reliably produced. Our collaborators in the Chazin lab have worked extensively with these model systems and can provide us with the necessary instruction in order to prepare the samples. It will also be necessary to perform chemical shift assignments for the side-chain 13 C and 1 H nuclei in the half-saturated states. This step will be time-consuming, but well-established lab protocols are now available that should expedite the process. Once these tasks have been accomplished, the 2 H methyl relaxation measurements should then proceed relatively quickly. There are several other experimental directions that might also be pursued. Previous 15 N relaxation studies have suggested that calbindin s backbone dynamics play an important role in the cooperativity observed between the two binding sites. It would be a valuable exercise to confirm and extend these findings using complementary methods, such as those developed by Zuiderweg and co-workers. 102,103,104 The Zuiderweg lab has developed a series of experiments to characterize the dynamics of the C'-C α bond vector. Interestingly, their work has demonstrated that the C'-C α bond vector is sensitive to conformational fluctuations that are not reported by the N-H bond vector. 65

72 We should also expect that important side-chain fluctuations are not reported by the protein s methyl groups. Two other targets for future side-chain investigations are the amide group, which is located in Asn and Gln side-chains, and the methylene group, which is present in 16 of the 20 amino acids. Both of these chemical groups are amenable to the 2 H-based methods employed in our methyl dynamics studies. In Appendix 7, we present the results obtained from an initial study of the side-chain amide dynamics in calbindin. The initial reports of 2 H-based amide ( 15 NHD) 105 and methylene ( 13 CHD) 106 dynamics have included only longitudinal and transverse relaxation rate measurements. We suspect, however, that the technique can be extended to include the quadrupolar order, transverse antiphase and double quantum relaxation rates as well. By increasing the number of available rates, spectral density mapping would be made possible at these sites for the first time. In addition to the 2 H-based methods, the methylene dynamics may also be characterized by various 13 C relaxation measurements. In Chapter 1, several technical factors were mentioned that have complicated the progress made thus far in the use of the 13 C nucleus as a dynamics probe. One of the complicating factors is the presence of 13 C- 13 C scalar and dipolar couplings in uniformly 13 C-labeled proteins. The alternate spin labeling strategy of LeMaster and Kushlan was mentioned as one way to prevent the occurrence of adjacent 13 C nuclei. 33 If this spin labeling strategy is combined with fractional deuteration, one would then have available a large number of 13 CHD groups that are free of 13 C- 13 C couplings. Use of the 13 CHD group also avoids the problem of dipole-dipole cross-correlated relaxation present in a 13 CH 2 group. These factors suggest that samples containing isolated 13 CHD spin systems would be ideally suited for a large 66

73 number of methylene studies, including both 2 H- and 13 C-based methods. These experiments would provide a very useful test of whether or not the 2 H- and 13 C-based approaches provide a consistent description of the fast timescale (ns-ps) methylene dynamics. The 13 C experiments might also include a R 1ρ rotating-frame relaxation measurement to detect slower timescale (µs-ms) conformational exchange. 107 This development would represent an important technical advance and is motivated by our observation of a large increase in the non-zero spectral density and τ f values in the methyl groups near calbindin s site I. We conclude by noting that calbindin presents several opportunities for computational studies that would complement the large body of experimental work on the protein. Marchand and Roux have published an insightful molecular dynamics simulation of calbindin that provides an important starting point for future investigations. 108 Calbindin presents several technical challenges for the computational scientist. For instance, it has a high surface charge density. Twenty-seven of its 75 residues are charged at neutral ph, resulting in a net charge of -7 for the apo state. The multiple charge-charge repulsions present under these conditions likely contribute to the large conformational fluctuations observed in the apo state. It is also important to note that the distance between the two Ca 2+ ions is only ~12 Å in the Ca 2+ -loaded state. These factors require that considerable attention be paid to the treatment of long-range electrostatic interactions in calbindin. Another technical challenge results from the fact that most force fields do yet include polarization effects, which are thought to be important in protein-metal interactions. 108 Although these technical concerns necessitate caution when interpreting 67

74 any computational results, they also indicate that calbindin may be a useful molecule for testing new modeling techniques. There are several existing applications that might be pursued initially, however. The basic approach would be to first demonstrate that the data extracted from an MD simulation is consistent with the known experimental data. The second step would then be to extract additional information from the simulation that is not experimentally accessible. In the study by Marchand and Roux, the backbone order parameters observed experimentally were well-reproduced by the simulation. Our objective would be to determine if it is also possible to reproduce the side-chain methyl order parameters. If future simulations were successful in this regard, they would then lead the way to morespecialized MD applications. Two of the more promising methods to pursue are the reorientational eigenmode dynamics (RED) method of Prompers and Brüschweiler 25,109 and the anisotropic thermal diffusion (ATD) method of Ota and Agard. 110 The RED method is used to identify correlated motions in proteins, while the ATD method is used to identify the paths by which conformational perturbations propagate through the core of a protein. It would very interesting to perform these calculations on each of calbindin s four binding states. The goal of these studies would be to identify new mechanisms by which cooperative binding is achieved. 68

75 Appendix 1: Methyl chemical shift assignments The chemical shift values reported in the following tables were referenced to an internal DSS standard at 0 ppm. These values were observed in the samples used for the 2 H relaxation experiments and therefore reflect the effects of a 2 H isotope shift. Apo Resonance 13 C [ppm] 1 H [ppm] A14CBHB A15CBHB I73CD1HD I73CG2HG I9CD1HD I9CG2HG L23CD1HD L23CD2HD L28CD1HD L28CD2HD L30CD1HD L30CD2HD L31CD1HD L31CD2HD L32CD1HD L32CD2HD L39CD2HD L40CD1HD L40CD2HD L46CD1HD L46CD2HD L49CD1HD L49CD2HD L53CD1HD L53CD2HD L69CD1HD L6CD1HD L6CD2HD M0CEHE M43CEHE T34CG2HG T45CG2HG V61CG1HG V61CG2HG

76 V68CG1HG V68CG2HG V70CG1HG V70CG2HG Ca 2+ -loaded Resonance 13 C [ppm] 1 H [ppm] A14CBHB A15CBHB I73CD1HD I73CG2HG I9CD1HD I9CG2HG L23CD1HD L23CD2HD L28CD1HD L28CD2HD L30CD1HD L30CD2HD L31CD1HD L31CD2HD L32CD2HD L39CD1HD L39CD2HD L40CD1HD L40CD2HD L46CD1HD L46CD2HD L49CD1HD L49CD2HD L53CD2HD L69CD1HD L69CD2HD L6CD1HD L6CD2HD M0CEHE M43CEHE T34CG2HG T45CG2HG V61CG1HG V61CG2HG V68CG1HG V68CG2HG

77 V70CG1HG V70CG2HG

78 Appendix 2: Representative spectra The following spectra were both obtained in the R(D + ) experiments at 600 MHz with a 0.2 ms relaxation delay. Figure 11: Representative apo spectrum 72

79 Figure 12: Representative Ca 2+ -loaded spectrum 73

80 Appendix 3: Relaxation rates Apo, 500 MHz Resonance R(D z ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ±

81 Resonance R(D + ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ±

82 Resonance R(3D 2 z -2) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D + D z +D z D + ) [s -1 ] 76

83 A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D 2 + ) [s -1 ] A14CBHB ±

84 A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Apo, 600 MHz Resonance R(D z ) [s -1 ] 78

85 A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D + ) [s -1 ] A14CBHB ±

86 A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(3D 2 z -2) [s -1 ] A14CBHB ± A15CBHB ±

87 I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D + D z +D z D + ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ±

88 I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D 2 + ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ±

89 I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD1HD ± L32CD2HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD1HD ± L53CD2HD ± L69CD1HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Ca 2+ -loaded, 500 MHz Resonance R(D z ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ±

90 I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ± L39CD1HD ± L40CD2HD ± L46CD1HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ± L69CD2HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D + ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ±

91 L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ± L39CD1HD ± L40CD2HD ± L46CD1HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ± L69CD2HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(3D 2 z -2) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ±

92 L39CD1HD ± L40CD2HD ± L46CD1HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ± L69CD2HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D + D z +D z D + ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ± L39CD1HD ± L40CD2HD ± L46CD1HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ±

93 L69CD2HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D 2 + ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ± L39CD1HD ± L40CD2HD ± L46CD1HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ± L69CD2HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ±

94 V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Ca 2+ -loaded, 600 MHz Resonance R(D z ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ± L39CD1HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ± L69CD1HD ± L69CD2HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ±

95 T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D + ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ± L39CD1HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ± L69CD1HD ± L69CD2HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ±

96 V61CG1HG ± V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(3D 2 z -2) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ± L39CD1HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ± L69CD1HD ± L69CD2HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ±

97 V61CG2HG ± V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D + D z +D z D + ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ± L39CD1HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ± L69CD1HD ± L69CD2HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ±

98 V68CG1HG ± V68CG2HG ± V70CG1HG ± V70CG2HG ± Resonance R(D 2 + ) [s -1 ] A14CBHB ± A15CBHB ± I73CD1HD ± I73CG2HG ± I9CD1HD ± I9CG2HG ± L23CD1HD ± L23CD2HD ± L28CD1HD ± L28CD2HD ± L30CD1HD ± L30CD2HD ± L31CD1HD ± L31CD2HD ± L32CD2HD ± L39CD1HD ± L39CD2HD ± L40CD1HD ± L40CD2HD ± L46CD1HD ± L46CD2HD ± L49CD1HD ± L49CD2HD ± L53CD2HD ± L69CD1HD ± L69CD2HD ± L6CD1HD ± L6CD2HD ± M0CEHE ± M43CEHE ± T34CG2HG ± T45CG2HG ± V61CG1HG ± V61CG2HG ± V68CG1HG ±

99 V68CG2HG ± V70CG1HG ± V70CG2HG ±

100 Appendix 4: Model-free parameters The complete set of LS2 fitting parameters is listed here. The LS3 parameters are also listed for those resonances assigned to the LS3 model. In each table, the symbol σ refers to the error in the corresponding parameter. Apo, S f 2 (LS2) Resonance 2 S f σ A14CBHB A15CBHB I73CD1HD I73CG2HG I9CD1HD I9CG2HG L23CD1HD L23CD2HD L28CD1HD L28CD2HD L30CD1HD L30CD2HD L31CD1HD L31CD2HD L32CD1HD L32CD2HD L39CD2HD L40CD1HD L40CD2HD L46CD1HD L46CD2HD L49CD1HD L49CD2HD L53CD1HD L53CD2HD L69CD1HD L6CD1HD L6CD2HD M0CEHE M43CEHE T34CG2HG T45CG2HG

101 V61CG1HG V61CG2HG V68CG1HG V68CG2HG V70CG1HG V70CG2HG Apo, τ f (LS2) Resonance τ f [s] σ [s] A14CBHB A15CBHB I73CD1HD I73CG2HG I9CD1HD I9CG2HG L23CD1HD L23CD2HD L28CD1HD L28CD2HD L30CD1HD L30CD2HD L31CD1HD L31CD2HD L32CD1HD L32CD2HD L39CD2HD L40CD1HD L40CD2HD L46CD1HD L46CD2HD L49CD1HD L49CD2HD L53CD1HD L53CD2HD L69CD1HD L6CD1HD L6CD2HD M0CEHE M43CEHE T34CG2HG T45CG2HG

102 V61CG1HG V61CG2HG V68CG1HG V68CG2HG V70CG1HG V70CG2HG Apo, S f 2 (LS3) Resonance 2 S f σ L40CD2HD L46CD2HD L53CD2HD L69CD1HD Apo, τ f (LS3) Resonance τ f [s] σ [s] L40CD2HD L46CD2HD L53CD2HD L69CD1HD Apo, τ c eff (LS3) Resonance τ f [s] σ [s] L40CD2HD L46CD2HD L53CD2HD L69CD1HD Ca 2+ -loaded, S f 2 (LS2) Resonance 2 S f σ A14CBHB A15CBHB I73CD1HD I73CG2HG I9CD1HD I9CG2HG

103 L23CD1HD L23CD2HD L28CD1HD L28CD2HD L30CD1HD L30CD2HD L31CD1HD L31CD2HD L32CD2HD L39CD1HD L39CD2HD L40CD1HD L40CD2HD L46CD1HD L46CD2HD L49CD1HD L49CD2HD L53CD2HD L69CD1HD L69CD2HD L6CD1HD L6CD2HD M0CEHE M43CEHE T34CG2HG T45CG2HG V61CG1HG V61CG2HG V68CG1HG V68CG2HG V70CG1HG V70CG2HG Ca 2+ -loaded, τ f (LS2) Resonance τ f [s] σ [s] A14CBHB A15CBHB I73CD1HD I73CG2HG I9CD1HD I9CG2HG L23CD1HD

104 L23CD2HD L28CD1HD L28CD2HD L30CD1HD L30CD2HD L31CD1HD L31CD2HD L32CD2HD L39CD1HD L39CD2HD L40CD1HD L40CD2HD L46CD1HD L46CD2HD L49CD1HD L49CD2HD L53CD2HD L69CD1HD L69CD2HD L6CD1HD L6CD2HD M0CEHE M43CEHE T34CG2HG T45CG2HG V61CG1HG V61CG2HG V68CG1HG V68CG2HG V70CG1HG V70CG2HG Ca 2+ -loaded, S f 2 (LS3) Resonance 2 S f σ L6CD2HD L30CD1HD L30CD2HD Ca 2+ -loaded, τ f (LS3) 98

105 Resonance τ f [s] σ [s] L6CD2HD L30CD1HD L30CD2HD Ca 2+ -loaded, τ c eff (LS3) Resonance τ f [s] σ [s] L6CD2HD L30CD1HD L30CD2HD

106 Appendix 5: Calbindin D 9k NMR sample preparation (step-by-step instructions) This protocol was initially developed in the labs of Walter Chazin and Eva Thulin. 83 It was later modified for use in Mark Rance s lab by Eric Johnson. Day 1: Transformation Thaw BL21-DE3 star cells (10 µl) and plasmid solution (~50 ng ml -1 ) on ice. -We ve obtained good results with these cells. One can also use regular BL21-DE3 cells (no star). I would suggest not using using BL21 plyss cells. Add 1 µl of the plasmid solution to the cells. Gently mix with a pipette tip. Incubate on ice for 30 min. Heat pulse for 30 s at 42 C. Put back on ice for ~2 min. Add 500 µl SOC medium. Incubate at 30 C for 1 hr with shaking (~250 rpm). Spread 100 µl of ampicillin stock (30 mg ml -1 ) on an LB agar plate. After 1 hr incubation, spread 100 µl of the SOC cell culture on the agar plate. Incubate at 30 C overnight (O/N) w/ shaking. -I generally perform the transformation later in the afternoon. If the cells are allowed to grow for too long a period of time on the agar plate, individual colonies may begin to coalesce. Day 2: Prepare minimal media and inoculate LB The next morning take the agar plate out of the incubator. Store the plate at 4 C until later in the afternoon. 100

107 Most of your time today is spent preparing minimal media (MM). You ll need to prepare ml MM and 4 25 ml MM. Here s what you initially add to the 500 ml MM. Use a large baffled flask ml M9 salts (5 ) ml 1M NaOH g K 2 HPO ml H 2 O Here s what you initially add to the 25 ml MM. Use a small baffled flask. -5 ml M9 salts (5 ) -531 µl 1M NaOH g K 2 HPO 4-18 ml H 2 O After adding these initial ingredients to the MM, autoclave the solutions. After autoclaving the solutions, add these ingredients to the 500 ml MM: -2 ml 1M MgSO 4-1 ml solution (soln) O -25 mg thiamine -25 mg niacin -270 µl 250 mm CaCl µl 2 mg ml -1 biotin µl 0.9 M ZnSO 4 Add these ingredients to the 25 ml MM: -100 µl 1M MgSO 4 101

108 -50 µl solution (soln) O mg thiamine mg niacin µl 250 mm CaCl µl 2 mg ml -1 biotin µl 0.9 M ZnSO 4 Before you leave for the day, inoculate some LB media with the transformed colonies. Put 25 ml LB in a baffled flask. Add 83 µl amp stock. Streak multiple colonies from the agar plate. Inoculate the LB with these colonies. Incubate the LB culture at 30 C O/N w/ shaking. Day 3: Protein expression This is a long day. You can either sleep in and get your rest, knowing that you ll be up late, or you can wake up early and try to get to bed at a reasonable hour. Know that if you re making deuterated protein, it s going to take close to 24 hours. Add these ingredients to the 500 ml MM: -0.5 ml 0.01M FeCl ml vitamin stock g glucose -0.5 g NH 4 Cl Add these ingredients to the 25 ml MM: -25 µl 0.01M FeCl 3 102

109 -25 µl vitamin stock -0.1 g glucose g NH 4 Cl Recipes for the vitamin stock and soln O have been written up by Kimber Baldwin and are in the lab s green notebook. These are common recipes that are also readily found on the web. Inoculate each of the 4x25 ml MM with 1 ml each of the O/N LB culture. Incubate the MM cultures at 30 C with shaking. Mark the time. Measure the OD 600 of the O/N LB culture. It is likely very high and will need to be diluted so that the measurement is in the linear range of the absorbance curve. Collect 2x1mL of the O/N LB culture for future analysis. Place the 2x500 ml MM in the incubator at 30 C in order to preheat the media. (They have not yet been inoculated.) After the 25mL MM cultures have had maybe 2 hr to grow, begin checking the OD 600 on an hourly basis. When the MM cultures reach an OD 600 of ~1, pour 2 of the 25 ml cultures into one of the 500 ml MM and the other two 25 ml cultures into the second 500 ml MM. Mark the time. -It will probably take the cells ~4 hr to reach an OD 600 of 1 if growing in H 2 O. If growing in D 2 O, this step will take ~6 hr. Increase the temperature (T) to 37 C. Measure the OD 600 of the 2x500 ml MM cultures on an hourly basis. 103

110 When the OD 600 reaches ~1, add 500 µl 1M IPTG to each of the 500 ml MM cultures. Before you do this, collect 2x1mL of each of the cultures for future analysis. These aliquots will serve as the pre-induction samples. Spin down the 1 ml aliquots in microcentrifuge tubes and decant the S/N. Freeze the pellets for later use. -I was initially told by someone in Walter s group that the expression in this system is T- inducible. I don t know if that s the case or not. Walter says that he knows nothing about it. Even if it is T-inducible, we still add IPTG. I probably add more IPTG than is necessary. I ve found that the results are fairly insensitive to changes in the IPTG concentration. -The time it will take to reach an OD 600 of 1 will vary. In H 2 O, it can take anywhere from ~3-7 hr following the inoculation of the 2x500 ml MM. In D 2 O, it s hard to say definitively, maybe ~6-10 hr. Regardless of how long it takes to get to the point where you induce the expression of the protein, harvest the cells 4 hr post-induction. Spin down the 1 ml aliquots in microcentrifuge tubes and decant the S/N. Freeze the pellets for later. Before you do anything else, collect 2x1mL of each of the cultures for future analysis. These aliquots will serve as the post-induction samples. In order to harvest the cells, dump one of the 500 ml culures into one 1L centrifuge jar and the other 500 ml culture into another 1L centrifuge jar. Balance the jars. Spin the cells down at 2500 g for 30 min at 4 C. 104

111 -As far as I know, the Lingrel lab is the only lab that owns a centrifuge that can accommodate 1L jars. If need be, you could split the cells up into smaller volumes. For instance, the centrifuge in our lab should take 250 ml jars. After the 30 min spin, dump out the supernatant (S/N). It should be clear. Collect the pellets from the 2 jars in a 50 ml conical tube. I ve found that the best way to do this is to scrape the pellets out of the jars into the tube using a spatula. You won t be able to get all the cells using a spatula, so rinse out the jars a couple of times with some water and pipette the wash from the jars into the tube with a Pasteur pipette. Keep the conical tube and jars on ice during this transfer. You should now have a suspension of the cells in a conical tube. Spin down the conical tube at 1800 g for 30 min at 4 C. This can be done in our centrifuge. After this spin step, dump out the S/N. You should then have ~4-5 ml pellet in the conical tube. Put the tube in the freezer until you re ready to begin purifying. Modification to the above protocol for making deuterated protein When making up the 4x25 ml and 2x500 ml MM, adjust the amount of H 2 O and D 2 O to the desired ratio. In all the work I ve done, this ratio has been 0.5/0.5. Make sure you account for all the different components. All the stock soln will contribute H 2 O. You can t autoclave these solutions. My understanding is that if you do autoclave them, much of the D 2 O, which is very expensive, will be lost. You can autoclave the baffled flasks before you put the solutions in them. That s probably a good idea. At the stage where you normally autoclave, sterile filter the solutions. After the MM has been prepared, parafilm it until ready for use. 105

112 Since the solutions aren t autoclaved, add amp (83 µl amp stock to each of the 25 ml MM, 1.66 ml to each of the 500 ml MM). I generally add the amp with the ingredients that are added the day of the expression. I ve only made deuterated protein a couple of times. Before you undertake the cost, talk to someone who has some experience with expressing in deuterated media. They may alert you of additional considerations of which I am unware. You should probably have performed this protocol a couple of times before using deuterated media. Although the use of D 2 O increases the amount of time needed to complete the protocol, in my experience, it has not impaired the protein yield. Day 4: Heat extraction and gel analysis Calbindin is heat stable. You can boil the cell pellet. Most of the proteins will precipitate out, leaving calbindin behind in soln. Make an imidazole buffer g imidazole g NaCl -Bring up to ~80 ml in H 2 O. -Adjust ph to 7.0 w/ HCl. -Bring up to 100 ml in H 2 O. Meanwhile, allow the cell pellet to thaw. Once it has thawed, keep it on ice if you re not ready to use it. Assuming your pellet volume is ~4.5 ml, suspend the pellet in 30 ml imidazole buffer. Adjust the volume accordingly if your pellet volume differs. Bring another 30 ml imidazole buffer to boiling in an Erlenmeyer flask. 106

113 Once the soln is boiling at ~95 C, pour the suspension into the boiling buffer. Make sure that the pellet is well suspended in the 30 ml before doing this. Make sure all the suspension gets into the boiling buffer. Continuously stir the ~60 ml suspension until the T comes back up to ~95 C. You want this to happen within a couple of minutes. You don t want the suspension to spend a long time in an intermediate T range where proteases will be very active. You will likely burn your fingers a little bit while stirring the suspension and checking the T. As soon as the suspension is back up at 95 C and boiling, pour it into a beaker. Place the beaker in an ice H 2 O slurry and stir the suspension until the T approaches that of the ice bath. You want this T transition to occur as rapidly as possible as well. Stirring should facilitate things. Once the suspension is well cooled, transfer it to a conical tube. Centrifuge the suspension at 1800 g for 20 min at 4 C. After centrifugation, pour the S/N into another conical tube. This S/N contains the majority of calbindin. There may be a minor fraction of calbindin left in the pellet. This can be assessed by gel analysis. If a significant amount of calbindin is left in the pellet, you can repeat this extraction step, but it shouldn t be necessary. Run an SDS-PAGE gel. The instructions in this section are not as detailed as in the previous and following sections. I ve included only those details about running an SDS-PAGE gel that are specific to calbindin. Be aware that calbindin runs ahead of its molecular weight (MW). In my experience, it is always the last protein band observed. 107

114 I generally run a 15% separating gel. I also use higher buffer concentrations than most other people. Carol Caperelli suggested that the higher buffer concentration may improve the resolution of proteins near the MW of calbindin. The separating gel consists of: ml 30% acrylamide/0.8% bisacrylamide ml 8X TrisCl, ph µl 10%(w/v) SDS ml H 2 O -dash of ammonium persulfate -5 µl TEMED The stacking gel consists of: ml 30% acrylamide/0.8% bisacrylamide ml 4X TrisCl, ph µl 10%(w/v) SDS -3 ml H 2 O -dash of ammonium persulfate -5 µl TEMED Another modification suggested to me by Dr. Caperelli is to run the gel w/ 2X SDS electrophoresis buffer (instead of 1X). This soln is made from a 5X stock soln whose recipe is in the Rance lab green notebook. During the expression and heat extraction, you can collect as many samples for gel analysis as you find useful. At a minimum, you should have: 108

115 -the pre-induction samples collected from the culture right before the induction with IPTG -the post-induction samples collected before harvesting the cells -a sample from the pellet following the heat extraction -a sample from the S/N following the heat extraction Prepare the pre- and post-induction samples in the following manner: Resuspend the cell pellets in 50 µl TE buffer. Sonicate using the following settings. (These settings pertain to the Heat Systems- Ultrasonic, Inc. W-225.) -output level: 7 -duty cycle: 50% -Pulse 5X. -Add 50 µl 2X SDS sample buffer. Again this recipe is common and is found in the green lab notebook. -If the samples are not suspended after vortexing a little bit, it may be necessary to go back and sonicate the samples with a few additional pulses. Sonicate each of the samples for the same number of total pulses. Prepare the sample from the heat extraction pellet in the following manner: Dip a 20 µl pipette tip in the thawed pellet. Smear the pellet in a microcentrifuge tube. Resuspend in 100 µl TE buffer. Add 100 µl 2X SDS sample buffer. Vortex. 109

116 Prepare the sample from the heat extraction S/N in the following manner: Add 15 µl S/N to a microcentrifuge tube. Add 15 µl 2X SDS sample buffer. Before loading any of these samples on the gel, boil them at 95 C for 5 min in the heat block. Load the samples immediately after boiling them. Here s how much to load in each case: For the pre- and post-induction samples, divide 40 by the A 600 reading for that cell pellet. Load that many microliters. For the sample from the heat extraction pellet, load 10 µl. For the sample from the heat extraction S/N, load 30 µl. I prefer to stain the gel w/ PhastGel TM Blue R from Amersham. N.B. Up until this point, it s been important to perform the steps across sequential days. The rest of the procedure does not need to be performed in sequential days. Run a DEAE Sephacel (anion exchange) column. Make 1L of a PIPES buffer g PIPES g CaCl 2 2H 2 O -Bring up to ~900 ml in H 2 O. -Adjust ph to 6.5. The PIPES will not go into soln until ml 5M NaOH have been added. -Bring up to 1L w/ H 2 O. 110

117 The column should be prepared and run in either a deli case or a cold room. First you need to pack the column. You want to use 20 ml of DEAE Sephacel resin. The separation is best performed in a 1.5 cm diameter Kontes column. You can determine how high to fill the column using this formula for the volume of a cylinder: V=π r 2 h h=v/( π r 2 ) =20 ml/[π (0.75 cm) 2 ] =11.3 cm Mark a height of ~11 cm on the column with a marker. Put a small cushion of H 2 O in the column. (Keep the valve closed.) Suspend the Sephacel resin in its packaging. Pour some of the suspended resin into the column. Open the valve and allow the resin to begin to settle. Repeat until the column is packed to the height that you marked on the side of the column. Next equilibrate the column by running 400 ml of the PIPES buffer over the column. (400 ml corresponds to 20 column volumes (CV).) I perform this step and those that follow using a gravity-fed siphon. You can certainly use a pump or a more elaborate system, if one is available. The siphon is likely to work best if there is no dead space in the column above the resin. For this reason, use a Kontes column that is not too long relative to the height of the resin, and fill any dead space with fluid. (If running buffer over the column, fill the dead space with buffer. If loading the 111

118 sample, fill the dead space with the sample.) The remainder of the fluid can be suspended above the column in an Erlenmeyer flask. It s a good idea to check the ph of the effluent towards the end of the equilibration in order to make sure that it matches the ph of the buffer being loaded on to the column (ph=6.5). Towards the end of the equilibration, centrifuge the sample at 8000 g for 30 min at 4 C. Again, I generally use a centrifuge in the Lingrel lab for this step. There should be just a small amount of pellet from this centrifugation step. Pipette the S/N into a graduated cylinder. Add an equal volume of the PIPES buffer. Then transfer the sample to an Erlenmyer flask. Adjust the ph of the sample to 6.5. (It may be a good idea to collect 500 µl for future gel analysis.) Prepare the gradient maker by putting 175 ml of the PIPES buffer into the inner chamber with a stir bar. To another 175 ml PIPES buffer, add g NaCl. Add this soln to the outer chamber. (Make sure that the valve separating the 2 chambers is closed until you are ready to start the gradient.) Next we want to load the sample on to the column. Allow all of the equilibration buffer to enter the column. Pipette the sample on to the column up to the top. Set up a siphon with the remaining sample. Allow the entire sample volume to enter the column. Collect the load. 112

119 Perform a wash step by running 100 ml (5 CV) of the PIPES buffer over the column. Collect the wash. Now you re ready to run the gradient. Fill the column up to the top with PIPES buffer. Set up the siphon to the gradient maker. Start running the gradient over the column. Collect 65 drops per fraction using the fraction collector. N.B. It is difficult to regulate the flow rate using a simple gravity-fed column. In general, the resolution will be improved by decreasing the flow rate, but if the flow rate is too slow it may take very long to complete this procedure. I generally perform the equilibration one day and the rest of the separation the next day. The resin can be used again if cleaned and stored properly. See the manufacturer s instructions. I haven t had much luck in cleaning the Sephacel resin. Cleaning the resin involves washing it with high salt soln that seem to dehydrate the resin, making it difficult to resuspend. If you aren t able to resuspend the resin (i.e. if it s clumpy), then I would recommend using new resin the next time. From the Sephacel column, you should have collected : -a load -an intial wash step -~80 fractions -possibly a clean step, if you attempted to clean the resin. I generally run a BCA assay on the load, wash, and every fourth fraction. 113

120 Using the BCA results, construct a chromatogram. You can find a representative chromatogram for the Sephacel column on page 70 of my lab notebook 5. The chromatogram normally has a dominant peak in the first 20 fractions corresponding to calbindin and then small, broad peaks in the later fractions corresponding to other protein impurities. The first couple of times that you perform this separation, I would also recommend that you perform an SDS-PAGE in order to determine where calbindin elutes. In general, I have found significant amounts of calbindin in the load, wash, and first 20 fractions. I combine the load, wash, and first 20 fractions and lyophilize that entire volume. Lyophilization typically takes a couple of days for a volume of this size. Run a Sephadex (size-exclusion) column. The Sephadex resin is a dry powder when first used and needs to be brought up in H 2 O and swell for a few hours. Put 15 g of Sephadex G-25 Medium in a beaker. Add ~100 ml H 2 O. Stir gently and allow the resin to swell for 3 hr at RT. At the start of the day, it s a good idea to put a container with 1 L H 2 O in it in the deli case for use during the course of this procedure. After swelling, pipette off the H 2 O. Add a small cushion of H 2 O to a 2.5 cm diameter Kontes column. Add enough H 2 O to the resin in order to resuspend it. Pour the suspension into the column. Open the valve to initiate packing. -This column should also be performed in either the deli case or in a cold room. 114

121 Equilibrate the column by running over it 250 ml H 2 O. I ve used a simple gravity-fed system for the both the Sephacel and Sephadex separations. The sample has been lyophilized. I generally scrape it out of the lyophilizer jar using a spatula and transfer it to a 50 ml conical tube. Then add 3 ml 10 mm CaCl 2. The sample will appear foamy and slightly yellow in color. The resolution of the separation will be improved if you add as little of the CaCl 2 soln as possible. If you add too little, however, it may be difficult to cleanly load the sample on to the column. Allow the entire equilbration volume to enter the column. Close the valve. Then gently load the sample on top of the resin. Open the valve again in order to let the sample enter the column. Close the valve once the sample has entered. Wash the conical tube that contained the sample with 1 Pasteur pipette-full of H 2 O. Load that volume on to the column by washing the inner sides of the column that had contacted the sample. Then allow the H 2 O to enter the column. Repeat this wash step. Elute the column with 225 ml (3 CV) of H 2 O. Collect 30 drops per fraction using the fraction collector. -I generally try to be more careful in controlling the flow rate during the Sephadex separation than I might have have been with the Sephacel separation. I try to keep the flow rate at ~15-20 drops per minute at the start of the separation. After the first 50 fractions have been collected, I might increase the flow rate to ~40 drops per minute. Make a chromatogram by measuring the A 280 of every fourth fraction. A representative chromatogram is found on page 30 of notebook 6. There are generally 2 dominant peaks. The first peak contains calbindin. The second peak contains mostly small molecule 115

122 impurities. The first couple of times that you perform this procedure, you should confirm this result with a BCA assay and by SDS-PAGE. From the results of the chromatogram, decide which fractions contain calbindin and pool them together. I generally combine fractions ~ I then dialyze the pooled fractions against 4L H 2 O O/N at 4 C. Following dialysis, I run an SDS-PAGE on the sample in order to determine if significant impurities are present. The sample will never be completely clean, but the SDS-PAGE should reveal a large, heavy band corresponding to calbindin and a few, very light bands at higher molecular weights. It s necessary to try and estimate the yield from this protocol. First, I estimate the concentration of my sample from the A 277. Mikael Akke indicates that the extinction coefficient for calcium-loaded calbindin is 1681 M cm -1. Next I perform a BCA assay. The BCA method generally provides a higher concentration value than the A 277 measurement. On the basis of these 2 methods, I estimate how many NMR samples I might have. I generally try and shoot for 4 mm NMR samples. I used to get about 2 samples from this protocol. I now typically get 5. Distribute the sample into as many conical tubes as you might have samples. Before this transfer step, weigh each of the tubes. Then lyophilize the samples. Following lyophilization, weigh the tubes again on the balance, which provides a third yield estimate. Simple weighing of the samples generally provides the largest values. Store the lyophilized samples in the freezer until ready for NMR. Preparation of Ca 2+ -loaded NMR samples 116

123 In most cases, I have found that the above protocol leaves the protein predominantly in the Ca 2+ -loaded state. If you intend to use the same sample in order to perform experiments on both the Ca 2+ -loaded and apo forms of calbindin, then I recommend first completing work with the Ca 2+ -loaded protein before proceeding on with any apo experiments. The Ca 2+ -loaded state is easier to prepare and is more stable. You will also likely lose some protein in the process of decalcification. I generally try to work at a calbindin concentration of 4 mm, although the concentration can be lower for certain applications. In estimating the protein concentration for NMR sample preparation, I rely upon the mass of the lyophilized protein on a balance. In making the Ca 2+ -loaded NMR samples, I add to the lyophilized protein: -10 mm imidazole-d 4-10 mm CaCl 2 (or 2.5 molar equivalents relative to the protein concentration) -1 µl of the lab stock NaN 3 soln -1 mm DSS The amount of D 2 O added to the sample will likely differ depending on the application. ph the sample to 7.0. Preparation of apo NMR samples This part of the protocol works, but you may be able to optimize it further. I originally suffered very large protein losses when decalcifying my samples. I now obtain anywhere from 60-80% yields for this procedure. Transfer the sample from the NMR tube to a microcentrifuge tube. If the Ca 2+ -loaded sample has been used for a longer period of time, it likely contains some precipitate at the bottom of the NMR tube. Try not to transfer this precipitate to the microcentrifuge tube. 117

124 Estimate the total number of moles of protein plus calcium in your sample. Then add a 20-fold molar excess EGTA. Adjust the ph of the sample to 8.0. I would recommend making the EGTA stock soln at ph 8.0 as well. EGTA is not soluble at lower ph. Transfer the sample to a clean NMR tube. Obtain a 1D 1 H spectrum. The upfield methyl region has a characteristic appearance for the Ca 2+ -loaded and apo protein. The most upfield resonance in the Ca 2+ -loaded spectrum is a triplet corresponding to Ile 73. For the apo-state, the most upfield resonance is a doublet belonging to Val 70. Note that the apo spectra will differ in appearance depending upon whether or not EGTA is present. The position of the most upfield doublet, however, is consistent. It s a good idea to collect reference 1D 1 H spectra for this region in both states. If the apo state is identified and there are no signs of the Ca 2+ -loaded state, then it is time to dialyze out the EGTA/Ca 2+ complex and any excess EGTA. I generally transfer the sample to a microdialyzer. If the sample volume exceeds ~700 µl, I would recommend distributing the sample across 2 microdialyzers. Dialyze against: -4L 0.5 mm EGTA O/N at 4 C -then against 4L 0.05 mm EGTA over the course of a work day at 4 C -then against 4L mm EGTA O/N at 4 C -and finally against 4L Chelex-treated H 2 O. The Chelex needs to be washed for a couple of days by running water over it until the effluent is neutral. The effluent will initially be basic. 118

125 It is necessary to prepare the container for the Chelex-treated H 2 O by first washing it with 50 ml 5mM EGTA and then ~2 L of the Chelex-treated H 2 O. Following these wash steps, collect an additional 4 L of the Chelex-treated H 2 O for the last dialysis step. Following the dialysis of the sample, use a syringe needle to puncture the membrane on the microdialyzer. Draw up the sample into the syringe and then transfer the sample to a conical tube. Before introducing the sample into the conical tube, weigh the tube. Then wash it w/ 2 ml 5mM EGTA, followed by several washes w/ Chelex-treated H 2 O. Lyophilize the apo sample. The apo NMR sample contains the same components as the Ca 2+ -loaded sample described above, except that CaCl 2 is not added. In order to remove any residual Ca 2+ from these components, I recommend that you prepare a stock apo NMR buffer and run it over the Chelex column. 119

126 Appendix 6: Tips to aid in future data analyses The spectral density mapping and model-free calculations were both performed within the Mathematica software. It should be possible to modify the existing Mathematica notebooks in order to perform similar calculations with future data sets. This appendix is intended to provide some hints for how this task might be best accomplished. It is highly recommended that anyone who aims to perform these calculations also consult references 94 and 95, which are both excellent. Spectral density mapping A good example of these calculations is found in the file 'spec_dens_ca_500_600_ nb'. The rates and rate errors are entered at the start of the notebook using the variables ca500resonancevsratevsrateerror and ca600resonancevsratevsrateerror. Modify these variables in order to reflect the current data set. The main result of the calculation is assigned to the variable caresonancevsspecdensalltenrates. In order to obtain this result, it is necessary to first construct the matrix caprefactormatrixalltenrates, which is an analogue of eq. 1 from Skrynnikov et al. 30 It is important to note that this matrix will inevitably be a rectangular matrix. In order to invert a rectangular matrix, the appropriate Mathematica function is PseudoInverse, not Inverse. The PseudoInverse function implements the singular value decomposition method mentioned in the Skrynnikov et al. reference. After the spectral density values are obtained, it is very useful to back-calculate the rates. This step involves the evaluation of a dot product involving the variables caprefactormatrixnoterrorweighted and caresonancevsspecdensalltenrates. The result is assigned to the variable caresonancevsrateexpvsratefit. 120

127 Model-free calculations A good example of the model-free calculations is found in the file model_free_ca_500_600_ nb. This notebook implements a least-squares minimization procedure. The method employed is a basic grid-search that includes Powell s quadratically convergent method described in Chapter 10 of reference 94. The first step is again to enter the rates and rate errors using the variables ca500resonancevsratevsrateerror and ca600resonancevsratevsrateerror. The variable sse (sum of squared errors) is minimized by optimizing the values of the model-free parameters, represented by the variable currentparam. Note that these calculations are sensitive to the starting values of the model-free parameters. This problem results from the presence of local minima on the χ 2 surface, which are commonly encountered in least-squares minimization procedures. It requires that the calculations be performed multiple times with different starting values of the variable currentparam. Each time the calculation is performed, the result is assigned to variables of the form summary1ls2, summary2ls2, etc. Following the completion of multiple runs, the results are searched for the lowest overall sse values. This search needs to be performed independently for each resonance. (i.e. Do not expect the same starting value of currentparam to produce the minimum sse values for all resonances.) The LS3 calculations are then performed in much the same manner. The primary difference encountered in the LS3 calculations is that the variable currentparam is modified to include the additional parameter and Powell s modification of the grid search is not included. Although Powell s modification has been found to increase the speed of the LS2 calculations, it does not appear to provide a benefit in the LS3 calculations. Once the 121

128 LS3 results have been obtained, the F-statistics for the two models and p-values are calculated within the Mathematica package Statistics`HypothesisTests`. At the end of the notebook, the model-free results are plotted along with the spectral density values. This step provides a graphical test of the efficacy of the least-squares minimization procedure. Monte Carlo simulation of errors Both the spectral density mapping and model-free calculations include Monte Carlo simulations to provide error estimates for the fitted parameters. 96 Examples of these calculations are provided in the files 'spec_dens_ca_500_600_ nb' and model_free_error_ca_500_600_ nb. The basic Monte Carlo procedure is outlined below. 1) Solve for the fitted parameters using the experimental rates and rate errors. (In this case, the fitted parameters refer to the spectral density values or model-free parameters.) 2) Report the result from step 1 as the best-fit values. 3) Construct normal distributions for each of the experimental rates. The mean value of each distribution is the back-calculated rate from the fitted parameters, while the standard deviation in the distribution corresponds to the error in the experimental rate. 4) Construct a synthetic data set by randomly selecting rates from each of these distributions. In Mathematica, this step is accomplished with the package Statistics`ContinuousDistributions`. 5) Solve for the fitted parameters using the synthetic data set. The best-fit values will vary slightly from those obtained in step

129 6) Repeat steps 4 and 5 some large number of times. We have found that 500 simulations are adequate for our purposes. (Perform the procedure with increasing numbers of simulations in order to determine the point at which the error estimates have stabilized.) 7) For each of the parameters, we then have 500 fitted values from the synthetic data sets. Calculate the standard deviation in each distribution of values and report it as that parameter s error. 123

130 Appendix 7: Side-chain amide 2 H relaxation rates Side-chain amide groups play an important role in Ca 2+ binding by calbindin D 9k. Calbindin has two Asn and four Gln residues, constituting a total of six side-chain amide groups. Two of these six groups (N21 and Q22) are located at site I, while one group (N56) is located at site II. Side-chain amide groups are also important mediators of binding specificity among DNA-binding proteins. Consequently, amide groups are important targets for future side-chain dynamics studies. Pervushin et al. have demonstrated that the 2 H relaxation experiments developed by Kay and coworkers for the study of side-chain methyl dynamics can be modified to measure relaxation rates in 15 NHD groups. 105 These experiments require uniformly 15 N- labeled protein that is dissolved in a buffer containing ~50% H 2 O/50% D 2 O. Under these conditions, a side-chain amide group exists in one of four possible states: 1) fully protonated (NH 2 ), 2) fully deuterated (ND 2 ), 3) deuterated at the Z position (NH E D Z ) or 4) deuterated at the E position (ND E H Z ). The pulse sequences used in these experiments are designed so that both the singly deuterated species are observed in the resulting spectra, while the fully protonated and fully deuterated species are filtered out. Longitudinal and transverse relaxation rates for calbindin s side-chain amide groups were measured before the start of the methyl experiments. These experiments were originally conducted in order to optimize the sample and spectrometer conditions for the methyl experiments. The results, however, indicate that future investigations of the amide dynamics are merited. Figure 13 indicates that the longitudinal relaxation rates undergo several changes with Ca 2+ binding. For example, the change at residue Q67 is particularly dramatic. The transverse relaxation rates in Figure 14 exhibit even larger 124

131 Figure 13: R(D z N z ) rate measurements at 500 MHz 125