C h a p t e r
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General Introduction: Protein dynamics by NMR
Chapter 1
Protein dynamics Dynamic processes in proteins cover a large timescale regime (for review see Gurd and Rothgeb, 1979; Frauenfelder and Gratton, 1986), ranging from very fast fluctuations of the individual atoms on the picosecond timescale, loop and domain motions on the nanosecond timescale, conformational rearrangements on the millisecond timescale to breathing modes on a timescale slower than seconds (for recent review see Kern and Zuiderweg, 2003). These motions are very important since functional processes, such as ligand binding, enzyme catalysis, and molecular recognition and signal transduction processes, often require a certain level of structural plasticity and flexibility. In addition, molecular motions can contribute to protein stability through the increases of the entropy of the system. While a number of experimental techniques are available for the detailed characterization of molecular motions, NMR spectroscopy is unique in yielding site-specific information on multiple timescales (for review see Palmer, 1997; Kay, 1998). In NMR spectroscopy motions are reflected in the averaging of NMR parameters, such as chemical shifts, scalar and residual dipolar couplings, as well as in relaxation of the nuclear spins (Figure 1). In particular, NMR relaxation rates are sensitive to internal motions on the ps-ns time-scale (for review see Daragan and Mayo, 1997); slower internal motions (ns-μs) are obscured by the global rotational reorientation of the protein in solution and are therefore undetectable. However, due to a modulation of the chemical shift, μs-ms exchange processes are detectable and result in an increased contribution to the effective transverse relaxation rate (for review see Akke, 2002; Wang and Palmer, 2003). �������������������������� ����������� ����������������
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Figure 1. Overview of protein dynamics on different time-scales. On top different NMR parameters are indicated with their corresponding time-scales.
This thesis will focus on the development and application of NMR relaxation methods for the study of protein internal dynamics and this first chapter will provide a basal introduction 8
General Introduction: Protein dynamics by NMR
in NMR relaxation theory (for review see Fischer et al., 1998; Luginbühl and Wüthrich, 2002), as well as an overview of NMR methods to measure protein dynamics (for recent review see Atkinson and Kieffer, 2004). In the first part of this chapter a brief introduction in the semi-classical theory of nuclear spin relaxation is given. Whereas the investigation of protein backbone motions from 15N relaxation rates as well as 13C relaxation rates is discussed in the second part of this chapter, the third part describes recent advances in utilizing either 13 C or 2H nuclei to study side chain dynamics. This is followed by a short description about the interpretation of NMR derived motional parameters in terms of thermodynamics. The last part will summarize the subsequent chapters in this thesis.
NMR relaxation Nuclear relaxation is the process by which an excited spin system returns to equilibrium and is a consequence of interactions between the nuclear spins and the rest of the system, referred to as the lattice. Classically the lattice can be represented by random molecular motions. Hence spin relaxation in solution can be treated as arising from spin interactions, where the quantum nature of the spins is taken into account, modulated by random rotational and translational motions of the molecules in which the spins are embedded. This forms the basis of the semi-classical relaxation theory as formulated by for instance Wangsness, Bloch (1953) and Redfield (1965). The random processes that represent the lattice can be characterized by spectral density functions, which describe the available energy at a certain frequency of the random rate processes that induce relaxation. Only random processes that have a frequency corresponding to an energy separation between levels of the spin-system can induce transitions and will lead to relaxation. In this description the lattice is considered to be an infinite thermostat, where the equilibrium state of the lattice is not affected by the exchange of energy with the spin system. Semi-classical relaxation theory The state of a spin system at a certain time can be described by the density matrix σ(t). The Liouville-von Neuman equation gives the time evolution of the density matrix: d (1.1) σ (t) = −i[H 0 (t),σ (t)] dt where the properties of the spin system are described by the Hamiltonian H(t):
H (t) = H 0 + H 1 (t)
(1.2)
with H0 the time independent part and H1(t) the time dependent random fluctuating part, the so called spin-lattice coupling. The effect of the spin-lattice coupling can be isolated through the use of an interaction representation (i.e. using a rotating frame when H0 is purely Zeeman), in which each operator Q will be replaced by: Q˜ = e iH 0 t Q(t)e−iH 0 t
(1.3)
9
Chapter 1
where the tilde indicates the interaction representation. Transformation of the Liouvillevon Neuman equation to the interaction representation and integration by successive approximations up to the second order (see for example Abragam, 1961; Goldman, 1988), gives the master equation for the density matrix in the interaction representation: t d σ˜ (t) = − ∫ H˜ 1 (t), H˜ 1 (t'),(σ˜ (t) − σ˜ eq ) dt' dt 0
[
]]
[
(1.4)
where σ˜ eq is the thermal equilibrium form of the density matrix and the overbars denote the ensemble average. The time-dependent Hamiltonian can be separated into spin operators Aq acting on the spin system only and spatial functions Fq(t) that describe the random fluctuations of the lattice:
H 1(t) = ∑ Aq Fq (t) = ∑ Aq† Fq* (t) q
(1.5)
q
where the index q defines the rank of the tensors as well as the different types of interactions. In the interaction representation the spin operators rotate with a characteristic frequency ωq: iω t A˜ q = e iH 0 t Aq e−iH 0 t = e q Aq
(1.6)
The spatial functions Fq(t) are assumed to be statistically stationary, meaning that Fq (t) is time-independent, and products of the spatial functions can be written in terms of correlation functions: Fq (t)Fq'* (t') = Gqq' ( t − t' )
(1.7)
where the correlation functions are such that Gqqʹ (τ) = Gqqʹ (- τ) and it is assumed that they decay with a characteristic correlation time τc such that Gqqʹ (τ) ≈ Gqqʹ (0) for τ < τc and Gqqʹ (τ) ≈ 0 for τ >> τc. Replacement of t-tʹ by τ and assuming that the correlation function decays much faster than the evolution time of the density matrix (τc pB), the Swift-Connick relationship gives the relaxation rate for a SQ coherence (Swift and Connick, 1962):
19
Chapter 1
⎡ R 0 R 0 + p k + Δω 2 ⎤ 2B ( 2B A ex ) 0 ⎥ R2 = pA R2A + pA pB k ex ⎢ ⎢ ( R 0 + p k ) 2 + Δω 2 ⎥ 2B A ex ⎣ ⎦
(1.26)
Three different chemical shift time-scales can be defined based on the relative magnitudes of kex and Δω : Slow exchange (I)
kex < Δω
0≤α Δω
1> 1 ) the cross-relaxation rate is dominated by J(0) and σCC becomes simply: σ CC = −d ⋅ J(0)
(1.32)
When the simple form of the Lipari-Szabo spectral density function (model 1; Table 2) is used, J(0) can be further simplified to be equal to S2τc and shows that the cross-relaxation rate between two carbon nuclei is to a good approximation directly proportional to the order parameter of internal motion.
From protein dynamics to thermodynamics In the last part of this chapter I would like to focus on the correlation between dynamical parameters derived from NMR relaxation rates and thermodynamic parameters. In many cases, changes in protein flexibility that occur upon protein, DNA, ligand or metal binding can be interpreted in terms of changes in conformational entropy (for review see Spyracopoulos and Sykes, 2001; Stone, 2001). Backbone or side chain flexibility can either decrease or 24
General Introduction: Protein dynamics by NMR
increase upon binding. Decreases are often associated with ‘enthalpy-entropy compensation’ and ‘induced fit’, whereas increased flexibility leads to an entropic stabilization of the complex. In order to interpret NMR derived order parameters in terms of entropy, a proper model that describes the motional behavior needs to be chosen. A motional model that is often used is referred to as ‘diffusion-in-a-cone’, which corresponds to diffusion in a square well potential (Akke et al., 1993; Li et al., 1996; Yang and Kay, 1996b). Using this model the change in conformational entropy (ΔSconf) can be calculated from the order parameters in free (F) and bound (B) states: ⎡ 3 − 1+ 8S ⎤ F,i ⎥ ΔSconf = −k ∑ ln⎢ 3 − 1+ 8S ⎢ ⎣ ⎦ B,i ⎥ i
(1.33)
where k is Boltzmann’s constant and Si are the square roots of the model-free order parameters S2. From the temperature dependence of the order parameter, the heat capacity contribution from ps-ns motions can be estimated: dS (T) C p,conf = conf (1.34) d(lnT) This approach to calculate the induced change in conformational entropy has still several limitations (Cavanagh and Akke, 2000; Wand, 2001): (i) the relative simple physical model used to describe the motional behavior is limited in accurately describing the rich and varied dynamics in proteins; (ii) the order parameter is insensitive to motions slower than overall rotational diffusion; (iii) the order parameter is only sensitive to motions that reorient the bond vector involved; (iv) possible correlations between motions of different bond vectors are not taken into account; and (v) only a subset of bond vectors are commonly characterized (i.e. only backbone amide bond vectors or side chain methyl axes). However, mainly the fast motions contribute to conformational entropy and even the slowest vibrational modes of proteins tend to fall within the ps-ns time window, which implies that limitation (ii) may not be severe. Additionally, the errors introduced by (iii) and (iv) tend to cancel one another to some extent. Furthermore, the agreement between entropy contributions estimated based on calorimetric measurements and NMR derived conformational entropy values, supports the validity of the aforementioned approach (Bracken et al., 1999; Zidek et al., 1999; Lee et al., 2000). A number of groups have studied changes in NMR order parameters of backbone as well as side chain bond vectors upon complex formation and related them to changes in conformational entropy. For illustration a few examples will be given below. Zidek et al. (1999) demonstrated the increase of protein flexibility upon target binding. This was reflected in a general decrease of backbone NH order parameters after binding of a small hydrophobic ligand to mouse major urinary protein. In contrast, a decrease in protein conformational entropy was observed upon binding of the dinucleotide inhibitor pTppAp to ribonuclease A (Kovrigin et al., 2003). A study of peptide binding to calcium-loaded calmodulin revealed that while backbone NH order parameters were essentially unaltered, the motions of several side chains were dramatically affected by peptide binding (Lee et al., 2000). Significant changes of methyl axis order parameters were shown to occur throughout 25
Chapter 1
the whole structure, indicating a loss as well as redistribution of side chain entropy upon peptide binding. Interestingly, the temperature dependence of methyl order parameters in the calmodulin-peptide complex revealed three classes of motions (Lee et al., 2002) and extrapolation to lower temperatures predicted the so called glass-transitions in proteins (Lee and Wand, 2001). Earlier studies of phosphopeptide binding to SH2-domains revealed mainly increases in methyl axis order parameters for methyl groups located in the peptide binding interface (Kay et al., 1998), where the observed changes were significantly smaller than the changes observed for calmodulin. These distinct changes in protein dynamics upon ligand binding highlight the great interest in more studies of backbone and side-chain motions and their changes upon interaction with ligands to provide a more coherent view of the functional importance of protein motions.
Overview of this thesis In the first chapter of this thesis I have highlighted the importance of protein dynamics and have given an overview of NMR relaxation theory and methods that can be used to probe these dynamics. In Chapter 2 we have studied the structure and backbone dynamics of the ubiquitin conjugating enzyme UbcH5B, where the observed dynamics could partly be related to the protein function. In the next chapter 15N relaxation rates of two C4C4 RING domains that each bind two zinc atoms have been measured in order to relate the protein backbone dynamics to observed differences in metal exchange kinetics. The fourth chapter relates observed protein backbone dynamics of a serine protease in different solvents to protein stability. In the last chapter a new method to measure side chain dynamics is discussed.
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