Solid State NMR Investigation of Protein-Based Biomaterials Resilin: An Extremely Efficient Elastomeric Protein
by
Andrew L. C. Reddin
B.Sc., Dalhousie University, 2008
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
in
The Faculty of Graduate Studies
(Physics)
THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April, 2012 © Andrew L. C. Reddin 2012 Abstract
Solid state nuclear magnetic resonance experiments were performed in order to investigate the microscopic properties of three resilin/resilin-like proteins: An16, rec1-resilin, and nat- ural resilin in dragonfly tendons. Three different types of experiments were performed: measurements of chemical shifts in 13C spectra, measurements of residual quadrupole cou- plings in deuterated water absorbed in the samples, and measurements of proton residual dipole couplings based on the buildup of multiple quantum coherences. The results suggest that the molecular chains in the materials tested are primarily randomly coiled and lacking in regular structure, and are able to easily change between many transient conformations. These conformations can vary significantly in terms of their structural characteristics, re- sulting in a broad distribution of localized dynamics. When stretched, An16 showed a slightly increased tendency to adopt β-sheet secondary structure. The natural resilin also exhibited slightly more rigid structure than the other materials, which may be related to greater efficiency in the natural crosslinking process.
ii Table of Contents
Abstract ...... ii
Table of Contents ...... iii
List of Tables ...... vi
List of Figures ...... vii
Acknowledgements ...... x
1 Introduction ...... 1
2 NMR Background ...... 3 2.1 Larmor Precession...... 3 2.1.1 Classical Picture...... 3 2.1.2 Quantum Mechanical Picture...... 4 2.2 Bulk Magnetization...... 6 2.2.1 Thermal Equilibrium Magnetization...... 7 2.3 Density Matrix Formalism...... 8 2.3.1 Thermal Equilibrium Density Matrix...... 9 2.3.2 Time Dependence of the Density Matrix...... 10 2.4 The Rotating Frame...... 10 2.4.1 Zeeman Truncation...... 11 2.5 Radio-Frequency Pulses...... 12 2.5.1 Pulse Propagators and Rotation...... 13 2.6 Relaxation...... 14 2.6.1 The Nuclear Overhauser Effect...... 15 2.7 NMR Spectroscopy...... 18 2.7.1 Fourier Transform Spectroscopy...... 19 2.7.2 2D spectroscopy...... 21 2.8 NMR interactions...... 23 2.8.1 Chemical Shielding...... 23
iii 2.8.2 Dipolar Interaction...... 25 2.8.3 Quadrupolar Interaction...... 27 2.8.4 Residual Couplings...... 31 2.9 Average Hamiltonian Theory...... 34 2.9.1 The Toggling Frame...... 35 2.10 Multiple Quantum Coherence...... 36 2.10.1 Excitation of Multiple Quantum Coherence...... 38 2.10.2 MQ Coherence in Extended Spin Systems...... 39 2.10.3 Detection of Multiple Quantum Coherence...... 39
3 Materials Background ...... 42 3.1 Proteins...... 42 3.1.1 Secondary Structure...... 44 3.1.2 Higher Order Structure...... 47 3.2 Elastomers...... 47 3.3 Resilin...... 50 3.3.1 Natural Occurrence and Function...... 51 3.3.2 Mechanical Properties...... 52 3.3.3 Molecular and Structural Properties...... 54 3.3.4 Amino Acid Composition and Sequence...... 57 3.4 Recombinant Resilin-Like Proteins...... 59 3.4.1 Production and Sequence...... 59 3.4.2 Stability and Purification...... 60 3.4.3 Crosslinking...... 61 3.4.4 Secondary Structure...... 62 3.4.5 Mechanical Properties...... 64 3.4.6 Applications of Resilin-like Polypeptides...... 65
4 Experimental Methods ...... 68 4.1 Materials Used...... 68 4.2 NMR Setup...... 69 4.3 Stretching Setup...... 70 4.4 Carbon-13 Spectroscopy...... 70 4.4.1 Pulse Sequence...... 71 4.5 Quadrupole Coupling Measurements...... 73 4.5.1 Pulse Sequence...... 74 4.5.2 Trial Materials...... 75 4.6 Multiple-Quantum Experiments...... 76
iv 4.6.1 Spectrometer Tuning...... 77 4.6.2 Pulse Sequence...... 79 4.6.3 Buildup Curves...... 81 4.6.4 Pre-selection Techniques...... 83 4.6.5 Tikhonov Regularization...... 86 4.6.6 Analytical Distribution Functions...... 87 4.6.7 PDMS Trial Experiments...... 87
5 Results & Discussion ...... 91 5.1 Carbon-13 Chemical Shifts...... 91 5.2 Residual Quadrupole Couplings...... 96 5.3 Residual Dipole Couplings...... 99 5.3.1 Pre-selection...... 99 5.3.2 An16...... 105 5.3.3 Rec1-resilin...... 109 5.3.4 Dragonfly Tendons...... 113 5.3.5 Discussion...... 117
6 Conclusion ...... 126
Bibliography ...... 129
v List of Tables
3.1 The twenty-one standard eukaryotic amino acids, including their structure and the polarity and pH of their side-chains...... 43 3.2 Mechanical properties of various elastic proteins...... 54 3.3 The (previously reported) mechanical properties of natural dragonfly resilin and recombinant resilin-like proteins: rec1-resilin, An16, Dros16, Cf-resB, and Hi-resB...... 64
4.1 Summary of the coherence orders (and their phases) generated in the two types of MQ experiment (reference and MQ-filtered)...... 81 4.2 Analytical distributions functions P (D) used for least-squares fitting of MQ buildup curves using numerical integration...... 88 4.3 Parameters obtained from various types of fits to the experimental PDMS buildup curve...... 89
5.1 Parameters for the α/β-carbon chemical shift differences for the different residues...... 94 5.2 The effect of the different pre-selection techniques on the total intensity of the different samples...... 103 5.3 Parameters obtained for An16 with a DQ pre-selection time of 2.5 ms.... 106 5.4 Parameters obtained for An16 with a DQ pre-selection time of 5.0 ms.... 109 5.5 Parameters obtained for rec1-resilin with a DQ pre-selection time of 2.5 ms. 109 5.6 Parameters obtained for rec1-resilin with a DQ pre-selection time of 5.0 ms. 112 5.7 Parameters obtained for the tendons with a DQ pre-selection time of 2.5 ms. 114 5.8 Parameters obtained for the tendons with a DQ pre-selection time of 5.0 ms. 116 5.9 The mean (D¯), mean deviation (∆D), and the plateau level (C) of the dis- tribution functions obtained for the different samples and pre-selection times. 120
vi List of Figures
2.1 A diagram of the energy levels and transitions for a system of spin-1/2 nuclei. 16 2.2 An illustration of a simple pulse sequence, with one ninety-degree pulse fol- lowed by acquisition of the free induction decay (FID)...... 19 2.3 Absorptive and dispersive line-shapes corresponding to the real and imagi- nary parts of a single Lorentzian...... 21 2.4 Illustration of a typical pulse sequence used for a two-dimensional experiment. 22 2.5 Illustration of the chemical shift powder spectrum for several values of η.. 25 2.6 The dipolar powder pattern, also known as a Pake pattern...... 27
2.7 Illustration of the quadrupolar powder spectrum for several values of ηQ... 31 2.8 Illustration of a pulse sequence used to produce and detect MQ coherence.. 40
3.1 An illustration of the formation of a peptide bond to link together two polypeptides (or amino acids) into one larger polypeptide...... 44 3.2 Molecular representations of α-helix and β-sheet secondary structures.... 46 3.3 An illustration of the difference in entropy between relaxed and extended states for a model polymer on a square lattice...... 48 3.4 Structural representation of a di-tyrosine crosslink between two polypeptides and some reactions used to produce di-tyrosine crosslinks from tyrosine residues. 55
4.1 A simplified block diagram of the spectrometer setup used in the current experiments...... 69 4.2 A photograph of the stretching apparatus used to control the length of the samples during testing...... 70 4.3 The pulse sequence used to acquire 13C spectra for chemical shift analysis.. 72 4.4 The pulse sequence used to investigate residual quadrupole couplings in stretched resilin samples...... 74 4.5 Contour plot of a two-dimensional spectrum of a deuterated glycine crystal for one particular orientation...... 75 4.6 Contour plot of a two-dimensional spectrum of a rubber band soaked in deuterated benzene and stretched to 75% strain...... 77
vii 4.7 Quadrupolar splitting spectra for a rubber band soaked with deuterated ben- zene at various levels of strain...... 78 4.8 An illustration of one cycle of pulses used for excitation and reconversion of MQ coherence...... 79 4.9 Plot of the original and improved kernel functions for comparison...... 83 4.10 Illustration of the MQ pulse sequence including the DQ pre-selection and delay period...... 84 4.11 Illustration of the MQ pulse sequence including both DQ pre-selection and
T1 pre-selection sequences...... 85 4.12 Plot of the MQ buildup curve for a PDMS sample used as a trial for the MQ experiment...... 88 4.13 Residual coupling distributions obtained using various methods as described in the text. The corresponding fits to the buildup curve are shown in the inset. 90
5.1 An example of a 13C spectrum obtained for a sample of An16...... 92 5.2 Plot of the deviations between the measured and predicted chemical shift values for different types of secondary structure for the different carbons in the sample...... 93 5.3 Plots of the chemical shift difference between Cα and Cβ for various residues as a function of sample extension...... 95 5.4 Contour plot of the two-dimensional deuterium spectrum obtained for a sam- ple of An16 at 80% strain...... 97 5.5 Contour plot of the two-dimensional deuterium spectrum obtained for a sam- ple of An16 at 80% strain with the transmitter frequency shifted to a point outside the spectrum...... 98 5.6 Plot of the one dimensional 1H spectra of the three samples with no pre-
selection and after both T1 and DQ pre-selection with τDQ = 2.5 ms..... 100 5.7 Several spectra of the An16 sample showing the effects of the different types of pre-selection...... 101 5.8 Plot of the intensity of different parts of the An16 spectrum as a function of
the T1 pre-selection time...... 102 5.9 Plot of the normalized sum intensity as a function of the excitation time, for a DQ pre-selection time of 2.5 and 5.0 ms...... 104 5.10 Distributions of residual couplings and corresponding fits to the buildup curve obtained for the An16 sample with a DQ pre-selection time of 2.5 ms.... 106 5.11 Distributions of residual couplings and corresponding fits to the buildup curve obtained for the An16 sample with a DQ pre-selection time of 5.0 ms.... 107
viii 5.12 Comparison of the distributions and buildup curves for An16 with a DQ pre-selection time of 2.5 and 5.0 ms...... 108 5.13 Distributions of residual couplings and corresponding fits to the buildup curve obtained for the rec1-resilin sample with a DQ pre-selection time of 2.5 ms. 110 5.14 Distributions of residual couplings and corresponding fits to the buildup curve obtained for the rec1-resilin sample with a DQ pre-selection time of 5.0 ms. 111 5.15 Comparison of the distributions and buildup curves for rec1-resilin with a DQ pre-selection time of 2.5 and 5.0 ms...... 112 5.16 Distributions of residual couplings and corresponding fits to the buildup curve obtained for the tendon sample with a DQ pre-selection time of 2.5 ms... 114 5.17 Distributions of residual couplings and corresponding fits to the buildup curve obtained for the tendon sample with a DQ pre-selection time of 5.0 ms... 115 5.18 Comparison of the distributions and buildup curves for the tendon sample with a DQ pre-selection time of 2.5 and 5.0 ms...... 116 5.19 Comparison of the buildup curves and distributions for the three samples, with a DQ pre-selection time of 2.5 ms...... 118 5.20 Comparison of the buildup curves and distributions for the three samples, with a DQ pre-selection time of 5.0 ms...... 119 5.21 Plot illustrating the simulated effect of differential relaxation on an example buildup curve...... 122
ix Acknowledgements
Any work such as this cannot be accomplished without the help of others, and therefore much thanks is in order. First and foremost, the one who made this work possible, my supervisor Dr. Carl Michal. For your expert guidance, timely assistance, and ongoing patience, I am deeply grateful. I would like to thank my fellow labmates, who kept me sane during the long hours spent in Hennings 100. Tom, for always keeping cool; Clark, for many shared bagel lunches and random noises; and David, for providing endless entertainment during our group meetings. I would also like to thank Dr. Ron Dong, for sharing his experience and helpful advice. Thanks is also due to everyone else who helped me at various stages in my research. Especially to Dr. John Gosline, Dept. of Zoology at UBC, for providing the recombinant resilin samples. I would also like to thank Dr. Jon Nakane and Berhard Zender in the engineering physics project lab for their help in solving experimental design problems. Dr. Stefan Reinsberg also deserves a big thank you for consenting to read my thesis in the final hour. I must also thank my fellow physics graduate students; for knowing what it’s like and providing an outlet for both commiseration toward and celebration of the academic process. The greater part of my thesis was completed remotely, in various parts of the world. I owe a big thanks to my sister Lindsey, for tolerating my erratic interruptions during our travels across New Zealand. Thanks is also due to everyone at Ponsonby Backpackers for providing some much needed stability in my life. Also thanks to my good friend Uta; for your patience with my work and for providing necessary distractions. In Indonesia, thanks is owed to the staff at Warung Gramin: for keeping me fed and providing a roof over my head. David deserves another mention here, for putting up with a boring travel partner and for his sympathetic understanding of the writing process. Thanks also to Tigrou and Pleng Peng, for keeping me company during those lonely days on Gang Kidul. I would also like to mention the couchsurfing communities in Auckland, Bali, and across Java, who have become like extended family to me. I am also grateful to my various hosts, who put up with my occasional interludes of frantic work. And also to everyone else I met along the road! Whether or not you had any direct involvement in my work, you all kept me going in one way or another. Finally, to my lovely parents, thank you for helping me get to where I am, and supporting me through all my endeavours.
x Chapter 1
Introduction
Although it combines elements from the fields of physics, biology, and chemistry, the present research is probably best categorized as biophysics. Biophysics, in general terms, can be described as the study of the physics of biological systems. Through evolution, organisms are able to develop highly specialized organs and systems which are well adapted to their specific functions. By studying such systems one can not only better understand the organism itself, but also more generally seek to understand the fundamental principles affecting its design (i.e. the relationship between its structure and function). One can also seek inspiration for new devices and materials, solutions to design problems, or other inventions through biomimicry. The extreme range of diversity in biological organisms lends itself to many possible subjects for investigation and opportunities for innovation. The present research is focussed on resilin, a structural protein found in arthropods which has highly specialized mechanical properties. These include a very high elastic effi- ciency (resilience), high extensibility (stretchiness), and long fatigue lifetime (resistance to wear). Recently, natural resilin-encoding genes have been identified in various insects, mak- ing possible the creation of recombinant proteins which incorporate resilin-based polypep- tide sequences. These can be used to reproduce the properties of natural resilin, or combine those properties with other protein sequences for specific applications. The potential ap- plications of these recombinant proteins range from tissue engineering applications and artificial muscle fibres to chemical sensing and drug delivery systems. Here we attempt to better understand the microscopic properties of a natural resilin (from dragonfly tendons) and two recombinant proteins based on resilin (An16 and rec1- resilin). In doing so, we can hope to understand what specific properties are important to the mechanical properties of these materials. We also hope to determine any similarities and differences between the different materials tested. Characterizing the microscopic properties specific to resilin may also be relevant to future applications of resilin-based recombinant proteins. The primary tool of investigation used in the present experiments was nuclear magnetic resonance (NMR) spectroscopy. NMR spectroscopy is a powerful analytical technique which can be used in a wide range of applications. NMR has been used extensively to study the structure and dynamics of polymers, and in proteins has also been used extensively in
1 determining three-dimensional structures (when such permanent structures exist). NMR spectroscopy of samples in the solid state is complicated by the effects of orientational dependence and relaxation of magnetization, however these can also be used to provide valuable information about the materials being studied. The depth of different techniques available using NMR makes it well suited to study the microscopic structure and localized dynamics of a material like resilin. The ability to study the properties of materials using different experimental approaches can help to alleviate the problems of ambiguity which can occur when studying systems such as these, which have relatively complex microscopic structure. Following this introduction, the next two chapters present a broad range of more de- tailed background information. The first deals primarily in the physics and underlying principles of NMR spectroscopy, especially those relevant to the experiments performed here. The next provides an overview of some relevant topics in soft condensed matter and biology, and also summarizes past research on resilin. The fourth chapter provides an overview of the technical details of the experiments performed and the data analysis techniques employed. Following that is a detailed description of the results obtained along with some interpretation, and then finally a conclusion containing a brief summary of the most important results and their implications.
2 Chapter 2
NMR Background
Nuclear Magnetic Resonance (NMR) spectroscopy is based on the study of the precession of nuclear magnetic moments in the presence of a magnetic field. The frequency of this precession depends on the type of nucleus and the strength of the field. The subject of NMR is rich in theoretical background and practical applications; here we will provide an overview of the key concepts relevant to this research.
2.1 Larmor Precession
Larmor precession is the term used for the precession of a magnetic moment in the presence of a magnetic field, which is the fundamental phenomenon studied in NMR spectroscopy. The details of Larmor precession will first be derived for a classical magnetic moment, and then for a fully quantum mechanical spin. The results of these two approaches are equivalent for simple systems, and the classical picture provides an intuitive understanding of the more complex quantum mechanical description.
2.1.1 Classical Picture
The basic interactions involved in magnetic resonance can be treated using a classical vector picture. The magnetic moment of a nucleus, denoted µ, is the product of its magnetogyric ratio γ (also called the gyromagnetic ratio)[1] and its spin I:[2]
µ = γI. (2.1.1)
The gyromagnetic ratio for a nucleus is given by[3]
ge γ = , (2.1.2) 2mp where e is the elementary charge (charge of the proton), mp is the proton mass, and g is the so-called g-factor, a dimensionless quantity which depends on the type of nucleus.
3 In the presence of a magnetic field B, a magnetic moment µ will experience a torque[4]
τ = µ × B. (2.1.3)
By definition, the torque causes a change in the angular momentum of the nucleus:
d τ = J = µ × B. (2.1.4) dt
The precession occurs because the angular momentum J is the same as the nuclear spin I. Thus, combining equations 2.1.1 and 2.1.4 gives
d µ = γµ × B. (2.1.5) dt
Choosing B to be along the z-axis, i.e. B = (0, 0,B0) , gives the solution
µ (t) = (µx,0 cos (γB0t) , −µx,0 sin (γB0t) , µz,0) , (2.1.6) where the initial magnetic moment is chosen to be in the x-z plane, i.e. µ (0) = (µx,0, 0, µz,0). This equation of motion describes precession of the magnetic moment around the magnetic
field with frequency ω0 = −γB0, where the negative sign indicates clockwise precession. This can be written in vector form as
ω0 = −γB0; (2.1.7) the direction of the vector ω0 indicates the axis of precession, which is aligned with the external magnetic field.
2.1.2 Quantum Mechanical Picture
For an isolated spin in a magnetic field B, the Hamiltonian is given by[2]
Hˆ = −µˆ · B, (2.1.8) which is commonly referred to as the Zeeman Hamiltonian. Similarly to the classical case above, the nuclear magnetic moment operator µˆ is given by
µˆ = γ~Iˆ. (2.1.9)
4 Note that the factor of ~ has been removed from the spin operator Iˆ, according to the convention commonly adopted in NMR literature.[2] Taking B along the z-axis gives1
ˆ ˆ H = −γ~B0Iz. (2.1.10)
The time evolution of the spin state can be derived using the time-dependent Schrödinger equation:[3] ∂ Hˆ |ψi = i |ψi . (2.1.11) ~∂t For a time-independent Hamiltonian the solution is of the form
ˆ |ψ (t)i = e−iHt/~ |ψ (0)i , (2.1.12) where the operator e−iHt/ˆ ~ is called the propagator and is denoted Uˆ (t). Inserting the Zeeman Hamiltonian from equation 2.1.10 gives
ˆ ˆ |ψ (t)i = eiγB0tIz |ψ (0)i = e−iω0tIz |ψ (0)i , (2.1.13) where we make the identification ω0 = −γB0 as above. This can be shown to correspond to precession by examining the time dependence of the expectation value of Iˆ:
D E Iˆ ≡ hψ (t)| Iˆ |ψ (t)i t (2.1.14) ˆ ˆ = hψ (0)| e−iω0tIz Iˆeiω0tIz |ψ (0)i . We can first examine the x-component:
D E −iω0tIˆz iω0tIˆz Iˆx = hψ (0)| e Iˆxe |ψ (0)i . (2.1.15) t
−iω tIˆz Since e 0 is the generator of a rotation by an angle ω0t about the z-axis,[5] it follows that
−iω0tIˆz iω0tIˆz e Iˆxe = Iˆx cos ω0t + Iˆy sin ω0t. (2.1.16)
This can be derived using the fundamental commutation relations of the angular momentum operators:[5] h i Iˆi, Iˆj = iεijkIˆk, (2.1.17) where εijk is the Levi-Civita symbol (again adhering to the convention of removing the
1 Note that in some texts the factor of ~ is removed from the Hamiltonian, i.e. the Hamiltonian is reported in frequency units: H˜ = H/ˆ ~. For the sake of clarity in comparison with standard quantum mechanics texts, this convention will not be adopted here.
5 ˆ factor of ~ from Ii). Making use of equation 2.1.16 in equation 2.1.15 gives D E Iˆx = hψ (0)| Iˆx |ψ (0)i cos ω0t + hψ (0)| Iˆy |ψ (0)i sin ω0t t D E D E (2.1.18) = Iˆx cos ω0t + Iˆy sin ω0t. 0 0 D E Again we can choose the initial spin state to be in the x-z plane so that Iˆy = 0, which 0 gives D E D E Iˆx = Iˆx cos ω0t. (2.1.19) t 0 D E Following the same process for Iˆy gives t D E D E Iˆy = Iˆx sin ω0t. (2.1.20) t 0
−iω tIˆz Since Iˆz commutes with the propagator e 0 , its expectation value doesn’t change with time: D E D E Iˆz = Iˆz . (2.1.21) t 0 Combining the three components gives
D E D E D E D E Iˆ = Iˆx cos ω0t, Iˆx sin ω0t, Iˆz , (2.1.22) t 0 0 0 or in other words
hµˆit = (hµˆxi0 cos ω0t, hµˆxi0 sin ω0t, hµˆzi0) . (2.1.23) This is similar to the result derived above using the classical vector picture, except that the magnetic moment vector has now been replaced with the expectation value of the magnetic moment vector operator.
2.2 Bulk Magnetization
In practice we are interested in the magnetization of a particular type of nucleus in a bulk sample of material. In this case the net magnetic moment is given by the sum of the magnetic moments arising due to each nucleus:
X M = µi. (2.2.1) i
This net magnetization will obey the same equation of motion as each individual magnetic moment (see equation 2.1.5): d M = γM × B. (2.2.2) dt
6 This result holds because the cross product and time derivative are distributive over addi- tion. Hence the net magnetization will also precess with frequency ω0 = −γB0.
2.2.1 Thermal Equilibrium Magnetization
The ensemble of nuclei will have the tendency to align with the external field because it is a lower energy configuration. In thermal equilibrium the population in each spin state will be proportional to the Boltzmann factor for that state:[6]
1 P (m) = exp (−E /kT ) , (2.2.3) Z m where m is the spin projection quantum number quantized along the field axis and Z is the normalizing partition function:[1]
X Z = exp (−Em/kT ) . (2.2.4) m
From the Zeeman Hamiltonian (see equation 2.1.10 above) we have
Em = −γ~B0m. (2.2.5)
The net magnetization for an ensemble of N spins is given by
I X M = N µmP (m) m=−I I P (2.2.6) γ~m exp (γ~B0m/kT ) = N m=−I . I P exp (γ~B0m/kT ) m=−I
Since γ~B0/kT 1 at normal temperatures and field strengths (the so-called “high- temperature limit”), it is usually acceptable to expand the exponential functions to first order (for example this factor is approximately 6.55 × 10−5 for protons resonating at 400
7 MHz at room temperature). Therefore we can evaluate
I P m (1 + γ~B0m/kT ) M ≈ Nγ m=−I ~ I P (1 + γ~B0m/kT ) m=−I I (2.2.7) P m2 Nγ2 2B = ~ 0 m=−I kT 2I + 1 Nγ2 2I (I + 1) = ~ B . 3kT 0
The factor relating M and B0 is called the static nuclear susceptibility:[6]
Nγ2 2I (I + 1) χ = ~ . (2.2.8) 0 3kT
2.3 Density Matrix Formalism
The full dynamics of a statistical ensemble of quantum spins can be described using the density matrix formalism of quantum mechanics. In a particular basis, every possible state
|ψni can be specified by a set of coefficients cni of the eigenstates |φii of that basis:
X |ψni = cni |φii . (2.3.1) i
In a statistical ensemble each possible state ψn has a probability pn of occurring. From this, the ijth element of the density matrix ρ is defined to be[2]
X ∗ ρij = pncnicnj (2.3.2) n such that the expectation value for an operator Aˆ is given by
D E X Aˆ = Tr (Aρ) = Aijρji, (2.3.3) i,j where A is the matrix representation of the operator Aˆ:
Aij = hφi| Aˆ |φji . (2.3.4)
8 From the elements of the density matrix ρij = hφi| ρˆ|φji we can identify the form of the density operator:[2] X ρˆ = pn |ψni hψn| . (2.3.5) n
2.3.1 Thermal Equilibrium Density Matrix
The ensemble occupation probability of a basis state |φii is given by
X 2 Pi = pn |hφi|ψni| = ρii, (2.3.6) n therefore in thermal equilibrium (in the energy eigenbasis) we would expect (see equation 2.2.3 above) ! 1 (ρ ) = Xp c c∗ = exp (−E /kT ) . (2.3.7) ii eq n ni ni Z i n eq
Since the phases of the coefficients cni should be uniformly distributed in equilibrium, the off-diagonal elements of ρeq will vanish[1] leaving
δ (ρ ) = ij exp (−E /kT ) . (2.3.8) ij eq Z i
Since, in this basis, H |φii = Ei |φii we can identify the equilibrium density operator as
1 ρˆ = exp −H/kTˆ , (2.3.9) eq Z and also Z = Tr exp −H/kTˆ . (2.3.10)
For a system of identical spins in equilibrium, interacting with a static magnetic field (i.e. the Zeeman Hamiltonian, see equation 2.1.10) the density matrix will be given by
1 ρˆ = exp γ B Iˆ /kT . (2.3.11) eq Z ~ 0 z
In the high-temperature limit (see equation 2.2.7) we can make the expansion[2]
! 1 γ B Iˆ ρˆ ' 1ˆ + ~ 0 z . (2.3.12) eq Z kT
The identity operator commutes with all other operators so it can be omitted without affecting the calculation of any observables. Furthermore, in this limit Z ' Tr (1) = 2I + 1,
9 which gives γ B ρˆ ' ~ 0 Iˆ . (2.3.13) eq (2I + 1) kT z Computing the expectation value for the magnetization gives:
! D E γ2 2B Iˆ2 hM i = N hµˆ i = N γ Iˆ = NTr γ Iˆ ρˆ = NTr ~ 0 z . (2.3.14) z z ~ z ~ z eq (2I + 1) kT
ˆ2 1 Since Tr Iz = 3 I (I + 1) (2I + 1)[1] this gives
Nγ2 2I (I + 1) hM i = ~ B , (2.3.15) z 3kT 0 which agrees with the result derived in equation 2.2.7 above.
In practice the prefactor in ρˆeq is not usually important so we can simply write
ρˆeq ∼ Iˆz. (2.3.16)
2.3.2 Time Dependence of the Density Matrix
The detailed dynamics of a spin system can in principle be elucidated by calculating the dynamics of the corresponding density matrix. The time evolution of the density matrix follows from the time-dependent Schrödinger equation (see equation 2.1.11). Applying the product rule and the definition of the density matrix elements (see equation 2.3.2) gives the well-known von Neumann equation:
dρˆ i h i = ρ,ˆ Hˆ , (2.3.17) dt ~ where Hˆ is the Hamiltonian of the system. For a time-independent Hamiltonian, the solu- tion is[2] ˆ ˆ ρˆ(t) = e−iHt/~ρˆ(0) eiHt/~ (2.3.18) = Uˆ (t)ρ ˆ(0) Uˆ −1 (t) , which follows simply from equations 2.1.12 and 2.3.5 above. Time-dependent Hamiltonians can in some cases be treated as effectively time-independent using average Hamiltonian theory, see section 2.9.
2.4 The Rotating Frame
It is common when considering NMR dynamics to perform a transformation into a frame of reference that rotates in the lab frame at a frequency of ωR ' ωo = −γB0, such that
10 Larmor precession due to the static magnetic field Bo is eliminated. This is equivalent to making the substitution[7]
iωRtIˆz −iωRtIˆz ρˆR (t) = e ρˆ(t) e , (2.4.1) where ρˆR is the density matrix in the rotating frame. An interaction described by a Hamil- tonian Hˆ1 in the laboratory frame is similarly transformed to
iωRtIˆz −iωRtIˆz Hˆ1,R (t) = e Hˆ1 (t) e . (2.4.2)
For a Hamiltonian Hˆ = Hˆ0 + Hˆ1 in the lab frame (where Hˆ0 is the Zeeman Hamiltonian) the von Neumann equation in the rotating frame (with ωR = ω0) becomes
dρˆR i h i = ρˆR, Hˆ1,R . (2.4.3) dt ~
Thus the effects of Hˆ0 are effectively eliminated. This is also equivalent to the introduction of a fictitious magnetic field Bf = −ωR/γ, which will cancel B0 if ωR = ω0. The transformation to the rotating frame is an example of an interaction representation, in which the effect of a Hamiltonian is removed by transformation into an appropriate reference frame.[2]
2.4.1 Zeeman Truncation
From equation 2.4.2 we can see that any parts of the Hamiltonian that are static in the lab frame, and which do not commute with Iˆz, will oscillate rapidly in the rotating frame. This suggests that their effects will tend to be averaged away. According to time independent perturbation theory, the first order energy corrections due to a perturbing Hamiltonian Hˆ1 are given by[3] D E (1) 0 ˆ 0 Em = ψm H1 ψm , (2.4.4) 0 where the ψm are the eigenstates of the unperturbed Hamiltonian. Thus, to first order in energy, only the parts of the perturbing Hamiltonian that commute with the unperturbed Hamiltonian will contribute. Taking the unperturbed Hamiltonian to be the Zeeman Hamil- ˆ ˆ tonian, H0 = −γ~B0Iz, gives agreement with the above suggestion that the oscillating components will have a relatively small effect and can usually be ignored (note that this requires that the Zeeman Hamiltonian be much larger than any other interactions, which is not always the case). Those parts that do commute with Hˆ0 are called the secular parts of the Hamiltonian, and the assumption that the non-secular parts can be ignored is referred to as Zeeman truncation.[7] After truncation, a time-independent Hamiltonian in the lab frame will also be time-independent in the rotating frame. Thus, from equation 2.4.3, the
11 time evolution of the density matrix in the rotating frame will be:[7]
ˆ ˆ −iH1t/~ iH1t/~ ρˆR (t) = e ρˆ(0) e . (2.4.5)
This is similar to the result in equation 2.3.18 above, except that the Zeeman interaction has been effectively removed from the Hamiltonian. The rotating frame is usually more convenient in NMR, so the subscript R will be dropped and ρˆ will subsequently refer to the density matrix in the rotating frame. One important exception to the rule of Zeeman truncation is in the case of magnetic fields which rotate in the plane perpendicular to the static magnetic field, B0. If the frequency of rotation is equal to the Larmor frequency, then transformation from the laboratory frame to the rotating frame will transform the Hamiltonian from being time-dependent to time-independent. Rotating fields such as this occur in the radio-frequency pulses used to manipulate magnetization in a typical NMR experiment, see section 2.5.
2.5 Radio-Frequency Pulses
In most NMR experiments, nuclear magnetization is manipulated by applying oscillating magnetic fields for discrete time intervals, i.e. in pulses. In the lab frame, a pulse is generated by a radio-frequency (RF) alternating current (AC) passing through an inductive coil in the NMR probe. This will create an alternating magnetic field[2]
B1 = 2B1 cos (ωrf t) x, (2.5.1) where x is the unit vector along the axis of the coil (assumed to be perpendicular to the static field along the z-axis) and ωrf is the AC frequency. The Hamiltonian (in the lab frame) for a nuclear spin in the presence of this field will be[2]
ˆ L ˆ H1,x = −2γB1 cos (ωrf t) Ix. (2.5.2)
The factor of 2 has been added for convenience so that the field can be decomposed into two counter-rotating components:
ˆ ˆ ˆ ˆ ˆ L iωrf tIz ˆ −iωrf tIz −iωrf tIz ˆ iωrf tIz H1,x = −γB1 e Ixe + e Ixe . (2.5.3)
The pulse is said to be on-resonance when its frequency is approximately the Larmor fre- quency of the nucleus of interest in the static magnetic field B0, and it is customary to take the rotating frame (see above) to be rotating at the frequency ωrf ' ω0.[7] In this case, one of the components of Hˆ1 will be static, while the other will rotate at twice the original
12 frequency. By an argument similar to that given in support of Zeeman truncation (see section 2.4.1) the rapidly rotating component can be ignored. In this case, the Hamiltonian (in the rotating frame) is simply given by[7]
ˆ ˆ H1,x = −γ~B1Ix. (2.5.4)
This corresponds to a static magnetic field directed along the negative x-axis.
2.5.1 Pulse Propagators and Rotation
According to equation 2.4.5 above, the time evolution of the density matrix under the
Hamiltonian Hˆ1,x given in equation 2.5.4 will be
ˆ ˆ ρˆ(t) = eiγB1Ixtρˆ(0) e−iγB1Ixt. (2.5.5)
Taking the initial density operator to be in thermal equilibrium according to equation 2.3.16 and defining ω1 = γB1 gives iω1tIˆx −iω1tIˆx ρˆ(t) = e Iˆze . (2.5.6)
This is equivalent to a rotation of the density operator by a tilt angle θ = ω1t around the x-axis (as one would expect for classical precession of magnetization around a magnetic field in the x-direction). In the lab frame, this rotation about the (rotating) x-axis corresponds to a nutation, that is a change in the angle of a precessing vector with respect to the axis about which it is precessing. The result of this rotation is[7]
ρˆ(t) = Iˆz cos (ω1t) + Iˆy sin (ω1t) . (2.5.7)
To show that this does in fact correspond simply to a rotation of the magnetization, one can make use of the identity
1 Tr IˆIˆ = I(I + 1)(2I + 1)δ . (2.5.8) i j 3 ij
Since the expectation value of the magnetization along the α-axis,
D ˆ E ˆ hMαi = N γ~Iα = NTr γ~Iαρˆ , (2.5.9) involves the trace of Iˆα with ρˆ, only the part of ρˆ proportional to Iˆα will contribute. Hence we can conveniently identify terms in ρˆ proportional to Iˆα as corresponding to magnetization along the α-axis.[7] Thus, in the example given in equation 2.5.7 above, the magnetization can easily be identified as rotating in the yz-plane (i.e. about the x-axis).
13 To rotate the magnetization around a different axis, it is sufficient to change the phase of the RF current relative to the consistently rotating frame of reference. For an arbitrary phase offset Φ, corresponding to a magnetic field
B1 = 2B1 cos (ωrf t + Φ) x, (2.5.10) the Hamiltonian in the rotating frame will be (from equation 2.5.3)
ˆ ˆ ˆ −iΦIz ˆ iΦIz H1,Φ = −γ~B1e Ixe . (2.5.11)
Thus the phase Φ intuitively corresponds to the angle of the field (and hence the rotation π axis) in the (rotating) xy-plane. For example, a phase of Φ = 2 would give an effective field along the y-axis:
Hˆ π = Hˆ1,y = −γ B1Iˆy, (2.5.12) 1,Φ= 2 ~ which would in turn lead to (clockwise) rotation around the y-axis:
ρˆ(t) = Iˆz cos (ω1t) − Iˆx sin (ω1t) . (2.5.13)
2.6 Relaxation
Relaxation is the process by which a spin system is returned to thermal equilibrium. In practice it determines the length of time for which an NMR signal can be acquired, and also the delay required before an experiment can be repeated. Relaxation dynamics can also be used to study the nature and dynamics of interactions within a spin system. Relaxation is an incoherent process driven by stochastic interactions in an NMR sys- tem. There are two primary types of relaxation which are relevant in most NMR systems.
The first is longitudinal relaxation, which is described by a relaxation time constant T1 and hence also referred to as T1 relaxation. Longitudinal relaxation is the relaxation of the z-component of magnetization towards its equilibrium value along the positive z-axis (aligned with the external magnetic field). The second type of relaxation is called transverse relaxation and describes the relaxation of magnetization in the xy-plane towards zero. It is described by the time constant T2 and hence, similarly to above, is also referred to as T2 relaxation. T2 is necessarily less than or equal to T1, since transverse magnetization states require coherence between positive and negative z-component magnetization. In a classical description of the evolution of magnetization, this phenomenological de- scription of relaxation can be combined with the effects of the external magnetic field in
14 equation 2.1.5 to give the Bloch equations. These can be written in vector form as[8]
d M = γM × B − R (M − M ) , (2.6.1) dt 0 where the relaxation matrix R is given by
1/T2 0 0 R = 0 1/T 0 . (2.6.2) 2 0 0 1/T1
The solutions of the Bloch equations for B along the z-axis and magnetization initially along the x-axis are then given by[9]
−t/T2 Mx = Mx,0 cos (ω0t) e ,
−t/T2 My = Mx,0 sin (ω0t) e , (2.6.3) −t/T1 Mz = M0 1 − e . The NMR spectrum is obtained from the Fourier transform of the x-component of the magnetization in the rotating frame (see section 2.7.1). For a single resonant component with simple T2 relaxation as described above, the spectrum will be a Lorentzian line shape (see equation 2.7.7) with a full-width at half-maximum (FWHM) of 1 (in ordinary fre- πT2 quency units). Hence a slower relaxation time will lead to a narrower line shape, and thus an increased signal-to-noise ratio (SNR). This is because the noise is generally evenly distributed in frequency, and for slower relaxation the same total signal intensity will be confined to a narrower frequency range.
Transverse (T2) relaxation is effected by stochastic dephasing of transverse components of magnetization, which is normally differentiated from reversible dephasing caused by ef- fects such as magnetic field inhomogeneities. These processes can have similar effects on a spectrum, however dephasing caused by non-stochastic effects can be eliminated by ap- plying an appropriate pulse sequence and acquiring data in a two-dimensional experiment (see section 2.7.2). Hence the apparent relaxation of a time domain signal is sometimes de- ∗ scribed by the time constant T2 to differentiate it from true irreversible relaxation described by T2.[9]
2.6.1 The Nuclear Overhauser Effect
The nuclear Overhauser effect (NOE) is the change in magnetization that occurs in one type of nucleus when another type of nucleus is saturated (i.e. its equilibrium magnetization is eliminated). It is also sometimes referred to as a nuclear Overhauser enhancement,
15 Figure 2.1: A diagram of the energy levels for a system of spin-1/2 nuclei, I and S. The spin state of the pair is indicated by the small arrows, with the I spin on the left. The Wα are the transition rates between the different states. The frequencies associated with the transitions are indicated on the right, with ω0 = ωI − ωS and ω2 = ωI + ωS. Note that the direction of the transitions and the sign of the frequencies is inconsequential. with the same initials. What is usually considered is the fractional change in the average magnetization of the spin I due to saturation of the spin S:
I0 − Iz fI {S} = , (2.6.4) I0 where I0 is the average equilibrium magnetization of spin I.[10] An NOE can occur when two coupled nuclei are able to change state simultaneously, with the net difference in energy either absorbed or released from the translational and rotational energy of the lattice (i.e. the whole system of particles).[10] This is called cross- relaxation, because it is driven by the same stochastic processes that give rise to conventional relaxation. The simplest case in which cross-relaxation can occur is for a system of two dipole-coupled spins-1/2. Figure 2.1 shows the energy levels of such a system, along with the associated transitions between the different two-spin states. As indicated in the figure,
Wα are the transition rates between the different states and ωα are the corresponding transition frequencies (with ω0 = ωI − ωS and ω2 = ωI + ωS). The equation of motion for the longitudinal magnetization of one spin I in such a system is given (from the Solomon equations) by[10]
dI z = −(I − I )(W + 2W + W ) − (S − S )(W − W ). (2.6.5) dt z 0 0 I 2 z 0 2 0
16 dIz If the spin S is saturated (so that Sz = 0) and steady state is reached (so that dt = 0), then making the substitution S0/I0 = γS/γI and re-arranging the equation gives for the fractional NOE I0 − Iz γS W2 − W0 fI {S} = = . (2.6.6) I0 γI W0 + 2WI + W2
Note that the NOE can be either positive or negative, depending on the relative size of W0 and W2. In order for a particular energy transition to occur, the molecular motions in the lattice must contain frequency components which cause fluctuating magnetic fields corresponding to the coupling frequency of that transition. For two spins interacting only via their mutual dipole-dipole coupling, the transition rates can be shown to be[10]
1 W = K2J(ω − ω ), 0 20 I S 3 W = K2J(ω ), (2.6.7) I 40 I 3 W = K2J(ω + ω ). 2 10 I S Here, the coupling constant K is given by
µ0 ~γI γS K = 3 , (2.6.8) 4π rIS and the function J(ω) is the spectral density of the spatial part of the time-dependent nuclear spin interaction. Note that the time dependence is assumed to arise primarily from angular re-orientations, such that the inter-nuclear distance rIS can be treated as a constant. If the molecular motion is isotropic, and can be described by a correlation function which decays exponentially with a time constant τc, then the spectral density can be written simply as[10] 2τc J(ω) = 2 2 . (2.6.9) 1 + ω τc
For rapidly tumbling small molecules, τc will be very short and ωτc 1. In this limit, J(ω) will be independent of the frequency ω, and the expression for the NOE will reduce to
1 γS fI {S} = . (2.6.10) 2 γI
On the other hand, for large, slowly tumbling molecules the motion will be concentrated at low frequencies. Therefore if the two nuclei have relatively similar resonant frequencies, only J(ωI − ωS) will make any significant contribution. In this case, the fractional NOE
17 will reduce to γS fI {S} = − . (2.6.11) γI Thus for a homonuclear system, the maximum fractional NOE would theoretically vary between −1 and 0.5 depending on the molecular dynamics in the system. Otherwise, the size of the NOE will depend on the ratio of the gyromagnetic ratios for the nuclei considered, and also the timescale of the molecular motions in the sample.
2.7 NMR Spectroscopy
In a typical NMR experiment, magnetization is excited into the xy-plane by applying one or more RF pulses via an inductive coil (see section 2.5). Pulses are usually specified by their tip angle, which depends on the AC power (which determines the field strength and hence the rate of nutation) and the duration of the pulse. The timing of pulses and delays used to manipulate magnetization (the pulse sequence) is programmed using a computer and effected by a hardware frequency synthesizer and waveform generator. The resulting signal is then passed through a high-power amplifier, which reduces the duration of the pulse, or pulse width, required to produce a given tip-angle. After (or, in some cases, during) the application of the pulse sequence, the behaviour of the magnetization must be recorded for subsequent analysis. In the lab frame, magnetization in the xy-plane will oscillate due to Larmor precession and any other relevant interactions (see sections 2.1 and 2.8 respectively). This will produce an oscillating magnetic flux Φ(t) in the same coil used for applying pulses. This will in turn induce a voltage V = −dΦ/dt in the coil.[7] Thus for magnetization oscillating with frequency ω0, for example Mx (t) =
M0 cos (ω0t), the induced voltage will behave like
d V (t) ∼ M (t) = ω M sin (ω t) . (2.7.1) dt x 0 0 0
The magnitude of this induced voltage depends on the frequency of oscillation, however relative differences between frequencies are typically small enough that this effect can be π ignored. Thus, aside from a phase shift of 2 , the induced voltage will provide an accurate representation of the behaviour of the magnetization. Note that increasing the external magnetic field strength will generally increase both ω0 and M0 and hence can lead to a significant increase in sensitivity. In general there will be a distribution P (ω) of frequencies present in the system, whence the induced voltage becomes
V (t) ∼ P (ω) ω cos (ωt) dω. (2.7.2) ˆ
18 Figure 2.2: An illustration of a simple pulse sequence, with one ninety-degree pulse followed by acquisition of the free induction decay (FID). The pulse is represented by a rectangle with the tip angle and phase indicated above. The acquisition period is illustrated as a decaying oscillation representing the FID.
This induced voltage is commonly amplified and filtered prior to measurement, then digi- tized using an analog-to-digital converter (ADC). The resulting time-domain signal is called the free induction decay (FID). An example of a simple pulse sequence, with just one pulse followed by the acquisition of the NMR signal, is illustrated in figure 2.2. Typically the measured signal is mixed with the original transmitter frequency, i.e. the AC frequency of the pulses, prior to acquisition. This effectively shifts all the measured frequencies by the transmitter frequency, and yields relative frequencies which are generally in the Hz or kHz range, rather than in MHz, which facilitates conversion to a digital sig- nal. The mixing process generally produces both positive and negative relative frequencies which must be differentiated. This is done by splitting the unmixed signal into two separate channels separated by a relative phase of ninety degrees. These are separately mixed and digitized and can then be combined as the real and imaginary parts of a complex time do- main signal which contains unambiguous information on the sign of the relative frequencies. This method is called quadrature detection.[7]
2.7.1 Fourier Transform Spectroscopy
In NMR spectroscopy one is generally interested in quantifying the populations of nuclei oscillating at different frequencies. This is equivalent to determining the frequency distribu- tion P (ω), which can be obtained from the FID by Fourier transform (FT). The continuous Fourier transform F (ω) of a function f (t) is given by[7]
∞ F (ω) = f (t) e−iωtdt, (2.7.3) ˆ−∞
19 and the inverse transform is given by
1 ∞ f (t) = F (ω)eiωtdω. (2.7.4) 2π ˆ−∞
In an NMR experiment, the time domain signal is made up of discrete data points sampled at a regularly spaced interval tdw (called the dwell time) and for a limited total acquisition time tacq = Ntdw, where N is the number of data points acquired. In this case one must use a discrete Fourier transform:[7]
N−1 1 X F (ω ) = f (t ) e−iωktj , (2.7.5) k N j j=0 where ωk = kωf , ωf = 2π/tacq, and tj = jtdw. For N complex time-domain points, N complex frequency domain points are obtained. Thus the bandwidth in the frequency domain, ωsw = Nωk, which is often called the spectral width, is inversely proportional to the dwell time. The frequency resolution or spacing between points in the frequency domain,
ωf , is inversely proportional to the total acquisition time tacq.[7] In a typical NMR experiment, the FID is represented by a complex signal whose overall phase is usually adjusted such that the real part starts at a maximum and the imaginary part starts at zero. For example, the time domain signal (after mixing) for nuclei with a single resonant frequency ω0 and transverse relaxation time T2 would be
S(t) = Aei(ω0−ωRF )te−t/T2 , (2.7.6) where ωRF is the transmitter frequency with which the signal is mixed. Defining the relative resonant frequency Ω = ω0−ωRf , the spectrum would then be a Lorentzian function (centred at ω = Ω):[7] ! T (ω − Ω) T 2 S(ω) = A 2 − i 2 . (2.7.7) 2 2 2 2 1 + (ω − Ω) T2 1 + (ω − Ω) T2 The real and imaginary parts of this signal are illustrated in figure 2.3. The real part of this signal is symmetric, always positive, and is an example of an absorptive line-shape. The imaginary part is antisymmetric, has wider tails, and is called a dispersive line-shape. Thus one can also write this spectral function in shorthand notation as
S(ω) = A(ω − Ω) + iD(ω − Ω), (2.7.8) where A and D are the absorptive and dispersive parts respectively. Typically only the absorptive part of a spectrum is of interest as it provides better
20 Figure 2.3: Absorptive and dispersive line-shapes corresponding to the real and imaginary parts of a single Lorentzian. resolution between different components and a better signal to noise ratio compared to a dispersive spectrum. In general there can be an overall phase shift of the signal that must be corrected so that the real part of the spectrum is fully absorptive.[7]
2.7.2 2D spectroscopy
Multidimensional experiments allow the correlation of resonant frequencies detected during different time periods. Generally a two-dimensional experiment is done in such a way that the FID is modulated by some interaction during a delay period, whose duration is then incremented. This delay period is usually called the evolution period, as opposed to the normal detection period. These are also called the indirect and direct dimensions, respectively, and are illustrated in figure 2.4. In some experiments particular interactions are selected during the evolution period, which may differ from those present during the detection period. In other experiments the evolution and detection periods are separated by some fixed mixing time, during which some type of evolution or mixing can occur, and correlations are then observed between the two periods. In either case data is acquired during the detection period for several different durations of the evolution period, such that several FIDs are obtained. A Fourier transformation can then be performed with respect to first one dimension, to give a series of spectra for different time values, and then the second dimension, to give a full two-dimensional frequency spectrum.[7] In order to obtain a fully absorptive spectrum in two dimensions, special considerations must be made. For a phase-modulated time-domain signal of the form
iΩ1t1 iΩ2t2 S(t1, t2) ∼ e e , (2.7.9)
21 Figure 2.4: Illustration of a typical pulse sequence used for a two-dimensional experiment. Such an experiment usually includes four distinct time periods: preparation, evolution, mixing, and finally detection. The experiment is repeated several times with the evolution period t1 increased incrementally. The mixing period might be made up of one or more pulses and/or delay intervals.
simply performing two successive Fourier transformations over t1 and t2 will yield a mix of absorptive and dispersive components in the real and imaginary parts of the two-dimensional spectrum:[7]
S(ω1, ω2) ∼ (A1A2 − D1D2) + i (D1A2 + A1D2) , (2.7.10) where Ai and Di for i = {1, 2} are the absorptive and dispersive parts, respectively, for the ith dimension. However, often in a two-dimensional experiment one is actually measuring an amplitude-modulated signal of the form
iΩ2t2 Sc(t1, t2) = cos (Ω1t1) e . (2.7.11)
In this case an absorptive spectrum can be obtained by first performing a Fourier trans- form with respect to t2, then setting the imaginary part of the spectrum to zero prior to performing the second FT over t1. The direct result of this procedure is[7]
1 S (ω , ω ) = (A (ω − Ω ) + A (ω + Ω )) A , (2.7.12) c 1 2 2 1 1 1 1 2 which unfortunately contains overlapping signals from positive and negative relative fre- quencies Ω1. This problem can be avoided if one sets the detection frequency slightly off-resonance, such that the spectral intensity is confined to regions of positive relative frequencies, which will yield an ideal absorptive spectrum of the form
1 S (ω , ω ) = A A . (2.7.13) 1 2 2 1 2
It is also possible to collect a fully absorptive spectrum without the need for the off- resonance condition by collecting an additional data set corresponding to
iΩ2t2 Ss(t1, t2) = i sin (Ω1t1) e . (2.7.14)
22 The combination of these two sets of data is called a hyper-complex data set. Performing the above procedure for Ss yields 1 S (ω , ω ) = (A (ω − Ω ) − A (ω + Ω )) A . (2.7.15) s 1 2 2 1 1 1 1 2
Subtracting these two parts gives
S (ω1, ω2) = Sc (ω1, ω2) − Ss (ω1, ω2) = A1A2, (2.7.16) which yields the desired result of a fully absorptive spectrum, however it can also leave artifacts due to imperfect cancelling of terms between the two components.[7]
2.8 NMR interactions
If the only relevant effect in NMR was the Zeeman interaction of the nucleus with the exter- nal magnetic field, then NMR spectroscopy would not be a very useful scientific tool. The frequencies observed would depend only on the field strength and type of nucleus. Luckily this is not true, and there are several types of interactions that affect the NMR frequencies in different cases. These interactions generally cause a change in the local magnetic field at the nucleus or otherwise affect the Zeeman energy levels of the system. Their effects will generally depend on the chemical and structural environment of the nucleus, as well as molecular orientation and dynamics. Thus, useful information about the microscopic properties of a material can be obtained by studying NMR frequencies. The interactions important to the current experiments are presented below. The chem- ical shielding (section 2.8.1) is relevant to the carbon-13 measurements described in section 4.4 and discussed in section 5.1. The dipolar interaction (section 2.8.2) is relevant to the multiple-quantum experiments described in section 4.6 and discussed in section 5.3, and is also mentioned in the section describing the pulse sequence used for the carbon-13 exper- iments (4.4.1). The section on quadrupole couplings (2.8.3) is relevant to the deuterium experiments described in section 4.5 and discussed in section 5.2. Finally, the general topic of residual couplings, presented in section 2.8.4, is especially relevant for the interpretation of both the quadrupolar measurements (section 5.2) and the multiple-quantum experiments (section 5.3).
2.8.1 Chemical Shielding
Chemical shielding describes the change in local magnetic field at the nucleus due to the interaction of nearby electrons with the external magnetic field B0. It can be decomposed
23 into diamagnetic components, which oppose the magnetic field, and paramagnetic compo- nents, which enhance it. The change in resonant frequency of the nucleus that results from this is called the chemical shift.[2] Since chemical shielding is a linear interaction generated by the external magnetic field, its full effect can be described by the Hamiltonian
ˆ L ˆ Hcs = γ~I · σ · B0, (2.8.1) where σ is a second-rank tensor, called the chemical shielding tensor, which depends on the chemical environment of the nucleus. Taking only the secular part of the Hamiltonian (see section 2.4.1) leaves[2] ˆ ˆ Hcs = γ~IzσzzB0. (2.8.2)
This will result in a chemical shift (in frequency) of
ωcs = −ω0σzz. (2.8.3)
The chemical shielding tensor is most easily characterized in a reference frame in which it is diagonal, called the principal axis frame or principal axis system (PAS). This frame is fixed relative to the chemical environment of the nucleus, which will in turn be fixed relative to a particular molecule or other arrangement of atoms. In this frame the diagonal P AS P AS P AS elements of σ are the eigenvalues σxx , σyy , and σzz . If the direction of B0 (the lab frame z-axis) in this frame is specified by a polar angle θ and azimuthal angle φ, then we can write[2]
P AS 2 2 P AS 2 2 P AS 2 σzz = σxx sin θ cos φ + σyy sin θ sin φ + σzz cos θ. (2.8.4)
It is customary to define a new set of parameters to describe the chemical shielding:[2]
1 σ = σP AS + σP AS + σP AS , iso 3 xx yy zz P AS δ = σzz − σiso, (2.8.5) 1 η = σP AS − σP AS , δ xx yy where σiso is the isotropic chemical shift, δ is the anisotropy, and η is the asymmetry. Adopting the convention
P AS P AS P AS σzz − σiso ≥ σyy − σiso ≥ σxx − σiso (2.8.6)
24 Figure 2.5: Illustration of the chemical shift powder spectrum for several values of η, as indicated in the figure. ensures that 0 ≤ η ≤ 1.[7] In terms of these new variables we can write
1 ω (θ, φ) = −ω σ − ω δ 3 cos2 θ − 1 + η sin2 θ cos 2φ . (2.8.7) cs 0 iso 2 0
In general a distribution of orientations will be present in a sample, which will give rise to a distribution of intensity at different frequencies. A sample containing a uniform distribution of orientations will give rise to a non-uniform distribution of frequencies called a powder pattern. Depending on the values of δ and η, different orientations will contribute to the same frequency, thus determining the shape of the frequency distribution. The shape of the powder patterns for several values of η are shown in figure 2.5. Note that the anisotropic part of the chemical shift will average to zero if the angles
θ and φ are averaged over a uniform distribution, leaving only ωiso = −ω0σiso. This type of averaging would occur in the case when a molecule is tumbling isotropically on a time scale that is fast relative to the width of the static chemical shift anisotropy, ∆ωcs ∼ ω0δ, as is the generally the case in liquids. See section 2.8.4 for further discussion of motional averaging of orientational dependence. The chemical shift is usually measured with respect to a reference molecule because chemical shifts cannot be directly differentiated from any shifts due to a change in the external magnetic field. Also, since the chemical shift scales with magnetic field strength, it is usually reported as a relative frequency ωcs/ω0 in units of parts per million (PPM).
2.8.2 Dipolar Interaction
Classically, the interaction energy of two magnetic dipoles µ1 and µ2 is given by[4]
µ 1 r r U = 0 µ · µ − 3 µ · µ · , (2.8.8) 4π r3 1 2 1 r 2 r
25 where r is the vector separating µ1 and µ2 and r = |r|. Substituting the quantum mechan- ˆ ˆ ical magnetic moment operators µˆ 1 = γI ~I and µˆ 2 = γS~S gives the Hamiltonian[7]
µ γ γ 2 r r Hˆ = 0 I S~ Iˆ · Sˆ − 3 Iˆ · Sˆ · . (2.8.9) D 4π r3 r r
Expressing the vector r in terms of the spherical coordinates θ and φ (in the lab frame), we can expand the Hamiltonian into several terms[2]
ˆ HD = −~d (A + B + C + D + E + F ) , (2.8.10) where we have defined the dipolar coupling constant (in frequency units)
µ γ γ d = 0 I S~ (2.8.11) 4π r3 and the following terms:
2 A = IˆzSˆz 3 cos θ − 1 , 1 B = − Iˆ Sˆ + Iˆ Sˆ 3 cos2 θ − 1 , 4 + − − + 3 C = Iˆ Sˆ + Iˆ Sˆ sin θ cos θe−iφ, 2 z + + z 3 (2.8.12) D = Iˆ Sˆ + Iˆ Sˆ sin θ cos θe+iφ, 2 z − − z 3 E = Iˆ Sˆ sin2 θe−2iφ, 4 + + 3 F = Iˆ Sˆ sin2 θe+2iφ. 4 − − We also use the standard raising and lowering operators:
Iˆ± = Iˆx ± iIˆy. (2.8.13)
To first order (see Zeeman truncation, section 2.4.1) only terms A and B will contribute, since all other terms will either raise or lower the Zeeman basis states.[6] Furthermore, term B will only contribute in the homonuclear case (where I and S are the same type of spin). Thus we are left with the following truncated forms of the Hamiltonian:
ˆ (homo) 1 2 ˆ ˆ ˆ ˆ HD = − ~d 3 cos θ − 1 3IzSz − I · S , 2 (2.8.14) 1 Hˆ (hetero) = − d 3 cos2 θ − 1 2Iˆ Sˆ . D 2~ z z For a single pair of interacting spins, a single resonance will be split into two lines
26 Figure 2.6: The dipolar powder pattern, also known as a Pake pattern, which is equivalent to the superposition of two chemical shift powder patterns with η = 0.[7] separated by a frequency difference of
3 ∆ω = d 3 cos2 θ − 1 . (2.8.15) D 4
For a full isotropic distribution of orientation angles θ, the resulting powder spectrum will be composed of a superposition of two powder patterns of opposite orientation, each having the same shape as the chemical shift anisotropy with asymmetry parameter η = 0 (see section 2.8.1). This powder spectrum, known as a Pake powder pattern and illustrated in figure 2.6, has a total width of 3d and two sharp horns separated by a frequency difference 3 of 2 d.[11]
2.8.3 Quadrupolar Interaction
Unlike chemical shifts and dipole couplings, the quadrupolar interaction is due to the elec- 1 trical properties of the nucleus. All nuclei with spin greater than 2 will have a nuclear electric quadrupole moment which will interact with local electric fields at the nucleus.[2] Starting from a classical picture, the energy of a charge distribution q (r) in an electric potential V (r) is[1] E = q (r) V (r) dr (2.8.16) ˆ Expanding V (r) in a Taylor series about the centre of mass of the nucleus gives:[1]
! ∂V 1 ∂2V V (r) = V (0) + Xx + Xx x + ..., (2.8.17) α ∂x 2! α β ∂x ∂x α α r=0 α,β α β r=0
27 where xα = {x, y, z} for α = {1, 2, 3}. Making the definitions
∂V V ≡ , α ∂x α (2.8.18) ∂2V Vαβ ≡ , ∂xα∂xβ gives for the energy
1 E = V (0) q (r) dr + XV x q (r) dr + XV x x q (r) dr + .... (2.8.19) ˆ α ˆ α 2! αβ ˆ α β α α,β
The first term is simply the electrostatic energy of the nucleus treated as a point charge at the origin. The second term is the electric dipole interaction of the nucleus, however it will vanish since the centre of charge coincides with the centre of mass of the nucleus.[1] The third term represents the electric quadrupole interaction in which we are interested, which we will call E(2). We can treat the charge distribution as a series of point charges (protons) at positions rk,[1] which gives X q (r) = e δ (r − rk) , (2.8.20) k where e is the elementary charge. Introducing the quantities
Q = 3x x − δ r2 q (r) dr αβ ˆ α β αβ (2.8.21) X 2 = e 3xαkxβk − δαβrk k gives[1] 1 E(2) = XV Q , (2.8.22) 6 αβ αβ α,β P since V must satisfy Laplace’s equation at the origin ( Vαα = 0). This can be changed α to a quantum mechanical expression by simply replacing all position variables with their corresponding operators, giving the quadrupolar Hamiltonian[1]
1 Hˆ = XVˆ Qˆ , (2.8.23) Q 6 αβ αβ α,β where Vˆαβ and Qˆαβ are the operator forms of Vαβ and Qαβ respectively. We can use the Wigner-Eckart theorem to change from position-space operators to angular momentum
28 operator, which gives[6,1]
eQ 3 Qˆ = Iˆ Iˆ + Iˆ Iˆ − δ Iˆ2 , (2.8.24) αβ I(2I − 1) 2 α β β α αβ where Q is the nuclear quadrupole moment:
X 2 2 Q = hψm=I | 3ˆzk − rˆk |ψm=I i . (2.8.25) k
Here |ψm=I i is the full state ket of the nucleus, including spatial dependence and any addi- tional quantum numbers required to specify the state of the nucleus (with the z-component spin projection quantum number m equal to the principal spin quantum number I).[1] The Hamiltonian then becomes
eQ 3 Hˆ = XV Iˆ Iˆ + Iˆ Iˆ − δ Iˆ2 , (2.8.26) Q 6I (2I − 1) αβ 2 α β β α αβ α,β where we have dropped the operator caret on Vαβ since it operates only in position-space. This can be written in a more compact form as[2]
eQ Hˆ = Iˆ · V · Iˆ, (2.8.27) Q 2I (2I − 1) where V is the electric field gradient tensor. As in the case of chemical shielding (see section 2.8.1) we can choose an axis system xP AS, yP AS, zP AS in which the electric field gradient tensor is diagonal, which we will again call the principle axis system. In this axis system the electric field gradient tensor P AS P AS P AS has diagonal elements Vxx , Vyy , and Vzz . We will again define a conventional set of parameters P AS eq = Vzz , P AS P AS (2.8.28) Vxx − Vyy ηQ = P AS , Vzz where ηQ is called the quadrupolar asymmetry parameter.[2] Note that unlike the case for chemical shielding, the quadrupolar coupling Hamiltonian has no isotropic part (except in higher order terms, see below). In this axis system the Hamiltonian becomes
eQ Hˆ = Iˆ2 V P AS + Iˆ2 V P AS + Iˆ2 V P AS Q 2I (2I − 1) xP AS xx yP AS yy zP AS zz (2.8.29) e2qQ = 3Iˆ2 − Iˆ2 + η Iˆ2 − Iˆ2 . 4I (2I − 1) zP AS Q xP AS yP AS
29 If the external magnetic field is along polar angles (θ, φ) in this axis system, transforming back into the lab frame gives:[2]
ξ Hˆ = (A + B + C + D + E) , (2.8.30) Q ~3 with the constant 3e2qQ ξ = , (2.8.31) 4I (2I − 1) ~ and the terms
1 A = 3 cos2 θ − 1 + η sin2 θ cos 2φ 3Iˆ2 − Iˆ2 , 2 Q z 1 B = sin θ (3 cos θ + η (cos θ cos 2φ − i sin 2φ)) Iˆ Iˆ + Iˆ Iˆ , 2 Q + z z + 1 C = (3 sin θ cos θ + η (cos θ cos 2φ + i sin 2φ)) Iˆ Iˆ + Iˆ Iˆ , (2.8.32) 2 Q − z z − 3 D = sin2 θ + η cos2 θ + 1 Iˆ2 + Iˆ2 , 4 Q + − i E = η cos θ sin 2φ Iˆ2 − Iˆ2 . 4 Q + − In the case where the quadrupole coupling is small compared to the Zeeman interaction, it is sufficient to consider only the secular parts of HˆQ, i.e. term A:
ˆ ξ 1 2 2 ˆ2 ˆ2 HQ ≈ ~ 3 cos θ − 1 + ηQ sin θ cos 2φ 3Iz − I , 3 2 (2.8.33) 1 = ω 3Iˆ2 − Iˆ2 , 3~ Q z where the coupling frequency ωQ is given by
1 ω = ξ 3 cos2 θ − 1 + η sin2 θ cos 2φ . (2.8.34) Q 2 Q
In intermediate cases the second order energy corrections may also need to be taken into account. Although the full expression for the second order term is not included here, it should be noted that it contains an isotropic part (independent of molecular orientation) which could be mistaken for chemical shift. The second order term depends inversely on the Larmor frequency, however, making it less important at higher field strengths.[2] To first order, the quadrupolar interaction will split a single resonance into 2I lines. For spin-1 nuclei such as deuterium (as studied here) each resonance will be split into a pair of lines, separated by the splitting 2ωQ.[12] In general a distribution of orientations will be present, which will result in a powder pattern spectrum. As in the case of the chemical shift anisotropy (see section 2.8.1), the shape of this powder pattern will depend on the
30 Figure 2.7: Illustration of the quadrupolar powder spectrum for several different values of ηQ, as indicated in the figure. The spectrum for ηQ = 0 has the same shape as the dipolar powder spectrum shown in figure 2.6.
value of the asymmetry parameter ηQ. For the quadrupolar interaction, however, there will be two superimposed powder patterns, as in the case of the dipolar interaction (see section 2.8.2). These component spectra will be symmetric about the original resonant frequency, and with opposite orientation. The relative separation of the two component spectra, and the full width of the powder pattern, will depend on the value of the asymmetry parameter
ηQ. Several examples of spin-1 powder patterns for different values of ηQ are shown in figure 2.7.
2.8.4 Residual Couplings
In a system which exhibits molecular motion that is fast relative to the strength of a particular interaction (in frequency units), the apparent angular and spatial dependence of a particular interaction will be given by the average (over the fast motions) of the static form of the interaction. Random motion processes can be described using the autocorrelation function, which provides a measure of the average correlation of a particular quantity at two times separated by an interval τ. Thus, the autocorrelation function provides a measure of the timescale of fluctuations in the random process. In general, for a normalized function which fluctuates randomly in time, F (t), the correlation function of F would be given by
C (τ) = hF (t) · F (t + τ)i , (2.8.35) where the angle brackets represent an average over the reference time t. In the case that the time average of F is zero, then it is found that the correlation function is often given
31 by an exponentially decaying function of the form[13]
C (τ) = e−τ/τc , (2.8.36) where τc is called the correlation time. As an example, consider the dipole coupling Hamiltonian as given in equation 2.8.14. This can be re-written in a generic form as
ˆ ˆ HD = ~ωD (r) F I, S , (2.8.37)
where ωD(r) is the effective coupling between the two spins and F Iˆ, Sˆ contains the spin-operator dependence of the Hamiltonian. We can account for the average over fast molecular motions by replacing ωD with the residual coupling:[14]
res ωD (r) = hωD(r)it * + µ 1 3 cos2 θ(t) − 1 = 0 γ γ 4π I S~2 r(t)3 (2.8.38) µ γ γ 1 D E ' 0 I S~ 3 cos2 θ(t) − 1 4π hr(t)i3 2 1 D E = hdi 3 cos2 θ(t) − 1 , 2 where d is the dipolar coupling constant defined in equation 2.8.11. The angle brackets here represent the time average over an interval t that is short relative to the static coupling frequency: −1 t ωD. (2.8.39)
Note that the averages over the separation r and the angle θ can be taken separately, as suggested in the above equation, as long as the time scales of reorientations and translations are well separated.[14] If this is not the case then the average must be performed over both quantities simultaneously. In the case of an isotropically varying θ, the average of 3 cos2 θ − 1 will vanish, and the correlation function will be an exponentially decaying function as in equation 2.8.36 above. If the correlation time is short relative to the static strength of the interaction, then the anisotropic parts of the interaction will be reduced to the point of being insignificant. This is typically the case in liquids or liquid-like samples. In this case dipole couplings will become negligible and chemical shifts will be reduced to their isotropic values (asymmetry terms of the form sin2 θ cos 2φ will also average to zero for isotropic motion). The full quadrupolar coupling Hamiltonian contains higher-order isotropic terms, however these are
32 often sufficiently small that their effects can be ignored. If the fast variations of θ are anistropic, then the average in equation 2.8.38 will not, in general, go to zero. In this case the correlation function will decay to a constant value of S2, where S is called the order parameter:[13]
C (τ) = 1 − S2 e−τ/τC + S2. (2.8.40)
The order parameter S provides a measure of the anisotropy of the molecular motion, and is equivalent to the ratio of the averaged vs. the static dipole coupling:[13]
1 D E hω i S = 3 cos2 θ(t) − 1 = D . (2.8.41) 2 ωD
In systems which contain an isotropic distribution of molecular orientations (i.e. a powder), fast anisotropic motions will reduce the width of the powder pattern which arises due to the orientational dependence of a particular interaction. For interactions containing a non-zero asymmetry parameter, such as the chemical shift, new effective anisotropy and asymmetry parameters η0 and δ0 can be derived by averaging the relevant interaction tensor over the fast motions, then analyzing the averaged tensor as above (see section 2.8.1).[7] A similar process can be followed for the quadrupolar interaction. The intermediate motional regime is that where motion exists with a correlation time −1 on the order of the static coupling, (τc) ≈ ωD. In this case simple analysis in terms of a dynamic order parameter is not generally possible. The time domain NMR signal can, in general, be written as * t !+ F (t) ∼ exp i ω t0 dt0 , (2.8.42) ˆ0 where the angle brackets now indicate an ensemble average over spins in the sample. The frequency ω (t) will vary randomly in time due to motional fluctuations. Various models and theoretical approaches exist which can be used to give solutions in different cases.[13] Rigid materials with little motion at sufficiently fast time scales will exhibit the full static range of spatially dependent coupling strengths, and will give a full powder pattern spectrum. In an inhomogeneous sample, regions within the sample may have variations in their dynamics, which would lead to different residual couplings. The resulting spectrum would then be composed of the ensemble superposition of spectra according to the distribution of residual couplings. This distribution can yield information on the strength and range of dynamics present in a sample, however it is often difficult to quantitatively determine the distribution from experimental results. This is due to the ill-posed nature of the problem, wherein many possible distributions can give rise to the same spectrum (see section 4.6 for further discussion). Nevertheless, modelling can be a useful tool to give information about
33 what kinds of distributions are consistent with particular results.
2.9 Average Hamiltonian Theory
Average Hamiltonian theory is widely used in NMR to understand and design specialized pulse sequences for investigating particular interactions. It is especially useful here for understanding the multiple quantum experiments described in section 4.6. For a time-independent Hamiltonian, the time evolution of the density matrix is given by (from section 2.3.2) ˆ ˆ ρˆ(t) = e−iHt/~ρˆ(0) eiHt/~. (2.9.1)
For a Hamiltonian that is piecewise time-independent over the intervals t1, t2, . . . , tn with t = t1 + t2 + ··· + tn this becomes
ˆ ˆ ˆ ˆ ˆ ˆ ρˆ(t) = e−iHntn/~ . . . e−iH2t2/~e−iH1t1/~ρˆ(0) eiH1t1/~eiH2t2/~ . . . eiHntn/~. (2.9.2)
We can re-write this series of exponential time propagation operators in terms of an average Hamiltonian H¯ˆ (t): ˆ ˆ ˆ ¯ˆ e−iHntn/~ . . . e−iH2t2/~e−iH1t1/~ = e−iHt/~. (2.9.3)
In general this average Hamiltonian will be time-dependent, however we are usually inter- ested in evaluating its form at particular points in time. To do this we make use of the Baker-Campbell-Hausdorff relation:[7]
ˆ ˆ 1 h i 1 h h ii hh i i eAeB = exp Aˆ + Bˆ + A,ˆ Bˆ + A,ˆ A,ˆ Bˆ + A,ˆ Bˆ , Bˆ + ... . (2.9.4) 2! 3!
This can be used to expand the average Hamiltonian in what is called the Magnus expansion:[2]
H¯ˆ (t) = H¯ˆ (0) (t) + H¯ˆ (1) (t) + H¯ˆ (2) (t) + ..., (2.9.5) where
1 H¯ˆ (0) (t) = Hˆ t + Hˆ t + ··· + Hˆ t , t 1 1 2 2 n n i h i h i h i H¯ˆ (1) (t) = − Hˆ t , Hˆ t + Hˆ t , Hˆ t + Hˆ t , Hˆ t + ... , 2 t 2 2 1 1 3 3 1 1 3 3 2 2 ~ (2.9.6) ˆ (2) 1 h h ii hh i i H¯ (t) = − Hˆ3t3, Hˆ2t2, Hˆ1t1 + Hˆ3t3, Hˆ2t2 , Hˆ1t1 6~2t 1 h h ii 1 hh i i + Hˆ t , Hˆ t , Hˆ t + Hˆ t , Hˆ t , Hˆ t + ... . 2 2 2 2 2 1 1 2 2 2 1 1 1 1
When the Hamiltonians for different time intervals commute or have only small non-
34 commuting parts, only the first term is required. This first term is just the simple time average of the Hamiltonian at different times. This expansion also provides a more rigorous justification of Zeeman truncation. If an interaction Hamiltonian is small compared to the Zeeman Hamiltonian then it is usually sufficient to consider only the first order term H¯ˆ (0), which for a continuously varying Hamiltonian becomes[2]
1 t H¯ˆ (0) (t) = Hˆ t0 dt0. (2.9.7) t ˆ0
Any parts of the Hamiltonian that do not commute with the Zeeman Hamiltonian will oscillate in the rotating frame and hence average to zero in this integral.
2.9.1 The Toggling Frame
In cases where the Hamiltonians do not commute at different times, it is often possible to transform to a new reference frame in which they do. This is called the toggling frame. The basic idea is that rather than directly calculating the time-evolution of the density matrix ρˆ, one instead transforms to a time-dependent reference frame in which ρˆ is subject to a relatively simple average Hamiltonian (we will assume that the transformation to the rotating frame has already taken place).[2] This is particularly useful in analyzing multi- pulse sequences used to manipulate a particular type of interaction Hamiltonian into a desired average Hamiltonian. In such cases, each pulse in the sequence will cause a change in reference frame. For short pulses we can approximate this change to be instantaneous (hence the term toggling frame). The form of the Hamiltonian will in general change when the reference frame changes, which will result in a Hamiltonian that is piecewise time- independent (as in equation 2.9.2). If this toggling-frame Hamiltonian commutes with itself during the times between pulses, then the average Hamiltonian can be calculated simply using equation 2.9.6. Suppose the Hamiltonian (in the rotating frame) is given by
Hˆ = Hˆint + Hˆp, (2.9.8)
ˆ ˆ ˆ where Hint is some spin interaction Hamiltonian and Hp = ~ωpIα is the Hamiltonian due to a pulse along the α-axis which is active for a time tp (with tip angle θp = ωptp). Then, the transformation to the toggling frame will give a new Hamiltonian[2]
ˆ ∗ ˆ−1 ˆ ˆ ˆ H = Rα HRα − Hp, (2.9.9)
−iθpIˆα where Rˆα = e is the standard rotation operator for a rotation angle of θp about the
35 α-axis. Since Hˆp is invariant under this rotation, this expression will reduce to
ˆ ∗ ˆ−1 ˆ ˆ H = Rα HintRα, (2.9.10) such that the pulse is effectively eliminated from the Hamiltonian. For a pulse sequence made up of a series of pulses p1, p2, . . . , pn separated by a series of delays t1, t2, . . . , tn−1, the interaction Hamiltonian in the toggling frame during the interval tk will be given by[8]
ˆ ∗ ˆ−1 ˆ−1 ˆ−1 ˆ ˆ ˆ ˆ Hk = R1 R2 ... Rk HintRk ... R2R1, (2.9.11)
ˆ ∗ noting that the rotations corresponding to later pulses are applied first. If these Hk commute for all k then the average Hamiltonian will be given by
1 H¯ˆ ∗ (t) = Hˆ ∗t + Hˆ ∗t + ··· + Hˆ ∗t . (2.9.12) t 1 1 2 2 n n
Finally, the density operator in the toggling frame at time t will be given by[2]
¯ˆ ∗ ¯ˆ ∗ ρˆ∗ (t) = e−iH (t)t/~ρˆ∗ (0) eiH (t)t/~. (2.9.13)
In many cases the pulse sequence is designed to be cyclic in nature, such that at the end of a cycle the toggling frame coincides with the normal rotating frame. In this case the density operator in the rotating frame will be equal to the density operator in the toggling frame.[2] If this is not the case, then another transformation would be required to obtain the expression for the density operator in the rotating frame.
2.10 Multiple Quantum Coherence
Multiple quantum coherence can be used in NMR experiments to investigate the dipole and quadrupole couplings of spins, which can in turn be used to characterize chain dynamics in mobile polymeric systems.[15] Here it was used to determine the distribution of proton residual dipole couplings present in different samples. The experimental description and results are presented in sections 4.6 and 5.3, respectively. Some theoretical background is presented below. Quantum coherence is a general term referring to an ensemble of states in which the relative phases φm of different Iˆz eigenstates |mi are constant (or at least asymmetrically distributed such that they have a non-zero average). The density matrix for such an en- semble will therefore have non-zero off-diagonal elements. For example, if the state of the
36 1 k-th spin in an ensemble of spin- 2 nuclei is
1 1 1 iφk |ψki = √ + e − , (2.10.1) 2 2 2 then the density matrix for an ensemble of N spins will be
−iφ ! 1 X 1 X1 1 e k ρˆ = |ψki hψk| = . (2.10.2) N N 2 iφk k k e 1
If the phases φk are asymmetrically distributed, then the off-diagonal terms of ρˆ will interfere constructively. Otherwise they will average to zero. Transverse magnetization states such as Iˆx and Iˆy are examples of coherent states, which in this case would correspond to phases π of φk = 0 and φk = 2 , respectively. Coherence can also exist in multi-spin states which are used to describe systems of coupled spins. The description is analogous to the single- spin case, except that coherence can exist between states in the full product-space of the multi-spin system. Multiple quantum (MQ) coherence generally refers to coherence between eigenstates with a difference in quantum number greater than one. Coherence between the m1 and m2 eigenstates with n = |m1 − m2| is called n-quantum (nQ) coherence or coherence of order n.[1] In terms of the density matrix, n-quantum coherence corresponds to n-off-diagonal elements (in the total z-component angular momentum basis) that are non-zero, that is
ρi,i±n 6= 0.[16] In terms of single-spin operators, nQ coherence will correspond to terms containing n raising or lowering (ladder) operators, or products of k ladder operators with n − k ladder operators of the opposite type. For example, double quantum (DQ) coherence can be ˆ+ ˆ+ ˆ− ˆ− represented by two-spin product terms such as I1 I2 + I1 I2 or multiple spin terms such ˆ+ ˆ+ ˆ+ ˆ− ˆ+ ˆ+ ˆ+ as I1 I2 I3 I4 . Triple quantum coherence can be represented by products such as I1 I2 I3 , and so on. The n-quantum nomenclature is also used to describe single quantum (SQ) ˆ+ ˆ− ˆ+ ˆ coherence (n = 1) which will contain terms such as I1 + I1 and I1 I2z. Zero quantum ˆ+ ˆ− ˆ− ˆ+ (ZQ) coherence corresponds to terms such as I1 I2 + I1 I2 which contain equal numbers of raising and lowering operators, and hence involve coherence between states with the same total z-component angular momentum.[16] This is usually differentiated from longitudinal magnetization (LM) or populations, which are represented by terms such as Iˆ1z +Iˆ2z.[17][16] Note that for a density operator representing a proper physical system, raising and lowering operators must appear in combinations such that the elements of the density matrix satisfy ∗ the relation ρij = ρji.
37 2.10.1 Excitation of Multiple Quantum Coherence
To see what types of Hamiltonians will give rise to multiple quantum coherence, we return to the formula for the time evolution of the density operator (equation 2.3.18). This equation is of the general form ˆ ˆ Cˆ = e−BAeˆ B, (2.10.3) which can be expanded using the Baker-Hausdorff relation[5] to give
h i 1 h h ii 1 h h h iii Cˆ = Aˆ − B,ˆ Aˆ + B,ˆ B,ˆ Aˆ − B,ˆ B,ˆ B,ˆ Aˆ + .... (2.10.4) 2! 3!
This expansion gives for the density operator
it h i 1 t2 h h ii ρˆ(t) =ρ ˆ(0) − H,ˆ ρˆ(0) + H,ˆ H,ˆ ρˆ(0) + .... (2.10.5) ~ 2! ~2
If we assume that the system starts in equilibrium, such that ρˆ(0) ∼ Iˆz, then we can determine what types of terms are required in the Hamiltonian to give rise to MQ terms ˆ+ ˆ+ (such as I1 I2 ) in the density operator. This can be accomplished using the commutation relations[16] h ±i ± Iˆz, Iˆ = ±Iˆ , , (2.10.6) h ± ∓i Iˆ , Iˆ = ±2Iˆz from which we can see that a Hamiltonian which includes terms involving ladder operators ˆ+ ˆ+ such as I1 I2 will lead to MQ coherence. 1 As an example, consider a system of two dipolar-coupled spins- 2 starting from equilib- rium (ρˆ(0) = Iˆ1z + Iˆ2z) and evolving under the effective Hamiltonian
ˆ ˆ+ ˆ+ ˆ− ˆ− HDQ = ~ω I1 I2 + I1 I2 . (2.10.7)
This Hamiltonian is labelled DQ because it will excite MQ coherence in multiples of two. A pulse sequence used to generate this effective Hamiltonian is described in section 4.6 below. Plugging this Hamiltonian into equation 2.10.5 and making use of the relations in equation 2.10.6 gives
ˆ ˆ ˆ+ ˆ+ ˆ− ˆ− 2 2 ˆ− ˆ+ ˆ ˆ+ ˆ− ˆ ρˆ(t) = I1z + I2z + 2iωt I1 I2 − I1 I2 − 4ω t I1 I1 I2z + I2 I2 I1z + .... (2.10.8)
Even terms in this expansion will contain ZQ or population terms, and odd terms will contain DQ terms. The nth term in equation 2.10.8 will contain products of n + 1 spin operators, however no terms with higher than DQ coherence can be excited for a system 1 composed of only two spins- 2 . In the absence of relaxation, intensity will oscillate between
38 these different coherence orders indefinitely. In a powder sample, however, the orientational dependence of the coupling strength will cause destructive interference leading to steady- state intensities of the different coherence orders. The time scale of MQ excitation will depend on the strength of the coupling between spins (ω), since the time-dependent density operator is made up of terms containing different powers of ωt. In a liquid or liquid-like sample where molecules undergo rapid isotropic orientational and translational diffusion, the dipole couplings between spins will be averaged away and no MQ coherence will develop.[18] In the intermediate motional regime, or for anisotropic reorientations, the rate at which MQ coherence develops will depend on the strength of the residual (partially averaged) couplings.
2.10.2 Multiple Quantum Coherence in Extended Spin Systems
In a system of many coupled spins, the Hamiltonian will contain terms for all spin pairs:
ˆ X ˆ+ ˆ+ ˆ− ˆ− HDQ = ωij Ii Ij + Ii Ij . (2.10.9) i For a system involving N spins, multiple quantum coherence can exist up to order N. Under a double quantum Hamiltonian such as HˆDQ, however, only even order coherences can be created. Higher order coherence terms in the density operator will appear only in higher order terms of the expansion in equation 2.10.5, which contain products of increasing numbers of spin operators.[19] These higher order expansion terms will in general also contain sub-terms corresponding to lower order coherences. When the density operator is expressed in terms of raising and lowering operators, the coherence order of a particular term can be recognized as the number of raising operators minus the number of lowering operators. For a given excitation time, the intensity of MQ coherences will decrease with increasing coherence order. This is because at a given level in the expansion there will be fewer terms corresponding to higher order coherences (due to simple combinatorics). As the excita- tion time increases higher order intensities will become more significant because intensity will transfer from lower to higher orders. For an infinite excitation time intensity should theoretically be equally distributed among all excitable coherence orders.[19, 15] 2.10.3 Detection of Multiple Quantum Coherence Multiple quantum coherence cannot be detected directly, so it must be converted into single quantum coherence in a way that retains some information about the relative intensity in different orders of coherence. Reconversion of multiple quantum coherence back into populations can be achieved by simply reversing the evolution that produced it in the first 39 Figure 2.8: Illustration of a pulse sequence used to produce and detect MQ coherence. Coherence is produced during the excitation period and then reconverted to ZQ coherence during the reconversion period. Changing the phase of the reconversion period allows coherence of different orders to be separated. After reconversion, ZQ coherence is converted to observable SQ coherence by a 90° read pulse. place. This can be done by subjecting the spin system to a Hamiltonian that is the negative of the original excitation Hamiltonian for a time equal to the excitation time (such that ˆ ˆ −1 Urec = Uexc). Populations and zero quantum coherence can then be converted to observable single quantum coherence with a single pulse.[16] Simple reversal would not contain any information on multiple quantum coherence, however, so another step is required. This step should cause different orders of coherence to evolve differently, so that the population intensity after reconversion is encoded with information about the different orders of coherence. Incrementing the relative phase of the entire set of pulses in the reconversion period is one way of accomplishing this, as illustrated in figure 2.8.[16] Changing the phase of the pulses in the reconversion period by a phase φ is equivalent −iHt/ˆ to changing the propagator for that period from Uˆ0 (t) = e ~ to[16] −iφIˆz iφIˆz Uˆφ (t) = e Uˆ0 (t) e . (2.10.10) To see how this affects the observed intensity we can calculate the expectation value of Iˆz after the reconversion: D E n o Iˆz (2texc) = Tr ρˆ(2texc) Iˆz n ˆ −1 ˆ ˆ −1 ˆ ˆ o = Tr Uφ U0ρˆ(0) U0 UφIz (2.10.11) n ˆ ˆ ˆ ˆ o −iφIz ˆ −1 iφIz ˆ ˆ ˆ −1 −iφIz ˆ iφIz ˆ = Tr e U0 e U0IzU0 e U0e Iz . Cyclic permutation of the operators inside the trace will not affect the result, so we can 40 write D E n ˆ ˆ ˆ ˆ o ˆ iφIz ˆ ˆ ˆ −1 −iφIz ˆ iφIz ˆ −iφIz ˆ −1 Iz (2texc) = Tr e U0IzU0 e U0e Ize U0 n ˆ ˆ o iφIz ˆ ˆ ˆ −1 −iφIz ˆ ˆ ˆ −1 = Tr e U0IzU0 e U0IzU0 (2.10.12) ˆ ˆ X iφIz ˆ ˆ ˆ −1 −iφIz ˆ ˆ ˆ −1 = hm| e U0IzU0 e U0IzU0 |mi . m We can also freely insert an identity operator of the form 1ˆ = P |m0i hm0| to give m0 D E ˆ ˆ ˆ X iφIz ˆ ˆ ˆ −1 −iφIz 0 0 ˆ ˆ ˆ −1 Iz (2texc) = hm| e U0IzU0 e m m U0IzU0 |mi. (2.10.13) m,m0 0 Finally, the states |mi and |m i can be taken to be eigenstates of Iˆz, which gives D E 0 2 ˆ X iφ(m−m ) ˆ ˆ ˆ −1 0 Iz (2texc) = e hm| U0IzU0 m . (2.10.14) m,m0 ˆ ˆ ˆ −1 Since ρˆ(texc) = U0IzU0 is the density matrix after the excitation period in which mul- 0 tiple quantum coherence was developed, and ρm,m0 = hm| ρˆ|m i corresponds to coherence of order k = m − m0, we can identify the collection of terms 2 X ˆ ˆ ˆ −1 Mk (texc) = hm| U0IzU0 |m − ki m (2.10.15) X 2 = |hm| ρˆ(texc) |m − ki| , m as the intensity of coherence of order k developed during the excitation period. This inten- sity is modulated by a phase proportional to the coherence order: D E X iφk Iˆz (2texc) = e Mk (texc) . (2.10.16) k This result can be used to separate different orders by incrementing the phase and then performing a Fourier transform with respect to φ.[16] A slightly modified approach is to add and subtract intensities with different phases to select a subset of coherence orders, without any additional Fourier transformation. This latter method was used here, and will be further explained in section 4.6. 41 Chapter 3 Materials Background The primary subject of the work presented here is the elastic protein, resilin. The term resilin refers to a family of proteins which occur naturally in insects. These are remarkable for their high reversible extensibility and extreme elastic efficiency; resilin has been described as one of the most resilient materials known.[20] The first two sections in this chapter present some background information on proteins and generic elastic materials. The last two sections present an overview of existing research into naturally occurring resilin and the recently developed recombinant proteins based on these natural resilins. The focus is on the molecular and mechanical properties of resilin, and the relationship of these to its natural function and potential applications. 3.1 Proteins Proteins are an essential part of all living cells and take part in a wide variety of functions, ranging from structural and mechanical roles to signalling and metabolism. A protein is a polymer made up of a sequence of amino acids, also called residues, linked by peptide bonds. Short amino acid polymers are also called peptides or polypeptides. The sequence of amino acids in a protein (called the primary structure) is stored in the DNA (deoxyribonucleic acid) of an organism. From the DNA it is transcribed to RNA (ribonucleic acid), whence the protein can actually be synthesized.[22] Each amino acid is comprised of an amine (N-H), a carbonyl (C=O), and a backbone carbon, or α-carbon, which has an attached side-chain (denoted R). The two carbon atoms are labelled Co and Cα, respectively. The peptide bond links together the nitrogen and the carbonyl carbon of two adjacent amino acids in a polypeptide, as shown in figure 3.1. The structure of the side-chain distinguishes the different amino acids. There are twenty-one standard amino acids which are incorporated into eukaryotic proteins, of which twenty are encoded directly by DNA.[22] By convention, the primary structure of a polypeptide is usually given from the amine end (N-terminus) to the carboxylic acid end (C-terminus). 42 Amino Acid Alanine Cysteine Aspartic Acid Glutamic Acid Abbrev. Ala (A) Cys (C) Asp (D) Glu (E) Structure polar, polar, polar, Properties non-polar acidic acidic acidic Phenylalanine Glycine Histidine Isoleucine Lysine Phe (F) Gly (G) His (H) Ile (I) Lys (K) polar, polar, non-polar non-polar non-polar weak basic basic Leucine Methionine Asparagine Proline Glutamine Leu (L) Met (M) Asn (N) Pro (P) Gln (Q) non-polar non-polar polar non-polar polar Arganine Serine Threonine Selenocysteine Valine Arg (R) Ser (S) Thr (T) Sel (U) Val (V) polar, polar, polar non-polar non-polar strong basic weak acidic Tryptophan Tyrosine Trp (W) Tyr (Y) non-polar polar Table 3.1: The twenty-one standard eukaryotic amino acids, including their structure and the polarity and pH of their side-chains. Note that non-polar side chains are generally hydrophobic, while polar side chains are hydrophilic. Of the standard amino acids, seleno- cysteine is the only one not encoded directly by DNA.[22] 43 H R C O H R cis N O N C H HR H H O OH + N C N C H H2O + O H O HR HR H N C N C trans H O HR Figure 3.1: An illustration of the formation of a peptide bond (indicated by the small arrow) to link together two polypeptides (or amino acids) into one larger polypeptide. The side chains are indicated by the letter R, and depend on the type of amino acid. Due to the nature of the peptide bond, there are two possible geometries for the peptide bond, which are labelled cis and trans (see section 3.1.1). The trans configuration is generally much more favourable than the cis, in part due to repulsive interactions between side chains (illustrated by the dashed line).[21] 3.1.1 Secondary Structure The backbone of proteins is often folded into regular, repeating patterns. These local confor- mations are referred to as the secondary structure of the protein. Most secondary structures are stabilized by hydrogen bonding between amino acids, usually between carbonyl (C=O) and amine (N-H) groups.[22] Secondary structural elements can be described by a sequence of torsion or dihedral angles, which define rotations about covalent bonds between atoms in the backbone. These angles range from −180◦ to 180◦, with an angle of ±180◦ (called trans) corresponding to the angles found in a fully extended chain. An angle of 0◦ is called cis. The peptide bond has a partial double bond characteristic due to resonance with the carbonyl double bond, hence its rotation (described by the angle ω) is generally restricted to either cis or trans; see figure 3.1 for an illustration. Between most pairs of amino acids the trans form is favoured by a factor of approximately 103, however in peptide bonds preceding proline residues the trans form is favoured only by a factor of around 4. The remaining two torsion angles are labelled φ for the N-Cα bond and ψ for the Cα-Co bond. The energetic favourability (due to repulsive interactions between non-neighbouring atoms) of the angles φ and ψ can be indicated on a 2-D map of the φ − ψ plane, which is called a Ramachandran plot.[22] Two of the most common secondary structures are the α-helix and β-strand, illustrated in figure 3.2. As the name suggests, in the α-helix the protein backbone is twisted into a right-handed helical shape with 3.6 amino-acid residues per turn. The structure is stabilized 44 by hydrogen bonds between the amine hydrogen and carbonyl oxygen of two amino acids separated by four positions along the backbone. Every amino acid in an α-helix is normally bonded to two others, however the last four residues at each end of the helix will only have one bond. The length of an α-helix is usually ten to fifteen residues, but can be as high as fifty.[22] In the β-sheet, segments of protein backbone (called β-strands) are aligned side-by-side, with hydrogen bonds between adjacent strands. The strands may be arranged in a parallel or anti-parallel fashion (shown in figure 3.2), and are generally not stable in isolation. Like the α-helix, the β-sheet involves hydrogen bonds between amine hydrogens and carbonyl oxygens, and each residue can be bonded to two others. Unlike an α-helix, however, the β-sheet does not necessarily involve bonds between residues that are in close proximity in the primary structure (although they frequently are). The β-sheet is a more extended configuration than the α-helix, because the backbone is more linear. The sheet can involve from two to several parallel strands, each usually between three and ten amino acids long. The strands can belong to different parts of the same polypeptide if it is able to loop back on itself, or separate polypeptides. Various types of secondary structure also exist which are based on the β-sheet. For example, a β-sheet can wrap around in a cylindrical shape to form what is known as a β-barrel, or two β-sheets can come together to form a sandwich-like structure.[23, 22] Another less common type of secondary structure, which is relevant to the present study, is known as polyproline. Proline residues have a distinctive cyclic side-chain and lack an amine hydrogen. Because of this, they are unable to form the normal hydrogen bonds that stabilize secondary structures such as α-helices and β-sheets. The nature of the side-chain also restricts the range of dihedral angles in a proline residue. In the polyproline-II (PPII) structure, the protein backbone can form a helix which is stabilized by interactions between solvent molecules and backbone polar groups. Other residues besides proline can also be part of PPII helical structures, especially glycine (which has no side-chain).[22] Proteins often form folded structures containing a mixture of stable secondary structures (see section 3.1.2). Due to the roughly spherical shape of these folded structures, these are commonly referred to as globular proteins. Because common secondary structure elements such as α-helices and β-sheets are relatively straight, globular proteins often contain sharp bends in the protein backbone near their surface. The most common types of stable turns are the β-turn, which has a hydrogen bond between the carbonyl oxygen of one residue and the amine hydrogen of the residue three positions further along the chain, and the less common γ-turn, which has a similar bond between residues separated by just two positions.[22] β-turns frequently involve glycine residues, because their lack of a side-chain allows them to more easily adopt the required dihedral angles. Regular recurring β-turns 45 Figure 3.2: Molecular representations of common secondary structures. In (A) is shown a short α-helix. In (B) and (C) are shown two β-strands bonded together in an anti- parallel (B) and parallel (C) fashion. In the colour image, amine nitrogen is shown in blue, and carbonyl oxygen in red. For simplicity all hydrogen atoms are hidden and only the first carbon atom of the side chain is shown. Hydrogen bonds are shown as dotted lines. Generated from PDB file 1ERT using RasMol.[25] can also lead to loose helical structures called β-spirals. Unlike α-helices these do not seem to rely on hydrogen bonds between turns in the helix, and are instead stabilized by hydrophobic interactions.[24] Protein segments that do not adopt any stable secondary structure are often categorized under the general term “random coil.” In a randomly coiled protein the conformational prop- erties of a particular residue will depend only on the conformation of its neighbours. Due to the lack of stable structure, a randomly coiled protein can rapidly change conformation (on a timescale on the order of micro-seconds) and can sample a wide range of configuration space. Because of this lack of order, random coils have higher conformational entropy than other types of regular secondary structure. The presence of glycine and proline residues tends to reduce the dimensions of randomly coiled polypeptides because they respectively allow greater flexibility and facilitate changes in direction in the protein backbone. Many unfolded proteins which were previously thought to be almost exclusively randomly coiled are now believed to contain significant amounts of ordered structure, such as polyproline- II.[22] 46 3.1.2 Higher Order Structure The overall three-dimensional shape of the protein backbone is called the tertiary structure, which in general can include a variety of secondary structural elements. It may specify the coordinates of all the atoms in the protein, including the side-chains.[26] Tertiary structure can be stabilized by a range of interactions, including ionic interactions between charged side-chains, hydrophobic interactions between side-chains and solvent molecules, and by hydrogen bonding or van der Waals forces. Often such interactions are part of regular secondary structures. Tertiary structure can also be stabilized by covalent bonds such as disulphide bridges between cysteine residues.[27] It may also include crosslinks between different protein strands, which may be bonded together to form extended three-dimensional networks.[28, 29] Many proteins adopt a single, well defined tertiary structure which is critical to their biological function. Such proteins often form compact domains made up of a few interacting secondary structural elements, and are called globular proteins (as mentioned previously in section 3.1.1). The tertiary structure of larger proteins may be divided into sections containing different structural elements, which may include distinct globular domains and randomly coiled segments. Further to tertiary structure is quaternary structure, which describes the aggregation and physical arrangement of multiple, distinct proteins. Quaternary structure occurs as the result of spatial and physical complementarity of the interacting surfaces of the constituent proteins. The surface properties of functional proteins must also prevent undesired binding to any other proteins that might be encountered.[22] A great variety of physical forms can result from protein sequences made up of only a small number of amino acids. Proteins may adopt stable folded structures, undergo triggered transitions between a limited number of possible configurations, or rapidly sample a wide range of disordered conformations. The physical structure of proteins is what allows them to perform such a multitude of specific functions. 3.2 Elastomers An elastomer is any polymeric material which exhibits rubber-like elastic properties, es- pecially that of high reversible extensibility. In other words, elastomers can normally be extended to several times their unstretched length without breaking and upon removal of stress will rapidly return to their original state. The other important characteristic of rubber-like elasticity is that restoring forces in elastomers are primarily associated with an increase in entropy rather than a decrease in internal energy, as would be the case for metals or other crystalline materials.[30] 47 Figure 3.3: An illustration of the difference in entropy between relaxed (A) and extended (B) states for a model polymer on a square lattice. In (A) a few states are illustrated for a polymer with a particular end-to-end separation (overall length). In (B) the length is doubled, with the chain-length of the polymer conserved, and it can be seen that there are fewer states consistent with the constraints. An ideal elastomer consists of a network of free chain segments connected by crosslinks, which can either be permanent covalent bonds or more labile physical entanglements. The deformability of physical entanglements can give rise to visco-elastic effects like stress re- laxation, where stress decreases over time at a fixed extension, and creep (also called flow), where permanent deformation occurs as the result of stress maintained for long periods of time. Ideal elastomeric behaviour requires that the crosslink density not be too high, oth- erwise the molecular matrix will be too restricted. The chain segments between crosslinks must also be flexible and relatively free of intermolecular interactions, so that they can adopt many possible conformations and hence have high entropy.[30] When an elastomer is deformed, the entropy decreases because the chain segments are restricted to statistically less favourable states. When stretched, for example, the molec- ular chains become more extended in the direction of the applied stress. There are fewer molecular configurations consistent with an extended state, as illustrated in figure 3.3, and hence the entropy is lower. Thermal fluctuations lead to an increase in the entropy of the molecular system (by driving, for example, lateral vibrations in molecular chains) and hence produce a restoring force.[31] Any realistic (non-ideal) elastomer will exhibit rubber-like mechanical behaviour only at sufficiently high temperatures and for sufficiently low rates of deformation. At low temper- atures there will be insufficient thermal energy to overcome attractive interactions between molecular chains, and the polymer will be frozen into a stiff glassy material, with high mod- ulus (ratio of stress to strain) and low extensibility (maximum strain before failure). As the temperature is increased, long-range coordinated molecular motion will become possible, and the modulus will typically decrease by a factor of around 1000. The temperature at which the transition from glass to rubber occurs is called the glass transition temperature and is denoted Tg. The value of Tg depends on the time scale over which measurements (e.g. 48 of modulus) are performed. For rapid deformations the molecular chains will not have suffi- cient time to change configuration in response to applied stress and the material will behave like a glass. In contrast, at long enough time scales, amorphous materials that appear to be stiff and glassy can slowly respond to stress, with strain increasing at constant stress or stress decreasing at constant strain. In fact, changes in time scale and temperature can be interchanged, with a shift in temperature related to a shift in the logarithm of time. This method can be used to combine measurements at different temperatures and frequencies, and is called time-temperature superposition.[32, 31, 33] The entropic nature of rubber-like elasticity gives rise to distinct thermodynamic effects. The most prominent is that the modulus (stiffness) increases with temperature, since the change in free energy associated with a given change in entropy is proportional to the abso- lute temperature. This means that tensile force at constant strain will increase linearly with increasing temperature. For example, at constant pressure and temperature, the differential change in free energy is given by[30] dG = dU + pdV − T dS, (3.2.1) where dU is the differential change in internal energy, pdV is the pressure-volume work, and dS is the differential change in entropy. The force exerted by the material under these conditions is given by[30] ∂G ∂U ∂S f = = − T , (3.2.2) ∂L p,T ∂L p,T ∂L p,T where L is the length or extension and we have assumed no change in volume during deformation. In an ideal entropic rubber there is also no change in internal energy upon deformation, so the first term can be dropped to give the expected result: ∂S f ≈ −T . (3.2.3) ∂L p,T Thus, measurements of the temperature dependence of the stress-strain relationship of a material can be used to elucidate the mechanism of elasticity. Because of the relationship between temperature and time scale (see above), the modulus will also increase as the rate of deformation is increased. Another property of elastomers that depends on the time scale (or, equivalently, tem- perature) is the capacity to store and release mechanical energy. This can be characterized in several ways, but the most relevant parameter here is the resilience. Resilience can be defined as the ratio of extractable work available during relaxation back to the unstrained state versus the work initially required for deformation, and is often reported as a per- 49 centage. Theoretically, for elastic materials it should be perfect (100%) for quasi-static deformations, and decrease with increasing amplitude and frequency (and decreasing tem- perature). However, once the material makes the transition from an elastic to a glassy material it should again increase.[34, 31, 35, 36] Polymer materials can often be softened with small molecules that are dissolved in the polymer, called plasticizers. They can be absorbed by a bulk polymeric material, causing the volume to increase, and function by increasing the separation between molecular chains and interfering with interactions between them. Hence, plasticizers will lower the glass transition temperature for amorphous polymers and reduce the degree of crystallinity for semi-crystalline ones. They can also reduce the elastic modulus and increase the resilience of elastomeric materials.[33] Elastomeric materials exhibit low modulus, high reversible extensibility, and high re- silience (elastic efficiency). These properties are all dependent on the mobility of the molec- ular matrix and its rapid response to applied stress. Because of this, all are suppressed at low temperatures and high frequencies, and can be enhanced by the presence of swelling agents. The mechanical properties of elastomers makes them ideally suited for applica- tions ranging from flexible parts and shock absorbers, to seals and adhesives, to biomedical devices. 3.3 Resilin Resilin is a type of elastic protein that was first discovered decades ago in elastic hinges and tendons of locusts and dragonflies.[37] It was found to rapidly return to its original shape even after being held stretched for days or weeks. It was also found to have very high resilience (see section 3.3.2). Because of these properties, it was given the name resilin, from the Latin word resilere: to spring back.[37] Further studies found resilin to be present in the cuticle of a wide range of arthropods.[34, 38] It can provide reversible elasticity and low-stiffness flexibility in tissues, and can store en- ergy by compression, extension, and bending. It occurs in extracellular deposits, usually in discrete combination with other fibrous proteins and with chitin, a hard non-proteinaceous polymer.[37, 39, 34] The impressive mechanical properties of resilin, detailed in section 3.3.2, could be highly desirable in industrial and biomedical applications. This has driven research into develop- ing synthetic materials with resilin-like properties. Recently, recombinant proteins based on naturally occurring resilins have been created and studied.[40, 41] Despite having sim- plified primary structure, these proteins showed mechanical properties similar to those of natural resilin and could be produced in quantities sufficient to make practical applications 50 a foreseeable possibility.[42, 43] Some of these applications will be described in section 3.4. 3.3.1 Natural Occurrence and Function The earliest observations of resilin were in the flight mechanisms of locusts[37, 39, 34] and dragonflies.[37, 34] It was originally thought to store energy during the upstroke of the wing and release it to help power the downstroke, hence increasing the efficiency of flight,[39] however it has been recently suggested that this may not be the case, and that it might instead act to passively stabilize the wing during flight.[36] It has also been proposed that it functions to reduce friction in the wing hinge, and restores mechanical flow in solid cuticle.[39] Resilin has also been identified in the wing veins of damselflies and dragonflies, where it provides flexibility to the wing and helps restore the shape of the wing during flight, and also acts as a damper to aerodynamic forces.[44, 45] It is also present in the wings of earwigs,[46] scarab beetles[47], and ladybugs,[47] where it is used to store energy during the folding or unfolding of the wings, and may help prevent fatigue damage in wing folds. Resilin has also been identified in the jumping mechanisms of fleas,[48, 49] froghoppers,[50, 51] cicadas,[52] and click beetles.[53] Powerful jumpers such as fleas and froghoppers must jump by storing energy in specialized regions of the cuticle, since the power output during jumping is greater than what can be directly provided by musculature.[48, 51] Due to the presence of large amounts of resilin in such insects, and the high mechanical efficiency of resilin, it was originally believed that most of the energy for jumping was stored directly in resilin deposits. However, it was later shown, by observing the deformation of cuticle during jumping of froghoppers, that energy is actually primarily stored in solid cuticle. The results suggest that resilin is used to store some energy, but more importantly provides flexibility, wear resistance, and helps restore the original shape of the insect after jumping.[51] Resilin has also been found as a key component of leg movement in cockroaches, wherein it can act as an antagonist to leg muscles.[54, 55] The high efficiency and long-range elasticity of resilin allows it to quickly restore leg position even for large displacements.[55] There are many other cases in which resilin-containing structures act as elastic antagonists in joints containing only one muscle. Examples include the pedipalps (pincers) of scorpions,[56] flagella in crabs and crayfish,[57] and leg joints in crayfish.[34] Resilin may also occur as an elastic antagonist in the leg joints of arachnids.[58, 59] Resilin plays an important role in the sound production organs of cicadas[60, 61, 62] and moths.[63] It was found to be part of resonant structures which operate at frequencies as high as 13 kHz in cicadas[62] and 52 kHz in moths,[63] which suggests that resilin retains its elastic properties even at very high strain rates. Resilin is also found in arthropods which must undergo rapid increases in body size, such 51 as ticks[64] and physogastric queen termites, which undergo a rapid increase in abdomen size following fertilization.[65] Under normal circumstances, arthropods must undergo ecdysis (moulting) to allow an increase in size of their rigid exoskeleton. The presence of resilin, however, can facilitate temporary changes in body size. In the case of the tick, for example, resilin is present in lamellar surface folds which allow the alloscutum to expand during feeding.[64] Resilin is also found in other specialized structures, such as the feeding mechanism of tsetse flies[66] and reduviid bugs,[67] the lens cuticle of dragonflies,[68] and (tentatively) in the venom apparatus of honey bees.[69] In summary, resilin occurs in a wide variety of organs with a wide variety of function. It may provide long range flexibility, mechanical energy storage, resistance to wear, and dynamic elasticity. It may occur in pure deposits of varying size (up to 0.7 mm long in dragonfly tendons), or may be interspersed with layers or filaments of chitin and/or solid cuticular proteins, leading to a variety of mechanical response in resilin-containing cuticular elements.[37, 70] 3.3.2 Mechanical Properties Resilin was initially found to be highly elastic, meaning it recovered its shape quickly and completely after deformation. Deposits of resilin were found in certain parts of insect cuticle exhibiting reversible extensibility and rubber-like mechanical behaviour.[37] Resilin was also found to be highly extensible, and could be stretched to over three times its unstretched length (200% strain) before breaking. It was also found to have very little mechanical flow; even when held stretched for extended periods (100% strain for two weeks), it returned to its original shape and regained its original mechanical properties within minutes of being released.[37, 39] When loaded, resilin samples were found to change their length in response to changes in stress within less than one second.[71] One of resilin’s most interesting and characteristic properties is its high degree of re- silience, which is a measure of elastic efficiency (see section 3.2). Several measurements were performed during the initial investigations on resilin. When stretched at at a bio- logically relevant rate of 600%/s (at a cyclic frequency of 50 Hz), the resilin-containing locust pre-alar arm was found to have a resilience of greater than 97%, meaning that for one half-cycle (extension followed by retraction), less than 3% of the energy used to deform the material is dissipated as heat.[37] For deformations of similar amplitude at up to 200 Hz the resilience did not decrease below 93%.[39] The resilience of the locust wing-hinge, another resilin containing organ, was found to be 97% at 20Hz.[34] Analysis of these early measurements suggested that the resilience began to decrease above 100Hz, indicating the beginning of the glass transition. However, this would make it 52 impossible for resilin to function as a rubbery elastic material in the sound producing organ of cicadas, which operate at frequencies up to 13 kHz, calling into question the validity of these results.[36] At approximately 4 kHz these sound producing organs were found to be at least 78% resilient despite being subjected to large amplitude deformations.[61] More recent dynamic mechanical testing on dragonfly tendons suggests that their resilience is even higher than previously reported. The glass-transition frequency was reported to be approximately 2 MHz, consistent with a material that could retain its elastic properties over the known range of biological frequencies.[36] Resilin was found to have relatively low stiffness (compared with other elements of solid cuticle), consistent with that of a rubbery material. The static elastic modulus (initial stress per unit strain) of resilin in the elastic tendon of dragonflies was found to be about 1.5 MPa, and in the pre-alar arm of locusts was found to be 4.7 MPa (possibly higher due to the presence of chitin lamellae).2 Only a small change in dynamic stiffness (a few percent) was observed at biological strain rates.[37, 71, 39] Another important mechanical property of resilin is its high fatigue lifetime. Resilin in fruit flies was found to be produced only during the pupal stage, meaning that resilin must survive for the entire adult lifetime of the insect.[40] In locusts, it was found that resilin was deposited during the first days or weeks of adult life, which supports the same conclusion.[37, 72, 73] As an elastic protein, resilin shares properties with a diverse group of other proteins. The mechanical properties of a few elastic proteins are presented in table 3.2 for comparison. Resilin is most similar to the vertebrate protein elastin, which also has high reversible exten- sibility, low stiffness, high resilience, and high fatigue lifetime. Given its similar mechanical properties, it is not surprising that elastin has a similar biological role to resilin; it also func- tions to store elastic energy, for example in connective tissues and arteries of vertebrates. Other elastic proteins share some, but not all, of resilin’s mechanical properties, depending on their functional roles. Viscid (flagelliform) silk from spiders and mussel byssus threads have very low resilience. They do, however, have high extensibility, which means they can absorb (and dissipate) a large amount of energy before failing (e.g. from an insect flying into a spider’s web). Conversely, collagen fibres have high resilience, but are very stiff and have low extensibility, allowing them to provide structural support and mechanical energy storage in connective tissues. Spider dragline silk is also very stiff and strong, but, like flag- elliform silk, also has low resilience, allowing it to act as a strong reinforcing structure while 2It should be noted that various conflicting values of the modulus of resilin have been reported in the literature (compare, for example, refs. [35], [70], and [20]). This appears to be due, at least in part, to an alternative definition of the term elastic modulus used especially in early papers (see [39] and [28]). Different units have also been used in different sources, and a range of actual measured values is to be expected. Numbers reported here are based on the author’s best attempts to choose representative measured values and convert them to the given measure and units. 53 protein modulus (MPa) extensibility strength (MPa) toughness (MJ/m3) resilience resilin 2 2 4 4 0.92 elastin 1.1 1.5 2 1.6 0.90 collagen 1200 0.13 120 6 0.90 proximal byssus 16 2 35 35 0.53 distal byssus 870 1.09 75 45 0.28 viscid silk 3 2.7 500 150 0.35 dragline silk 10,000 0.3 1100 160 0.35 Table 3.2: Mechanical properties of various elastic proteins. Modulus (stiffness) is given as the ratio of stress over strain for small deformations. Extensibility is the maximum strain before failure, given as a multiplicative factor of the unstretched length. Strength is the stress at maximum strain and toughness is the energy required to cause failure (obtained from the integral of the stress-strain curve). Resilience is the ratio of energy returned during return to equilibrium vs. the work done during deformation. All measurement are presumed to have been performed in conditions (e.g. temperature and humidity) that reflect the natural in vivo environment of each material and at appropriate rates of deformation. Reproduced with permission from reference [35]. simultaneously helping to dissipate energy. Dragline silk normally functions in a relatively dry aerial environment, however when immersed in water it swells and is transformed into a rubber-like material. Abductin and titin are two other elastic proteins that function to store mechanical energy in the shell-opening ligament of bivalve molluscs and in striated muscle fibres, respectively. All these materials have well-tuned mechanical properties that are specific to their biological function, and which could potentially be useful in industrial, biomedical, or other applications.[35, 74, 20] 3.3.3 Molecular and Structural Properties Initial studies indicated that resilin closely resembles an ideal rubber.[37, 28] By comparison to theoretical models, it was suggested that resilin is composed of a swollen network of flexible molecular chains connected by stable chemical crosslinks with little stable secondary structure.[28] X-ray diffraction and electron microscopy studies showed little evidence for regular structure or crystallization, even when resilin samples were stretched and slowly dried.[75] The rubber-like nature of resilin was confirmed by measurement of its thermodynamic properties. When held at constant extension, the stress was found to increase linearly with temperature, as expected for an entropic rubber (see section 3.2). The elastic contribution due to changes in internal energy was found to be positive, but small compared to the entropic contribution.[71] Elasticity in rubber-like materials is associated with high conformational entropy in the 54 A O OH N O O N OH O B Figure 3.4: (A) Structural representation of a di-tyrosine crosslink between two polypep- tides. The covalent bond created during the crosslinking process is shown as a dotted line. (B) Reactions used to produce di-tyrosine crosslinks from tyrosine residues. From top to bottom: enzyme-catalyzed crosslinking, as believed to occur for natural crosslink- ing of resilin in organisms;[79] photo-Fenton reaction, which can also create other oxidation products;[80] ruthenium-bipyridyl mediated photo-crosslinking method.[81] These reactions have all been used to successfully crosslink recombinant resilin-like polypeptides (see section 3.4.3). relaxed state, which is reduced upon deformation (see section 3.2 above). In resilin, it was originally proposed that the molecular chains between crosslinks were randomly coiled and free of intermolecular interactions. Entropy in the relaxed state would thus arise from the freedom of chains to dynamically sample many possible conformations.[71] More recent studies on synthetic peptides and recombinant proteins, however, suggest that some degree of regular secondary structure, especially β-turns and poly-proline II helices, may be formed. In the model proposed, configurational entropy in the unstretched state arises from dynamic interchange between different conformations.[76] For more details, see section 3.4 below. The precise nature of the crosslinks in resilin was originally unknown, although evidence (in particular the complete lack of mechanical flow) suggested that they were composed of stable covalent bonds, not merely physical entanglements.[37] Not long after the initial dis- covery, they were positively identified as being the result of di- and tri-tyrosine (see figure 3.4).[77] In nature these are believed to be formed as a result of the joining of tyrosine residues in resilin by peroxidase catalysis,[78, 29] which occurs as resilin is deposited.[72] The uncrosslinked protein was termed pro-resilin to distinguish it from the solid crosslinked material.[34] Early studies suggested that molecular chains between crosslinks were com- posed, on average, of approximately sixty residues.[28] The presence of di- and tri-tyrosine gives resilin characteristic fluorescence properties 55 that have been used to positively identify resilin proteins in a variety of organisms. At neutral and higher pH, resilin absorbs UV radiation with a peak in absorbance near 330 nm. At acid pH this absorption maximum shifts to approximately 285 nm. This pH dependence is believed to be the result of the ionization of phenolic groups in tyrosine compounds at pH > 7. The emission spectrum has a maximum near 420 nm regardless of pH. Therefore, analyzing the pH dependent fluorescent properties of insect cuticle allows resilin containing regions to be readily detected.[82, 55] Another property of resilin which is frequently used to to aid in identification of resilin containing structures is that it is stained by both methylene blue and toluidine blue.[37, 34] Resilin was shown to be optically isotropic when relaxed, but birefringent when de- formed, with positive birefringence in the direction of extension and negative birefringence in the direction of compression.[37, 28] Birefringence occurs when light passing through an anisotropic material is decomposed into two beams due to a difference in index of refraction for light with different polarizations. In rubbery materials such as resilin, stress induced birefringence is believed to be caused by alignment of molecular chains upon extension.[28] Resilin was found to be heat stable, retaining its properties even when heated to 125◦ Celsius and only starting to disintegrate between 140 and 150◦C. It was found to be readily digested by proteinases, but resistant to a variety of coagulants, fixatives, and tanning agents which normally have profound effects on proteins.[37, 28] This supports the proposition that the elastic properties of resilin are independent of any stable secondary structures such as α-helices or β-sheets, which would be disrupted by the high temperatures and chemical reagents tested. It also substantiates the stable, covalent nature of the intermolecular crosslinks. Resilin was found to be a highly swollen material, absorbing water up to a composition of 50-60% at neutral pH. It could also absorb other polar solvents, such as glycerol, ethylene glycol, and formamide, but was not penetrated by any non-polar solvents tested. The degree of swelling at equilibrium is determined by the balance between swelling forces and elastic restoring forces, and changes slightly with strain and pH changes. When dried, resilin becomes solid and glassy, but upon rehydration it was found to regain its original properties within minutes or even seconds. Therefore, absorbed water acts as a plasticizer, separating and preventing interactions between molecular segments (see section 3.2).[37, 28] Because resilin appears to be a homogeneous and isotropic material at the macroscopic scale, its interesting mechanical properties must arise as a result of its microscopic structure. The entropic nature of elasticity in resilin is believed to be an important factor in its high elastic efficiency and high fatigue lifetime.[74] The precise nature of the molecular structure of natural resilin, and the specific characteristics required for its mechanical behaviour, are not known, however recombinant resilin-like polypeptides have provided new insights as 56 well as opportunities for further study. 3.3.4 Amino Acid Composition and Sequence Early amino acid analysis of locust resilin found it to be composed of approximately 1/3 glycine residues, a feature common in other structural proteins including elastin, collagen, and silk fibroin (a protein found in silkworm silk). Resilin was also found to have moderate amounts of proline, alanine, aspartic acid, and serine. Elastin also contains a comparable amount of proline, which, due to its unique amine ring (see section 3.1), can impose con- siderable constraints on the protein backbone conformation. This might contribute to the lack of crystal formation observed in both elastin and resilin; fibroin, conversely, contains very little proline and has a high tendency to crystallize. Resilin was also found to be com- pletely lacking in sulfur-containing residues, ruling out the possibility of disulphide bond formation.[83, 34] Of the residues in resilin, 34% have hydrophilic side-chains; this is distinctly different from elastin, which contains only 5% hydrophilic residues.[83] Because resilin and elastin are both naturally swollen to similar degrees and have similar mechanical properties, this difference in composition suggests that either hydrophobic effects are not important for their elastic behaviour or that they have different mechanisms of elasticity. The identification of a resilin-encoding gene in the fruit fly Drosophila melanogaster (CG15920) revealed for the first time the full sequence of a resilin protein. It was found to have similar amino acid composition to locust and dragonfly resilins, with a high content of glycine and proline, especially in the elastic regions. The gene was identified by searching for gene products similar to polypeptide fragments extracted from locust resilin. The protein was found to be 620 amino acids long, and it includes an N-terminal signalling peptide (which suggests that the protein is secreted into the extracellular matrix) and a chitin binding domain (the RR2 variant of the Rebers–Riddiford consensus sequence) flanked by two elastomeric domains. Both of the elastomeric domains contain repeat sequences, in which a consensus sequence of amino acids occurs several times with slight variations. The N-terminal domain contains 18 repeats of the sequence GGRPSDSYGAPGGGN, in which the YGAP motif is fully conserved. This may be related to the role of the tyrosine (Y) residue in the formation of di- and tri-tyrosine crosslinks. The C-terminal elastomeric domain contains 11 repeats of the sequence GYSGGRPGGQDLG. The repeats from the N- and C-terminal repetitive regions are called type A and type B repeats, respectively.[84] The proline and glycine residues are highly conserved, suggesting that they may be structurally important. The recurring PGG motif suggests the presence of β-turns, and the regular spacing of this motif suggests the formation of an extended β-spiral structure.[84] Because glycine lacks a side chain, it tends to confer flexibility to the protein backbone, 57 making the formation of β-turns possible. Conversely, proline residues have a restrictive effect, and while they usually disrupt the formation of secondary structure such as α-helices and β-sheets, they can lead to the formation of the more extended (and hydrated) poly- proline II structure. The combination of proline and glycine, which has been observed in other elastic proteins, has been suggested to result in relatively flexible molecular chains that are conformationally free but do not form stable, compact structures.[85] The putative resilin-encoding role of the D. melanogaster gene was confirmed by the observation of characteristic resilin-like properties in recombinant proteins based on its se- quence (see section 3.4). Structural studies on these recombinant proteins have confirmed that they are highly disordered and appear to form some β-turn and poly-proline II struc- ture, however no evidence supporting the presence of β-spirals was found.[80] Putative resilin-encoding genes have also been identified in other species of Drosophila, as well as in other insects. Most putative resilin sequences identified to date contain one or more types of elastic repeat sequences which vary in both length and number of occur- rences. The relative content of amino acids in the proteins and the sequence details of the elastic repeats also show significant variance.[86, 87] Computer simulations of resilin-like polypeptides based on these gene sequences suggest that the proteins are highly disordered and should exhibit typical rubber-like elastic behaviour.[87] Structure prediction algorithms have also been applied to these sequences and support the disordered nature of the proteins, with the exception of the chitin binding domain, which was suggested to have more stable secondary structure.[88] The diversity of sequence properties of possible resilin-encoding genes has raised the question of what characteristics are important for their proper classification as such. Genes have been identified in several species whose products resemble other resilin proteins except for the lack of a chitin-binding domain. Because of this, it is possible that they do not occur in cuticle, in which case they should not be classified as true resilins.[86] Other properties (aside from the presence of a chitin binding domain) which are characteristic of resilin proteins include the presence of N-terminal repetitive domains containing YGAP or YGPP motifs, C-terminal repeats containing GYSGG, GYSSG, GYPGG or GYPSG motifs, an N- terminal signal peptide (indicating that the protein is secreted into extracellular regions), an N-terminal EPPVSYLPPS-like domain, and generally a high glycine content.[89, 86] The sequences of other elastic proteins (including abductin, byssus, gluten, spider silks, elastin, and titin) also contain elastomeric domains comprised of repeat sequences. Inter- molecular crosslinks (which may be covalent or otherwise) usually occur in separate domains, however resilin and abductin are exceptions, with crosslinks forming between residues within the elastic repeats. Many of these proteins also contain significant amounts of glycine and proline, and many contain significant amounts of β-turn structure. Spider silk and elastin, 58 for example, both have short repeats containing glycine and proline, which lead to regular β-turns which can form β-spirals. Gluten, which is an exceptional elastic protein because its elasticity is not directly related to its biological function, is also rich in glycine and proline, and contains β-turns and PPII structure.[90, 74, 84, 20] 3.4 Recombinant Resilin-Like Proteins The identification of a resilin encoding gene in the genome of the fruit fly Drosophila melanogaster has opened up the possibility of creating recombinant proteins based on natu- ral resilin.[84] A putative resilin-encoding gene (although later disputed) was subsequently identified in the genome of the mosquito Anopheles gambiae,[41, 43] and more recently such genes were identified in the flea Ctenocephalides felis and buffalo fly Haematobia ir- ritans exigua.[89] Putative resilin-encoding genes have been identified in several species of Drosophila and in other insects by searching for sequences similar to the previously identi- fied resilin genes, however these have not yet been produced experimentally.[86, 87] Proteins containing elements based on the products of the resilin-encoding genes have been termed resilin-like proteins or resilin-like polypeptides (RLPs). 3.4.1 Production and Sequence Synthetic polypeptides or proteins can be created by direct chemical reactions involving amino acids, or by creating genetically engineered organisms which express the desired molecules as gene products; the latter method is normally used for larger proteins. Creating a protein in this way is a multi-step process. First, the desired sequence of the protein must be decided, often based on the sequence of one or more naturally occurring proteins. Next, recombinant DNA must be created which contains the gene which will encode the desired protein. This recombinant DNA is then incorporated into a host organism, such as Escherichia coli. The modified organism is then grown in such a way that transgenic cells are selected and can be separated from non-transgenic cells. For a single-cell host, a population of modified organisms is created, whereas for a multi-cellular host the modified gene is usually incorporated into all cells of the organism (for example, by creating a modified embryo). The modified organism (or population) is then induced to express the desired protein, which must subsequently be extracted and finally purified.[91] The first recombinant resilin-like protein was created from exon 1 of the resilin-encoding fruit fly gene, and was accordingly called rec1-resilin. The protein was produced by cloning and expressing the gene in E. coli, and the resulting pro-resilin precursor protein was crosslinked (see section 3.4.3) to form a solid material with similar properties to natural resilin.[40] A simplified protein called An16 was created in a similar way from the puta- 59 tive resilin-encoding mosquito gene. An16 is comprised of sixteen copies of an 11-residue consensus repeat identified from the gene sequence, and, despite differences in amino acid composition and simplified primary structure, it was found to exhibit resilin-like mechanical behaviour.[92, 41] A repetitive RLP based on the fruit fly resilin sequence, called Dros16, was also created, and although it was found to have more stable secondary structure than the other resilin-like proteins, it also exhibited rubber-like properties.[92, 43] These results show that the elastic repeat sequences play a crucial role in determining the mechanical properties of resilin proteins.[43] The full length D. melanogaster protein, containing both of the elastic domains and the chitin binding domain, was also produced using recombinant methods.[93] Recently, additional resilin-encoding genes have been identified and cloned from a flea (Ctenocephalides felis) and buffalo fly (Haematobia irritans exigua), to produce two new RLPs called Cf-resB and Hi-resB, respectively.[89] Another new RLP has also been created from exon 3 only of the D. melanogaster gene, as well as one from exon 1 (similar to rec1-resilin).[80] Rec1-resilin contains 310 residues, most of which are part of the N-terminal repet- itive region of the full resilin protein. This region contains 17 approximate copies of the type A (N-terminal) elastic repeat sequence GGRPSDSYGAPGGGN.[40] Dros16 con- tains exactly 15 copies of the same repeat sequence.[92] The unnamed RLP created from exon 3 of the D. melanogaster gene contains 11 copies of the type B (C-terminal) re- peat GYSGGRPGGQDLG.[80] An16 is comprised of 16 copies of the type A elastic repeat AQTPSSQYGAP.[41] Cf-resB and Hi-resB contain both N-terminal and C-terminal repet- itive regions, and therefore contain both type A and type B repeats. The type A elastic repeats from all the proteins created so far all contain the highly conserved YGAP motif, which could potentially play a role in the formation of crosslinks between tyrosine residues. The An16 sequence contains significantly less glycine than the Drosophila resilins, and is missing the PGG motif which has been suggested to potentially play a role in the deter- mination of secondary structure (see section 3.3.3).[92] Although An16 was found to have resilin-like mechanical properties, it has recently been suggested that the product of the gene upon which it is based should not be classified as a true resilin. The gene lacks a chitin binding domain, and was found to be more similar to a D. melanogaster gene whose product has been identified as a mucin protein.[94, 89] For simplicity, however, it will still be treated as a resilin-like protein (RLP) here. 3.4.2 Stability and Purification Like natural resilin, all the recombinant proteins produced were found to be heat stable. For An16, Dros16 and rec1-resilin, no deterioration was detected after being kept at a temperature of 95◦C for 10 minutes. After 1 hour at 95◦C An16 and rec1-resilin were 60 still not significantly affected, whereas Dros16 resilin became partially degraded. After 4 hours at 95◦C, An16 was found to have some degradation, rec1 was significantly degraded, and Dros16 was completely degraded.[92] The heat stability of Cf-resB and Hi-resB was not tested in detail, but both were found to be unaffected by heating to 80◦C for 10 minutes.[89] The exon 1 and exon 3 proteins were also found to be stable at 100◦C for 30 minutes.[80] The high heat stability of synthetic RLPs has been used to aid in purification, since at high temperatures many other undesired proteins will denature and precipitate, facil- itating removal. The recombinant proteins were also found to selectively precipitate in relatively low concentrations of ammonium sulfate: approximately 20% for An16, Cf-resB, Hi-resB, and rec1-resilin, and 30% for Dros16. This property was used, along with high temperature treatment, to rapidly and easily achieve high purity protein solutions.[92, 42] An efficient induction method was also developed to achieve high-yield expression of RLPs, giving maximum volumetric productivity of 300 mg/L for rec1-resilin and 400 mg/L for An16. Unoptimized production of Dros16 was significantly less than for An16 and rec1- resilin, giving a yield of only 20 mg/L. Because of this, Dros16 has only been produced in small quantities, and has not been studied as extensively as An16 and rec1-resilin.[43] Yields of the exon 1 and exon 3 RLPs was also relatively low, at approximately 25 mg/L.[80] 3.4.3 Crosslinking In resilin, crosslinks occur as di- and tri-tyrosine complexes which join two or three tyro- sine residues, respectively. These stable, covalent crosslinks allow resilin to form a solid, mechanically active material, and also give it characteristic fluorescence properties (see sec- tion 3.3.3 above). The structure of di-tyrosine, and the reactions used to produce it, are illustrated in figure 3.4. Full length recombinant resilin was successfully crosslinked into a solid material using a peroxidase catalyst, in a method similar to that proposed for natural resilin.[93] Initial attempts to crosslink rec1-resilin using this method were successful but impractical; the reaction was found to proceed too quickly, resulting in the formation of oxygen bubbles.[40] It was, however, later used to crosslink the exon 1 and exon 3 RLPs in a controllable manner, and the exon 1 protein was found (by fluorescence measurements) to have a significantly higher di-tyrosine content than that from exon 3.[80] An alternative photo-chemical crosslinking method, previously found to catalyze di- tyrosine formation, was successfully adapted to allow crosslinking of rec1-resilin in a rapid and controllable manner.[40] This method requires the introduction of ruthenium-bipyridyl 2+ ions ([Ru(bpy)3] ) and an electron acceptor such as ammonium persulfate, followed by irradiation with an ultraviolet light source.[81] The exon 1 protein was also crosslinked using a citrate-modified photo-Fenton reaction, 61 in which Fe2+ ions are used to generate hydroxyl (OH) radicals from hydrogen peroxide. These in turn lead to di-tyrosine formation, as well as other tyrosine oxidation products which were found to give the crosslinked material strong adhesive properties.[80] Analysis of the extent of crosslinking in rec1-resilin found it to form a solid material when at least 17% of total tyrosine residues where involved in crosslinks, and to reach a maximum of 21% crosslinking. This is similar to the estimation of 25% tyrosine crosslinking in natural resilin.[40] Cf-resB, Hi-resB, An16 and Dros16 were also crosslinked using the photo-chemical method, with estimated di-tyrosine contents of 19.2%, 17.1%, 14% and 46%, respectively.[89, 43] The much higher degree of crosslinking in Dros16, without a significant increase in modulus, suggests a much higher degree of intramolecular crosslinking. This might be related to the higher molecular order observed in Dros16 (see section 3.4.4 below).[43] Crosslinking methods involving amino acids other than di- and tri-tyrosine have also been used in novel recombinant proteins which incorporate resilin-derived elastic repeat sequences. These are described further in section 3.4.6. 3.4.4 Secondary Structure Because the primary structure of the various RLPs created to date varies considerably, it is expected that there should be some variation in their secondary structure as well. Many different techniques have been employed to study the secondary structure of RLPs, including circular dichroism (CD), Fourier transform infrared spectroscopy (FTIR), Raman spectroscopy, small angle X-ray scattering (SAXS), and NMR. Measurements have been performed on both crosslinked and uncrosslinked materials, and the structural content ap- pears to be relatively unaffected by the degree of crosslinking. An16 and rec1-resilin were the first RLPs to be created and studied. CD spectra for An16 and rec1-resilin were found to be similar, despite the significant differences in their primary structure. The data suggests that the proteins are both highly disordered, with very little α-helical content and some β-sheet and β-turn structure. Standard analysis techniques for CD spectra suggest the presence of modest amounts of polyproline-II (PPII) structure, however a more detailed analysis showed that it might be more significant. In both An16 and rec1-resilin, the data suggests that PPII structures exist in equilibrium with some other conformation.[43] Similar results were obtained for Cf-resB and Hi-resB, with the data suggesting an equilibrium between random coil and partially ordered conformations (likely a combination of PPII and β-turns); the randomly coiled conformations were found to be more prevalent at higher temperatures.[89] These results are generally consistent with CD, FTIR, and NMR studies of polypeptides based on the repetitive sequences of D. melanogaster resilin, which showed a mixture of PPII, random coil, and β-turn structure.[76, 85] More 62 recent CD and FTIR studies on the products of exon 1 and exon 3 of D. mel. resilin showed a greater contribution from folded structures, such as α-helices and β-strands, but generally agreed with the description of a disordered protein able to sample many different conformations.[80] Interestingly, preliminary CD data for Dros16 suggests that it is more ordered than the other RLPs (with significantly more β-sheet structure), despite being based on the same elastic repeat found in rec1-resilin.[43] As mentioned above, this may be related to the higher degree of crosslinking in this material. NMR, Raman, and SAXS studies of uncrosslinked An16 pro-resilin also indicate a high degree of heterogeneity and dynamic disorder. NMR experiments detected minimal α-helix and β-strand structure, with the exception of the glycine (G) and alanine (A) residues in the YGAP motif, which appear to be slightly more ordered. This may facilitate self-association and subsequent crosslinking between tyrosine residues. The data does not support the presence of static β-turns and β-spirals, however it does not rule out the presence of “slid- ing β-turns,” in which isolated β-turns are able to form and reform along the molecular chain. It is also consistent with a random network model, in which the chains are highly disordered.[41] FTIR and CD spectroscopy were also used to study the conformational properties of full length recombinant resilin. It was found to have more folded secondary structure than An16 and rec1-resilin, containing a mix of α-helices, β-strands, and β-turns, however it still exhibited a significant percentage of random coil. As has been suggested for the other recombinant resilins, it is likely that the full length protein can dynamically sample many different conformations.[93, 95] Despite the evidence supporting significant amounts of disordered conformations, many of the recombinant resilins were found to coacervate (assemble or clump together) at tem- peratures below 4◦C, which indicates some degree of energetic self-association.[92, 89] Rec1- resilin and various polypeptides based on repetitive sequences in resilin were also found to self-assemble into fibrils.[85] The results all suggest that the elastic properties in resilin and resilin-like polypeptides are the result of relatively free molecular chains. Although some secondary structure, es- pecially β-turns and PPII helices, is formed, it appears to be highly unstable, so that a given molecular segment can dynamically sample many different conformations (especially randomly coiled ones). Elastin appears to behave in a similar way.[20] The distribution of secondary structure varies somewhat between the different RLPs, suggesting that the precise details are less important (to the elastic properties) than the general dynamic disorder. 63 Property (units) Natural Rec1 An16 Dros16 Cf-resB Hi-resB Modulus (kPa) 2000 25.5 5.7 - - - SPM penetration (nm) - 324 495 235 654 347 Extensibility (%) 200 250 350 - - - Resilience (%) 92 97 98 91 88 87 Di-tyrosine (%) 25 18.8 14.3 46 19.2 17.1 Swelling (%) 50-60 80 80 - - - Table 3.3: The (previously reported) mechanical properties of natural dragonfly resilin[37, 35] and recombinant resilin-like proteins: rec1-resilin,[40, 43] An16,[41, 43] Dros16,[43] Cf- resB,[89] and Hi-resB.[89] The modulus is given by the ratio (derivative) of stress vs. strain for short extensions. The scanning probe microscope (SPM) penetration should be inversely related to the modulus. Extensibility is given as the average maximum strain before failure. Resilience values quoted for the recombinant proteins are for full-cycle SPM compression measurements performed at a strain rate of 1 Hz. The di-tyrosine content is reported as a percentage of total tyrosine. Swelling is given as the percent concentration of water at equilibrium in unstretched samples. All measurements reported are presumed to be taken at room temperature. 3.4.5 Mechanical Properties The mechanical properties of the recombinant proteins resemble those of natural resilins, as shown in table 3.3. The recombinant resilins have been found to have much lower modulus than natural resilin, and, although this discrepancy has not been widely discussed, it could be due to their higher degree of swelling, the lack of the more structure-forming chitin binding domain, or due to more intramolecular crosslinks due to either increased proximity of crosslinking sites in the primary structure or the different crosslinking process used. The scanning probe microscope (SPM) penetration measurements can be used to compare the modulus (stiffness) of different materials, with a higher penetration indicating a lower modulus. These suggest that Dros16 has a higher modulus than An16 and rec1-resilin, which would be consistent with its higher degree of crosslinking and greater composition of stable secondary structure. A few different measurements of resilience using various methods and at various deformation rates are available, and, although the results vary slightly, all show that the resilience of recombinant resilins should meet or exceed that of natural resilin. Full-range dynamic mechanical testing has yet to be published for the recombinant resilins. The effects of hydration and temperature on the molecular mobility and elastic prop- erties of rec1-resilin have recently been studied in greater detail.[96] The glass transition temperature for the dehydrated protein was found to occur at around 180◦C, where a change in specific heat capacity was found to occur (at a heating rate of 5◦C/min). This transition temperature was found to decrease rapidly as the level of hydration was increased. As the water content was increased from 0 to 10%, the mechanical properties were found to change 64 dramatically, with the hardness decreasing significantly and the hysteresis between mechan- ical loading and unloading curves increasing significantly; this is consistent behaviour for a material in transition from the glassy regime to the elastomeric regime. The resilience of fully hydrated resilin was measured as a function of temperature (for cyclic deformations at a frequency of 1 Hz). At room temperature it was found to be approximately 92%, and did not vary significantly between 7 and 70◦C.[96] Despite the differences in primary and secondary structure among the different RLPs, their mechanical properties have been found to be relatively similar. This supports the idea that these are the result of generic molecular features rather than specific sequences. Studying a variety of RLPs can help shed light on which aspects are important for imparting particular elastic properties. The flexibility allowed by recombinant techniques enables the tweaking of properties for applications and also facilitates bulk production of materials. 3.4.6 Applications of Resilin-like Polypeptides The unique nature of the mechanical properties of resilin and other elastic proteins has motivated research into their potential applications. Elastin and elastin-like polypeptides have been studied extensively, and show promise for use in bioengineering and biomedicine. Rec1-resilin has also been the subject of several studies to evaluate the feasibility of using RLPs in various applications. The protein showed interesting surface binding and self- organization properties and the responsiveness of the protein to stimuli such as temperature and pH could potentially be useful for things such as chemical sensing and drug delivery systems. More recently, recombinant proteins have been created which combine resilin- derived elastic repeats with other polypeptide sequences to create materials with specific mechanical and/or biological properties.[20, 95] The surface adsorption properties of rec1-resilin have been studied using atomic force microscopy (AFM) and scanning thermal microscopy (SThM); it was found that the surface coverage and nanostructure of the protein depends on both pH and on the properties of the adsorbed surface, such as hydrophobicity.[97] The surface binding ability and morphology of rec1-resilin on a gold substrate were also found to depend on pH.[98] In a separate study, it was found that rec1-resilin could be used to stabilize and control the growth of gold nanoparticles. Bioconjugates of rec1-resilin and nanoparticles were created, and the particle size, photophysical properties, and interfacial interaction were all found to depend on pH.[99] Uncrosslinked rec1-resilin was also found to exhibit rare dual phase behaviour, having both an upper and lower critical solution temperature (UCST and LCST). At intermediate temperatures the protein was soluble, whereas at low temperatures (< 6◦C at pH 7.4) ◦ it formed a high density network and at high temperatures (& 70 C) it formed discrete 65 spherical aggregates. The UCST was found to be tunable by changing the pH of the protein solution, which also affects the protein’s fluorescence properties.[100] The self-assembling properties of polypeptides based on D. melanogaster resilin have been studied, and they were found to form fibrous structures, with some variation de- pending on the particular sequence. This behaviour is promising for the development of high-performance resilin-based elastic materials.[85] A resilin-like protein called RLP12 was created containing elastic repeats from the pre- viously studied D. melanogaster gene product. It also included sequences for cell binding, proteolytic degradation, and heparin binding (to permit sequestration and delivery of cell growth factors). These elements were chosen to create a protein that could mimic the na- tive extracellular matrix and act as a scaffold for tissue engineering applications. Lysine residues were added outside the elastic repeats to allow an alternative crosslinking mecha- nism. The tyrosine residues in the elastic repeats, which are used for crosslinking in nature, were replaced with phenylalanine to allow another possible method of crosslinking. Despite the change in sequence of the elastic domain, the difference in crosslinking method, and the addition of biologically active domains, the resulting protein resembled other recom- binant RLPs in terms of mechanical behaviour and conformational properties.[101] It was found that the modulus of the protein could be controlled by changing the concentration of uncrosslinked polypeptide and the amount of chemical crosslinker added.[102] The results suggest the feasibility of using resilin-like polypeptides in tissue engineering applications for such things as vocal folds, blood vessels and cardiac tissues.[101, 102, 95] A few recombinant proteins were created which incorporate resilin-based elastic repeat sequences and globular domains to mimic the composition of the muscle protein titin. At low strains the restoring force arises from the decreased conformational entropy of the resilin elastic repeats and folded globular domains, hence the material exhibits high resilience. At high strains the globular domains can be reversibly unfolded, lowering the resilience and in- creasing the capacity to dissipate mechanical energy. The multi-modal elastic behaviour and other mechanical properties of the material are comparable to the passive elastic properties of muscle fibres, and could be useful for engineering artificial muscles.[103] A recombinant protein was also created which contained sequences derived from resilin, elastin and collagen, in order to combine the elasticity of the first two with the resistance to deformation of the latter. The resulting protein was found to self-aggregate into fibrillar structures, and had a modulus comparable to that of the constituent elastic proteins.[104] These studies show that the mechanical properties of polypeptides which incorporate resilin-like repeat sequences are relatively insensitive to the presence of other types of do- mains. The modular design of these recombinant proteins enables the creation of materials which combine particular mechanical and biological functions. Currently a few designs have 66 been explored, and show promising features relevant to a variety of applications. Given the extensive catalogue of proteins and polypeptide domains to choose from, the possibilities when designing new materials are virtually endless. 67 Chapter 4 Experimental Methods In this chapter the practical aspects of the experiments performed will be described. Refer- ences to previous chapters will be made as necessary, especially in regards to the theoretical NMR background. Some results of experiments performed for verification and calibration will be presented here, while the more relevant results will be presented in the following chapter. 4.1 Materials Used Samples of resilin were obtained from Dr John Gosline, Department of Zoology, University of British Columbia. Carbon-13 (chemical shift) and deuterium (quadrupole coupling) experi- ments (see sections 4.4 and 4.5 respectively) were performed on sections cut from previously created, crosslinked rings of An16. These had originally been subjected to mechanical test- ing and then kept frozen in storage for several months before the NMR experiments were performed. Quadrupole coupling measurements were also performed on resilin-containing elastic tendons dissected from dragonflies (Aeshna sp.) which were extracted and then stored in the freezer for approximately 18 months before testing. These tendons were also used in the multi-quantum (MQ) experiments (see section 4.6), along with pieces of An16 and rec1-resilin extracted from samples created approximately 12 months prior. Over the course of the present study, the samples were stored in the refrigerator in a deuterated aqueous solution containing phosphate-buffered saline (PBS) and a protease inhibitor (PI) cocktail (Sigma Aldrich P2714). The tendons were kept frozen when not in use. In the MQ experiments the samples were blotted dry prior to testing to reduce unwanted background signals. During testing, the samples showed signs of minor mechanical degradation, possibly as a result of mechanical damage, proteolytic digestion, or other damage during storage. The An16 ring samples had previously been tested to failure, which may have contributed some microscopic damage. The maximum extensibility of the An16 rings was found here to be under 100% strain, compared to over 300% strain observed by others. Attempts to perform measurements on stretched dragonfly tendons also showed lower than expected maximum strain. When held stretched for extended periods, the An16 samples were found 68 power amplifier pulse RF computer programmer transmitter synthesizer receiver probe A/D DSP converter mixer receiver pre-amp Figure 4.1: A simplified block diagram of the spectrometer setup used in the current ex- periments. The pulse program is designed using computer software and effected by the hardware pulse programmer and RF transmitter. Multiple channels with different carrier frequencies and discrete amplifiers can be used for simultaneous manipulation of magnetiza- tion from different nuclei. During detection, the signal induced in the probe coil is amplified and passed to the receiver. The resulting data is ultimately acquired and analyzed using the computer. For more details, see reference [105]. to slowly become permanently deformed; although no published experiments on the effects of long-term extension on An16 are known, natural resilin was previously found to recover completely even after extended periods of strain. The degree of crosslinking (which was not tested) could also affect the mechanical properties of the samples and possibly contribute to the deficiencies observed. 4.2 NMR Setup All experiments were performed on a home-built solid-state NMR spectrometer with a proton resonant frequency of approximately 363 MHz and a 89 mm vertical bore. The magnetic field was approximately 8.4 tesla (from a liquid helium cooled superconducting coil) and a shim-set of 13 shim coils was available for reducing magnetic field inhomogeneity (plus one uniform field coil to correct for drift). A diagram of the setup is shown in figure 4.1. Various probes, both commercially available and home-built, were used in the different experiments (see individual sections below). Pulse programming and spectral analysis was performed using a software package called XNMR (previously developed in the lab). 69 o-ring sample o-ring internal thread fixed adjustable end end clamps Figure 4.2: A photograph of the stretching apparatus used to control the length of the samples during testing. The sample is held in place by clamps and o-rings are used to seal both ends so that the sample can be kept hydrated. One end is fixed by a protrusion, while the other end can be adjusted to control the length (strain) of the sample. 4.3 Stretching Setup A special holder, shown in figure 4.2, was designed and built to allow the length (strain) of a sample of resilin to be held fixed while collecting NMR spectra. The apparatus was made to fit inside a standard 5mm glass NMR tube (approximately 4mm internal diameter). The sample is held in place by a normally-closed clamp cut into Delrin (polyoxymethylene) rod with a laser cutter, which can be pried open for loading. One end is terminated by a protrusion, allowing it to be held in a fixed position by tension, while the other end is adjustable. Both ends are sealed with o-rings made of Viton (a fluorinated rubber) to allow samples to be kept hydrated during testing. At low strains (and hence low forces), the friction of the o-ring was sufficient to maintain a fixed extension. At higher levels of force, a screw and washer could be inserted into an internal thread on the adjustable end of the holder to fix it in place. 4.4 Carbon-13 Spectroscopy The purpose of these experiments was to investigate the secondary structure of resilin samples, and also any changes that occur as the material is stretched. This was done by measuring the (natural abundance) carbon-13 (13C) chemical shift spectrum of An16 samples as a function of extension, which was controlled using the apparatus described in section 4.3 and measured using a ruler. The samples were cut from previously created rings, with dimensions of approximately 10 mm long by 4 mm wide by 1 mm thick. Chemical shifts for the α-carbons and side-chain carbons were calculated using the SimPred[106] web server for pure α-helix, β-sheet, and random coil secondary structures. These were then compared to the observed peak positions. Because an absolute chemical 70 shift reference was not available, the spectra were first aligned such that for each type of secondary structure the average deviation of peak positions was zero. The standard deviation of the difference between observed and calculated peak positions could then be used to compare the consistency of the data with the different types of secondary structure. The difference between α-carbon and β-carbon peak position for each residue was also compared to calculated values, since this method also requires no absolute reference. In addition, this helped to increase sensitivity as these peaks generally move either farther apart or closer together for different types of secondary structure. A double resonance probe tuned for protons (1H) and 13C was used to allow NOE en- hancement (see section 2.6.1) and decoupling on the 1H channel. A 5 mm (hand-wound) horizontal probe coil was used for excitation and detection. The primary transmitter fre- quency was centered in the 13C spectrum near 91.4 MHz. A modified pulse sequence, described in section 4.4.1 below, was used to acquire the spectra. Shims were adjusted (by minimizing the line-width in the proton spectrum) whenever a sample was installed in the probe. An automated algorithm, described in reference [107], was used to facilitate opti- mization of the shim settings. Resonant peaks were fit using the XNMR software package, and assignment of peaks was done by comparison to previously assigned spectra.[108] 4.4.1 Pulse Sequence The pulse sequence used for the 13C experiments is illustrated in figure 4.3. The first part of the sequence, labelled saturation in the figure, is made up of a series of alternating pulses and delays on the proton (1H) channel. These serve to saturate the 1H magnetization, or in other words eliminate the equilibrium magnetization normally aligned with the static magnetic field. Due to dipole couplings between 1H and 13C nuclei, this will result in an increase in the equilibrium 13C magnetization, leading eventually to greater detected signal. This is called the nuclear Overhauser effect (NOE), and is described in section 2.6.1. The second part of the sequence, labelled bg-elim, is included to reduce signals from materials outside the probe coil (e.g. from the probe itself). This is done by applying π 13 two 2 pulses back-to-back on the C channel. The phase difference between the pulses is alternated, such that nuclei in the coil undergo a net rotation of either zero or π radians. The pulses are followed by a fixed dephasing period to eliminate any transverse magnetization. The receiver phase is adjusted so that the two opposing cases add constructively. Conversely, signals from nuclei which undergo smaller rotations (due to their distance from the coil) will be cancelled out by this phase cycling. π After a standard 2 excitation pulse, a fixed duration spin echo is included. This creates a delay between the last pulse and detection, which eliminates interference due to probe ring-down. The spin echo sequence is composed of a single π pulse flanked before and after 71 Figure 4.3: The pulse sequence used to acquire 13C spectra for chemical shift analysis. The sequence starts with a series of pulses to the proton (1H) channel which saturates the 1H spins and enhances the 13C magnetization via the nuclear Overhauser effect (NOE). This is followed by a pair of pulses whose phase is cycled in order to reduce background signal π from materials outside the probe coil. Carbon magnetization is then excited with a 2 pulse, which is followed by fixed duration spin echo to allow signal acquisition without interference from probe ring-down. During detection the proton channel is continuously irradiated to eliminate the effects of 13C- 1H dipole couplings. The phase cycle of the various pulses and of the receiver are indicated in the figure. 72 by delay periods of equal duration. The pulse inverts the magnetization so that evolu- tion in the proceeding delay period is the reverse of evolution during the preceding delay (for interactions such as chemical shielding, dipole couplings, and magnetic field inhomo- geneities). The net result, in the absence of any interactions that are not refocussed, is that the magnetization will return to its original state (hence the echo) at the end of the sequence. After the spin echo sequence, the carbon NMR signal is acquired. During detection, the proton channel is continuously irradiated, causing the proton magnetization to nutate. This causes the dipolar interaction between 13C and 1H nuclei to be averaged away, reducing any line-broadening or splitting that might otherwise occur. This technique is an example of heteronuclear decoupling. π The phase of the sequence on the carbon channel is also increased by 2 radians, and the entire experiment repeated. This results in a four-step phase cycle, as indicated in the figure. This cycle is then repeated many times to increase the signal to noise ratio (SNR) to acceptable levels. Experiments performed here were typically repeated 100,000 times, in blocks off 10,000 scans each (to check for any temporal variation). The transmitter power π was set so that the pulse width required for a rotation of 2 radians was 3.5 µs in the carbon channel and 8.5 µs on the proton channel. The delay period in the saturation part of the experiment (τH ) was set to 100 ms, while the delays in both the bg-elim and spin echo sequences (τbg and τe) were set to 1 ms. 4.5 Quadrupole Coupling Measurements Experiments were performed to measure the residual quadrupole couplings experienced by deuterium nuclei, both in exchangeable hydrogen sites in the sample and in deuterated water molecules in the solution absorbed into the material. The distribution of residual couplings will depend on the degree of alignment of molecular chains in the protein matrix, due to incomplete averaging of the quadrupole coupling Hamiltonian (see sections 2.8.4 and 2.8.3 for more details). Experiments were performed on samples of An16 stretched to various levels of extension using the stretching apparatus described in section 4.3 above. It would generally be expected that at higher strains the size of the residual couplings will increase because the solvent molecules (and other molecular segments) will experience an increasingly anisotropic environment.[109] Attempts were also made to perform experiments on individual dragonfly tendons, which were stretched using an alternative setup due to the small size of the tendons (approximately 500 µm long by 50 µm in diameter). In this setup the 5 mm NMR tube was replaced with a 1 mm (approximate external diameter) glass capillary. Gorilla Glue (a commercially 73 Figure 4.4: The pulse sequence used to investigate residual quadrupole couplings in π stretched resilin samples. The sequence is made up of a normal 2 excitation pulse fol- lowed by a spin echo sequence composed of a single π pulse flanked by two delay periods of equal duration. The echo refocusses the effects of chemical shielding and magnetic field inhomogeneity, but not those of the quadrupolar interaction. The experiment is repeated with the echo delay (t1) increased incrementally, yielding a two-dimensional spectrum. available polyurethane glue) was used to attach the sample at both ends to short pieces of glass fiber optic cable (approximately 100 µm in diameter) to facilitate manipulation. After drying, the sample and attached fiber optics were inserted into the glass capillary. Both ends were sealed using silicone glue, with one end pre-dried and punctured to allow the fiber optic to slide in and out to control the length of the sample. Friction between the fiber optic and silicone was sufficient to maintain constant extension. Samples were wet with deuterated solution containing PBS and PI and blotted dry before being sealed into the capillary. Small pieces of An16 (with dimensions of approximately 1.0 by 0.2 by 0.2 mm) were cut from rings and also used in the same setup during testing. A single resonance probe (tuned to the deuterium frequency at approximately 55.8 MHz) was used for all experiments. Horizontal probe coils were hand-wound for the two different sizes of glass tube used. 4.5.1 Pulse Sequence The pulse sequence used for quadrupole coupling measurements is illustrated and described in figure 4.4. It requires only a single channel tuned for deuterium (2H), and is made up π of a simple 2 excitation pulse followed by a spin echo sequence (described in section 4.4.1 above). To cancel signals from ring-down following the π pulse (and to help eliminate any effects due to variations in pulse angle) the phase of the π pulse is alternated between being in phase and 180◦ out of phase with the excitation pulse. This is also phase cycled between x and y (in conjunction with the receiver phase), giving a four-step phase cycle overall. The spin echo refocusses effects such as chemical shift, dipole couplings, and magnetic field inhomogeneity, but does not refocus the quadrupolar interaction. This means that if the duration of the delay periods (t1) is changed, signal components from nuclei with non- 74 40 30 intensity 20 30 10 20 0 10 -10 5 -20 -30 -40 30 20 10 0 -10 -20 -30 Figure 4.5: Contour plot of a two-dimensional spectrum of a deuterated glycine crystal for one particular orientation. The direct (f1) dimension contains quadrupole couplings as well as chemical shifts and field inhomogeneities, which in this case are both small compared to the full static quadrupole couplings. The indirect (f2) dimension should contain only quadrupole couplings. As expected, the signal intensity is concentrated on the diagonals. The signal intensity scale is given in arbitrary units (here and in all following spectra). zero residual quadrupole couplings will be modulated at the quadrupole frequency. In the current experiments, the delay period was incremented for several experiments, producing a two-dimensional data set from which a two-dimensional spectrum could be generated. In the indirect dimension, spectral intensity at non-zero frequencies is indicative of non-zero residual quadrupole couplings. For a single coupling frequency (i.e. for a single crystal orientation) the spectrum should be split in both dimensions by the same frequency, pro- ducing intensity at the four corners of a square. A continuous distribution of coupling frequencies (i.e. a powder pattern), should result in a roughly x-shaped spectrum. In both cases, the direct dimension might be broadened or skewed by any interactions which do not appear in the indirect dimension because they are refocussed by the spin echo sequence. Because no hyper-complex data set is collected, the indirect dimension is necessarily sym- metric for positive and negative frequencies, however this is not a serious concern since π the quadrupole splitting is by nature symmetric. The 2 pulse width was 4.00 µs for the experiments performed on An16 samples. 4.5.2 Trial Materials To verify that the experiments were working correctly, two trial materials were tested. The first was a deuterated glycine crystal, produced by dissolving glycine in a deuterated aqueous solvent and then allowing the solvent to slowly evaporate. A crystal approximately 2 mm 75 long by 1 mm wide by 1 mm thick was produced using this method. A two-dimensional spectrum was collected (shown in figure 4.5), and, as expected, showed equal splitting in both dimensions. Also, as expected, the magnitude of the splittings was different for a different orientation of the crystal (with respect to the static magnetic field). The second test sample was a commercially available rubber band soaked in perdeuter- ated benzene. This was mounted in the stretching apparatus described in section 4.3, and two dimensional spectra were collected for different extensions. The resulting spectra, an example of which is shown in figure 4.6, showed some deviation from the results expected for a deuterated swelling agent in a stretched material. The spectra contained significant signal intensity off the diagonal, most likely resulting from physical exchange between ab- sorbed and vapourous benzene. Despite their irregular shape, the spectra did show an increase in splitting with increasing extension, as shown in figure 4.7. This increased split- ting corresponds to an increase in alignment of molecular chains in the rubber band. The size of the splitting was found to follow a roughly linear relationship with respect to the strain, as shown in the figure, however a more detailed model predicts a linear dependence of the splitting on λ2 − λ, where λ = 1 + is the extension ratio and is the strain. This model yielded a slope of 50.6 Hz, which is consistent with previously reported results for a strained rubber with a relatively low absorbed volume fraction of deuterated benzene.[109] The one-dimensional spectra shown in the figure were generated by performing a Fourier transform only in the indirect dimension and examining the slice corresponding to the first data point in the direct dimension. 4.6 Multiple-Quantum Experiments Experiments were performed to measure the distribution of proton (1H) residual dipole cou- plings in resilin samples by studying the development of multiple-quantum (MQ) coherence (see section 2.10 for more theoretical background on this subject). Experiments were per- formed on three samples for comparison: An16, rec1-resilin, and dragonfly tendons. For the recombinant resilin-like proteins, small pieces (estimated to be approximately 5-10 mg dry protein) were extracted from previously created crosslinked samples. These were inserted into a specially built holder made of Kel-f, a fully chlorinated/fluorinated polymer plastic (polychlorotrifluoroethylene - PCTFE) to decrease background signal from protons. The holder was built with a threaded cap which could be closed and sealed with Teflon tape to prevent the sample from drying out. The tendon sample was a collection of tendons (esti- mated to be approximately 0.5-1.0 mg total dry mass) in a small glass tube approximately 1 mm in diameter. All samples were stored in a deuterated aqueous solution containing PBS and PI, and blotted dry prior to testing. This was slightly more difficult with the 76 100 intensity 50 10 20 30 40 0 50 60 -50 -100 -200 0 200 400 600 Figure 4.6: Contour plot of a two-dimensional spectrum of a rubber band soaked in deuter- ated benzene and stretched to 75% strain. The particular shape was not as expected for a strained elastomer, but was consistently produced under various experimental conditions. For an unstrained elastic band the spectrum collapses into a single peak. tendon sample, but efforts were made to remove as much solution as possible. Experiments were performed in a single-resonance probe tuned for protons with a resonant frequency of approximately 363.6 MHz. 4.6.1 Spectrometer Tuning Because of the large number of pulses involved in the sequences used for the MQ experi- ments (see section 4.6.2 below), steps were taken to fine tune the spectrometer setup prior to π performing the experiments. The first process was to carefully optimize the 2 pulse widths. π This was done using a flip-flip sequence, in which a series of 2 pulses are applied (with constant phase) and data is sampled between the pulses. For an ideal sample with ideally tuned pulses, the resulting data points should repeat the cycle positive-zero-negative-zero, with the magnitude of the positive and negative signals decreasing over time due to de- phasing and relaxation. In reality there will be some additional oscillations which must be minimized to find the best pulse width.[110] The second step was to tune the phase transients which arise due to a small shift in phase during the rise and fall of the pulse envelope. These must be balanced so that the shift during the rise is cancelled by the shift during the fall, otherwise an overall phase shift will accumulate as more pulses are included. This is done using a flip-flop sequence, which is similar to the flip-flip sequence except that every second pulse has its phase inverted. In 77 8 0% 150 25% 50% 0 Hz 100 75% 6 50 0 0 25 50 75 48 Hz 4 82 Hz 2 122 Hz 0 -200 -100 0 100 200 Figure 4.7: Quadrupolar splitting spectra for a rubber band soaked with deuterated benzene at various levels of strain. The one-dimensional spectra were generated by performing a Fourier transform in the indirect dimension and taking only the slice corresponding to the first time point in the direct dimension. As expected, the spectrum is split to an increasing degree at higher extensions. The inset shows the magnitude of the splitting as a function of strain, which exhibits an approximately linear relationship with a slope of 165 Hz (per 100 percent). The peak at zero frequency is likely due to unabsorbed benzene vapour. 78 Figure 4.8: An illustration of one cycle of pulses used for excitation and reconversion of multiple-quantum (MQ) coherence. The general scheme of MQ excitation is explained in section 2.10.1. Assuming that spins interact primarily via dipole couplings, the sequence yields an effective Hamiltonian that will excite quantum coherence in multiples of two (double-quantum excitation Hamiltonian). The four 180◦ (π) pulses are included to help compensate for imperfect pulse angles. The pulses are equally spaced, separated by delay intervals with a duration of tc/12 (excepting the first and last half intervals of tc/24). The sequence is adapted from reference [17]. theory only a single data point need be acquired before each pulse, however the sequence used the data was oversampled to make the results easier to read. The ideal result would be a repeating cycle from positive to zero and back to positive, but in reality there will be some oscillations due to frequency dispersion which must be minimized (as in the flip-flip sequence).[110] The phase transients can be optimized in a few different ways: by changing the length of the cables used between the amplifier and the probe, by adjusting slightly the tuning capacitors in the probe, and by adjusting the output coupling of the amplifier. To allow the latter adjustment, an RF tube amplifier was added in series with the normal high power amplifier (see figure 4.1). Both types of calibration were initially performed with a small glass bulb filled with water and then refined for the actual samples tested. 4.6.2 Pulse Sequence A generic method for creating and detecting multiple-quantum (MQ) coherence was de- scribed in section 2.10.1 and figure 2.8. The pulse sequence used to create a DQ Hamiltonian in the present experiments is described in this section. Figure 4.8 shows the basic building block of the full pulse sequence. The cycle is made up of twelve pulses separated by delay intervals, with rotation angles and phases as indicated in the figure. The toggling frame approach (described in section 2.9.1) can be used to easily calculate the effective average Hamiltonian over the course of one cycle. We can assume that the unmodified Hamiltonian corresponds to the homonuclear dipolar interaction (see section 2.8.2), ˆ (homo) ˆ ˆ ˆ ˆ HD = −DIJ 3IzJz − I · J , (4.6.1) 79 1 2 where DIJ = 2 ~d 3 cos θ − 1 . Then, the resulting average Hamiltonian over one full cycle (as shown in figure 4.8) will be given by D H¯ˆ = IJ Iˆ+Jˆ+ + Iˆ−Jˆ− . (4.6.2) DQ 6 This corresponds to the DQ excitation Hamiltonian already mentioned in equation 2.10.7, with ω = DIJ /6~. The total excitation time is given by the product of the time for one cycle by the number of cycles, texc = tcnc. The excitation time is usually scaled by a duty cycle correction factor to account for the duration of the pulses, yielding an effective evolution time of[17] 3 τ = t 1 − ψ , (4.6.3) exc 4 where ψ = 16 tp is the duty cycle and t is the π pulse width (ranging from 2.60 to 3.35 µs tc p 2 in the present experiments). As described in section 2.10.3, MQ coherence of different orders can be distinguished by changing the phase φ of the reconversion period relative to the excitation period. This is equivalent to performing a rotation of angle φ to the angle of all the pulses shown in figure 4.8. The result of this (from equation 2.10.16) is that (after excitation and reconversion) the intensity of different coherence orders will be modulated by a phase proportional to the coherence order: D E X iφk Iˆz = e Mk (τ) , (4.6.4) k where the scaled excitation time τ has been used in place of texc. As explained in sections 2.10.2 and 2.10.3, the coherence intensity Mk will initially increase faster for spins with stronger couplings and for lower coherence orders. In the present experiments, the phase shift φ was incremented by 90◦ between scans. Two types of experiment were performed: the reference experiment and the MQ-filtered experiment. These differ by a change in the phase cycle of the receiver: in the reference spectrum the receiver is in phase with the excitation sequence, whereas in the MQ-filtered experiment the phase of the receiver is alternated as the phase shift is changed. The coherence orders which are selected by the different phase shifts and receiver phases are shown in table 4.1, which takes into account the fact that the coherence order k is restricted to even numbers due to the DQ excitation Hamiltonian. Adding together the results for the different phase shifts gives for the reference and MQ-filtered experiments: X Sref = M4n, n X (4.6.5) SMQ = M4n+2. n 80 π 3π phase shift 0 2 π 2 P P n P P n reference M2n (−1) M2n M2n (−1) M2n n n n n P P n+1 P P n+1 MQ-filtered M2n (−1) M2n M2n (−1) M2n n n n n Table 4.1: Summary of the coherence orders (and their phases) generated in the two types of experiment (reference and MQ-filtered), based on equation 4.6.4. The phase shift is the difference in phase between the excitation and reconversion parts of the pulse sequence. Because the excitation Hamiltonian is double-quantum in nature, only even orders of co- herence are produced (k = 2n). During data collection, the terms from the different phase shifts are added together for each type of experiment. This means that the reference experiment will contains coherence of orders 0, 4, 8, 12, . . . whereas the MQ-filtered experiment will contain coherence of orders 2, 6, 10, 14, . . . . The MQ-filtered experiment is named as such because it does not contain zero-quantum (or single-quantum) coherence. Note that the entire phase cycle of the excitation/reconversion sequence is repeated with the phase of the 90◦ read pulse (see figure 2.8) shifted by 180◦. The phase cycle of the receiver is also shifted by the same amount. This is done to cancel any anomalous signal components due to imperfections such as pulse angle dispersivity and materials outside the probe coil. 4.6.3 Buildup Curves The MQ buildup curve is obtained by evaluating the ratio SMQ BMQ (τ) = (4.6.6) Sref + SMQ as a function of the (scaled) excitation time, τ. Here the excitation time was changed by incrementing the number of cycles during the excitation period (nc) with the time for one cycle (tc) kept fixed at 0.5 ms. The reason for calculating the ratio in equation 4.6.6 (rather than just examining the MQ-filtered experiment directly) is that the total intensity will decrease as a function of excitation time due to relaxation during the excitation/reconversion process. The sum Sref + SMQ will contain all coherence orders, and hence its decrease with time will reflect the overall relaxation progress. Normalizing the MQ-filtered intensity by this sum can effectively correct for the effects of relaxation (especially at short excitation times). An example of a buildup curve, including the original MQ-filtered and reference data, is shown in section 4.6.7. The intensities Sref and SMQ were obtained by integrating the same frequency range in the spectra, which was chosen to avoid the part of the spectrum 81 which contained more weakly coupled nuclei (as described in section 5.3) Initially (for zero excitation time), all of the magnetization will exist as longitudinal magnetization (with zero coherence order). As the excitation time is increased, this will be converted first into double-quantum coherence, and later into higher order coherence. The buildup curve should theoretically approach a value of 0.5 at long times, because as higher coherence orders are excited, the number of coherence orders present in the MQ- 1 filtered intensity will approach 2 of the total number of coherence orders in the sum. This is also based on the assumption of equal partition of coherence among all excited coherence orders.[111] However, samples will often contain either solvent molecules or other spins whose residual couplings are very small due to efficient, isotropic averaging. Because of this, relaxation effects will prevent magnetization from these spins from evolving into MQ coherences. To ensure proper normalization, the signal from these spins can either be subtracted directly (as described in reference [17]) or various pre-selection techniques can be used (as was done here: see section 5.3.1). In earlier reports, an inverted Gaussian was used as a fitting function for the buildup curve (based on a static second-moment approximation and the assumption of Gaussian statistics):[15] 1 2 I(τ, D) = 1 − exp − (Dτ)2 , (4.6.7) 2 5 Later, an improved fitting function containing an oscillatory component was introduced to allow more quantitative fitting:[111] 1 n h 3 i o I (τ, D) = 1 − exp − (A Dτ) 2 · cos (A Dτ) . (4.6.8) 2 1 2 The constants A1 = 0.378 and A2 = 0.583 were determined by calibrating the new function to match the results of fits using the original inverted Gaussian.[111] Figure 4.9 shows a plot of the two kernel functions for comparison. Note that in equation 4.6.8, D is given in angular units (radians/s), but in literature it is often reported in frequency units (i.e. as D/2π); this convention will also be used here. Both of the above fitting functions assume a single coupling constant (D), however in general a distribution of couplings, P (D), will be present within the sample. In this case, the theoretical buildup curve can be calculated from the distribution integral of I (τ, D): ∞ g(τ) = I (τ, D) P (D) dD. (4.6.9) ˆ0 The single coupling curve I (τ, D) is called the kernel function of the integral transformation. A typical experiment would involve measuring an experimental buildup curve BMQ (τ), and then comparing that to a generic function of the form given for g (τ). One is generally 82 0.6 0.5 0.4 improved original 0.3 0.2 0.1 0 0 2 4 6 8 10 Figure 4.9: Plot of the original and improved kernel functions for comparison (as in equa- tions 4.6.7 and 4.6.8, respectively). The improved function contains an oscillatory compo- nent introduced to provide better fits to experimental buildup curves. interested in the distribution P (D), since it can give information about the chain dynamics and degree of crosslinking or entanglement in an elastomeric material. However, even given a suitable expression for the kernel function I (τ, D), determining P (D) from g (τ) is an ill-posed problem which generally has many possible solutions. If a particular form for P (D) can be justified a priori (for example a Gaussian function for a narrow distribution of couplings), then least-squares fitting can be performed numerically (see section 4.6.6). In the case where no suitable closed expression for P (D) can be determined, solving the inversion problem can still be attempted using a regularization algorithm (see section 4.6.5). 4.6.4 Pre-selection Techniques Background signal from solvent and other molecular segments containing nuclei with very small (highly averaged) residual dipole couplings will interfere with the shape of the buildup curve because they will not develop MQ coherence before relaxation has made them unde- tectable. This will result in a higher than normal reference spectrum, and hence a decrease in the buildup curve (due to improper normalization). Several techniques can be used to address this issue. The simplest method is to simply subtract this component from the ref- erence spectrum by fitting an exponential curve to the slowly relaxing tail in the reference spectrum. This method is only viable, however, if the signal components in the tail all relax at the same rate (i.e. they can be characterized by a single relaxation time).[17, 15] A more robust method, which will be referred to as DQ pre-selection, is to precede the 83 Figure 4.10: Illustration of the MQ pulse sequence including the DQ pre-selection and delay period. The pre-selection sequence is similar to the normal excitation/reconversion sequence but with a fixed duration. The delay period τz is included to allow redistribution of magnetization to ensure a proper representation of the full distribution of residual couplings. The phase cycle and timing used in the present experiments are described in the text. incremented excitation/reconversion process by another similar sequence of fixed duration. This is illustrated in figure 4.10. The pre-selection sequence functions to select only mag- netization that has been converted into DQ (actually (4n + 2)-quantum) coherence within the pre-selection time tpre. This is accomplished by incrementing the phase shift of the pre-selection reconversion sequence (θ) in 90◦ increments and alternating the receiver phase for alternating phase shifts, much the same as for the phase cycling of the original MQ detection scheme described in section 4.6.2. Note that for each phase of the pre-selection sequence the entire phase cycle of the detection sequence must be effected. The receiver phase must be alternated with respect to both phase shifts (i.e. the changes in receiver phase are cumulative). The end result is a 32-phase cycle, including the alternating phase of the 90◦ read pulse.[17, 15] The choice of pre-selection time tpre will affect the results in a few different ways. Firstly, it should be selected such that it is long enough for sufficient MQ coherence to build up, but not so long that relaxation effects have eliminated a significant amount of the signal intensity (in other words, it should be close to the maximum observed in the MQ-filtered experiment without pre-selection). Additionally, a shorter pre-selection time will create MQ coherence only in molecular segments with relatively large residual couplings, since more weakly coupled spins evolve more slowly under the excitation Hamiltonian. Finally, the choice of tpre will also affect the coherence orders that are filtered, since for longer times the higher order (4n + 2)-quantum coherences will make up a higher fraction of the intensity. These last two effects, especially the dependence on coupling strength, can be partially controlled for by adding a delay period τz between the pre-selection and the normal excitation/reconversion sequences. This will allow magnetization to diffuse (through spin- spin interactions as well as physical displacement and reorientation) to more weakly coupled spins, giving a more faithful representation of the full distribution of couplings. Those spins 84 Figure 4.11: Illustration of the MQ pulse sequence including both DQ pre-selection and T1 pre-selection sequences. The initial π pulse inverts that magnetization which then relaxes back toward equilibrium during the delay period, as illustrated in the figure. The delay period τ1 is chosen so that the unwanted signal (which has a slower relaxation rate) will be close to zero, while the remaining signal will have already relaxed to a greater positive amount. with very small residual couplings will exchange magnetization at a correspondingly slower rate, so they should not re-acquire as much magnetization in this way. Physical diffusion could lead to the reintroduction of signal components with very small couplings, if both regimes can arise from different states of the same molecule. This might occur, for example, if molecular segments can dynamically sample different conformations with very different dynamics. However, if the weakly coupled nuclei reside in different molecules, for example mobile solvent molecules in a more constrained matrix, this will not be an issue. The delay τz must be chosen so that it is long enough for diffusion to take place, but not so long that longitudinal relaxation negates the effects of the pre-selection. In the present experiments a delay period of 100 ms was found to provide good balance. The experiments were performed for a pre-selection time of 2.5 and 5.0 ms for comparison, composed of 5 and 10 cycles, respectively (with a constant cycle time of tc = 0.5 ms). A second type of pre-selection technique was used to further reduce the impact of un- wanted signals from weakly coupled elements. This involves an additional 180◦ pulse fol- lowed by a delay period preceding the DQ pre-selection and the normal excitation/reconversion sequence, as illustrated in figure 4.11. The technique takes advantage of the fact that the characteristic longitudinal relaxation time T1 is generally longer for weakly coupled spins than for more strongly coupled spins. The 180◦ pulse inverts the equilibrium magnetization, which then recovers during the delay period. The delay time τ1 is chosen to coincide with the approximate time at which the longitudinal magnetization from the weakly coupled spins crosses zero. Because the more strongly coupled spins relax quicker, they will already have recovered to a positive amount by this time. This can greatly reduce the signal from weakly coupled spins, but at the expense of also reducing somewhat the signal from the 85 more strongly coupled spins. This technique was found to be useful for reducing the unwanted signal component which was presumed to arise from absorbed solvent molecules. It was found that the signal intensity of these weakly coupled spins was sufficiently large (compared to the signal from spins which actually develop MQ coherence) that even with DQ pre-selection, artifacts remained due to imperfect cancelling between different scans in the phase cycle. This problem increases with increasing excitation time because the relative magnitude of the MQ intensity will decrease due to relaxation. Figure 5.7 in section 5.3.1 shows the one dimensional proton spectrum of an An16 sample with and without the DQ pre-selection and the T1 pre-selection. The weakly coupled signal was identified as a relatively narrow peak in the spectrum, and a series of one-dimensional inversion/recovery experiments was performed to determine the optimal duration of the delay period τ1 (such that this peak was minimized). 4.6.5 Tikhonov Regularization Regularization is a general technique which can be used to attempt to solve ill-posed prob- lems by imposing additional criteria which must be met by potential solutions. Tikhonov regularization is a particular type of regularization which can be applied to the distribu- tion integral inversion problem described in section 4.6.3 above. Here, a computer program called FTIKREG (Fast TIKhonov REGularization) was used for the calculations.[112] The calculation amounts to a minimization (with respect to the distribution function) of the sum of the squared deviations plus an additional term which ensures a smooth estimate: n 1 V (λ, P (D)) = X (B (τ ) − g (τ ))2 + λ kLP (D)k2 . (4.6.10) σ2 MQ i i i=1 i Here BMQ (τi) is a data point from the buildup curve in equation 4.6.6 for a scaled excita- tion time τi (with uncertainty σi), g (τi) is the distribution integral given in equation 4.6.9 evaluated at τi, and P (D) is the distribution of residual couplings. The additional term contains the operator L, which provides some inverse measure of smoothness (usually the identity operator or the second derivative) and λ, a weighting multiplier called the regu- larization factor. The value of of λ determines the importance of having a smooth solution relative to how closely the solution fits the data. The double vertical brackets indicate the Euclidean norm: v u nD uX 2 kLP (D)k := kQ (D)k = t Q (Di) , (4.6.11) i=1 86 for some discrete set of values of the independent variable, {D1,D2, ..., DnD }. The op- timum value of the regularization factor λ can be calculated in the program using the self-consistence (SC) method described in reference [113]. For the present analysis, the operator L was chosen to be the identity operator, so that the modulus of the distribu- tion is minimized in the second term. The improved single-coupling buildup curve given in equation 4.6.8 was used for the kernel function, I (τ, D). The other parameters in the software that can affect the outcome of the calculation are the limits and number of points to be used in the distribution P (D) as well as the upper limit of the buildup curve to be considered in the fit. 4.6.6 Analytical Distribution Functions Choosing a particular analytical form for the distribution of residual couplings P (D) allows the inversion problem described above to be solved using conventional least-squares fitting methods, as long as a suitable functional form can be found. This will reduce the number of degrees of freedom to the number of free parameters in the chosen form for the distribution function. The procedure requires the implementation of a function g (τ) which can be fit to the experimental buildup curve. Here, the fitting function was calculated from the distri- bution P (D) by numerically evaluating the integral in equation 4.6.9 using the improved kernel function given in equation 4.6.8. The result can then be fit to the buildup curve using a non-linear least-squares fitting algorithm, which was done here with the Levenberg- Marquardt method as implemented in the Gnu Scientific Library (GSL). Four different distribution functions (each with two shape parameters and an overall scaling factor) were used, as detailed in table 4.2. This was done to allow a range of different fits and to check for consistency. All the distribution functions are implicitly normalized as given, however they were also normalized numerically to account for any discrepancies arising from the numerical integration procedure. 4.6.7 PDMS Trial Experiments A sample of polydimethylsiloxane (PDMS - GE Silicones RTV615B 01P) was used as a trial material to test the experiment, since PDMS samples had previously been studied using MQ techniques.[17, 15] Figure 4.12 shows an example of the results acquired, including the buildup curve, MQ intensity, reference intensity, and sum intensity. The buildup curve did not exhibit a completely stable plateau for longer excitation times (likely due to some weakly coupled components which are not converted to MQ coherence in the available excitation time), however the data was otherwise generally consistent with previous results. The Tikhonov regularization approach and numerical integration techniques (as described 87 Model P (D)/2 A 2 Log-Normal ln (D/D0) C √ 1 A exp − 2S2 2πS √ 2 (D−D0) 2/π Gaussian A exp − 2 C √ 2S 1+erf(D0/ 2S) 1 S Cauchy A 2 2 C π −1 (D+D0) +S 2 +tan (D0/S) S−1 −D/D 1 Gamma AD e 0 C S D0 Γ(S) Table 4.2: Analytical distributions functions P (D) used for least-squares fitting using nu- merical integration. A is the normalization factor for the distribution, taking into account the fact that the domain of P (D) is restricted to non-negative D. The variable C effec- 1 tively replaces the factor of 2 in the kernel function given in equation 4.6.8, and allows for parametrization of the final plateau level in the buildup curve. Each function has two additional free parameters, D0 and S, which have different effects depending on the model. Figure 4.12: Plot of the MQ buildup curve BMQ for a PDMS sample used as a trial for the MQ experiment. Also shown are the directly measured MQ intensity curve (coherence orders 4n+2), reference intensity curve (coherence orders 4n), and the sum curve containing all coherence orders. The buildup curve was collected with a DQ pre-selection time of τDQ = 2.5 ms. The slow decline of the buildup curve at later excitation times is likely due to some residual weakly coupled components which were re-introduced (via diffusion) following the DQ pre-selection sequence. 88 2 Model χ D¯ ∆D C D0 S Gaussian 194 0.194 0.065 0.52 0.193 0.084 Gamma 192 0.207 0.088 0.53 0.062 3.325 Log-Normal 124 0.208 0.076 0.51 0.186 0.460 Cauchy 198 0.197 0.054 0.51 0.180 0.025 FTIKREG 6.47 0.256 0.157 0.50 – – Table 4.3: Parameters obtained from various types of fits to the experimental PDMS buildup curve. The corresponding distributions are shown in figure 4.13, along with the resulting buildup curves. The quality of the fits can be compared using χ2, the weighted average squared deviation between the fit and the experimental data. D¯ is the calculated average of the distribution obtained using each method, and ∆D is the mean absolute deviation (MD). The MD was used as a measure of dispersion because it is less sensitive to long, small tails and any other outlying components in the distribution. The parameters D0 and S are dependent on the model (see table 4.2). above) were used to obtain distributions of residual couplings from the buildup curve. The resulting fit parameters are given in table 4.3, and the resulting distributions are illustrated in figure 4.13 along with their corresponding best-fit buildup curves. The buildup curve from the distribution obtained from regularization (FTIKREG) fits the data much better than the other functions, however this is to be expected because this fitting method has many more effective degrees of freedom. The log-normal distribution fit is slightly better than the other functions, which are very similar. Despite having dif- ferent shapes, the distributions obtained are generally similar in terms of their mean and dispersion (mean deviation). The regularization fit gives a larger mean and deviation than the others because of small contributions from large couplings. Note that the error bars in the experimental data are likely underestimates because they take into account only the experimental noise, not systematic deviations from the expected buildup curve shape. Only the first twenty points in the buildup curve were used for fitting, since beyond that point the curve begins to decrease significantly. 89 15 0.6 0.4 10 0.2 0 0 2 4 6 8 5 Data FTIKREG Log-Normal Gaussian Cauchy Gamma 0 0 0.2 0.4 0.6 0.8 1 Figure 4.13: Residual coupling distributions obtained using various methods, as described in the text and in table 4.3. The corresponding fits to the buildup curve are shown in the inset. Comparison of the distributions obtained to previously reported results suggests that the sample tested here is composed of mid- and short-length chains (see, for example, references [17] and [111]). 90 Chapter 5 Results & Discussion The results of the three types of experiments performed and their implications are discussed in this chapter. References will be made to previous chapters for background information and experimental procedures, and the final chapter will contain a summary of the results presented here. 5.1 Carbon-13 Chemical Shifts Figure 5.1 shows an example of a typical 13C spectrum for An16, collected as described in section 4.4 above. The peak assignments for the carbon atoms from each type of residue are also indicated in the figure. Although the material is a solid (i.e. it retains its own shape), the sharp peaks in the spectrum are reminiscent of a typical solution-state NMR spectrum, with no evidence of anisotropic line broadening. This shows that the chain segments in An16 are highly mobile, and able to sample many different orientations on a time scale that is fast compared to the coupling frequencies of any interactions present. Note that the width of the peaks observed will also be affected by magnetic field inhomogeneity due to sample and probe geometry, even with optimal shim settings. Thus the inherent mobility-limited line-widths of the peaks in the spectrum may be significantly less than those observed experimentally. Figure 5.2 shows the difference between predicted chemical shifts for different types of secondary structure (for an unstretched An16 sample) and the experimentally observed peak positions. The results show that the randomly coiled structure fits the data much better than either the β-sheet or α-helix structures. The dependencies of the chemical shift separations (between Cα and Cβ) on sample extension are shown in figure 5.3 and summarized in table 5.1. The results show only small shifts for each residue, however in all cases except for the upfield alanine peaks the trend line shows a shift toward the result predicted for the β-sheet conformation and away from that predicted for α-helix. Interestingly, the residues which exhibit the most significant changes are Y (9.7%), P (8.4%), and A1 (-7.1%), which are all part of the conserved YGAP motif (which likely plays a role in the formation of di-tyrosine crosslinks). The next most significant is Q2 (6.2%), which immediately precedes the tyrosine residue. Thus, it may be 91 6 4 2 0 180 160 140 80100120 60 2040 6 4 2 0 170180 150160 140 120130 6 4 2 0 70 5060 40 2030 chemical shift [ppm] Figure 5.1: An example of a 13C spectrum obtained for a sample of An16. (A) shows the full spectrum, while (B) and (C) present an expanded view of the downfield and upfield parts of the spectrum, respectively. The peaks are labelled according to their corresponding carbon atom position and residue type. Note that some residues showed different shifts for different positions in the repeat sequence, but there was no evidence of any differences between equivalent residues in different repeats. The carbonyl carbon peaks had too much overlap to be reliably assigned. The small peaks on either side of the threonine β-carbon (Tβ) were not observed in previous experiments and are presumed to be due to some component in the solution. Compared to typical (static) solid-state experiments, the spectrum is remarkably sharp, with line-widths ranging from approximately 0.3 to 0.5 PPM in width. This is indicative of the high molecular mobility of chain segments in the protein matrix. The x- axis is the frequency shift given in PPM; this was estimated from predicted peak positions and is an approximate reference only. 92 4 Random Coil Alpha Helix 3 Beta Sheet 2 1 0 -1 Deviation [ppm] -2 -3 -4 Peak Figure 5.2: Plot of the deviations between the measured and predicted chemical shift values for different types of secondary structure for the different carbons in the sample. In order to minimize any uncertainty due to chemical shift referencing, the absolute chemical shift in each case was set so as to minimize the deviations from the shifts predicted for that structure; the standard deviations for each type of structure could be then be compared. These are (in PPM) 0.22, 1.95 and 1.67 for the random coil, α-helix and β-sheet conformations, respectively. Thus it can be seen that the random coil conformation provides a substantially better fit than the others (which is also visually apparent from its smaller deviations in the figure). 93 amino acid coil pred. slope intercept† α-helix† β-sheet† % change T 9.5 -0.082 0.459 1.4 -4.8 1.7 S 5.5 -0.112 -0.219 2.4 -3.8 2.9 P 31.2 0.268 -0.216 -1.7 3.2 8.4 Y 19.1 0.378 -0.104 -3.9 3.9 9.7 Q1 26.3 0.120 0.180 -3.5 3.9 3.1 Q2 26.3 0.240 -0.396 -3.5 3.9 6.2 A1 33.4 -0.227 -0.316 -4.7 3.2 -7.1 A2 31.9 0.083 0.031 -4.7 3.2 2.6 Table 5.1: Parameters for the α/β-carbon chemical shift differences for the different residues. The second column in the table gives the difference predicted for the random coil conforma- tion. The slope and intercept of the linear trend lines fit to the data are also given, as well as the predicted differences for the α-helix and β-sheet. These three parameters (indicated with the symbol †) are given relative to the random coil prediction. The last column is the shift difference predicted by the trend line (at 100% strain) expressed as a percentage of the difference between the predicted shifts for the random coil and β-sheet conformations. It is evident that all pairs except A1 show a trend toward β-sheet and away from α-helix with increasing strain. All parameters except the percent change are given in units of PPM. that the constraints imposed by the crosslinks affect the formation of secondary structure when the material is stretched, and this may also account for the discrepancy in the trend for the alanine chemical shifts. No significant change in the line widths was observed as the material was stretched. These results suggests that the protein may have a slightly higher tendency to sample β-sheet conformations when the material is stretched, but is still highly dynamic and predominantly randomly coiled. This behaviour might be expected since the β-sheet conformation is a more extended configuration of the protein backbone. The small size of the shifts (predicted to be between 1 and 10% at 100% strain) suggest that β-sheet formation would be incomplete even at very high strain (although it is also possible that at higher strains there could be a change in the trends observed here). In summary, the 13C results show that An16 has a high molecular mobility, has a predominantly randomly coiled secondary structure, and seems to show a slight tendency toward increased β-sheet content at higher extensions. 94 0.6 0 0.5 -0.1 0.4 -0.2 0.3 -0.3 0.2 -0.4 0 0.15 0.3 0.45 0.6 0 0.15 0.3 0.45 0.6 0.1 0.2 0 0.1 -0.1 0 -0.2 -0.1 -0.3 -0.2 0 0.15 0.3 0.45 0.6 0 0.15 0.3 0.45 0.6 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 0 0.15 0.3 0.45 0.6 0 0.15 0.3 0.45 0.6 Figure 5.3: Plots of the chemical shift difference between Cα and Cβ for various residues (as indicated in each plot), as a function of sample extension. The y-axis in each plot is the difference between the measured peak separation and that predicted for a random coil conformation. The x-axis is the strain of the sample given as a fraction of the unstretched length. For glutamine (Q) and alanine (A), two peaks were observed for each carbon atom, corresponding to the two different positions in the repeat sequence; note that the range of the y-axis is larger for these two sub-plots. The error bars on the experimental data points are based on a combination of calculated uncertainty in the fit of the peak position and statistical variation from multiple measurements. Also shown for each pair of peaks is a linear trend line fit to the data. The changes observed are small compared to the error estimates and scatter, however there is a consistent trend toward more β-sheet structure at higher extensions (see table 5.1). 95 5.2 Residual Quadrupole Couplings Results from the attempts to measure residual quadrupole couplings in stretched An16 samples suggest that the majority of the nuclei in the material exhibit residual couplings that are small to the point of being undetectable. Two-dimensional spectra of a sample of An16 stretched to 80% strain are shown in figures 5.4 and 5.5, for different transmitter frequencies. The only intensity detected off the frequency origin in the indirect dimension occurs in the form of small dispersive elements along the frequency diagonal (relative to the transmitter frequency). The results are likely due to imperfect refocussing of the magnetization during the spin echo sequence, because of variation in the tip angle of the π pulse. The extraneous signal intensity was found to increase slightly with sample extension, however this is likely caused by the change in geometry of the sample; when the sample is stretched it will extend over a larger extent of the probe coil and experience an overall increase in the inhomogeneity of the B1 magnetic field associated with the pulses. It was also found that there was more extraneous signal aligned with the upfield peak in the direct dimension; this is because the duration of the π pulse was calibrated to the larger downfield peak. It should also be noted that the extraneous signals observed are much smaller than the main peaks along the indirect frequency origin (by a factor of around 100 in maximum amplitude). Attempts to measure residual quadrupole couplings in individual stretched tendons were for the most part unsuccessful due to difficulties in handling and stretching the small, fragile tendons. The results that were obtained (not included) also showed no detectable sign of residual quadrupole couplings, however further testing would be required to more rigorously characterize the properties of this material. The lack of detectable residual quadrupole couplings (even at relatively high sample strain) indicates that the deuterated solvent molecules (and exchangeable hydrogen sites in the protein) are not significantly affected by molecular alignment upon extension of the sample. Although the alignment is naturally expected to increase with sample extension, the results suggest that the molecular matrix and absorbed solvent molecules are still largely unconstrained and sufficiently dynamic to maintain very small residual couplings. The high degree of swelling in the material (approximately 80% water by weight at equilibrium in An16) likely contributes significantly to the small size of the couplings, as seen in other experiments involving extended materials swollen with deuterated solvents.[109] Also, only D2O was used as a solvent, which would be expected to have relatively weak residual couplings compared to more anisotropic molecules. Although the results do not allow a quantitative characterization of the properties of the material, they are consistent with the picture of a highly dynamic, predominantly randomly coiled protein which can easily sample different conformations on short time scales. 96 500 400 500 50 10 300 5 -5 -10 200 100 0 -100 0200400 -600-400-200 Figure 5.4: Contour plot of the two-dimensional spectrum obtained for a sample of An16 at 80% strain. There is little intensity off the frequency origin in the indirect dimension (f1). The notable signal components that do occur lie on the transmitter frequency diagonal (as indicated by the dashed line) and appear to be out of phase with the main peaks in the spectrum. These likely occur due to improper refocussing of magnetization during the spin echo sequence, caused by variation in the tip angle of the π pulse for different parts of the spectrum. Note that the peaks along the origin are broadened in the indirect dimension due to relaxation during the spin echo sequence. 97 500 Intensity 400 500 50 10 300 5 -5 -10 200 100 0 -100 0 -600-400-200 -1200-1000-800 Figure 5.5: Contour plot of the two-dimensional spectrum obtained for a sample of An16 at 80% strain, as in figure 5.4 but with the transmitter frequency shifted to a point outside the spectrum. The results are similar, and the spurious signal components again lie on the frequency diagonal. Note that the signal aligned with the upfield peak in the direct dimension would have occurred at a frequency outside the spectral window in the indirect dimension, and so is aliased back into a lower frequency. The dashed line showing the expected position of signals on the frequency diagonal reflects the effects of this aliasing. 98 5.3 Residual Dipole Couplings Multiple Quantum (MQ) experiments were used to study the distribution of residual dipole couplings in An16, rec1-resilin, and dragonfly tendon samples. Figure 5.6 (A) shows the (un- modified) one-dimensional proton spectra of the different samples tested. All three spectra contain a sharp peak which appears to arise predominantly from nuclei with small resid- ual dipole couplings (likely in kinetically free solvent molecules). Pre-selection techniques (see section 5.3.1) were used to reduce these signal components. The An16 and rec1-resilin spectra have similar features, whereas the dragonfly tendon spectrum is noticeably differ- ent, with a more broad, featureless spectrum (other than the sharp solvent peak). This may be due to the more heterogeneous nature of the tendon sample, which was made up of many individual tendons packed together. Furthermore, although the tendons are com- posed primarily of resilin, they might also be expected to contain small quantities of other materials. The relative broadness of the tendon spectrum may also be related to the faster relaxation observed in the tendons compared to the other materials, as discussed in section 5.3.1 below. For each material, buildup curves were obtained for DQ pre-selection times of 2.5 and 5.0 ms. In each subsection (5.3.2 to 5.3.4 below), a plot of the distributions obtained using different methods is shown, with their corresponding buildup curves inset. See sections 4.6.5 and 4.6.6 for descriptions of the methods used to obtain the distributions from the buildup curves. Tables summarizing the results obtained from the fits are also included, with parameters as described in table 4.3. The relative error of the points in the experimental buildup curves are based on the signal to noise ratio in the original spectra (and statistical analysis when multiple independent measurements were performed), carried through to the buildup curve using standard error propagation methods. This method, however, does not account for errors from artifacts due to incomplete cancelling between different phases in the phase cycle, so the error bars were also scaled by an additional factor estimated using the error scaling algorithm present in the FTIKREG software. Note that this overall scaling affects the χ2 estimates and visual appearance of the graphs only, not the best fit distribution curves (except for the regularization fits). 5.3.1 Pre-selection Figure 5.6 shows the spectra of the three different sample materials without pre-selection (A) and with both T1 and DQ pre-selection (B), for comparison. The most noticeable difference between the spectra with and without the pre-selection is that the relative magnitude of the sharp peak is substantially reduced, supporting the idea that it arises primarily from nuclei with small residual couplings. Figure 5.7 shows in greater detail the effects of the two 99 1.6 1.2 An16 rec1-resilin tendons 0.8 0.4 0 0.5 0.4 0.3 0.2 0.1 0 4000 02000 -4000-2000 Frequency [Hz] Figure 5.6: Plot of the one dimensional 1H spectra of the three samples with no pre-selection (A) and after both T1 and DQ pre-selection with τDQ = 2.5 ms (B). The relative frequency axis is approximate; the plots were aligned visually to facilitate comparison since no absolute chemical shift reference was available. The total intensity was normalized for each material. In (A), it can be seen that the An16 and rec1-resilin spectra have relatively similar features, whereas the spectrum of the tendon sample is generally more broad and featureless. All three spectra contain one large sharp peak, which is most pronounced for the tendon sample. By comparison with (B), it is apparent that the large downfield peak arises mainly from nuclei with weak residual couplings, since its relative magnitude is greatly reduced by the pre-selection. 100 3000 600 2000 400 1000 200 0 0 60 18 40 12 20 6 0 0 4000 -200002000 20004000-4000 -4000-20000 Figure 5.7: Several spectra of the An16 sample showing the effects of the different types of pre-selection. (A): No pre-selection. As can be seen by comparison to (C), the largest peak contains significant contribution from spins with small residual couplings. (B): T1 pre-selection with a relaxation time of τ1 = 800 ms. The peak from the weakly coupled spins is still partially inverted, whereas the rest of the spectrum has already relaxed back to positive intensity. (C): DQ pre-selection with τDQ = 2.5 ms (solid line) and 5.0 ms (dashed line). (D): Combination of both pre-selection sequences as in (B) and (C). In each plot the intensity scale (y-axis) is given in arbitrary but consistent units, with different axis ranges for each spectrum; the substantial differences in total intensity between the different spectra is discussed further in table 5.2. The x-axis is the relative resonant frequency in Hz. types of pre-filters used, both alone and in combination, on the one-dimensional spectrum of the An16 sample. It was found that the sharp peak had a slower relaxation rate than the rest of the spectrum, as illustrated in figure 5.8. This supports the idea that this peak arises from a different type of molecule (most likely from the residual concentration of 1H atoms in the absorbed solution). This difference in relaxation rate is necessary for the T1 pre-selection technique to be viable. Table 5.2 lists the effects of the various pre-selection techniques on the total spectral magnitude for the different samples, as well as estimates of the fraction of the sample represented after pre-selection. Increasing the DQ pre-selection time is expected to increase the fraction of the sample represented in the buildup curves, yielding a more accurate representation of the underlying distribution of residual couplings by introducing more 101 1 fast slow 0.5 0 -0.5 -1 0 500 1000 1500 T1 pre-selection time [ms] Figure 5.8: Plot of the intensity of different parts of the An16 spectrum as a function of the T1 pre-selection time (τ1). The curve labelled “slow” in the legend corresponds to the larger downfield peak, which is presumed to arise from solvent molecules. The curve labelled “fast” is from the smaller upfield peak. The range over which the spectrum was integrated in each case is indicated in the inset. It was found that the sharp downfield peak had a slower relaxation rate then the other parts of the spectrum; the solid lines are single exponential fits to the data, which yield time constants of 970 and 600 ms for the slow and fast curves respectively. The T1 pre-selection time used in subsequent experiments was chosen to be slightly longer than the time when the slow curve crosses zero, to account for any contribution from fast components in the range considered for the slow curve. 102 (A) T1 DQ (2.5 ms) DQ (5.0 ms) T1+DQ (2.5) T1+DQ (5.0) An16 4.1 40 47 150 190 rec1-resilin 5.8 14 17 63 75 tendons 4.2 48 110 130 280 (B) T1 DQ (2.5 ms) DQ (5.0 ms) T1+DQ (2.5) T1+DQ (5.0) An16 44 25.1 34.9 12.2 15.7 rec1-resilin 31 37.3 47.1 15.1 19.4 tendons 43 20.7 9.8 13.9 7.0 Table 5.2: The effect of the different pre-selection techniques on the spectral intensity of the different samples. The first table (A) gives the total factor by which the intensity for each material was reduced compared to the original spectrum. The second table (B) gives an estimate of the percentage of the sample represented after each type of pre-selection. This takes into account reduction of the slow component during the T1 pre-selection pro- cess (estimated from figure 5.8), T1 relaxation during the redistribution period, transverse relaxation during the DQ pre-selection (estimated from figure 5.9), and the final plateau levels of the buildup curves (as determined in sections 5.3.2 to 5.3.4 below). Note that the precise effects of the pre-selection techniques will have some dependence on experimental factors such as the amount of unabsorbed solution present in the samples. smaller couplings. However, this comes at the expense of increasing signal loss due to relaxation. Because the relaxation rate is expected to depend on the residual coupling strength, this is also expected to cause the stronger couplings to be more under-represented for the longer pre-selection time; the potential effects of differential relaxation are discussed further in section 5.3.5 below. The results show that the total spectral intensity is greatly reduced by the pre-selection techniques, however once the effects of relaxation are taken into account, the actual sample fractions represented are reasonably good. Note that these would only be expected to represent a fraction of the total populations to begin with because of the significant size of the (undesired) solvent peak observed in the unfiltered spectra. It should also be noted that the fractions estimated likely represent a lower limit, because the effects of relaxation are likely greater for the stronger couplings in the distribution; the curves shown in figure 5.9 are for the sum of all coherence orders after the redistribution delay, so they will include more contributions from weakly coupled nuclei and hence may underestimate the relaxation of the MQ coherences alone during the pre-selection process. Comparing the different samples in table 5.2 yields two interesting results. The first is that the rec1-resilin spectra show slightly less reduction due to the DQ pre-selection than the other samples, which suggests that this material contains a smaller fraction of nuclei with very small residual couplings compared to the others. The second is that the effect of increasing the DQ pre-selection time from 2.5 to 5.0 ms has a much greater impact on 103 1 An16 rec1 tendons 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 1 An16 rec1 tendons 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 Figure 5.9: Plot of the normalized sum intensity (SDQ +Sref ) as a function of the excitation time, for a DQ pre-selection time of 2.5 ms (A) and 5.0 ms (B). The solid lines are single exponential fits to the data, and yield relaxation constants for the An16, rec1-resilin, and tendon data of 3.32 ms, 3.42 ms, and 2.45 ms (respectively) in (A) and 4.15 ms, 4.10 ms, and 3.71 ms in (B). In both plots, it is apparent that the tendon curve relaxes faster than the others, although the difference is smaller for the longer pre-selection time. All curves show faster relaxation for the shorter pre-selection time, which is expected since the shorter pre-selection sequence will select nuclei with larger residual couplings. 104 the spectrum of the tendon sample than on those of the recombinant resilins, as seen by comparing the numbers in table (A). This is due in part to faster relaxation in this sample, as observed from the dependence of the sum intensity (SMQ + Sref ) on the excitation time (see figure 5.9). As can be seen in table (B), however, the observed relaxation is insufficient to fully account for the effect of increasing the pre-selection time. This results in a smaller estimate of the sample fraction for the longer pre-selection time, which is the opposite of the expected behaviour (as seen in the other samples). This might be due to the dependence of the relaxation rate on coupling strength, as mentioned above. The distributions obtained for the tendon sample (see section 5.3.4 below) suggest that this material contains more nuclei with larger residual couplings compared to the others. These would be expected to relax at a faster rate than more weakly coupled nuclei, which could lead to an underestimate of the transverse relaxation of the tendon signal during the DQ pre-selection process. One advantage of the MQ techniques over the quadrupole coupling measurements is that the former show only contributions from nuclei with sufficiently large residual couplings; as seen from the estimates of the sample fractions remaining after application of the pre- selection techniques, these represent only a fraction of the total population in the sample. 5.3.2 An16 For the An16 samples a T1 pre-selection time of 800 ms was used, corresponding to a point near where the sharp peak arising from weakly coupled nuclei was approximately zero (as illustrated in figure 5.8). Figure 5.10 shows the distributions obtained for the An16 sample with a DQ pre-selection time of 2.5 ms, and figure 5.11 for a pre-selection time of 5.0 ms. Tables 5.3 and 5.4 show the respective parameters obtained from the different fitting methods. The results are described in the captions of the respective figures and tables. Figure 5.12 shows a comparison of the results obtained for the different DQ pre-selection times. Both show a similar asymmetric shape, with a high concentration of weaker residual couplings and a decreasing tail in the higher coupling range. The regularization fits (as shown in the figure) show a small amount of additional couplings at the upper limit of the range considered, which may be the result of over-fitting to the initial data points in the buildup curves. Because the fit of this range is based on only a few initial data points, the details should not be considered highly reliable. As expected, the distributions obtained (both from the analytical functions and the regularization method) show a shift to lower couplings as the DQ pre-selection time is increased. Both buildup curves show a final buildup level significantly less than the predicted value of 0.5 (instead estimated to be around 0.2); this discrepancy will be discussed further in section 5.3.5. 105 3 0.4 Data FTIKREG 2.5 Cauchy Gamma 0.3 Gaussian 2 Log-Normal 0.2 1.5 buildup curve 0.1 1 Probability Density [1/kHz] 0 0 2 4 6 8 10 excitation time τ [ms] 0.5 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.10: Distributions of residual couplings obtained for the An16 sample with a DQ pre- selection time of 2.5 ms. The corresponding fits to the buildup curve are shown in the inset. All analytic distribution functions, as well as the regularization result (FTIKREG), show good agreement with each other and provide reasonable fits to the data. The distribution obtained from regularization includes more intensity at the upper range of coupling strengths considered (above 1.75 kHz), however this is probably due to overfitting of the initial rise in the buildup curve. 2 Model χ D¯ ∆D C D0 S Gaussian 0.82 0.31 0.20 0.26 -0.44 0.54 Gamma 0.81 0.33 0.21 0.25 0.23 1.42 Log-Normal 0.79 0.36 0.23 0.23 0.26 0.84 Cauchy 0.79 0.35 0.25 0.25 0.16 0.17 FTIKREG 0.93 0.53 0.45 0.22 – – Table 5.3: Parameters obtained for the An16 sample with a DQ pre-selection time of 2.5 ms (see table 4.3 for the parameter definitions). The results from the analytical distribution functions are highly consistent, which is a good indicator of reliability. The regularization result shows a higher mean and absolute deviation because the distribution predicts more couplings at the upper end of the range considered. As can be seen by comparing the χ2 values for the different fits, however, this feature is not necessary to obtain a good fit to the experimental buildup curve. 106 8 0.3 Data FTIKREG Cauchy 0.25 6 Gamma Gaussian 0.2 Log-Normal 0.15 4 buildup curve 0.1 0.05 Probability Density [1/kHz] 0 2 0 2 4 6 8 10 excitation time τ [ms] 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.11: Distributions of residual couplings and corresponding fits to the buildup curve obtained for the An16 sample with a DQ pre-selection time of 5.0 ms. There is some degree of variation among the different analytical models, with the log-normal distribution exhibiting much greater probability density for couplings below 0.1 kHz. This difference is also apparent in the corresponding fit obtained for the buildup curve, which shows a continuing slow rise at longer excitation times. The regularization result again predicts more couplings in the upper range compared to the analytical functions. 107 5 0.4 2.5 2.5 5.0 5.0 4 0.3 3 0.2 buildup curve 0.1 2 Probability Density [1/kHz] 0 0 2 4 6 8 10 1 excitation time τ [ms] 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.12: Comparison of the distributions and buildup curves for An16 with DQ pre- selection times of 2.5 and 5.0 ms (as indicated in the legends). As expected, the result for the longer pre-selection time shows more weak couplings because these require a longer excitation time to develop MQ coherence. This is also reflected in the buildup curve, which shows a slower buildup. Both curves show a lower plateau level than expected, at around 0.2 rather than 0.5, and the plateau level appears to be lower for the longer pre-selection time. 108 2 Model χ D¯ ∆D C D0 S Gaussian 1.51 0.19 0.14 0.22 -2.58 0.75 Gamma 1.54 0.20 0.14 0.21 0.18 1.12 Log-Normal 0.99 0.22 0.24 0.30 0.058 1.90 Cauchy 1.54 0.29 0.22 0.19 0.12 0.14 FTIKREG 0.98 0.50 0.58 0.19 – – Table 5.4: Parameters obtained for the An16 sample with a DQ pre-selection time of 5.0 ms. Despite having a distinctly different shape, the average residual coupling strength from the log-normal distribution is similar to those obtained from the other analytical functions. Its most significant difference is in the plateau level C, which is predicted to be significantly higher. Interestingly, the log-normal buildup curve gives a better χ2 than the other analytical models. Thus it can be concluded that the data is consistent with a range of possibilities, including intermediate cases between the different models. As in the case of τDQ = 2.5 ms, the regularization result shows a higher mean and absolute deviation than the analytical functions due to its increased probability density in the upper range of couplings. 5.3.3 Rec1-resilin For the rec1-resilin samples a T1 pre-selection time of 800 ms was also found to be optimal (the same as for the An16 samples). Similarly to the elements of section 5.3.2, figures 5.13 and 5.14 show the distributions obtained for the experiments with a DQ pre-selection time of 2.5 and 5.0 ms, respectively, and tables 5.5 and 5.6 show the corresponding fit parameters. Figure 5.15 shows a comparison of the results obtained in the rec1-resilin experiments with different DQ pre-selection times. As expected, there is a general shift to weaker couplings with increasing pre-selection time. Compared to the An16 results shown above, however, the shift appears to be less pronounced. The final plateau level is also much closer to the expected level of 0.5, and there appears to be little difference in the plateau level 2 Model χ D¯ ∆D C D0 S Gaussian 3.12 0.20 0.15 0.47 -14.8 1.76 Gamma 2.26 0.20 0.15 0.49 0.23 0.83 Log-Normal 1.01 0.24 0.20 0.46 0.13 1.12 Cauchy 0.86 0.25 0.22 0.47 0.056 0.13 FTIKREG 0.80 0.37 0.38 0.42 – – Table 5.5: Parameters obtained for the rec1-resilin sample with a DQ pre-selection time of 2.5 ms. As in the An16 results, the regularization fit shows a slightly higher average and mean deviation than the other fits. Otherwise the fits generally show good agreement with each other. The estimated plateau level (C) is predicted to be close to the expected value of 0.5. 109 7 0.5 Data 6 FTIKREG Cauchy 0.4 Gamma 5 Gaussian Log-Normal 0.3 4 0.2 buildup curve 3 0.1 Probability Density [1/kHz] 2 0 0 2 4 6 8 10 excitation time τ [ms] 1 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.13: Distributions of residual couplings and corresponding fits to the buildup curve obtained for the rec1-resilin sample with a DQ pre-selection time of 2.5 ms. The dis- tributions obtained are for the most part quite similar, and all provide good fits to the experimental data. The regularization result shows a small shoulder and some small in- tensity at higher couplings, but these do not appear to be highly significant. The gamma distribution fit shows more very small couplings than the others, however the fit is relatively insensitive in this range because these components will not have sufficient time to build up in the maximum excitation time considered. 110 15 0.5 Data FTIKREG Cauchy 0.4 Gamma Gaussian 10 Log-Normal 0.3 0.2 buildup curve 0.1 5 Probability Density [1/kHz] 0 0 2 4 6 8 10 excitation time τ [ms] 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.14: Distributions of residual couplings and corresponding fits to the buildup curve obtained for the rec1-resilin sample with a DQ pre-selection time of 5.0 ms. The gamma distribution and Gaussian function appear to not fit the data as well as the other functions, and underestimate slightly the buildup curve at longer excitation times. This is indicative of the shape of the corresponding distributions, which show slightly less probability density in the lowest couplings and more for slightly larger couplings (around 0.2 to 0.3 kHz). The regularization result again contains a shoulder, which is more pronounced here than for the shorter pre-selection time. 111 2 Model χ D¯ ∆D C D0 S Gaussian 9.59 0.13 0.10 0.47 -9.96 1.18 Gamma 7.67 0.13 0.10 0.48 0.16 0.81 Log-Normal 1.23 0.14 0.15 0.61 0.045 1.58 Cauchy 1.00 0.16 0.17 0.57 -0.010 0.060 FTIKREG 0.85 0.22 0.22 0.42 – – Table 5.6: Parameters obtained for the rec1-resilin sample with a DQ pre-selection time of 5.0 ms. As seen in the corresponding figure, the Gaussian and gamma functions don’t fit the data as well as the other functions. The log-normal and Cauchy distribution provide better fits, but estimate the plateau level to be higher than expected. This may be because the buildup curve appears to have not yet reached a plateau level in the experimental data, and is unlikely to be a real feature of the buildup curve. The regularization result is the only one to show both a good fit and a reasonable plateau level, suggesting that the additional shoulder seen in the figure might be a real feature of the distribution. 0.5 8 2.5 2.5 5.0 5.0 0.4 6 0.3 0.2 buildup curve 4 0.1 Probability Density [1/kHz] 0 0 2 4 6 8 10 2 excitation time τ [ms] 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.15: Comparison of the distributions and buildup curves for rec1-resilin with DQ pre-selection times of 2.5 and 5.0 ms. Although there is a general shift to smaller residual couplings, the precise changes in the shape of the distributions obtained do not easily lend themselves to a simple interpretation. This suggests that some features of the distributions, such as the shoulders and densities at higher couplings, may not be significant. Note that although the shape of the buildup curves are different, with the data for the shorter pre- selection time showing a steeper initial rise (as expected), the final plateau buildup level appears to be similar between the two experiments. 112 for the different pre-selection times. This may be due to a narrower distribution of residual couplings in rec1-resilin compared to the other materials, and will be discussed further in section 5.3.5. 5.3.4 Dragonfly Tendons For the dragonfly tendons a T1 pre-selection time of 575 ms was found to be optimal (com- pared to 800 ms for the An16 and rec1-resilin samples). The shorter optimal pre-selection time is likely related to the faster relaxation for the tendon sample as described in section 5.3.1. As above, figures 5.16 and 5.17 show the distributions obtained for the experiments with a DQ pre-selection time of 2.5 and 5.0 ms, respectively, and tables 5.7 and 5.8 show the corresponding fit parameters. Figure 5.18 shows a comparison of the results obtained in the tendon experiments with different DQ pre-selection times. As in the An16 results, the buildup curves show a plateau level that appears to be significantly less than the expected value of 0.5. Note that the mean coupling strength shows a substantial shift when the pre-selection time is increased (as seen in the tables), however this effect is not immediately apparent from visual inspection of the distributions or buildup curves. See the following section for additional discussion and comparisons between the samples. 113 10 0.7 Data FTIKREG 0.6 8 Cauchy Gamma Gaussian 0.5 Log-Normal 0.4 6 0.3 buildup curve 0.2 4 0.1 Probability Density [1/kHz] 0 0 2 4 6 8 10 2 excitation time τ [ms] 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.16: Distributions of residual couplings and corresponding fits to the buildup curve obtained for the tendon sample with a DQ pre-selection time of 2.5 ms. The distributions vary significantly in their estimates of the small residual couplings, and hence the corre- sponding fits also show a range of predictions for longer excitation times. All of the models, however, can be considered reasonable fits to the buildup curve because of the high uncer- tainties in the experimental data in this region. Once again, the regularization result shows more intensity at higher couplings, which could be due to overfitting of the steep initial rise in the data. 2 Model χ D¯ ∆D C D0 S Gaussian 1.04 0.41 0.30 0.28 -30.3 3.59 Gamma 0.91 0.39 0.37 0.33 0.88 0.42 Log-Normal 0.88 0.40 0.44 0.38 0.14 2.75 Cauchy 0.91 0.44 0.41 0.31 -0.25 8.1 × 10−5 FTIKREG 0.89 0.79 0.76 0.25 – – Table 5.7: Parameters obtained for the tendon sample with a DQ pre-selection time of 2.5 ms. Despite the differences in shape of the different distributions (and their corresponding fits to the buildup curve), all provide fits of similar quality (as seen by the χ2 values). The results are relatively consistent in terms of their mean and standard deviation, other than the regularization result which again shows higher values than the others. The estimates of the plateau level (C) show some variation, as can be seen from the behaviour of the fits at later excitation times (see figure 5.16). 114 10 0.7 Data FTIKREG 0.6 8 Cauchy Gamma Gaussian 0.5 Log-Normal 0.4 6 0.3 buildup curve 0.2 4 0.1 Probability Density [1/kHz] 0 0 2 4 6 8 10 2 excitation time τ [ms] 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.17: Distributions of residual couplings and corresponding fits to the buildup curve obtained for the tendon sample with a DQ pre-selection time of 5.0 ms. The experimental buildup curve has very large error bars because of the smaller sample size and low signal intensity (which was more significantly reduced by the change in DQ pre-selection time here than for the other samples). Aside from the regularization result, all of the distributions show a monotonic decrease in couplings from lower to higher values, with different balance between smaller and larger couplings. The fits to the experimental data show a correspond- ing variation in their predictions for the buildup curve, which are especially evident at longer excitation times. The distribution obtained with the regularization method shows a pronounced bimodality, however the lack of this feature in the other fits to the data suggest that it is not a necessary feature. 115 2 Model χ D¯ ∆D C D0 S Gaussian 1.74 0.27 0.20 0.30 -19.9 2.35 Gamma 1.42 0.25 0.22 0.35 0.45 0.50 Log-Normal 1.11 0.20 0.23 0.55 0.024 2.38 Cauchy 1.14 0.24 0.26 0.42 -0.094 3.8 × 10−5 FTIKREG 0.99 0.30 0.24 0.27 – – Table 5.8: Parameters obtained for the tendon sample with a DQ pre-selection time of 5.0 ms. The fit quality and shape parameters (mean and mean deviation) are relatively similar between the different models, but there is significant variation in the predicted values of the plateau level. This is not unreasonable given the large error bars in the experimental data at longer excitation times. The value of 0.55 obtained for C from the log-normal distribution is likely an overestimate. The predicted values are for the most part intermediate to those obtained in the An16 and rec1-resilin experiments. 5 0.7 2.5 2.5 5.0 0.6 5.0 4 0.5 0.4 3 0.3 buildup curve 0.2 2 0.1 0 Probability Density [1/kHz] 0 2 4 6 8 10 1 excitation time τ [ms] 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.18: Comparison of the distributions and buildup curves for the tendon sample with DQ pre-selection times of 2.5 and 5.0 ms. Despite the very significant change in total signal intensity observed between the two experiments (see table 5.2 above), the buildup curves obtained are relatively similar. Although it appears from the regularization fits that the plateau level is higher for the longer pre-selection time, this is not highly significant because of the large error bars in the experimental data at longer excitation times. The distributions obtained follow the expected behaviour of shifting to smaller couplings for the longer pre-selection time. 116 5.3.5 Discussion Figures 5.19 and 5.20 show for all three samples the distributions (obtained using the regularization method) and buildup curves for DQ pre-selection times of 2.5 and 5.0 ms, respectively. Table 5.9 lists the average values of the distribution parameters obtained from the analytical models. Differences and similarities were observed between all three samples, which are likely related to differences and similarities in their structure and origin. The An16 sample might be expected to differ from the other two samples, since it has a simplified, repetitive primary structure and also may not be based on a true resilin protein (as discussed in section 3.4.1). However, it did not seem to show any major differences, and often represented the intermediate case. The tendon sample, as the only natural resilin investigated, showed some distinct features, however it is difficult to form any conclusions about fundamental differences between natural and recombinant resilins based on the single example studied here. One potentially important result in this respect is that the natural resilin seems to contain more strong couplings than the other samples (as discussed below). This may be related to the difference in crosslinking processes between the natural and synthetic resilins, as described in chapter3. If the natural resilin contains more efficient inter-molecular crosslinks, this might lead to more constrained chain dynamics, and also the formation of more stable secondary structure (which would both yield larger residual couplings). The relationship between crosslink density and residual couplings is discussed further below. In previously published results (see section 3.4.5), it was found that natural resilin has a much higher modulus than either An16 or rec1-resilin. This may also be indicative of more effective crosslinking or the presence of more stable secondary structure. The suggestion that the tendon distributions contain more larger residual couplings than the other materials is supported by the steeper initial rise in the buildup curve, the higher average couplings from the analytical functions, and the shape of the distributions obtained using the regularization method. This would also be consistent with the faster relaxation observed in this material (as described in section 5.3.1). Based on the average results from the analytical functions, the tendons also appear to show the most significant change in the mean residual coupling size when the pre-selection time is increased. However, this is not obviously reflected in the buildup curves themselves, which, compared to the other materials, appear to be relatively similar for the two pre-selection times (as seen in figure 5.18). The more significant disparity is likely due to a decrease in the larger couplings. As mentioned above, these would be expected to relax at a faster rate (during the excitation/reconversion process) than weaker couplings, and their decrease would also support the explanation offered in section 5.3.1 for the more significant effect of increasing the pre-selection time on the total intensity of the tendon spectrum. As mentioned in section 5.3.2, however, the 117 6 An16 An16 Rec1 Rec1 0.6 5 Tendon Tendons 0.5 4 0.4 0.3 3 buildup curve 0.2 0.1 2 Probability Density [1/kHz] 0 0 2 4 6 8 10 excitation time τ [ms] 1 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.19: Comparison of the buildup curves and distributions (obtained using the reg- ularization method) for the three samples, with a DQ pre-selection time of 2.5 ms. The results shows that the distribution of couplings in rec1-resilin is notably different from the other materials, and also that the final buildup level differs significantly; the distribution appears be narrower than the other materials, and shows more weaker couplings. 118 8 An16 An16 Rec1 Rec1 0.6 Tendon Tendons 0.5 6 0.4 0.3 buildup curve 0.2 4 0.1 Probability Density [1/kHz] 0 0 2 4 6 8 10 2 excitation time τ [ms] 0 0 0.5 1 1.5 2 2.5 3 Residual Coupling D/2π [kHz] Figure 5.20: Comparison of the buildup curves and distributions (obtained using the reg- ularization method) for the three samples, with a DQ pre-selection time of 5.0 ms. Other than the secondary peaks in the distribution (which are possibly just artifacts from the regularization process), the distributions appear to be more similar between the three sam- ples here than for the shorter pre-selection time. Because of the more significant change in the plateau level of the An16 sample, the tendon buildup curve appears in this case to be intermediate between the An16 and rec1-resilin results. However, this pattern is not obvi- ously reflected in the corresponding distributions, where the tendons and An16 distributions appear to be relatively similar. 119 tpre = 2.5 ms D¯ ∆D C An16 0.34 ± 0.02 0.22 ± 0.02 0.25 ± 0.01 rec1-resilin 0.23 ± 0.02 0.20 ± 0.03 0.47 ± 0.01 tendons 0.41 ± 0.02 0.38 ± 0.05 0.33 ± 0.04 tpre = 5.0 ms D¯ ∆D C An16 0.22 ± 0.04 0.19 ± 0.05 0.24 ± 0.04 rec1-resilin 0.15 ± 0.01 0.15 ± 0.02 0.58 ± 0.04 tendons 0.24 ± 0.02 0.23 ± 0.02 0.42 ± 0.09 Table 5.9: The mean (D¯) and mean deviation (∆D), given in Hz, and the plateau level (C) of the distribution functions obtained for the different samples and pre-selection times. The results given are the average of the results from the different analytical functions, weighted by their χ2 values. The uncertainty intervals given are the standard deviations (not the standard deviations of the means) from each set of four distribution functions. The results suggest that the rec1-resilin sample has more small couplings than the other materials for both pre-selection times, which is consistent with the regularization results shown in the figures above. The estimate of 0.58 for the plateau level (C) for the rec1-resilin sample with a pre-selection time of 5.0 ms is likely an overestimate resulting from the experimental buildup curve having not yet reached a stable plateau at the maximum excitation time. strong couplings are based on only a few initial points in the buildup curve, and so the details of the fit cannot be considered to be very reliable. One of the most interesting effects that can be observed in the results is that both the An16 and tendon buildup curves showed a plateau level much smaller than the expected value of 0.5, whereas the rec1-resilin buildup curves did not. We offer two hypothetical explanations for this effect, both of which are related to the wide distributions of couplings observed in the samples tested. The first hypothesis is that the lower plateau levels could be caused by differences in the relaxation rates of nuclei with different residual coupling strengths. As mentioned previously, it would normally be expected that nuclei with stronger residual couplings would relax at a faster rate than those with smaller couplings. Figure 5.21 shows an example of the simulated effect of differential relaxation on a typical buildup curve, as explained in the figure caption. It can be seen that in addition to some changes in the shape of the buildup curve, the apparent plateau level is also significantly reduced. This simple example assumes a linear dependence of the relaxation rate on the coupling strength, with a proportionality constant chosen to give a change in plateau level comparable to that observed in the present experiments. Note that because adding differential relaxation effects changes the shape of the buildup curve, it also affects the apparent underlying distribution of couplings (as shown in the inset). The relaxation causes the distribution to broaden slightly, with a small shift 120 toward larger couplings, and also causes the plateau to decrease slightly at later excitation times. This example is intended as a proof-of-concept only, and the effects of relaxation may depend strongly on the original distribution of couplings and also the precise dependence of the relaxation rate on the residual coupling strength. The effects of differential relaxation on the zero-quantum reference curve Sref (see section 4.6.3) would also require further investigation. It should also be noted that a strong variation in the relaxation rate should cause the shape of the sum curve (SMQ + Sref ) used for normalization of the experimental buildup curve to deviate from that of a single exponential. However, as seen in figure 5.9 above, single exponential curves still provide reasonably good fits to the experimental results. Thus it is possible that differential relaxation is only partially responsible for the changes in the plateau levels observed. Another possible explanation for the lower than expected plateau levels is that diffusive exchange (both physical and as a result of internuclear interactions) could occur during the excitation/reconversion process, allowing the transfer of magnetization between states with different residual couplings. This would be expected to proceed toward an equilibrium which contains more nuclei with small residual couplings, because these would have been previously removed by the pre-selection techniques. This might cause intensity to start building up MQ coherence in a strongly coupled state before transferring to a more weakly coupled state in which coherence develops at a slower rate, thus resulting in a lower plateau level. The precise effects would depend on the precise dynamics of the exchange; simulations would be required to verify this hypothesis. Note that in the absence of signal reduction caused by relaxation, equipartition among different coherence orders would still be expected to occur after an infinitely long excitation time, yielding the expected plateau level of 0.5. It should also be noted that exchange during the excitation/reconversion process would also likely affect the shape of the buildup curve, for example by introducing more weaker couplings at longer excitation times. However, since these weaker couplings would not have been highly significant at shorter excitation times, this should not have too deleterious an effect on the distributions obtained. It is possible that both explanations might be partially responsible for the effects ob- served. In either case, it would be expected that a wider distribution of couplings would cause a more noticeable effect. This would increase the effects of differential relaxation be- cause of the dependence on the coupling strength, and also increase the effects of diffusive exchange if the distribution is further from equilibrium prior to the excitation/reconversion period. Compared to the results obtained for the PDMS sample (see section 4.6.7), the resilin/resilin-like samples showed much broader distributions, but with similar average residual couplings. This may explain why the plateau was found to be close to the ex- pected value for the PDMS samples. A narrower distribution may also explain why the 121 1 3 original with relaxation 0.8 sum relaxation 2 renormalized 1 0.6 0 0 1 2 3 coupling strength (D/2π ) [kHz] 0.4 0.2 0 0 2 4 6 8 10 excitation time [ms] Figure 5.21: Plot illustrating the simulated effect of differential relaxation on an example buildup curve. The original buildup curve was generated from a Gaussian distribution based on the best fit obtained for the An16 sample with a pre-selection time of 2.5 ms (see figure 5.10 and table 5.3), but corrected to have a plateau level close of 0.5. Differential relaxation was then simulated by adding a relaxation factor of exp (−Dt/2) to the kernel function (equation 4.6.8) and re-integrating, yielding the curve labelled “with relaxation” in the plot. The sum relaxation curve was obtained by integrating the relaxation factor over the original distribution, and the renormalized curve by dividing the “with relaxation” curve by the “sum relaxation” curve. It can be seen that the predominant effect is to lower the plateau level, in addition to changing the shape slightly. The inset shows the original distribution and the distribution derived from the renormalized curve using regularization. It can be seen that in this case the relaxation causes the distribution to broaden slightly, leading to an overestimate of the larger couplings. 122 plateau level was not reduced in the rec1-resilin results, for the same reasons. This might be consistent with the smaller average coupling strength observed for this material, which could result from an initially narrower distribution that would then also be less affected by differential relaxation and/or diffusive exchange. It would also be supported by the slightly smaller observed effect of the DQ pre-selection on the spectral intensity/sample fraction for rec1-resilin compared to the other materials, as well as the smaller shift in the residual coupling distribution as the pre-selection time is increased. As mentioned above, the buildup curves (and hence the underlying distributions) for all materials show a shift toward more smaller couplings when the pre-selection time is increased. This suggests that the redistribution of couplings is incomplete even after the redistribution delay. If this were the case, it would also support the possibility of net diffusive exchange toward smaller couplings during the excitation/reconversion process. The incomplete redistribution would also be consistent with a relatively broad distribution of couplings (which should take longer to reach equilibrium). The precise effects of differential relaxation and/or diffusive exchange could be investigated more thoroughly by repeating the experiments for different values of the redistribution delay τz and/or by performing additional simulations of both effects. In addition to helping to determine the underlying distributions of couplings, investigating the effects of exchange could also provide useful information regarding the time scale of conformational changes in the materials. Despite the delay period included to allow redistribution, the pre-selection sequences will necessarily change the residual coupling distributions observed. This is actually done intentionally to reduce interference due to weakly coupled components which do not develop MQ coherence in experimentally accessible time scales. From the sample fraction estimates in table 5.2, as well as the effects of increasing the pre-selection time, one can speculate on the properties of the full underlying distribution. The distributions would be expected to contain significant intensity at zero residual coupling strength arising from solvent molecules and kinetically free molecular segments which would have been removed by the pre-selection techniques. It is also likely that the distributions contain slightly more relatively weak couplings which would be re-introduced if the DQ pre-selection time was increased further. The DQ pre-selection also likely causes a relative reduction of the strongest couplings in the distribution, because these would be expected to relax at a faster rate during the pre- selection process. So, it is likely that the true distributions of couplings are even more heterogeneous than the results suggest directly. The high concentration of couplings in the smallest experimentally accessible range, as well as the significant impact of the pre-selection techniques on the represented sample fractions, suggests that the majority of conformations in the samples tested are highly dy- namic and relatively unconstrained. However, the broad distributions suggest that their 123 structure is also much more heterogeneous than that of the PDMS samples, with many different types of localized dynamics present. This might be expected given the freedom of the proteins to adopt many different types of secondary and tertiary structures. Of the ma- terials tested, it would appear that the rec1-resilin samples are slightly more homogeneous in terms of their molecular structure. The presence of diffusive exchange from strong to weak couplings would suggest that molecular segments in the materials tested can change between conformations having different dynamics. However, the fact the redistribution is still incomplete after the redistribution delay places some constraints on the time scale of these conformation changes. In previously published studies on PDMS samples it was shown to be possible to estimate the molecular chain weight, and hence the crosslink density, from the residual couplings obtained using MQ techniques.[15] This was found to be not consistently reliable, however, and also requires an estimate of the static coupling strength (which can be obtained using, for example, spin dynamics simulations). Because of this, quantitative analysis was not attempted here; however, qualitative analysis of the residual coupling distributions can still provide valuable information about local chain dynamics in the materials tested. The present case is also complicated by the presence of (impermanent) secondary structure in the proteins, which will result in a more constrained structure (and hence less motion and higher residual couplings) without any difference in crosslink density. As mentioned above, however, it is possible that the density of intra- and inter-molecular crosslinks might be correlated with the presence of secondary structure in the material, since the crosslinks could help to stabilize the molecular matrix. It might also be expected that inter-molecular crosslinks would be more effective in this regard, since these could constrain the dynamics of the matrix without imposing as many restrictions on the conformation of individual protein chains. They might also be more effective at constraining molecular dynamics, and hence increasing the size of residual couplings in the protein, independently of any effects on secondary structure. It should be noted, however, that in previous studies it was found that crosslinked and uncrosslinked An16 appeared to have similar secondary structure content (see section 3.4). As seen from the present experiments, MQ spectroscopy can provide a wealth of informa- tion which can be used to study the molecular dynamics of materials at many different lev- els, including fast dynamic averaging and potentially also relatively slow exchange/diffusion processes. There are many parameters available which can be changed to analyze these dy- namics from different angles, however this can also make the experiments intensive in terms of both experimental resources (e.g. spectrometer time) and data analysis. Because of the uncertainty inherent in the inversion problem, the interpretation of the various available results can also be somewhat unclear. This was especially true in the present experiments, 124 which appear to involve a wide distribution of couplings with a high dynamic range of probability densities and appear to be further complicated by exchange and/or differential relaxation. As such, in addition to providing useful information about the materials stud- ied, the current study also has value as a study of the capabilities and limitations of MQ spectroscopy. The depth of possible experiments available using this technique also gives it good potential for further study, for example by investigating the effects of changing differ- ent parameters or performing simulations to study the effects of diffusive exchange and/or differential relaxation on the buildup curves. 125 Chapter 6 Conclusion The important results from the previous chapter are summarized here, with a focus on their implications in terms of the microscopic properties of the materials tested. From the lineshapes in the 13C spectra, it was found that the An16 samples exhibit a significant degree of kinetic freedom and molecular mobility, both when stretched and unstretched. It was also found, by comparison to predicted chemical shifts, that the material appears to exhibit primarily randomly coiled secondary structure when unstretched. Upon extension, there appears to be a distinct tendency toward increased β-sheet structure, with the most dramatic changes occurring for the residues in the conserved YGAP motif (which likely plays a role in di- and tri-tyrosine crosslink formation). The lack of any observable change in the linewidths, however, suggests that the protein is still able to dynamically change between different conformations. The attempts to measure residual quadrupolar couplings in deuterated water absorbed in An16 showed (within the limits of precision imposed by the experiment) only the absence thereof. However, this can be considered consistent with a material that is generally dynamic at a microscopic level, and in which relatively little orientational order is imposed upon extension. More sensitive experiments would be required to characterize this aspect in greater detail. From the measurements of residual dipole couplings, a few conclusions can be drawn. For all three materials tested (An16, rec1-resilin, and natural resilin from dragonfly ten- dons), the results are consistent with the picture of a predominantly dynamic, randomly coiled structure. However, the results also suggest that the materials exhibit a significant degree of heterogeneous structure, corresponding to broad distributions of structural order. This indicates that the materials can sample a wide range of different types of secondary structure (and localized dynamics). Compared to similar experiments performed on sam- ples of PDMS, the current materials had similar average residual couplings, but with much broader distributions. The results obtained also support the occurrence of a certain amount of conformational exchange between different structural states in the resilin/resilin-like pro- teins. The results for rec1-resilin suggest that it may have a more homogeneous structure compared to the other materials, with generally less constrained dynamics. Conversely, the results for the natural resilin from dragonfly tendons suggest that this material may contain 126 more rigid structural elements and more long-lived secondary structure. This may be due to the difference in crosslinking method for this material; the natural crosslinking process may result in more effective inter-molecular crosslinks, which might more effectively stabilize the molecular matrix of the protein and also promote the formation of secondary structure. This is also likely related to the much higher stiffness (modulus) previously observed for natural resilin compared to the recombinant proteins. The picture of resilin and resilin-like proteins as being highly dynamic and predominantly randomly coiled is consistent with previous results, as described in section 3.4.4. It has previously been suggested that the elasticity of resilin and resilin-like proteins arises from the fact that many possible conformations can occur in the unstretched state, leading to a high entropy. The ability of the molecular chains to easily change between different states has also been suggested to contribute to the high conformational entropy of the unstretched state.[76] When strained, the conformations (and the interchange between them) might become more biased toward extended states, hence causing a reduction in the entropy. This picture is supported by the present results, which show both a high degree of conformational freedom in the unstretched state (for all three materials tested), and tendency toward more extended β-sheet structure upon extension (for An16). The current research also suggests many possibilities for further research. It would be interesting to see how the 13C results would be affected by increasing the sample extension to even greater strains; as seen in previous mechanical tests, strains up to 300% should be possible. It would also be interesting to repeat the experiments with different RLPs, such as rec1-resilin, to check for any differences or consistencies in behaviour. The measurements of residual quadrupole couplings could be repeated with more anisotropic (deuterated) solvent molecules, which would give better sensitivity to molecular alignment upon sample exten- sion, and could hopefully lead to more meaningful results. The measurements of residual dipole couplings using MQ techniques suggest the most possibilities for further investiga- tion. More detailed simulations would be beneficial to verify and quantify the hypothetical effects of differential relaxation and dynamic exchange during the excitation/reconversion process. Some theoretical treatment of both effects could also be beneficial in directing the simulations and interpreting the results in terms of meaningful parameters. Further exper- iments could also be performed to investigate the effects of changing some experimental parameters (as described in section 5.3.5), which could provide more detailed information about the structural and dynamic properties of the materials tested. Attempts could also be made to quantitatively interpret the residual dipole coupling distributions in terms of order parameter distributions or more specific quantities such as molecular chain weights and crosslink densities. Although it would present some experimental challenges, it would also be interesting to investigate the effects of sample extension on the residual coupling 127 distributions, by repeating the MQ experiments with the samples strained to varying de- grees. 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