Computational and Theoretical Modeling of Two Dimensional Infrared Spectra of Peptides and Proteins

University of Nevada Reno

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemical Physics

David. G Hogle

Matthew J. Tucker/ Dissertation Advisior

May 2019

Abstract

Two-dimensional infrared (2D-IR) spectroscopy evolved out of the theoretical underpinnings of nonlinear methods to provide a means of investigating detailed molecular structure on ultrafast timescales. This opens up new possibilities for the study of protein folding and molecule-solvent interactions through greater spatial and temporal resolution. To study biomolecules, functional groups sensitive to the local environment known as infrared reporters are used to investigate activity at a particular site or sites of interest. Through these probes the analysis of dynamic species and site-specific structural information, previously inaccessible to direct observation, can elucidate key details of conformational changes that exert major influences on the activities of complex molecular systems.

The broad objective of the research of the Tucker group is to obtain a description of the dynamic processes that lead to conformational changes at a chemical bond scale. A major component of

2D IR methods is the choice of vibrational probes used, which impacts the structural information provided about the larger surrounding environment. The type and placement of infrared reporter is highly system-specific, and developments within this area continue to open up new areas of inquiry for 2D IR experiments. My work has been towards providing an understanding of how these probes interact with their surroundings for the purpose of rational infrared probe design and insertion.

The use of non-natural amino acid side chain probes has shown promise for use in tracking the dynamics of the side chain region of proteins. Studies performed in this work have demonstrated the ability of local amino acid modes to capture changes in local electric fields and the degree of hydration at distinct locations in proteins. The ring mode of tyrosine has particular utility as a naturally occurring probe that shows a high degree of sensitivity to its environment through

solvent broadening, giving rise to applications in studies of folded and unfolded peptides/proteins. This is a promising step towards expanding the tools available to observe solvation and van der Waals forces during enzymatic and membrane-binding processes, amongst others.

We have investigated the interaction between two potentially useful reporters for use in in nucleotides and small peptides: the cyano- and the azido- moieties. Theoretical models developed using this data demonstrate opportunities for monitoring structural changes within oligonucleotides and peptides. The relations between the intensity of the coupling of these reporters with spatial orientation and distance allows mapping of groups on the sub-nanometer scale with a sub-picosecond time frame, taking advantage of the powerful time-resolution of infrared methods.

The advance of two-dimensional infrared spectroscopy depends upon the development of new techniques that take full advantage of new technology and theoretical methods. My work with the Tucker groups has been towards the application of computational work towards understanding the relationship between the behavior of the molecular groups of interest and the data collected from the infrared beam. This will hopefully continue to open up new and exciting possibilities for the further expansion of infrared techniques, which are speculated on in the final section. These findings represent a step forward in the development of 2D IR probe groups for the investigation of sub-picosecond processes in biomolecules.

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Acknowledgements

This journey has been an adventure, and like most actual adventures, I’m mainly thankful that it is almost over. I would like to affirm that graduate school is indeed rewarding, in its own way, although I believe the reward is mostly in moving on to other things. However, before I do, I would like to take the time to thank a few special people. Group members Amy Cunningham, Mat

Roberson, Natalie Fetto, and Andrew Schimtz have been there with me and have been supportive over the years, and I doubt I would have achieved half as much without their assistance and support.

While I took a year to make the decision to join the group, I was there near the start of the Tucker lab, so I had the opportunity to be there for much of the setup of the lab equipment. A major draw for this group was the chance to see the setup of a spectroscopy lab firsthand and in doing so learn about the techniques used in Two-dimensional infrared (2D-IR) spectroscopy, which I entered knowing next to nothing about except that infrared spectroscopy measures molecular vibrations. That seems so distant now; I have gained a broad knowledge base on the subject after working with it for almost five years, as well as the humbling realization of how much I still have yet so learn.

One interesting thing about molecular movement is that all vibrational modes have a harmonic component, an idealized motion where the molecule moves in perfect periodicity. This is called a normal mode; and when undergoing periodic motion the molecule is said to move sinusoidally, which I thought sounded like something that’s gotten stuck up your nose. Interestingly, the etymological conflation here is due to a misunderstanding involving a Latin translation of a paper on trigonometry written in Arabic with Sanskrit words. Sinus in Latin means cavity (such as up

your nose) and the common 'sinuses' are correctly referred to as the paranasal sinuses. So your nose has nothing to do with sine functions, which are sinusoidal, and hence nothing to sneeze at.

A love of mathematics and a love of (allegedly terrible) puns have probably helped to me to persevere at my work when things didn’t seem to be terribly engaging. One of the truisms of science is that progress is often far more frustration and boredom than flashes of brilliant insight.

The small victories in science, both personal and professional, are seldom celebrated or appreciated. Whether that is the modest success of me getting a particular code to run properly, or the slow, thankless progress of countless research groups developing the equipment and techniques over decades that I would base my work on, science is often a long, laborious process requiring countless setbacks and failed attempts before progress is made. Hopefully my own failed attempts and setbacks over the years have led to progress toward my own goals as well.

I would be remiss if I did not thank my friends and family. During my years spent at UNR, several close associates offered words of wisdom, their friendship, and both academic and emotional support. Some of these people have since moved on to bigger and better things, while others are still waiting to finish their studies and their chance to move on. Jen Schimdt, Dylan Jones, and Keveen Flieth, thank you for your friendship and support. On perhaps a more personal level,

I would like to thank Baron and Tyson, whose constant companionship has provided me the motivation to see this through. You guys mean more to me than most of those around me know, and I’ll always remember the time we spent together here in Nevada, and I look forward to doing more exciting science with both of you over the years

Abstract i

Acknowledgements iii

Table of Contents v

List of Tables vii

List of Figures viii

Chapter 1: An Introduction to Infrared Spectroscopy 1

1.1 Basics of Infrared Spectroscopy 1.2 Emergence of Multidimensional Infrared Methods 1.3 2DIR Methods Allow for the Greater Exploration of Peptides

Chapter 2: Models of Vibrational Interactions 17

2.1 Laser-Dipole interaction 2.2 Explanation of peak broadening and coupling 2.3 Computational Software Used 2.3.a Hessian and Gaussian Basis Sets 2.3.b Molecular Modeling Force Fields 2.4 Methods for measuring coupling 2.4.a Finite Difference Method 2.4.b Transition Dipole Coupling 2.4.c Transition Charge Model 2.4.d Transition Dipole Density Distribution 2.5 Solvent models 2.5.a Onsager model 2.5.b Conducting Polarizable Continuum Model

Chapter 3: Azide and Nitrile Probes- The Best of Both Worlds 43

3.1 Theoretical Modeling of Novel Probe Behavior 3.2 Investigation of Conformational Preferences in 2’-azido-5-cyano-2’-deoxyuridine. 3.3 Modeling the Coupling Mechanism 3.4 Simulations of Azido and Nitrile Groups in a Model Peptide System 3.5 Conclusions and Future Directions

Chapter 4: Molecular Dynamics Simulations with the Goal of Rational Probe Design 61

4.1 Development of Intrinsic Infrared Probes 4.2 Solvachromatic Effect of the Tyrosine Ring Mode 4.3 Theory and NAMD Simulations 4.4 Solvent Broadening Effects on the Tyrosine Ring Mode

4.5 Studies of Tyrosine as a Probe Involving the trp-Cage Peptide 4.6 Future Directions

Chapter 5: Fermi Resonance in Para-azido-cyano Benzene 82

5.1 Introduction 5.2 Fermi resonance theory 5.3 Calculation of Mode Frequencies for Anharmonic Coupling.

Chapter 6: Future Directions in 2D IR 96

6.1 Phenyalanine Derivatives 6.2 Future Probe Design 6.3 Transient 2D IR Methods 6.4 Conclusion

Appendix A: Supplemental Information for the Theoretical Computation of Fermi Resonance 106

Appendix B: Matlab Code 108

Appendix C: Fortran Code 113

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List of Tables

Table 4.1 Solvatochromic effect on the tyrosine ring breathing vibrational transition

Table 5.1 Results from simulated spectra to obtain cubic couplic terms

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List of Figures

Figure 1.1 Typical transient 2D IR optical design with pulse shaping module

Figure 1.2 Harmonic and anharmonic two-state excitation

Figure 1.3 Typical 2D IR spectrum with coupled peaks

Figure 2.1 Visual representation of Bloch dynamics

Figure 2.2 A sample molecule of 2′-azido-5-cyano-2′-deoxyuridine molecule

Figure 2.3 Two simple dipoles exerting forces on one another

Figure 2.4 False color representation of two groups in a Transition Dipole Density Distribution

Figure 2.5 Visualization of the Onsager field model

Figure 3.1 Structure of the 2′-azido-2′-deoxyuridine molecule

Figure 3.2 Comparison of Onsager and CPCM model.

Figure 3.3 Coupling of the groups is directly proportional to the splitting of the peaks

Figure 3.4 Absorptive IR spectra of N3CNdU in water.

Figure 3.5 Two-dimensional IR spectra of N3CNdU in water.

Figure 3.6 Structure of sample peptides used.

Figure 3.7 Ramachandran plots of the coupling data.

Figure 4.1 IR spectrum of the breathing mode of the localized primarily on the tyrosine ring

Figure 4.2 The bandwidth of the peak diminishes with the strength of the solvent’s electric field.

Figure 4.3 Relationship between the vibrational bandwidth and the solvent electron donor ability

Figure 4.4 The infrared mode of Tyrosine in water and DMSO

Figure 4.5 The Optimized configurations of solvent molecules and their dependence on angle of approach

Figure 4.6 Computer rendering of folded Trp-cage showing the position of the tryptophan residue in the hydrophobic pocket.

Figure 4.7 The relation of the infrared spectra to bandwidth of tyrosine in Trp-cage miniprotein.

Figure 5.1 Overtones in an anharmonic system

Figure 5.2 Explanation of Fermi Resonance

Figure 5.3 Components of a two state excitation

Figure 5.4 Experimental and simulated linear spectrum of p-azidobenzonitrile

Figure 5.5 Experimental and simulated 2D IR spectrum of p-azidobenzonitrile

Figure 6.1 A flowchart depicting the situations where various infrared probe types are of use.

Chapter 1: An Introduction to Infrared Spectroscopy

1.1 Basics of Infrared Spectroscopy

Historically, infrared spectroscopy is well known for its utility in compound identification and quantitative analysis, especially for organic compounds. All molecules contain a set of normal modes that vibrate at a characteristic frequency, and it is these modes that are the basis of the infrared spectrum. As these modes are not moving in isolation but interacting with the reminder of the molecule and its environment, these characteristic frequencies will shift depending upon local conditions. In conjunction with theoretical models of how these shifts behave, infrared spectra convey detailed information about the mechanics of a given molecule. Periodic functions such as the sine and cosine functions repeat over some time and are also referred to as harmonic functions. If an arbitrary function can be represented by a set of functions added together, those functions form an independent basis. If a given system is periodic, it is intuitive to represent this by some combination of simpler periodic functions.

Normal modes are defined by patterns that have simple oscillatory motion, meaning the restoring force is proportional and opposite to the displacement, resulting in wave-like motion. An analogy in macroscopic physical systems is the simple harmonics of a drum or radio waves. When a normal mode is excited, the resulting vibration can involve all the nuclei of the molecule to some extent, as there is no reason for a normal mode to be localized to a single bond or a single-bond angle.

Improvements in lasers and detection equipment, in conjunction with the rise of sophisticated computational modeling, have led to the rise of new, advanced multidimensional techniques using infrared for fast, precise monitoring of molecular processes. The normal modes are defined as linear combinations of the Cartesian displacement coordinates and are often complex to write out in geometrically defined coordinates.

A local mode is a mode where the absorbed energy is used to excite the stretching of a single bond or the bending of a single-bond angle (or dihedral angle). If a local mode picture can be developed, then relating a structure to its vibrational signatures becomes more intuitive. Local modes can be expressed by combinations of normal modes, as each normal mode in a molecule has a unique given frequency and they are mutually orthogonal, meaning they oscillate independent of one another. Relating normal modes to local modes requires knowing how the local modes interact with each other; in the case of vibrational spectroscopy, this means knowing how the local modes are vibrationally coupled. Thus, it is important to develop tools that can be used to measure the coupling between these local modes directly and develop models to understand the nature of the coupling.

In reality, neither macroscopic nor molecular vibrations are anharmonic oscillators subject to damping and elastic forces, which harmonic models provide only a first order approximation.

Multidimensional modes depend upon measuring this anharmonicity to obtain additional information about the system being investigated. Furthermore, vibrational modes do not exist in isolation because of this anharmonicity. Anharmonic effects can be due to anything that introduces a dampening force, whether this is in the atomic bonds themselves, electromagnetic effects, or transfer of energy from other vibrating bonds. This can come from the rest of the molecule via both through space and through bond effects. Electrostatic repulsion and collisions

(through space and through bond effects) can come from other molecules as well; whether from other molecules of the same species such as a dimer, other species in the surrounding area if applicable, or the solvent. The last one is especially important in the study of biological activity, as all interactions in vivo must take into account the presence of water molecules as solvent. If vibrational modes were excited in perfect isolation, multi-pulse methods would be an interesting

curiosity, but little more. Fortuitously, this is not the case. The effect from other modes, their environment, and the motion of the bulk molecule lead to vibrational coupling, and the observation of coupled peaks gives useful information on these interactions.

The astute reader may be familiar with these notions in a different context, as multidimensional

Nuclear Magnetic Resonance (NMR) methods that use multiple radio frequency pulses preempted multi-pulse IR methods by several decades due to the lack of a source of high-powered, fast pulsed infrared pulses. Improvements in lasers and detection equipment, in conjunction with the rise of sophisticated computational modeling, have led to the rise of new, advanced multidimensional techniques that use infrared light for precise monitoring of molecular processes. The development of fast pulsed infrared lasers moved the field from the theoretical to the practical at the beginning on the 21st century.1,2 Methods such as COSY,3,4 NOESY,5,6 and

EXSY7,8 known long before the arrival of 2D IR have paved the way for the development of similar methods using IR pulse sequences. Over the last two decades, multi-dimensional infrared spectroscopy has evolved from a theoretical curiosity to a practical method with versatile applications due to the rapid time frame of data acquisition and observation of interactions between coupled signals.9

Static and transient species such as rotational isomerism and solvent dynamics have previously been impractical or outright impossible to directly observe before the advent of femtosecond pulsed lasers. Hindered internal rotation takes place on the order of 10 picoseconds, and solvent bonds can form and break within picoseconds. The ultrafast time regime is accessed by the intrinsically very slow NMR spectroscopy indirectly through relaxation methods and residual dipolar couplings. For instance, in an EXSY experiment, an initial part of a pulse sequence

(preparation phase) is used to generate an excited spin or vibrational state, which is read out after

a some predetermined waiting period.6 EXSY has an upper limit of the excitation lifetime of the spin or vibrational state used, therefore the indirect nature of the relaxation techniques cannot validate the model assumption made to interpret the data and, hence, tend to oversimplify complex systems. The important difference between IR and NMR spectroscopy is this difference between direct and indirect measurement. The use of ultrafast laser in the infrared region allowed for the first time direct application of multidimensional methods in measuring ultrafast processes.

This work focuses on phenomena that cannot be measured without ultrafast methods, and how these detailed pictures of fast, small processes can be vital to understand why a given conformation or folding mechanism is preferred.10,11

1.2 Emergence of Multidimensional Infrared Methods

Transient pump–probe spectroscopy is the predecessor of 2D IR spectroscopy. In the hole-burning method, a broadband femtosecond pulse is used as a pump, placing the system in a vibrationally excited state. The broadband pump spans multiple vibrational modes, placing the system is an excited state. Once the system has been allowed to relax for a given time, the system is probed using a second pulse. This technique can be performed with incoherent light, such as from a bulb, or coherent light, as from a laser. The use of coherent light results in a polarization which persists during vibrational excitation.

A 2D IR experiment involves a specific sequence of three ultrashort mid-IR pulses in time. The times between the first and second excitation pulses, and between the second and third excitation pulses, are commonly referred to as the coherence time (τ) and the waiting time (T), respectively. The evolution of the system during τ labels the system with the initial frequencies.

Detection of the heterodyned signal after the set waiting time T interrogates the final frequencies

of the system. Thus, the resulting 2D IR spectrum reports on the correlation between the initial and final frequencies of the system after a time separation of T.

To acquire a single 2D IR spectrum, τ is scanned for a fixed value of T. At a time ≤ τ after the third pulse, a third-order signal is generated by the sample in a phase-matched direction. This can be filtered and enhanced for detection via heterodyning, usually this is done with a fourth IR pulse called the local oscillator. The combined pulses are frequency-dispersed onto an array detector, providing a direct measurement of the spectrum along the final detected frequency axis, which is the horizontal axis, or a Fourier transform of the detection time (t). At each detected frequency, scanning τ produces an interferogram. Numerical Fourier transformation of these interferograms generates the spectrum along the initial excitation frequency axis, which is the vertical axis. Note that the designation of τ and t as vertical and horizontal is arbitrary and varies amongst research groups.

Two-dimensional infrared uses coherent light, which places the system in a superposition, or coherent state. The system is then placed in a population state via a second pulse, after a given time. The phases decay due to interaction with their environment, which we shall cover in detail in Chapter 2, as this dephasing is vital to the strength of 2D IR as a system-sensitive technique. If another pulse hits, it will interfere constructively or destructively depending on the phase of the system. If the pulse arrives after a given time delay, one can measure the extent of the system that is still in phase.

2D IR evolved from the double-resonance experiment;12,13 a conventional pump-probe method where an intense ultrashort (<100 fs) broadband infrared laser pulse covering the whole spectral range of interest is split into a pump and a probe beam. The difference between the double-resonance experiment and a conventional pump-probe experiment is that the pump

pulse passes through a filter consisting of two partial reflectors that slices out a narrow-band pump pulse (5–15 cm-1) on a given frequency.14,15 The 2D-IR spectrum is thus a collection of these ‘slices’ where each horizontal cut in the probe frequency direction represents a transient absorption spectrum obtained by pumping at the frequency on the pump frequency vertical axis.

Heterodyne-detected photon echo spectroscopy is based instead on a three-pulse photon echo experiment.14,16 An intense ultrashort IR laser pulse, which spectrally covers the whole range of interest, is split into three pulses that are directed onto the sample with variable delay times.

The interferometric superposition of the generated third-order field is combined with a fourth replica of the initial ultrashort laser pulse, in a process known as heterodyning. The interferometric superposition can be either done in the time domain by scanning the time and phase of the local oscillator or in the frequency domain by spectrally dispersing both beams in a spectrograph. We must know the phase of the initial pulse, as the superposition depends on the optical phase between the first and second pulses as well as between the third and local oscillator pulses.

The box CARS configuration is a typical setup that allows separation of the ingoing (k1, k2 and k3) and the two outgoing (2k1+k2+k3 and k1+2k2+k3) beams by changing their respective phase- matching directions.17 Figure 1 depicts a sample transient 2D IR setup with pulses originating from two independent OPAs providing 4-7 µm mid IR pulses and a tunable excitation/triggering pulse. The infrared beam is split into two arms for the pump and probe IR beams. The first arm generates an IR probe beam which often has a time delay range of 120 ps. The second arm is diverted into an acoustic optic modulating pulse shaper to generate two pump pulses. The generated third-order field is 2D-Fourier transformed with respect to the time between the first

and the second pulses and the time after the third pulse. This generates a 2D-IR spectrum as a

18 function of two frequencies v1 and v3.

These echo signals of the 2D IR measurement allow the structural determination of the peptides for transient 2D IR measurements and the pulse shaping speeds up data acquisition and facilitates control of phase, pulse sequences, and peak shapes to optimize the experiment. In order to perform the Fourier transform, one needs to know the electric field irradiated by the third-order polarization, rather than the time-integrated intensity, which is what ‘‘normal’’ square law detectors measure. By determination of coupling through 2D IR pump probe, changes of angles and bond distances within a structure during the time evolution of the structure can be determined.

The 2D IR echo signal is due to the third-order macroscopic polarization generated by the excitation of vibrational energy states allowing the full three-dimensional response of the system to be measured at the picosecond or faster timescale. The combination of high resolution and ultrafast timescale is of critical importance in observing chemical and physical processes such as solvent-molecule dynamics, structural changes, chemical exchange and more. If the system is restricted to excite a single vibrational mode, two peaks arise in the 2D IR spectrum yielding information about the 0 to 1 transition of the target vibrational mode and 1 to 2 transition of the same mode.9,19 Figures 2 and 3 show a linear and two dimensional spectra of a two state excitation, respectively. In both figures, the 0 to 1 transition is red and the 1 to 2 transition is in blue.

Excited states can affect each other as well. This is an instance of coupling, and occurs if many rotational states are present, causing rotovibrational coupling. This is primarily observed in the gas phase, where the excited rotational states are predominant. When the nuclei move during an

electronic excited state, it is vibronic coupling. We are interested what happens when two normal modes interact. This is termed vibrational coupling and this process conveys a wealth of information about the system of interest.

Each excited oscillator will maintain a certain frequency distribution until several dephasing processes occur, leading to spectral diffusion which is captured by the frequency-frequency correlation function (FFCF). By introducing a waiting time into the pulse sequence, the FFCF resulting from significant spectral diffusion can be recovered. In this way, 2D IR diagonal signals separate out the inhomogeneous components from the homogeneous component present in the system.20 Spectral diffusion is affected by intermolecular and intramolecular surroundings and their induced electric fields on the vibrational mode, therefore measuring spectral diffusion can reveal solvent-solute effects and coupling to larger molecular motions. When more than one vibrational mode is excited, cross peaks result from the interaction between vibrational modes, interconversion between chemical species or environments, and intra- or intermolecular vibrational relaxation. These cross peaks are the hallmark of two-dimensional spectra due to their distinct appearance, and these relationships are inaccessible to linear methods.

The advancement of modern optics allowed linear IR to observe protein dynamics on the subpicosecond scale, while the application of multidimensional methods allowed the desired information to be extracted out of the overlapping signals from the system. It is difficult and time- consuming, if not outright impossible to accurately deconvolute linear IR spectra into homogeneous and inhomogeneous distributions. Conversely in a two-dimensional infrared spectrum, any inhomogenity in the system is rephased to create a vibrational echo, whereas the homogeneous components are not, causing the inhomogeneous contributions to be elongated along the diagonal. By introducing a waiting time into the pulse sequence, the 2D lineshape can

be viewed as a function of time that gradually widens because of spectral diffusion. By looking at systematic trends in the elongation and widening of the 2D IR peaks, this information can be correlated to the position of the labeled residue in the target molecule.

1.3 2D IR Methods Allow for the Greater Exploration of Peptides

Peptides and proteins play crucial roles in biological processes, folding into a variety of complex forms to accomplish their specific tasks. Due to the massive number of final shapes that proteins can take, protein folding is a highly complex process. All proteins fold as a relaxation process towards an energy minimum, this process is guided by short-lived intermediate states that help determine the available relaxation pathways. Proteins are large structures with thousands if not hundreds of thousands of atoms, and protein folding is a long, involved process that can take seconds or more, an eternity on atomic time scales. Yet transient states lasting only hundreds of picoseconds play critical roles in directing the subsequent steps. Given the large number of proteins found in living organisms, it would be prohibitively time consuming to understand each protein from first principles alone. Biology shows strong conservation of various structural motifs that play a number of versatile roles, as nearly identical secondary structures can behave in different ways depending on molecular context. They are a set of building blocks that can be best understood as having a profile that can fill a variety of roles. The greater understanding of these structural motif profiles allows for better rational understand of both protein function and folding.21,22

The initial steps from an unfolded chain into a helix or sheet occur on an extremely fast time-scale

(under a nanosecond) yet are instrumental in determining how the motif behaves relative to the larger structure. Current simulations of these structural features are often forced to rely upon crude approximations of fundamental forces that diverge from actual biology in unpredictable

ways. Knowing how secondary structures are affected by their environment from both intramolecular and solvent effects would allow the generation of algorithms with greater predictive power. Some of the most conserved simple motifs found in proteins are various types of helices and the β-sheets.

Understanding the behavior of proteins involves determining the relevant kinetics and observing short-lived intermediate states in folding processes,23,24 how proteins interact with membranes,25 and undergo dimerization.26 The ultrafast pulses of multidimensional infrared methods provide a means to freeze out these fast motions and capture sub-picosecond and picosecond dynamics within proteins. It is important to note that 2D IR and other multidimensional infrared methods are extremely versatile, with applications far beyond the analysis of biological systems including biotechnology, medicine, and nanotechnology.27–29 However, even before 2D IR was widely known, the need for precise time resolution for large scale molecular motions was highly desired in the field of structural biology, and one of the promising potentials for developing multidimensional methods was the understanding the underlying physical principles behind these processes.30,31

Our research is centered on understanding the earliest steps in protein folding which may lead to observing the dynamics of disordered and misfolded states that often result in disease.28,32,33 This writing will focus exclusively on biological applications of 2D IR, as other fields are well beyond the scope of this dissertation.

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Fourier Transform Two-Dimensional Infrared Spectroscopy: An Experimental and Theoretical Comparison. J. Chem. Phys. 2004, 121 (12), 5935–5942. https://doi.org/10.1063/1.1778163. (19) Le Sueur, A. L.; Horness, R. E.; Thielges, M. C. Applications of Two-Dimensional Infrared Spectroscopy. Analyst 2015, 140 (13), 4336–4349. https://doi.org/10.1039/c5an00558b. (20) Fenn, E. E.; Fayer, M. D. Extracting 2D IR Frequency-Frequency Correlation Functions from Two Component Systems. J. Chem. Phys. 2011, 135 (7), 74502. https://doi.org/Artn 074502 10.1063/1.3625278. (21) Petty, S. A.; Decatur, S. M. Intersheet Rearrangement of Polypeptides during Nucleation of -Sheet Aggregates. Proc. Natl. Acad. Sci. 2005, 102 (40), 14272–14277. https://doi.org/10.1073/pnas.0502804102. (22) Buchanan, L. E.; Dunkelberger, E. B.; Tran, H. Q.; Cheng, P.-N.; Chiu, C.-C.; Cao, P.; Raleigh, D. P.; de Pablo, J. J.; Nowick, J. S.; Zanni, M. T. Mechanism of IAPP Amyloid Fibril Formation Involves an Intermediate with a Transient β-Sheet. Proc. Natl. Acad. Sci. U. S. A. 2013, 110 (48), 19285–19290. https://doi.org/10.1073/pnas.1314481110. (23) Baiz, C. R.; Reppert, M.; Tokmakoff, A. Ultrafast Infrared Vibrational Spectroscopy; Fayer, M. D., Ed.; Taylor & Francis: Boca Raton, 2013. (24) Ganim, Z.; Chung, H. S.; Smith, A. W.; Deflores, L. P.; Jones, K. C.; Tokmakoff, A. Amide I Two-Dimensional Infrared Spectroscopy of Proteins. Acc. Chem. Res. 2008, 41 (3), 432– 441. https://doi.org/10.1021/ar700188n. (25) Mukherjee, P.; Kass, I.; Arkin, I. T.; Zanni, M. T. Picosecond Dynamics of a Membrane Protein Revealed by 2D IR. Proc. Natl. Acad. Sci. U. S. A. 2006, 103 (10), 3528–3533. https://doi.org/10.1073/pnas.0602988103. (26) Zhang, X. X.; Jones, K. C.; Fitzpatrick, A.; Peng, C. S.; Feng, C. J.; Baiz, C. R.; Tokmakoff, A. Studying Protein-Protein Binding through T-Jump Induced Dissociation: Transient 2D IR Spectroscopy of Insulin Dimer. J. Phys. Chem. B 2016, 120 (23), 5134–5145. https://doi.org/10.1021/acs.jpcb.6b03246. (27) Ramakers, L. A.; Hithell, G.; May, J. J.; Greetham, G. M.; Donaldson, P. M.; Towrie, M.; Parker, A. W.; Burley, G. A.; Hunt, N. T. 2D-IR Spectroscopy Shows That Optimized DNA Minor Groove Binding of Hoechst33258 Follows an Induced Fit Model. J. Phys. Chem. B 2017, 121 (6), 1295–1303. https://doi.org/10.1021/acs.jpcb.7b00345. (28) Middleton, C. T.; Marek, P.; Cao, P.; Chiu, C. C.; Singh, S.; Woys, A. M.; de Pablo, J. J.; Raleigh, D. P.; Zanni, M. T. Two-Dimensional Infrared Spectroscopy Reveals the Complex Behaviour of an Amyloid Fibril Inhibitor. Nat. Chem. 2012, 4 (5), 355–360. https://doi.org/10.1038/Nchem.1293. (29) Donaldson, P. M.; Hamm, P. Gold Nanoparticle Capping Layers: Structure, Dynamics, and Surface Enhancement Measured Using 2D-IR Spectroscopy. Angew. Chemie-International Ed. 2013, 52 (2), 634–638. https://doi.org/10.1002/anie.201204973. (30) Hunt, N. T. 2D-IR Spectroscopy: Ultrafast Insights into Biomolecule Structure and Function. Chem. Soc. Rev. 2009, 38 (7), 1837–1848. https://doi.org/10.1039/b819181f. (31) Hochstrasser, R. M. Two-Dimensional Spectroscopy at Infrared and Optical Frequencies. Proc. Natl. Acad. Sci. U. S. A. 2007, 104 (36), 14190–14196. https://doi.org/10.1073/pnas.0704079104. (32) Shim, S.-H.; Gupta, R.; Ling, Y. L.; Strasfeld, D. B.; Raleigh, D. P.; Zanni, M. T. Two- Dimensional IR Spectroscopy and Isotope Labeling Defines the Pathway of Amyloid Formation with Residue-Specific Resolution. Proc. Natl. Acad. Sci. 2009, 106 (16), 6614–

6619. https://doi.org/10.1073/pnas.0805957106. (33) Lomont, J. P.; Rich, K. L.; Maj, M.; Ho, J. J.; Ostrander, J. S.; Zanni, M. T. Spectroscopic Signature for Stable β-Amyloid Fibrils versus β-Sheet-Rich Oligomers. J. Phys. Chem. B 2018, 122 (1), 144–153. https://doi.org/10.1021/acs.jpcb.7b10765.

Figure 1.1 Typical transient 2D IR optical design with pulse shaping module. Figure by M. Tucker, used with permission.

Figure 1.2 (a) Theoretical depiction of overtones for a harmonic oscillator. All the levels are evenly spaced, and selection rules make overtones impossible. (b) In an anharmonic oscillator, higher levels move closer together, and overlap between energy levels allows for multilevel transitions.

Figure 1.3 Depiction of hypothetical 2D IR spectrum for two excited vibrational. The blue state represents bleaching of the ground state, and the red circles are absorption of the excited state. The off-diagonal peaks represent coupling between the two modes.

Chapter 2: Models of Vibrational Interactions

2.1 Laser-Dipole Interaction

Under most circumstances, a vibrating molecule has a permanent dipole. One dipole affects the other via its own electromagnetic field, and vice versa. This leads to splitting of energy and depends strongly on local environment. Most coupled systems have some combination of both.

This leads to vibrational modes observed with different frequencies than what would be predicted via each mode in isolation.

Coupling is greatest when the coupled groups have approximately equal energies. This can mean two vibrational states that are close in energy. The lower mode becomes lower, and the higher frequency in raised higher, resulting in greater splitting.

In a 2D IR pulse sequence, the first pulse places the vibrational modes of the system in coherence, storing phase information. If we are dealing with an isolated oscillator, this only occurs along the diagonal. Let the transition from the ground state to the first excited state be 휀푖. While the harmonic excited state would be exactly 2휀푖, the effect of anharmonicity is to lower this state by a given amount, which we designate ∆푖푖, and the second excited state has frequency 2휀푖 − ∆푖푖.

We shall consider two coupled vibrational modes in a single system, with the first mode is mode designated as i, and the second mode as j. These modes have harmonic frequencies 휖푖,휀푗 and anharmonicities of 2휀푖 − ∆푖푖 and 2휀푗 − ∆푗푗 for their respective second excited states.

Anharmonicity allows for the possibility of coupling between the two oscillators, with the coupling value between two locally identical oscillators is defined as follows:

∆푖푗≡ 휀푖푗 − 휀푖 − 휀푗 (1푎)

If they are not locally identical, the more general solution can be found by taking the geometric sum of the two modes, giving the coupling term as

∆푖푗≡ √∆푖푖∆푗푗 (1푏)

The general solution can always be found using matrix diagonalization, but a first order perturbation solution is valid when the frequency separation is over 100 cm-1 and the anharmonicity and individual coupling terms, are ≲ 10 cm-1.1

Most molecules have multiple potential excited vibrational states. Two excitations can excite no, one, or two states. Assume that all the molecules are in their ground vibrational state of 0 before the laser pulse interacts with them. The time the first laser pulse interacts with them is always called time 0. Let’s start with linear spectra, where there is only one pulse. We will need to consider two vibrational levels.

1 1 0 Consider a ket |0〉 = that gives the density matrix: |0〉〈0| = [ ], assuming a two-state 0 0 0 system. Represent the original state of this system using the notation H0. If nothing happens, we

2 have a steady-state system, and the Hamiltonian H = H0 for all times t.

At time t = 0 the system encounters a perturbation, represented by some off-diagonal matrix element. Represent this perturbation with matrix W. W is symmetric, because it can occur in either direction. So 퐻 = |0〉〈0| becomes 퐻′ = |0〉〈0| + 푊, written as equation (2).

1 휔 |0〉〈0| + 푊 = [ ] (2) 휔∗ 0

푤11 푤12 퐻(훾) = 퐻0 + 훾푊; 푊 = [ ] (3) 푤21 푤22

For these assumptions to be valid for some given H0, the value W must be small compared to H0.

The eigenvalues of H can be represented as a two-state system on a Bloch sphere analogous to spin such that the two eigenstates of the original 퐻0 are at the poles as seen in Figure 1. After the perturbation is applied 퐸′1 ≠ 퐸1, and the initial states are no longer eigenstates.

The new states are described as equations (4) and (5):

휃 −푖휑 휃 푖휑 |휑′ ⟩ = cos 푒 2 |휑 ⟩ + sin 푒 2 |휑 ⟩ (4) 1 2 1 2 2

휃 −푖휑 휃 푖휑 |휑′ ⟩ = −sin 푒 2 |휑 ⟩ + cos 푒 2 |휑 ⟩ (5) 2 2 1 2 2

휃 is termed the ‘mixing angle’ and defined in equation 6:

2|퐻 | tan 휃 = 21 (6) 퐻11 − 퐻22

1 1 1 0 Let 퐻 = (퐻 + 퐻 )퐼 + (퐻 − 퐻 )퐻′, where I is the identity matrix 퐼 = [ ] The 2 11 22 2 11 22 0 1 eigenvectors are the same, since I is invariant under an eigenbasis.

2퐻 1 12 퐻 − 퐻 −푖휑 ′ 11 22 1 tan 휃 푒 ( ) 퐻 = = [ 푖휑 ] 7 2퐻21 tan 휃 푒 −1 −1 [퐻11 − 퐻22 ]

Equation (7) is solved to obtain the eigenvalues: 휆2 − 1 − tan2 휃 ; 휆 = ±√1 + tan2 휃 = ± sec 휃.

This gives the linear equations (8) and (9):

−푖휑 (1 − sec 휃)휑푎 + tan 휃 푒 휑푏 = 0 (8)

−푖휑 tan 휃 푒 휑푎 − (1 + sec 휃)휑푏 = 0 (9)

휃 For the next step, we can use the half angle formulas from trigonometry and let 휃′ = 2

1 + tan2 휃′ 2 tan 휃′ 1 − tan2 휃′ 1 + tan2 휃′ −2 tan2 휃′ (1 − ) 휑 + 푒−푖휑휑 ; − = 1 − tan2 휃′ 푎 1 − tan2 휃′ 푏 1 − tan2 휃′ 1 − tan2 휃′ 1 − tan2 휃′

휃 휃 − tan2 휃′ 휑 + tan 휃′ 휑 푒−푖휑 = − sin 휑 + cos 휑 푒−푖휑 (10) 푎 푏 2 푎 2 푏

휃 −푖휑 휃 푖휑 Equation (10) can be written as cos 푒 2 |휑 ⟩ + sin 푒 2 |휑 ⟩ because the eigenbasis is invariant 2 푎 2 푏 under rotation. If we define 퐻 = 퐻0 + 푊, then

2|푊 | tan 휃 = 21 (11) 퐸1 + 푊11 − 퐸2 − 푊22

2|푊 | tan 휃 = 21 (12) 퐸1 − 퐸2

1 Equation (11) simplifies to (12) when 푊 = 푊 = 0. Defining ∆= (퐸 − 퐸 ) and solving for E’, 11 22 2 1 2

1 this gives 퐸′ = (퐸 + 퐸 ) ± √∆2 − |푊 |2. 2 1 2 12

Strong coupling of stretching vibrations occurs when there is a common atom between the two vibrating bonds. Coupling of bending vibrations occurs when there is a common bond between vibrating group. Coupling between a stretching vibration and a bending vibration occurs if the stretching bond is one side of an angle varied by bending vibration. They are bound by the covalent bonds of the molecule. The coupling separates the frequencies on a hyperbola. Define

2 4훽12 푎 = ( ) Equation (13) then gives the solution: 퐸2−퐸1

1 (퐸 − 퐸 ) 퐸′ = (퐸 + 퐸 ) ∓ 2 1 √1 − 푎 (13) 2 1 2 2

푎 푎2 When coupling is large, the a term can be expanded via Taylor series as √1 − 푎 = 1 − − − 2 8

푎3 . 16

훽 푎 When coupling is small. ( 12 ≪ 1) the a term can be approximated as √1 − 푎 ≈ 1 − ∆ 2

| |2 ′ 1 1 4 훽12 퐸 = (퐸1 + 퐸2) ± { (퐸2 − 퐸1) − } (14) 2 2 퐸2 − 퐸1

Lasers are an optical electromagnetic field, and can be reasonably thought of in this context as classical electromagnetic waves.3 Therefore, now that we have an equation for the coupling we define our arbitrary perturbation W as an electromagnetic wave using the following time - dependent function:

푊(푡) = 휇̂퐸(푡) = 퐸′(푡)푐표푠휔

The transition dipole moment must still be considered a quantum state ⟨1|휇̂|0⟩. As this pulse perturbs the system

푖푡 휕 푖 − (퐸(1)−퐸(0)) 푐 (푡) = − ∑ 푐 (푡)푒 ħ ⟨1|휇̂|0⟩ 퐸(푡) (15) 휕푡 1 ħ 0

푖푡퐸 푖푡퐸 − 0 − 1 |훹〉 = 푐0푒 ħ |0〉 + 푐1푒 ħ |1〉 (16)

c0 and c1 do not depend on time. This is a combination of states. All molecules in our sample are vibrating in phase. This is the coherence. Dephasing and relaxation eliminate coherence.

Dephasing is due to individual normal modes destructively interfering with each other. Relaxation is the elimination of excited states via energy transfer.

Relaxation is T1, written as equation (17)

2 2 −푡/푇1 −푡/2푇1 2 2 2 −푡/푇1 푐1 (푡) = 푐1 (0)푒 ; 푐1 = 푒 ; 푐0 (푡) = 푐0 (0) − 푐1 (0)푒 (17)

푖휔푡 −푡/푇2 휌01 = 푖푐0푐1푒 푒 . As the excited state relaxes, 푇2 ≥ 2푇1 resulting in homogenous dephasing. The system will also dephase over time; this is the cause of inhomogenous broadening.

1 1 1 ≡ + ∗ (18) 푇2 2푇1 푇2

Note that dephasing doesn’t change the ‘height’ of the Bloch vector, as it moves closer to the z- axis and is pulled ‘inside’ the Bloch sphere. Relaxation pushes the vector up.

2.2 Explanation of peak broadening and coupling

A laser can be naively considered to be a single transition between an excited state and some lower one, emitting as a single beam of coherent light. While this is a convenient abstraction, this is never the case, and to understand spectroscopic process in the real world, we must consider the imperfections in the emission and absorption process and how they will cause the results to deviate from this ideal. The Heisenberg limit establishes a theoretical minimum that frequency and time, thus creating a theoretical limit to which a given coherence can be produced. While practical instruments will be limited instead by the design, the issue still comes into play; a narrow frequency gives a wider time range, and vice versa. The more narrow limited the time is, the greater the inevitable broadening of the pulse frequency. The range of frequencies in a given pulse is the linewidth. Each pulse is different, with the shape depending upon the conditions it has been subjected to.

All contributions toward broadening change the line shape of the emission profile. Homogeneous broadening is when all atoms radiating from a specific level under consideration radiate with equal opportunity, giving a Lorentzian profile. Inhomogeneous broadening is due to some emitting particles being in a different local environment from others, and therefore emitting at a different frequency, resulting in a Gaussian profile.

Thus we see that homogeneous dephasing is the result of the off-diagonal elements of the density matrix representation, making the Bloch sphere a useful means of mathematically conceptualizing the physical process involved. However, this is just an abstraction without an adequate understanding of the contributions involved connecting the physics with the math.

Broadening changes the spectroscopic line shape of the emission profile. The laser emission is due to the difference between an excited state and a lower one. In real life, this energy difference has a fluctuation, so the frequency/wavelength of the beam will have a certain width. The limit is usually instrument design, but it can never be better than Heisenberg uncertainty, so a narrow frequency gives a wider time range, and vice versa. The range of frequencies in a pulse is, of course, the linewidth. The shape of the pulse could be Gaussian, but it need not be. Each pulse is different, with the shape depending upon the conditions it has been subjected to.

The initial broadening effect is due to the uncertainty principle in natural broadening. This broadening effect results in an unshifted Lorentzian curve. Molecules are constantly in motion, causing collisions of particles within the lasing medium. As the duration of the collision is much shorter than the lifetime of the emission process, when a collision occurs during emission, it will up or down shift the emitted energy, and depends on design of the laser. The broadening effect is described by a Lorentzian profile and there may be an associated shift. These are inherent to the instrument, and must be accounted for in interpretation.4,5

The sample is also in motion, and in solution this motion is a significant source of broadening, causing each photon when it is absorbed to be red or blue shifted depending on the velocity of the atom relative to the apparatus. This velocity depends on temperature and is appropriately termed thermal Doppler broadening. This broadening effect is described by a Gaussian profile and there is no associated shift.6,7 Solvent bonding and dielectric effects will cause inhomogeneous

broadening. These can also be an associated peak shift. These inhomogeneous effects are system- specific, as this can convey useful information about the behavior of our sample.8

2.3 Computational Software Used

2.3.a Hessian and Gaussian Basis Sets

The determination of the wavefunction for the ground state of a complex system is essential for accurate modeling of the system in question and an extraordinarily difficult problem to solve exactly. Fortunately, for most chemical reactions, we do not need to know the electron distribution exactly, solving for an arbitrary cutoff is often ‘good enough’. Electrons are fermions, and thus are antisymmetric particles. The wavefunctions for multiple electrons must therefore switch sign upon exchange to satisfy the antisymmetry property and can be represented by a matrix leading to the familiar Slater determinant.9

If a function gives a vector, the Jacobian is represented by equation 19:

휕푓 휕푓 1 1 휕푥1 휕푥푚 퐽 = (19) 휕푓 휕푓 푛 푛 [휕푥1 휕푥푚]

If the function gives a constant, this is a row matrix, or the gradient. If the map between the two spaces is linear, it is differentiable. We decompose the derivative of a function along some vector into its components. These are usually the Euclidean parts, but the equation is general.

Note that this means 퐷푓(푥) is the greatest increase. Taylor series for a multivariable function comes from repeated application of the Fundamental Theorem of Calculus:

푥0+ℎ 푘 푓′′(푥0) (푥0 + ℎ + 푡) 푓(푥 + ℎ) = 푓(푥 ) + 푓′(푥 ) ∙ ℎ + ℎ2 + ∫ 푓푘(푡)푑푡 (20) 0 0 0 2! (푘 + 1)! 푥0

The Hessian is defined only for a single-values function, it is

휕2푓 휕2푓 1 푛 1 휕푥2 휕푥 푥 퐻푓(푥 )(ℎ) = ℎ 1 푛 1 ℎ푇 (21) 0 2 2 2 휕 푓푛 휕 푓푛 [휕푥1푥푛 휕푥푛 ]

푓(푥0 + ℎ) = 퐷푓(푥0) ∙ ℎ + 퐻푓(푥0)(ℎ) + 푅(푥0, ℎ) (22)

At a saddle point or extreme (a critical point) 퐷푓(푥0) = 0.

The determinant of this Matrix can be used to obtain the functions for each electron within the system. The Hartree-Fock method does this but neglects electron-electron correlation for the sake of simplicity in doing so. While it is possible to take linear combinations of these functions to increase accuracy, there may be better methods. Hohenberg and Kohn’s work uses the fact that if we know the electron density exactly, we have the wave function. They developed a functional that relates the wavefunction to the electron density, and from there the variational principle can be used to minimize the energy.9–11 This is often more straightforward than added terms to the HF-equation. We would like to know the functional that will solve this system.

Alas, there is no way to know the functional exactly, as if we knew the functional, we would know the exact wavefunctions / electron density already. We have just moved the problem from dynamic electron correlation to exchange correlation. Using parameters taken from chemical experiments, approximations of this exchange correlation can be added for greater accuracy and speed. This can be further improved by combining both methods to take advantage of the specifics of both. B3LYP (named for its inventors- Becke Lee Yang and Parr) uses a 3 parameter exchange correlation functional based upon Hartree-Fock with Lee Yang and Parr’s correlation functional for dynamic electron correlation. B3LYP is popular as it is well understood, has

parameters taken from experiment, and was one of the first DFT methods that was a significant improvement over Hartree-Fock. It is fast and robust, with a small number of parameters, and well-established in the literature. There are more accurate methods developed since then but require more parameters and many are highly system-specific.12–14

Atomic orbitals are approximations of electron wave functions centered at the nucleus and are often a simple Gaussian function. They are used for ease of rapid computation. Slater-type orbitals are based on mathematical solutions for the Schrodinger equation, and more accurately model the physics of the system involved. Gaussian-type basis sets are common in the literature because of their advantage in speed, and additional parameters can be added as necessary for more complex system. Because of this, they exist in hierarchies of increasing size, giving a means to increase accuracy at the expense of computational cost. 6-31G is a split valence set developed by John Pople, 6 represents the 6 primitive Gaussians functions for each core atomic orbital, while 3 and 1 are two basis functions that represent the valence electrons.9,15 More parameters can be added as follows:

 6-31G* - Polarization functions on heavy atoms

 6-31G** - Polarization functions on heavy atoms and hydrogen

 6-31+G - Diffuse functions on heavy atoms

 6-31++G - Diffuse functions on heavy atoms and hydrogen

 6-31+G* - Polarization and diffuse functions on heavy atoms

 6-31+G** - Polarization functions on heavy atoms and hydrogen, as well as diffuse

functions on heavy atoms

2.3.b Molecular Modeling Force Fields

The molecular modeling term force field (not to be confused with the concept in physics) are a series of functions and parameters used to describe interatomic potentials for molecular mechanics and molecular dynamics simulations. The parameters of the energy functions may be derived from experimental observation, calculations from more detailed quantum mechanical models, or a combination of both. The Chemistry at Harvard Macromolecular

Mechanics (CHARMM) is a widely used set of force fields for molecular dynamics and shares the name with the software package developed by the same group. In this work, the term CHARMM will refer to the force field alone. The potential function uses the equation below:

2 2 푉 = ∑ 푘푏(푏 − 푏0) + ∑ 푘휃(휃 − 휃0) + ∑ 푘휑[1 − cos(푛휑 − 훿)] + 푅 12 푅 6 푞 푞 2 2 푚푖푛(푖푗) 푚푖푛(푖푗) 푖 푗 ∑ 푘휔(휔 − 휔0) + ∑ 푘푢(푢 − 푢0) + ∑ 휖 [( ) − ( ) ] + (23) 푟푖푗 푟푖푗 휖푟푖푗

푘푏 is the bond force constant and 푏 − 푏0 is the displacement from equilibrium. Similarly, 푘휃 is the angle force constant and 휃 − 휃0 is displacement. 푘휑 is the force constant for the dihedral angles (torsion angles) n is multiplicity, 훿 is phase shift, and 휑 is the dihedral angle. 푘휔 is for out of phase bending (improper) and 휔 − 휔0 is displacement. 푘푢 is the Urey-Bradley force constant describing interactions between nonbonding atoms. Finally, the Lennard Jones potential and coulombic potential terms describe the van der Waals interactions between the atoms.16,17

The CHARMM force fields for proteins include united-atom (sometimes termed extended atom)

CHARMM19, all-atom CHARMM22 and the dihedral potential corrected variant

CHARMM22/CMAP. All-atom force fields provide parameters for every type of atom in a system,

while united-atom interatomic potentials treat the hydrogen and carbon atoms in a methyl group and methylene bridge as one interaction center. In this work, the CHARMM22 force field is used exclusively, and the water model used is the TIP3P explicit water model, for which

CHARMM22 is parameterized.

2.4 Methods for Measuring Coupling

2.4.a Finite Difference Method

The Finite Difference Method (FDM) is the analytical differentiation of the potential energy with respect to each contributing local mode. In cases where the origins of the coupling is not known, for example, when mechanical coupling is suspected or electrodynamics may be present, using

FDM is the method of choice for calculating the vibrational coupling constant. FDM requires four single point energy (SPE) calculations to be done after perturbing along the vibrational coordinate of the local modes and placing them into the molecular geometry of interest.18,19

The molecule is moved along the normal mode in the positive or negative direction, causing each bond involved to be stretched or compressed in the new perturbed structures, as seen in Figure

2. By taking perturbations along two eigenmodes, four such structures can be created using the positive and negative directions, each structure moving the atoms by two modes (+,+) (+,-) (-,+) (-

,-). Four single point energies are calculated from these systems, and then used to compute the coupling using equation (24)

2 휕 (푞푖, 푞푗) 훽푖푗 = (24) 휕푞푖휕푞푗

The unit associated with this calculation is that of energy per change in the local mode squared.

The unit of the local mode is A*(amu1/2). In our code, we take output from the Gaussian 09

software package to calculate the single point energy in Hartrees and convert to cm-1. All coupling constants are given in cm-1.20

2.4.b Transition Dipole Coupling

The Transition Dipole Coupling model (TDC) was originally applied to amide I vibrations in polypeptides by Krimm et al.21–23 This models the modes as dipoles comprised of two point charges, a concept familiar to those who have studied electrostatics, shown in Figure 3a. The coupling between the two dipoles is calculated according to equation (25)

1 |휇⃑⃑⃑ 푖| ∙ |휇⃑⃑⃑푗 | − 3(휇⃑⃑⃑ 푖 ∙ 푟̂푖푗)(휇⃑⃑⃑푗 ∙ 푟̂푖푗) 훽푖푗 = 3 (25) 4휋휀0 푟푖푗

The two vibration modes i and j are represented by point charges with magnitude and direction

휇⃑⃑⃑ 푖 and 휇⃑⃑⃑푗 with the two sites connected by rij, with the unit vector 푟̂푖푗. The electronic structure of the molecule results in the dipole derivative with respect to a particular vibrational mode, called the transition dipole vector. In the case that the local mode approximation gives a reasonable picture, the dipole vector can be often taken to be along the relevant atomic bond. Figure 3b shows that for the amide I mode common to amino acids, the TDC model provides a good approximation of the carbonyl bond. However, findings of Torii et. al. show that the model works consistently only for the nearest neighbor couplings, as additional factors become more prominent as the dipole becomes farther away or is complicated with side chains.24–26

2.4.c Transition Charge Model

The Transition Charge Model (TCM) is a model of multipoles and allows for both fixed partial charges and dynamic partial charges to exist at each atom involved in the vibrational motion. The electrostatic energy has been found to be the primary stabilizing term medium-strength

hydrogen-bonded systems, and so often approximates the total interaction energy, without a need for additional calculations.27,28 The equation used to calculate the coupling falls from taking the mixed second derivative of the intermolecular Coulombic potential energy as given in Krimm,

Cheam, and Dybal.29

푁 ,푁 2 퐴 퐵 1 휕 푞푎푞푏 훽푖푗 = ∑ (26) 4휋휀0 휕푄퐴휕푄퐵 푟푎푏 푎,푏=1

This may seem familiar, as the second derivative method for FDM does indeed share the same form, however the terms used are what make this method distinct. The terms are summed over

푞푎푞푏 the intermolecular Coulombic potential energy between each pair of atoms, written as 푉푎푏 = 푟푎푏

with qa and qb representing the charges on the atom a or b. NA and NB are the number of relevant atoms in molecule or group A and B. The distance between them is rab. The local modes are written as Qi and Qj to avoid confusion with charges, which use the lowercase Q. All these terms depend upon the displacement Q, and the general solution is:

푁 ,푁 푁 ,푁 퐴 퐵 퐴 퐵 −1 1 1 휕푞푎 휕푞푏 휕푞푏 푞푎 1 휕푞푎 휕푞푏 휕푟푎푏 훽푖푗 = ∑ ( + ) + ∑ (푞푏 + 푞푎 ) + 4휋휀0 푟푎푏 휕푄퐴 휕푄퐵 휕푄퐴 휕푄퐵 4휋휀0 휕푄퐵 휕푄퐵 휕푄퐴 푎,푏=1 푎,푏=1 푁 ,푁 푁 ,푁 퐴 퐵 −1 퐴 퐵 2 −1 1 휕푞푎 휕푞푏 휕푟푎푏 1 휕 푟푎푏 ∑ (푞푏 + 푞푎 +) + ∑ 푞푎푞푏 (27) 4휋휀0 휕푄퐴 휕푄퐴 휕푄퐵 4휋휀0 휕푄퐴휕푄퐵 푎,푏=1 푎,푏=1

considering only first order charge interactions. If multipole terms are required, then the equation can be expanded as needed. Note that Krimm et. al. neglect cross terms because charge induction from the unrelated vibrational mode is unlikely when two separate molecules are considered,

such as in their work on the formic acid dimer. It should be noted that when groups on the same small molecule, and these cross terms are not necessarily trivial.30,31

휕푞 To evaluate the charge fluxes 푎 , we use an electronic structure calculation via perturbation 휕푄퐴

휕푟−1 along the local model of interest. To evaluate first and second order terms of 푎푏 we must change 휕푄퐴 the coordinate basis from Cartesian to normal mode mass weighted coordinates, which is trivial but tedious.

2.4.d Transition Dipole Density Distribution

The Transition Dipole Density Distribution (TDDD) model was applied to coupled electronic transitions by Graham Fleming and coworkers and applied to vibrational transitions in peptides by Moran and Mukamel.32,33 The TDDD model does not require the electron density to be collapsed onto atom centers, but instead each isolated site is enclosed in a cube that is divided into many points; the transition density is calculated at each point and then the electrostatic coupling constant can be calculated by summing the Coulombic interaction between the transition densities of the isolated sites. (Figure 4). These volumes are divided into a set of points and the electron density gradient is calculated at each point with respect to a displacement in one of the local modes. The transition density at point i within volume A is given by:

푁퐴,푁퐵 푟푛̂ +훿푟⃗푛 푀푖푀푗 훿휌(푟⃗푛) 훽푖푗 = ∑ ; 푀퐴 = 훿푉푛 ∫ 푑푟⃗푛 푠푑 (28) 푟푖푗 훿푄퐴 푖,푗=1 푟푛̂

where rij is the separation between point n in volume A and point m in volume B. The volumes can be any desired shape that is convenient but are typically rectangular prisms for ease in calculation. Again, the isolated bases need to be perturbed along the carbonyl mode and set into

the basepair configuration. In this case, the SPE calculations are done with one base perturbed and the second base held at equilibrium. Gaussian 09 is capable of generating volumetric data for arbitrary parameters using a utility known as a cubegen file; we make use of it here for charge density calculations. The data from the cubegen file can then be fed into the TDDD method to calculate the coupling constant. In our case, one cubegen file is generated for each SPE calculation with same dimensions and origins. These charge density cubes are subtracted and added together to generate one cubegen difference file that is used to obtain the TDDD coupling.

2.5 Solvent Models

In simulations, we often do not wish to consider molecules residing within gas-phase states; we want molecules that are found within solvents. This is because the interaction between the solvent and the solute will impact the chemistry of the molecule being studied, as solvent effects can alter energy, stability, and molecular orientation. This will cause results relating to energy, such as vibrational frequency, spectrum, etc. to change.

Therefore we need a way to model the chemistry of these molecules in a solvent like state without simulating the actual solvent atoms, because of the exponential increase in complexity this would entail. This can be accomplished using implicit solvation models, which differ from explicit models by instead treating the solvent as a continuous medium that acts upon the solute, leading to a significant reduction in complexity.

2.5.a Onsager Model

This is one of the first implicit solvation models, and it is named for the fact that it was designed by Lars Onsager in 1936, based upon the Born model (1920) which used a dielectric continuum around a spherical cavity to simulate the solute.34 The idea was that within the solvent, the solute

would interact with the solvent in a uniform spherical radius based upon the net charge on the solute as shown in Figure 5. A problem with the Born model was that it was arbitrarily accurate depending on the choice of the ion radii which was used to determine the size of the sphere, and the model did not allow for mutual polarization between the solute and the solvent. The Onsager model improved upon the Born model by looking at the dipole moment within the molecule instead of the net charge. This model considers a polarizable dipole with polarizability α at the center of a sphere. The solute dipole induces a reaction field in the surrounding medium which in turn induces an electric field in the cavity (reaction field) which interacts with the dipole.

Caution is needed when applying this, however. The Onsager method can give poor results for compounds where the electron distribution is poorly described by the dipole moment. Systems with a zero net dipole moment will not exhibit solvation at all by this model. It only takes into account the polarizability and does not really account for the cavitation energy or the electrostatic energy. Finally, Onsager needs a spherical molecule, as non-spherical molecules are modeled very poorly by a spherical cavity.

2.5.b Conducting Polarizable Continuum Model

The PCM is based upon the idea of generating multiple overlapping spheres for each of the atoms within the molecule inside of a dielectric continuum. This differs from the Onsager methodology which uses a single sphere (or an ellipse) to surround the whole molecule and thus allows for a greater amount of accuracy in determining the solute-solvent interaction energy. This method treats the continuum as a polarizable dielectric and thus is sometimes referred to as dielectric

PCM (DPCM). The PCM model calculates the free energy of solvation by attempting to sum over three different terms: electrostatic forces, dispersion-repulsion, and cavitation.35

This is in turn used to create a Hamiltonian in vacuo. A potential based upon the solute-solvent molecule interaction is added, and a time-dependent perturbation on the solute molecule is used to describe the cavity-field effect and the response of the solvent to the external field after creation of the solute cavity in the solvent. This allows for the direct calculation of the effective polarizabilities of the molecule in the solvent. The PCM method attempts to give a complete answer to the free energy of solvation but it fails to directly calculate the energy of cavitation which is the energy defined by the surface of the van der Waals-spheres and the dispersion- repulsion energy. The free energy of solvation for any PCM calculation is primarily the electrostatic energy. It is more "realistic" than Onsager and gives good electrostatic energy results, but does not account for the cavitation or dispersion-repulsion energies

The CPCM model is a variation of the DPCM model that uses a group of spheres to define a cavity within a dielectric continuum. The important difference between this model and the DPCM model is that the CPCM model treats the solvent like a conductor. This conductor will impact the polarization charges of the accessible surface area between the solvent and the solute. The CPCM model attempts to solve the nonhomogeneous Poisson equation for an infinite dielectric constant with scaled dielectric boundary conditions to approximate the result for a finite dielectric constant. In the CPCM model the permittivity of the solvent will impact the results of the model as solvents with higher permittivity will behave more like an ideal conductor and return better results.

This reduces the outlying charge errors caused by portions of the electron density which are actually outside of the cavity. The CPCM method is commonly employed for this reason. The math involved in calculating the integrals simplifies considerably by assuming that the dielectric constant is infinite. Although the results of the CPCM are improved when the dielectric constant

of the solvent is high, it has been shown that when the dielectric constant of the solvent is low the results are still equal to the results of a DPCM solvation model. It generally will not be worse than PCM, despite the simplifying assumptions. But it still does not accurately account for cavitation energy or dispersion-repulsion energy, as the equations are the same as DPCM only simplified.

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Chem. Phys. 1971, 54 (2), 724–728. (16) Phillips, J. C.; Braun, R.; Wang, W.; Gumbart, J.; Tajkhorshid, E.; Villa, E.; Chipot, C.; Skeel, R. D.; Kalé, L.; Schulten, K. Scalable Molecular Dynamics with NAMD. J. Comput. Chem. 2005, 26 (16), 1781–1802. (17) MacKerell, A. D.; Bashford, D.; Bellott, M.; Dunbrack, R. L.; Evanseck, J. D.; Field, M. J.; Fischer, S.; Gao, J.; Guo, H.; Ha, S.; et al. All-Atom Empirical Potential for Molecular Modeling and Dynamics Studies of Proteins †. J. Phys. Chem. B 1998, 102 (18), 3586– 3616. (18) Krummel, A. T.; Zanni, M. T. DNA Vibrational Coupling Revealed with Two-Dimensional Infrared Spectroscopy: Insight into Why Vibrational Spectroscopy Is Sensitive to DNA Structure. J. Phys. Chem. B 2006, 110 (28), 13991–14000. (19) Krummel, A. T.; Zanni, M. T. Evidence for Coupling between Nitrile Groups Using DNA Templates: A Promising New Method for Monitoring Structures with Infrared Spectroscopy. J. Phys. Chem. B 2008, 112 (5), 1336–1338. (20) M. J. Frisch, G.; Trucks, W.; Schlegel, H. B. .; Scuseria, G. E. .; Robb, M. A. .; Cheeseman, J. R.; Scalmani, G.; Barone, V. .; Mennucci, B. .; Petersson, G. A.; et al. Gaussian 09, Revision E. 01; Gaussian; 2009. (21) Moore, W. H.; Krimm, S. Transition Dipole Coupling in Amide I Modes of Betapolypeptides. Proc. Natl. Acad. Sci. U. S. A. 1975, 72 (12), 4933–4935. (22) Krimm, S.; Abe, Y. Intermolecular Interaction Effects in the Amide I Vibrations of Polypeptides. Proc. Natl. Acad. Sci. 1972, 69 (10), 2788–2792. (23) Cheam, T. C.; Krimm, S. Vibrational Analysis of Crystalline Diketopiperazine-II. Normal Mode Calculations. Spectrochim. Acta Part A Mol. Spectrosc. 1984, 40 (6), 503–517. (24) Torii, H.; Tasumi, M. Ab Initio Molecular Orbital Study of the Amide I Vibrational Interactions between the Peptide Groups in Di- and Tripeptides and Considerations on the Conformation of the Extended Helix. J. Raman Spectrosc. 1998, 29 (1), 81–86. (25) Kubelka, J.; Kim, J.; Bour, P.; Keiderling, T. A. Contribution of Transition Dipole Coupling to Amide Coupling in IR Spectra of Peptide Secondary Structures. Vib. Spectrosc. 2006, 42 (1), 63–73. (26) Průša, J.; Bouř, P. Transition Dipole Coupling Modeling of Optical Activity Enhancements in Macromolecular Protein Systems. Chirality 2018, 30 (1), 55–64. (27) Tokmakoff, A.; Sauter, B.; Kwok, A. S.; Fayer, M. D. Phonon-Induced Scattering between Vibrations and Multiphoton Vibrational up-Pumping in Liquid Solution. Chem. Phys. Lett. 1994, 221 (5–6), 412–418. (28) Grubbs, W. T.; Dougherty, T. P.; Heilweil, E. J. Vibrational Energy Redistribution in Cp*Ir(CO)2(Cp*=η5-Pentamethylcyclopentadienyl) Studied by Broadband Transient Infrared Spectroscopy. Chem. Phys. Lett. 1994, 227 (4–5), 480–484. (29) Dybal, J.; Cheam, T. C.; Krimm, S. CO Stretch Mode Splitting in the Formic Acid Dimer: Electrostatic Models of the Intermonomer Interaction. J. Mol. Struct. 1987, 159 (1–2), 183–194. (30) Hamm, P.; Woutersen, S. Coupling of the Amide I Modes of the Glycine Dipeptide. Bull. Chem. Soc. Jpn. 2002, 75 (5), 985–988. (31) Ham, S.; Cho, M. Amide I Modes in the N -Methylacetamide Dimer and Glycine Dipeptide Analog: Diagonal Force Constants. J. Chem. Phys. 2003, 118 (15), 6915–6922. (32) Krueger, B. P.; Scholes, G. D.; Fleming, G. R. Calculation of Couplings and Energy-Transfer Pathways between the Pigments of LH2 by the Ab Initio Transition Density Cube Method.

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Figure 2.1 Bloch sphere two-system dynamics represented in three-dimensional coordinate space.

Figure 2.2 A sample 2′-azido-5-cyano-2′-deoxyuridine molecule with normal modes chosen to perturb a cyano and azido group, respectively.

Figure 2.3 Symbolic representation of dipole-dipole coupling (a) for a sample system (b). https://chem.libretexts.org and Site-specific detection of protein. Taken from Zanni et al. Chem. Sci., 2018, 9, 463-474

Figure 2.4 False color representation of two separate groups in a cubegen file for use in a Transition Dipole Density Distribution (TDDD) calculation.

Figure 2.5 Visualization of the Onsager field model showing the approximation of solvent molecules and dipoles and the solute molecule as an electrically charged hollow sphere. Taken from Boxer J. Phys. Chem. B, 2012 116 (35) 10470-10476

Chapter 3: Azide and Nitrile Probes- The Best of Both Worlds

This work has been published in part from: A. J. Schmitz, D. G. Hogle, X. S. Gai, E. E. Fenlon, S. H. Brewer, M. J. Tucker, 2D IR Study of Vibrational Coupling Between Nitrile and Azide Reporters in a RNA Nucleoside, J. Phys. Chem B, 120, 9387-9394.

3.1 Theoretical Modeling of Novel Probe Behavior

The use of non-natural amino acid side chain probes, such as cyano- (CN), azido- (N3), and isonitrile, selenocyanate, and thiocyanate have been used for tracking dynamics of the side chain region of proteins.1–4 Thermal unfolding can be used to capture critical steps in the processes that guide the molecule towards its folded state such as changes in local electric fields and the degree of hydration, observed at distinct locations in nucleotides, soluble proteins and membrane binding proteins. One such group that has shown promise are members of the azide moiety, with success being reported in a variety of biological systems.5–9 Findings indicate that the absorption cross section of the N3 transition is one order of magnitude greater than that of CN, allowing lower concentrations to be utilized. Aliphatic azido- substitutions are stable under most conditions and often avoid accidental Fermi resonances. Thus, it would seem that the azido- reporter is a more efficient means of detecting the local environment in target molecules.

Studies done on the nucleoside 2’-azido-5-cyano-2’-deoxyuridine investigated the utility of azido- reporters in comparison with an alternative nitrile probe, as both the N3 transition and the CN transition have many similar spectral properties.8,10 The synthesis of azidonucleotides has allowed their incorporation into DNA and RNA for the study of folding behavior and interactions with other proteins. For example, linear and 2D IR measurements on the interaction of nucleic acids with ubiquitin and ribonuclease A show important such as local environmental and site-specific dynamics that lead to the loss of β-sheet content and an increase in random coil structure. In contrast, measurements made using the amide I region can confirm these general features during

thermal unfolding, but do not capture activity in the region where most peptide/protein changes transpire.

While this would seem to be the case in isolation, this neglects potentially useful information that arises from the interaction of the probe groups when from are employed together. As mentioned, both probes report within the same spectral region, thus coupling should be observable between the two. Existing studies shows coupling in the linear IR for two isotopomeric CN probes, and energy transfer has been observed for azido- and nitrile anions in solution by 2D IR, thus providing a precedent for the study of direct vibrational coupling between these two different reporters within a single biological system.7-9 We chose the nucleoside 2’-azido-5-cyano-2’-deoxyuridine, with structure shown in Figure 1, as it is a simple model system with its behavior known from the comparative studies mentioned previously.

3.2 Investigation of Conformational Preferences in 2’-azido-5-cyano-2’-deoxyuridine.

Confirming which configuration of the target system was preferred under ambient temperature was done by optimizing the energy of a 2’-azido-5-cyano-2’-deoxyuridine (N3CNdU) molecule in an arbitrary starting configuration. As the configuration space could contain multiple local minima and the system can rotate around the dihedral bond connecting the rings, albeit in a hindered fashion, this bond was investigated by computational studies to determine the degree of hindered rotation. Conformers were generated by rotating about the N-C bond between the base and sugar rings 360 degrees, the dihedral angle was then held rigid and the rest of the molecule was allowed to optimize, giving the energy, angle, and distance between the azide and nitrile dipoles for each structure.11,12

(The following is reproduced from the paper Two-Dimensional Infrared Study of Vibrational

Coupling between Azide and Nitrile Reporters in a RNA Nucleoside. Used with permission.)

As expected, the azido- and cyano- normal modes were the only two transitions observed in the spectral region corresponding to the N3 asymmetric mode and the CN symmetric mode. The transition dipole strengths were calculated to be 0.474 D for N3 and 0.198 D for CN vibrational modes with an angle of 60° between the transition dipoles. Transition densities were determined by calculating the difference in the electron density from the equilibrium structure after a displacement of ±0.03 Å along each normal mode. From these plots of transition densities with gradient of 0.002 esu/bohr, it is clear that the modes are localized with respect to the N3 and CN atoms.

A map of the energy versus dihedral angles was generated in this fashion, as seen in Figure 2. The geometry-optimized structure has a dihedral angle of 10.21°. Three energetic minima arise along the potential energy surface (PES) located at 1, 2, and 3, with the original geometry optimized structure existing in minimum 2. Minima 2 and 3 exist within thermal equilibrium, while 1 is significantly higher in energy along this PES. A second potential energy surface (PES) was constructed for different N3CNdU conformations using the conducting polarizable continuum model (CPCM) model to compare the relative energies to the Onsager model in aqueous solution.

(Figure 2) For both solvent models, three energetic minima, 1, 2, and 3 exists along the PES. The most noticeable difference between the two implicit water models is at minimum 3.

Upon inspection, the once lowest energy minimum in the CPCM water model is higher in relative energy compared to minimum 2 and is above the thermal distribution S2 for the Onsager water model. The energy well located at minimum 2 decreases in the Onsager model relative to minimum 3 causing the thermal distribution to be much narrower. Minimum 1 also exhibits a

change in the relative energy. The energy is similar to minimum 2 in the Onsager water model as opposed to being higher in energy for the CPCM model. This suggests that the energy well found at minimum 3 is dependent upon the solvent model used, and may be an artifact of the CPCM model. Several functionals (BLYP, B3LYP, B3PW91) with different basis sets (6-31G, 6-31+G**, 6-

311+G**) were tested to verify the curvature of the potential energy surface in the region of minima 2 and 3.

At each dihedral angle, the coupling constants between the modes were calculated using the finite difference method. The angle between the dipoles was also computed for each point along the surface. In minima 1, 2, and 3 of the PES, the average value of the coupling constant was determined to be 47.6, 62.0, and 35.7 cm-1, respectively. The corresponding average angle between the dipoles for minima 1, 2, and 3 were measured as 83.7° ± 10.2°, 64.3° ± 8.9°, and

77.72° ± 7.8°, respectively. Although only the relevant values are reported at these important regions of the PES, the coupling constants and the angles between the dipoles were calculated for each dihedral angle. It should be noted that a large variance was observed in the values of the coupling constants and corresponding angles along the PES. The distance between the dipoles was also measured for each dihedral angle, and the range varied from 5.0 to 8.5 Å.

A value of 32 ± 1.0 cm-1 was calculated for the geometric average of the diagonal anharmonicity,

Δ. Using the energy difference between the coupled vibrations (118 cm-1) and the measured off- diagonal anharmonicity of 44 ± 3.3 cm-1, the resulting coupling found to be is 66.4 ± 5.4 cm-1. The angle was estimated to be 53° ± 1° from the relative intensities of the coupled transitions in the

FTIR spectrum. In an attempt to determine the distance between the reporters, transition dipole coupling was implemented, a simple model that assumes coupling is solely due to electrostatic coupling of point dipoles. Using transition dipole coupling (TDC) theory, we calculated a distance

of 2.9 Å from the coupling constant and the transition dipole strengths along with the angle between the two dipoles. It is clear by inspection of the molecule that this distance is an underestimate. However, it is known that transition dipole theory underestimates the coupling strength because it breaks down at small distances. Thus, a more detailed coupling model is required to obtain a coupling closer to the experimental value.

3.3 Modeling the Coupling Mechanism

An FTIR spectrum of 2′-azido-5-cyano-2′-deoxyuridine in water was taken, as shown in Figure 4,

-1 and two transitions in the region of 2050–2300 cm arising from the asymmetric stretch of N3 and the symmetric stretch of CN were observed. Significant coupling interactions between the dipoles occur due to their close proximity. As a result, the relative intensities of the infrared transitions are sufficiently different than expected from prior literature as well as the calculated transition dipole ratio.

The coupling, β12, is directly measured from the anharmonicity of the off-diagonal peaks, Δ12, using perturbation theory. As experiment has shown our system to be within the weak coupling regime, equation (1) and (2) apply. As shown in Figure 3, β12 represents the coupling between the azido-

(εN3) and cyano-(εCN) vibrational modes. This bilinear coupling model predicts a shift in the observed absorption peak maxima and a change in the relative intensity of the two transitions dependent on the coupling strength.

훽2 12 ( ) ∆12= 4∆ ( 2 2) 1 (휀푁3 − 휀퐶푁) − ∆

To model this phenomenon, density functional theory (DFT) calculations using Gaussian09 were performed using B3LYP/6-31+G** level of theory with implicit solvation model of water calculated

13 by a self-consistent reaction field (SCRF) on the N3CNdU solvent system. (The SCRF solvation model with CPCM was used for all experiments.) For all N3CNdU structures, an optimization was first performed using B3LYP/6-31+G** in the given solvent model, thereby obtaining frequency and energy data. These optimizations were used to determine the coupling constant using the finite difference method (FDM) as explained by Torii and Tasumi.14 Using FDM, the molecule was displaced along the normal modes of the nitrile and azido- functional groups of N3CNdU by a small amount (0.03 Å) along both vibrational modes and a single point energy calculation was performed. These energies were used to calculate the vibrational coupling through the second derivative of the energy with respect to the normal modes discussed below.

The off-diagonal region of the spectrum shows the presence of vibrational signatures resulting from coupling between the CN and the N3 transitions. This coupling appears at T = 0 and decreases as a function of waiting times, T, indicative of direct coupling between the dipoles. By fitting the data to a single exponential function, we determined the decay of the cross peaks intensity to be

3.5 ± 1.0 ps as a function of the waiting time. After 1 ps the coupling signal is less pronounced, due to the dependence of the signal strength on the average value of the vibrational population decays, T10, and the transition dipole strength. Since the azide reporter has a shorter lifetime and a larger transition dipole magnitude, the decay of the cross peaks is somewhat dominated by these factors. It should be noted that the overall cross peak intensity is easily measurable despite the weak absorption cross section of the nitrile reporter.15

3.4 Simulations of Azido and Nitrile Groups in a Model Peptide System

(the following work was done by Tucker Group member Ryan Gustafson)

A simple two-member peptide alanoazide-cyanotryptophan (AlaN3–TrpCN) was chosen as a sample coupling model for our cyano-azide system.The choice of peptide and method used was based on a previous study investigating coupling with two azide groups.7 To compare the effects of angle and distance on coupling, the alanine side chain was functionalized with an azido- group

(N3), while the cyano-group (CN) was attached to the fifth and sixth carbon on the indole ring of tryptophan. These two systems are henceforth referred to as AlaN3–Trp5CN (Figure 6a) and

AlaN3–Trp6CN (Figure 6b), respectively.

Simulations for the two Ala-Trp peptides were performed using the Gaussian 09 software package using DFT with B3LYP/6-31+G** and the CPCM solvent model, conditions identical to those used for the calculations of N3CNdU. Calculations and coupling data were collected for 144 orientations of AlaN3–Trp5CN and AlaN3–Trp6CN by fixing phi and psi angles at 30-degree increments and allowed to optimize, again under the same conditions used for N3CNdU. A

Ramachandran plot was generated from the data collected by Ala-Trp peptide simulations, shown in Figure 7.

The data for AlaN3–Trp5CN is given in Figure 7a and 7c; the contour in Figure 7a illustrates that the low coupling regions in the bottom left hand corner and the top center contain minima around

-1.5 to -1.2 corresponding to angles of 18 – 29 degrees. High coupling points are found in the top right corner and top left corner with maxima around 1.0 to 1.59. These correspond to angles of

92 – 152 degrees. An interpretation of the TDC contour gives similar regions of high and low coupling data (Figure 7c). The data for AlaN3–Trp6CN was gathered in the same manner as the

data for AlaN3–Trp5CN. The coupling data for AlaN3–Trp6CN reveal minima in the top middle moving to the out to the left with a low point at -2.5 at an angle of 119 degrees. The maximum coupling data are found on either side of the bottom middle and in the top right reaching up to

1.7 corresponding to an angle of 111 degrees (Figure 7b). Figure 7d shows the data corresponding to the TDC calculation for AlaN3–Trp6CN

There is a strong correlation between angle and FDM coupling. There is also a good correlation between the FDM and TDC models, better than what we would have expected based upon prior experience with this model.16,17 This suggests that the coupling is predominantly through space and depends primarily upon the orientation of the dipoles, as opposed to the nucleotide system containing both through bond and through space coupling.

The vibrational Stark effect is the combination of the extinction coefficient and sensitivity to changes in local environment. It can be used to calibrate the sensitivity of any vibrational frequency to the internal electric field in an organized system. Protein local electric fields are predicted to be large (+10MV/cm) and vary tremendously in different locations. Understanding of VSE for nitrile lags behind other probes. The primary effect of an electric field on a molecular vibration is the interaction of the field with the change in dipole moment of the vibration, with the change in frequency is a linear function of the applied field. Stark shift is complicated by multiple bands arising from Fermi resonant interactions, varying degrees of hydrogen bonding, and multiple sample conformations.18

3.5 Conclusions and Future Directions

The difficulties in incorporating azide reporters into nucleic acids have including aggregation, poor selectivity, and unwanted side reactions. Common synthesis procedures, such as UV-aggregation

or the addition of azide in excess quantities proved impractical with the delicate biomolecular structure.19–21 This issues made the general use of azido groups seem unattractive for their potential in infrared methods, but further developments in cycloaddition chemistry and catalysis have shown promise in overcoming theses challenges22–24, meanwhile the utility of azide as both a highly sensitive reporter and versatile functional group has been demonstrated, thus we believe that azido groups will continue to play an expanded role in the study of nucleotides and other biomolecules.25 Our combination “best of both worlds” approach takes advantage of the high extinction coefficient and high sensitivity of nitrile along with be combined with the strong signal of azido groups.

This can be used with 2D IR intermode coupling in a way analogous to traditional spectroscopic

26,27 rulers such as EPR and FRET. Vibrational energy transfer and subsequent coupling has been observed for nitrile groups using 2D IR in previous studies,28–30 but there is at present little information in the literature regarding vibrational interactions between these two reporters. We have demonstrated the potential of azido and nitrile in conjunction using the N3 and CN region of

N3CNdU to capture the direct vibrational coupling between these probes in water. In addition to demonstrating the utility of this for biomolecules, we show the capability to detect site-specific dynamics at two positions simultaneously. Further advances in synthetic approaches show promise for the placement of N3 and CN probes as a useful monitor of site-specific dynamics in environments such as the major groove, minor groove, and phosphate–sugar region, as well as the incorporation in peptides and protein systems.

(1) Levin, D. E.; Schmitz, A. J.; Hines, S. M.; Hines, K. J.; Tucker, M. J.; Brewer, S. H.; Fenlon, E. E. Synthesis and Evaluation of the Sensitivity and Vibrational Lifetimes of Thiocyanate and Selenocyanate Infrared Reporters. Rsc Adv. 2016, 43 (6), 36231–36237. (2) Park, K.-H.; Jeon, J.; Park, Y.; Lee, S.; Kwon, H.-J.; Joo, C.; Park, S.; Han, H.; Cho, M. Infrared Probes Based on Nitrile-Derivatized Prolines: Thermal Insulation Effect and

Enhanced Dynamic Range. J. Phys. Chem. Lett. 2013, 4 (13), 2105–2110. (3) Oh, K.-I.; Choi, J.-H.; Lee, J.-H.; Han, J.-B.; Lee, H.; Cho, M. Nitrile and Thiocyanate IR Probes: Molecular Dynamics Simulation Studies. J. Chem. Phys. 2008, 128 (15), 154504. (4) Maj, M.; Ahn, C.; Błasiak, B.; Kwak, K.; Han, H.; Cho, M. Isonitrile as an Ultrasensitive Infrared Reporter of Hydrogen-Bonding Structure and Dynamics. J. Phys. Chem. B 2016, 120 (39), 10167–10180. (5) Oh, K. I.; Lee, J. H.; Joo, C.; Han, H.; Cho, M. β-Azidoalanine as an IR Probe: Application to Amyloid Aβ(16-22) Aggregation. J. Phys. Chem. B 2008, 112 (33), 10352–10357. (6) Tucker, M. J.; Gai, X. S.; Fenlon, E. E.; Brewer, S. H.; Hochstrasser, R. M. 2D IR Photon Echo of Azido-Probes for Biomolecular Dynamics. Phys. Chem. Chem. Phys. 2011, 13 (6), 2237–2241. (7) Bazewicz, C. G.; Liskov, M. T.; Hines, K. J.; Brewer, S. H. Sensitive, Site-Specific, and Stable Vibrational Probe of Local Protein Environments: 4-Azidomethyl- l -Phenylalanine. J. Phys. Chem. B 2013, 117 (30), 8987–8993. (8) Taskent-Sezgin, H.; Chung, J.; Banerjee, P. S.; Nagarajan, S.; Dyer, R. B.; Carrico, I.; Raleigh, D. P. Azidohomoalanine: A Conformationally Sensitive IR Probe of Protein Folding Protein Structure and Electrostatics. Angew. Chemie - Int. Ed. 2010, 49 (41), 7473–7475. (9) Nagarajan, S.; Taskent-Sezgin, H.; Parul, D.; Carrico, I.; Raleigh, D. P.; Dyer, R. B. Differential Ordering of the Protein Backbone and Side Chains during Protein Folding Revealed by Site-Specific Recombinant Infrared Probes. J. Am. Chem. Soc. 2011, 133 (50), 20335–20340. (10) Gai, X. S.; Coutifaris, B. A.; Brewer, S. H.; Fenlon, E. E. A Direct Comparison of Azide and Nitrile Vibrational Probes. Phys. Chem. Chem. Phys. 2011, 13 (13), 5926–5930. (11) Krummel, A. T.; Zanni, M. T. DNA Vibrational Coupling Revealed with Two-Dimensional Infrared Spectroscopy: Insight into Why Vibrational Spectroscopy Is Sensitive to DNA Structure. J. Phys. Chem. B 2006, 110 (28), 13991–14000. (12) Moran, A.; Mukamel, S. The Origin of Vibrational Mode Couplings in Various Secondary Structural Motifs of Polypeptides. Proc. Natl. Acad. Sci. 2004, 101 (2), 506–510. (13) M. J. Frisch, G.; Trucks, W.; Schlegel, H. B. .; Scuseria, G. E. .; Robb, M. A. .; Cheeseman, J. R.; Scalmani, G.; Barone, V. .; Mennucci, B. .; Petersson, G. A.; et al. Gaussian 09, Revision E. 01; Gaussian; 2009. (14) Torii, H.; Tasumi, M. Ab Initio Molecular Orbital Study of the Amide I Vibrational Interactions between the Peptide Groups in Di- and Tripeptides and Considerations on the Conformation of the Extended Helix. J. Raman Spectrosc. 1998, 29 (1), 81–86. (15) Schmitz, A. J.; Hogle, D. G.; Gai, X. S.; Fenlon, E. E.; Brewer, S. H.; Tucker, M. J. Two- Dimensional Infrared Study of Vibrational Coupling between Azide and Nitrile Reporters in a RNA Nucleoside. J. Phys. Chem. B 2016, 120 (35). (16) Barth, A. Infrared Spectroscopy of Proteins. Biochim. Biophys. Acta - Bioenerg. 2007, 1767 (9), 1073–1101. (17) Chalyavi, F.; Hogle, D. G.; Tucker, M. J. Tyrosine as a Non-Perturbing Site-Specific Vibrational Reporter for Protein Dynamics. J. Phys. Chem. B 2017, 121 (26). (18) Suydam, I. T.; Boxer, S. G. Vibrational Stark Effects Calibrate the Sensitivity of Vibrational Probes for Electric Fields in Proteins. Biochemistry 2003, 42 (41), 12050–12055. (19) Polushin, N. N.; Smirnov, I. P.; Verentchikov, A. N.; Coull, J. M. Synthesis of Oligonucleotides Containing 2′-Azido- and 2′-Amino-2′-Deoxyuridine Using

Phosphotriester Chemistry. Tetrahedron Lett. 1996, 37 (19), 3227–3230. (20) Anup M. Jawalekar, †; Nico Meeuwenoord, ‡; J. (Sjef) G. O. Cremers, †; Herman S. Overkleeft, ‡; Gijs A. van der Marel, ‡; Floris P. J. T. Rutjes, † and; Floris L. van Delft*, †. Conjugation of Nucleosides and Oligonucleotides by [3+2] Cycloaddition. 2007. (21) Sylvers, L. A.; Wower, J. Nucleic Acid-Incorporated Azidonucleotides: Probes for Studying the Interaction of RNA or DNA with Proteins and Other Nucleic Acids. Bioconjug. Chem. 1993, 4 (6), 411–418. (22) Aigner, M.; Hartl, M.; Fauster, K.; Steger, J.; Bister, K.; Micura, R. Chemical Synthesis of Site-Specifically 2′-Azido-Modified RNA and Potential Applications for Bioconjugation and RNA Interference. ChemBioChem 2011, 12 (1), 47–51. (23) Mullis, U. S. A.; Faloona, K. B.; Verma, F. A.; Eckstein, S.; Tyagi, F.; Kramer, S.; Nat, F. R.; Hood, L. E.; Kahl, J. D.; Greenberg, M. M.; et al. Click Chemistry to Construct Fluorescent Oligonucleotides for DNA Sequencing. Proc. Natl. Acad. Sci. U.S.A 1995, 92 (3), 609–612. (24) Lu, G.; Burgess, K. A Diversity Oriented Synthesis of 3′-O-Modified Nucleoside Triphosphates for DNA ‘Sequencing by Synthesis.’ Bioorg. Med. Chem. Lett. 2006, 16 (15), 3902–3905. (25) Gai, X. S.; Fenlon, E. E.; Brewer, S. H. A Sensitive Multispectroscopic Probe for Nucleic Acids. J. Phys. Chem. B 2010, 114 (23), 7958–7966. (26) Stryer, L. Fluorescence Energy Transfer as a Spectroscopic Ruler. Annu. Rev. Biochem. 2003, 47 (1), 819–846. (27) Rabenstein, M. D.; Shin, Y. K. Determination of the Distance between Two Spin Labels Attached to a Macromolecule. Proc. Natl. Acad. Sci. 2006, 92 (18), 8239–8243. (28) Chen, H.; Wen, X.; Li, J.; Zheng, J. Molecular Distances Determined with Resonant Vibrational Energy Transfers. J. Phys. Chem. A 2014, 118 (13), 2463–2469. (29) Li, J.; Bian, H.; Chen, H.; Wen, X.; Hoang, B. T.; Zheng, J. Ion Association in Aqueous Solutions Probed through Vibrational Energy Transfers among Cation, Anion, and Water Molecules. J. Phys. Chem. B 2013, 117 (16), 4274–4283. (30) Krummel, A. T.; Zanni, M. T. Evidence for Coupling between Nitrile Groups Using DNA Templates: A Promising New Method for Monitoring Structures with Infrared Spectroscopy. J. Phys. Chem. B 2008, 112 (5), 1336–1338.

Figure 3.1 Structure of the 2’-azido-5-cyano-2’-deoxyuridine molecule with interactions highlighted between the cyano and azido group.

Figure 3.2 Rotating around the dihedral bond, single point energies are calculated to find the energy minima Comparison of the Onsager (lower left) and the CPCM model (lower right). A. J. Schmitz, D. G. Hogle et. al., J. Phys. Chem B, 120, 9387-9394. Used with permission.

Figure 3.3 Coupling of the cyano and azido group is directly proportional to the splitting of the peaks and is governed by the Hamiltonian at right. A. J. Schmitz, D. G. Hogle et. al., J. Phys. Chem B, 120, 9387-9394. Used with permission.

Figure 3.4 Linear IR spectra of N3CNdU in water for (left) N3 and (right) CN. A. J. Schmitz, D. G. Hogle et. al., J. Phys. Chem B, 120, 9387-9394. Used with permission. Used with permission.

Figure 3.5 Clockwise from top left. Two-dimensional IR spectra of N3CNdU in water at waiting time T = 0, T = 400 fs, and T = 1.0 ps. A. J. Schmitz, D. G. Hogle et. al., J. Phys. Chem B, 120, 9387-9394. Used with permission.

Figure 3.6 Structure of sample peptides used (a) AlaN3-Trp5CN (b) AlaN3-Trp5CN.

Figure 3.7 Ramachandran plots of the coupling data over rotations around phi and psi dihedral axes of a protein chain in increments of 30 degrees. (a) FDM on AlaN3-Trp5CN (b) FDM on AlaN3-Trp6CN (c) TDC on AlaN3-Trp5CN (d) TDC on AlaN3-Trp5CN.

Chapter 4: Molecular Dynamics Simulations With the Goal of Rational Probe Design This work has been published in part from: F. Chalyavi, D. G. Hogle, M. J. Tucker, Tyrosine as a Non-Perturbing Site-Specific Vibrational Reporter for Protein Dynamics, J. Phys. Chem B, 121, 6380-6389.

4.1 Development of Intrinsic Infrared Probes

We have established the utility of extrinsic probes in monitoring dynamics in active sites of enzymes, membrane proteins, and nucleotides. Another approach for monitoring proteins is to use an intrinsic group as specific to a particular amino acid a side chain probe. This would avoid unnecessary perturbation to the system as well as sidesteps to process of insertion, both of which allow for easier use in large proteins. Arginine has been used as a site-specific 2D IR probe1,2 and tyrosine has shown promise in as a site-specific probe in fluorescence, absorption, and EPR spectroscopy.3–5 The phenyl side chain of tyrosine is present at many sites that are pivotal towards dictating folding behavior and biological activity. Tyrosine plays a pivotal role in the gating mechanisms of membrane proteins as well as the catalyst in enzymatic activity and performs a major role in proton and electron transfer for different biological reaction pathways. several biological functions and activities6–8

One of the normal modes of Tyr that acts as a potentially promising vibrational reporter is one of the ring breathing modes of the phenyl ring observed at 1517 cm-1. Ge and coworkers measured spectral diffusion of several phenolic ring modes using 2D IR spectroscopy and suggested that the observed changes in peak shape and center related to hydrogen bonding interactions of the ring mode in bulk water.9 Thus, tyrosine is believed to be sensitive to the degree of solvation at the probe site and could function as a reporter on solvation if a relationship between this and peak shift could be established.

4.2 Solvachromatic Effect of the Tyrosine Ring Mode

(all experimental work was done by Tucker Group member Farzaneh Chalyavi)

Linear IR spectra were collected on Tyrosine in the aforementioned solvents. In water a single vibrational transition was observed at 1517 cm-1, while simulations indicated two vibrational modes with corrected frequencies of 1499 and 1574 cm-1. The IR intensity of the mode at 1499 cm-1 was greater by a factor of 10 and was closer to the experimental frequency, thus we concluded this mode corresponding to the breathing mode of the phenyl ring. This measured transition dipole strength was in good agreement with the DFT calculated value of 0.215 D, and experiment confirmed this was within the range of commonly used vibrational probes utilized for protein dynamics such as cyano and amide transitions.

Simulations of tyrosine in solvent were performed to assess the sensitivity of the ring mode to the local environment. As shown in Table 1, several solvents were employed to characterize the subtle interplay between the weak hydrophobic interactions and the direct repulsive interactions between the lone pairs and the ring. Unlike other vibrational probes, such as cyano moieties, the dielectric constant of the solvent was uncorrelated to the observed changes in the bandwidth

(i.e., the frequency distribution). DMSO and diethyl ether exhibit the largest partial negative charge on the oxygen atom containing lone pairs resulting in the largest bandwidth due to their interaction with the electron-rich ring. The hydrophobicity of the two ethyl groups of diethyl ether influences the overall interaction with the ring system, resulting in a slightly increased in bandwidth compared to DMSO. Because of the high electronegativity of the fluorine atoms on

TFE, the lone pairs on the oxygen are less partial charge, causing the bandwidth to decrease significantly compared to the other organic solvents. As mentioned above, water is hydrophilic and cannot solubilize the ring well, thereby exhibiting the smallest bandwidth. The other organic

solvents have characteristics between these extreme cases. It should be noted for all solvents that no significant changes in maximum peak position were observed.

This was in good agreement with experiment, where significant charge in bandwidth was seen depending upon the solvent used. The solvents used can be partitioned into two major groups: mostly hydrophobic (ACN, TFE, Et2O, and CHCl3) and hydrophilic (water, MeOH, and DMSO). This data suggests the lack of hydrophobicity of water and methanol preclude their ability to interact with the ring mode transition as effectively as the other solvents, thus explaining their deviation predicted by the electron-donating ability alone. This change in bandwidth is somewhat unexpected based upon the behavior of other vibrational probes. Usually, a larger inhomogeneous distribution of states is observed within water due to the direct hydrogen bonding to the probes themselves and the indirect electric field distribution, both resulting in a larger number of observed vibrational states with various frequencies.

Initially, we attempted to determine a relationship between the changes in bandwidth and the dielectric constant, such as the commonly used Onsager model. However, since no correlation was found between the vibrational bandwidth and the dielectric constant, further calculations were performed to determine the role of the local electric field on the changes in bandwidth.

Thus, following the work of Boxer and coworkers, we utilized MD simulations to directly quantify the electric field experienced by the tyrosine ring mode.

(This work has been reproduced in part from: F. Chalyavi, D. G. Hogle, M. J. Tucker, Tyrosine as a

Non-Perturbing Site-Specific Vibrational Reporter for Protein Dynamics, J. Phys. Chem B, 121,

6380-6389. Used with permission.)

Briefly, the electric field was directly calculated by averaging the necessary Coulombic interactions. The resulting force determined from the interactions between each atom on the ring and the solvent molecules was projected on the dipole moment to calculate the electric field. The electric field contribution of each atom was added together to find the total electric field acting on the vibrational mode in each solvent. To calculate the charge–charge interactions, the atoms selected to represent the net charge of the ring were approximated as the 6-carbons of the aromatic ring. A clear trend was observed regarding the electric field and bandwidth (frequency distribution) of the experiment. A correlation in the spectral bandwidth with the electric field was determined, shown in Figure 2.

Boxer et. al. have shown that the Stark effect results in a linear relationship between the applied electric field and the peak shift of a given mode. It is well established in the literature that the lineshape of the amide I mode changes with the solvent environment along with the frequency.10–

13 Believing that a similar relation could exist between the bandwidth and the dielectric field of the solvent, the electric field value of each solvent was compared to the bandwidth. Figure 2 shows that the largest bandwidth is observed for the lowest electric field value, diethyl ether, as seen in Table 1. Water is the least hydrophobic solvent with the most positive electric field strength, giving the narrowest bandwidth. DMSO and methanol also show large bandwidths collate to low electric field values, whereas acetonitrile and chloroform are middle range, with water and trifluoroethanol are in the high range, and accordingly correspond to narrow bandwidth. Based on this assumption, the positive field does not interact with the ring mode strongly, and this may be due to stabilization of the phenyl-ring with the strong positive field.

To explain the observed trend, we compared the bandwidth with the Kamlet−Taft solvatochromic parameter (β), which reflects the electron donating capability of the solvent.14 A clear linear trend

was identified for the experimental bandwidth as a function of the electron donating capability of each solvent. (Figure 3). These observations suggest that the lone pairs of each solvent create a distribution of structures with varying vibrational frequency resulting from the repulsive interaction with the ring, similar to that of the electric field strength. This is substantiated by the results from our DFT calculations. However, this trend does not hold for water and methanol, suggesting that other solvent characteristics must be considered, such as the hydrophobicity, measured as the log of the solvent partition coefficient (log P).15 While there is not a linear relationship between the log P value of each solvent and the experimental bandwidth observed as with β, the solvents with negative log P values are hydrophilic in nature, such as water and methanol, while the more positive values represent the hydrophobic solvents. Thus, it helps explain the deviation from the electron donating ability of water and methanol. The two solvents are not hydrophobic enough to solvate the ring leading to less significant interactions. Although

DMSO also has a negative log P, the large value of β, electron donating ability, compensates causing the large observed bandwidth.

As suggested by our DFT calculations, the lone pairs of each solvent can influence the variation in the vibrational frequency caused by the repulsive interactions with the ring. This observed trend in the β parameter also seems to correspond to changes in the bandwidth as a function of electric field strength discussed above. As the electric field becomes more negative, the interactions with the electron-rich phenol ring become more significant. Such types of lone pair−π electron repulsive interaction have been extensively documented in prior literature.16–18 The effect of the beta parameter on the bandwidth does not hold for water and methanol, where a significantly larger bandwidth would be expected if this was the sole factor at work.

To assess the sensitivity of the ring mode to its surroundings, the infrared spectrum was measured in two solvents with dielectric constants varying by 40, water (εr ∼ 80.1) and DMSO (εr ∼ 46.7), seen in Figure 4. The solvation in bulk water is representative of the environment of exposed residues in a water-soluble peptides, while DMSO is comparable to the hydrophobic environment found within a membrane environment. No spectral shift in frequency is observed, but there is a significant change in bandwidth. The FWHM of this vibrational transition in DMSO was determined to be ∼1.5 times larger than the corresponding bandwidth in water.

4.3 Theory and NAMD simulations

From prior work on azide probes, past studies have established a relationship between the vibrational frequency and the calculated electric field using Onsager field theory.19 However, this only takes into account interactions due to the charge-dipole approximation, and does not account for direct electrostatic interactions, such as H-bonding, limiting its success in describing protic solvents. Using molecular dynamics (MD) simulations as well as linear stark spectroscopy,

Boxer et al. established a method for calculating the influence of the electric field on a vibration resulting from its local environment via its infrared spectrum, shown by the equation 1:

ℎ푐(휐표푏푠̅ − 휐0̅ ) 퐹⃗ ∙ 휇̂푝푟표푏푒 = (1) |∆휇⃗푝푟표푏푒|

where 휐표푏푠̅ is the observed vibrational frequency, 휐0̅ is a reference frequency calibrated to zero electric field, and |휇⃗푝푟표푏푒| is the magnitude of the probe’s difference dipole, which is defined by measuring the vibrational Stark effects and the sensitivity of vibrational shifts to electric field. 퐹⃗ ∙

휇̂푝푟표푏푒 is the electric field experienced by the vibration projected onto the difference dipole

20,21 vector, ∆휇⃗푝푟표푏푒. Pazos and coworkers extended the method to determine electric field effects

on the vibrational transitions of ester groups by correlating the vibrational frequencies within both protic and aprotic solvents.22 Others have shown such correlations between the electric field and the frequency for other oscillators.20,23,24

Density function theory (DFT) simulations of tyrosine with an acetylated N-terminus and an amidated C-terminus were performed using B3LYP/6-31+G** level of theory with implicit solvation model of water via the conducting polarizable continuum model (CPCM) using the

Gaussian 09 software package.25 Frequencies were obtained following structural optimization.

Gaussian 09 calculations were also performed with the same level of theory using a single explicit

DMSO molecule or water molecule at various distances from the tyrosine ring to determine the overall effect on the vibrational frequency.

All MD simulations were performed with Nanoscale Molecular Dynamics (NAMD) 2.9 utilizing a tyrosine amino acid with the N-terminus acetylated and the C-terminus amidated with a 30 Å solvent box with periodic boundary conditions using force field parameters from CHARMM36 all- hydrogen topology and the CGenFF topology.26 Following an initial 2 ps equilibration run at 298 K and 1 atm in the NPT ensemble, a production run was performed for 10 ns at 298 K in the NVT ensemble. The temperature was then reassigned every 500 steps until reaching a preset value, simulating a slow heating of the molecule. Our settings increase the temperature by 20 degrees each time to a maximum of 298K. Once the box was heated to 298K, a production run was carried out for 500000 steps (1 ns) and 5000000 steps (10 ns). MD simulations were performed for the following solvents: water, methanol, dimethyl sulfoxide (DMSO), trifluoroethanol (TFE), diethyl ether (Et2O), acetonitrile (ACN), and chloroform (CHCl3). For the electric field calculations, the direction and the magnitude of the ring-breathing vibrational mode dipole were obtained via

Guassian09 calculations mentioned above. The direction was determined to be roughly parallel to the vector connecting the 1,4 carbons on the ring.

To perform the electric field calculation, the force due to the coulombic interaction between a given atom in the tyrosine ring and each solvent atom within a given radius were added together.

The force due to the coulombic interaction was then projected along the dipole of the ring mode.

For our calculations, we utilized all six atoms on the ring-breathing mode of Tyrosine measuring the solvent effect over a 20 Å radius as the interactions beyond this distance should be negligible.

The computed electric fields were then plotted as a function of the bandwidth of the experimental peaks. The slope of the resulting line correlates to the Stark effect. Due to the external field applied to a sample, the local field on a probe due to the external charges will differ from the accurately known external (Maxwell) field by a factor called the local field correction factor. The local field will generally be larger because of extra contributions arising from polarization of the medium surrounding the probe induced by the external field. Unfortunately, the extent to which the local field is greater than the external field is not precisely known, and thus cannot be calculated from force field data. Using a combination of Stark spectroscopy and DFT simulations,

Boxer estimated a correction factor between 1.4 and 1.8 for calculated electric fields in different solvents. Following previous work, a correction factor of 1.8 was applied to the resulting electric field values calculated from the MD simulations.20,22

4.4 Solvent Broadening Effects on the Tyrosine Ring Mode

(the calculations below were performed in collaboration with Farzaneh Chalyavi)

To examine the nature of the broadening mechanisms within DMSO, DFT calculations were performed on the tyrosine molecule in the presence of a DMSO and water molecules. As shown

in Figure 4, a DMSO molecule was positioned at a distance of 2.4, 5.2, and 8.3 Å between the oxygen atom of DMSO and the center of the tyrosine ring, and the vibrational frequencies were calculated. A difference in vibrational frequency of ∼8–10 cm-1 was calculated between the different molecular configurations with varying distances. These simulations suggest that the observed changes in bandwidth are likely due to a direct charge–charge interaction between the lone pairs of the oxygen on the DMSO and the electron-rich tyrosine ring. The repulsive lone pair−π interactions create a larger inhomogeneous distribution of frequencies. Yet, tyrosine is also fully capable of forming weakly polar interactions/hydrophobic interactions with other compounds, which influence the overall inhomogeneous distribution.

A similar calculation was performed for water showing no significant change (<1 cm-1) in vibrational frequency until the closest distance (2.3 Å). At this distance, the vibrational frequency differed by >45 cm-1, and the overall configuration was energetically unfavorable, suggesting that the interaction of water with the ring is to a much lesser extent. Although a large change in bandwidth in the observed spectrum would be expected for water molecules within 2.3 Å, the observed bandwidth is quite small, suggesting that water does not have such close proximity likely due to the hydrophobicity of the ring.

4.5 Studies of Tyrosine as a Probe Involving the trp-Cage Peptide

Upon establishing the linear correlation of the bandwidth and electric field strength, a model peptide system, Trp-cage, was chosen to test the efficacy of the trend line and measure the electric field in the hydrophobic core region of a peptide. Shown in Figure 6, Trp-cage is a synthetic

20-residue miniprotein (NLYI-QWLK-DGGP-SSGR-PPPS), containing an α-helix, 310-helix, and a polyproline II helix; it is often considered the “hydrogen atom” of the folding and unfolding studies

Trp-cage is an ideal model system to test the behavior of tyrosine as it has a strong hydrophobic core containing the aromatic side chains of Tyr3 and Trp6 packed against Gly11, Pro12, Pro18, and Pro19.27,28 The IR spectrum of this tyrosine containing peptide folded state in buffer solution at room temperature exhibits a vibrational bandwidth (∼6.2 cm-1) similar to acetonitrile and chloroform.

Figure 7a depicts a normalized infrared spectra of tyrosine in Trp-cage miniprotein under chemical denaturation conditions (green) as well as the spectrum of the folded miniprotein in water (red).

The FTIR spectra of Trp-cage miniprotein in water and chemical denatured (GdnHCl) state of Trp- cage in solution at room temperature (~25°C) are shown in Figure 1. Each spectrum was fit to a following gaussian function. Although both infrared spectra have a gaussian shape, the observed vibrational bandwidth of the tyrosine ring mode in the folded state was ~1.3 cm-1 larger than the chemically denatured state. The corresponding value of the electric field was found to be 20.52

MV/cm. Upon temperature or chemical (guanidine hydrochloride) denaturation of the miniprotein, the vibrational bandwidth decreases corresponding to an increase in the electric field of approximately 7.9 and 12.9 MV/cm, respectively, seen in Figure 7b. The chemical denaturant has the largest electric field value of 33.37 MV/cm.

4.6 Future Directions

Tyrosine has been shown to be sensitive to the local environment through the ring breathing mode. It is an intrinsic and therefore non-perturbing infrared probe that occurs in a wide variety of protein systems. A solvation field model was developed and shown to be effective at monitoring the structural change of a Trp-cage miniprotein through changes in the bandwidth of the infrared peak under temperature denaturation and chemical denaturation. This change is due to the electric field within the hydrophobic core of the protein differing under native and

denatured conditions within the. The electric field strength around the tyrosine in the hydrophobic pocket of the protein was determined to be value (20.52 MV/cm), similar to the field generated by acetonitrile or chloroform and significantly lower than the field observed in bulk water solution. Upon denaturation, the field strength approached the bulk water solution.

DFT calculations illustrate that the observed inhomogeneous bandwidth is due to a repulsive charge–charge interaction between the lone pairs of DMSO and the electron-rich ring of tyrosine. The observed trend can be described by using a combination of only two solvent parameters: electron donating ability and the partition coefficient (measure of hydrophobicity).

The variation in the bandwidth arises from the interplay between the weak hydrophobic interactions and the direct repulsive interactions between the lone pairs and the ring. Further studies on larger protein systems will be needed to see if this fully characterizes the solvent-ring mode interactions, but the preliminary results in this study as well as the utility of tyrosine in other spectroscopic methods suggests that tyrosine has strong potential as an intrinsic non- perturbing site-specific probe.

References

(1) Ghosh, A.; Tucker, M. J.; Hochstrasser, R. M. Identification of Arginine Residues in Peptides by 2D-IR Echo Spectroscopy. J. Phys. Chem. A 2011, 115 (34), 9731–9738. https://doi.org/10.1021/jp201794n. (2) Mark S. Braiman, *,‡; Deborah M. Briercheck, § and; Kriger, K. M. Modeling Vibrational Spectra of Amino Acid Side Chains in Proteins: Effects of Protonation State, Counterion, and Solvent on Arginine C−N Stretch Frequencies†. 1999. https://doi.org/10.1021/JP983011B. (3) Stich, T. A.; Myers, W. K.; Britt, R. D. Paramagnetic Intermediates Generated by Radical S- Adenosylmethionine (SAM) Enzymes. Acc. Chem. Res. 2014, 47 (8), 2235–2243. https://doi.org/10.1021/ar400235n. (4) Rehms, A. A.; Callis, P. R. Two-Photon Fluorescence Excitation Spectra of Aromatic Amino Acids. Chem. Phys. Lett. 1993, 208 (3–4), 276–282. https://doi.org/10.1016/0009- 2614(93)89075-S. (5) Ferreira, S. T.; Stella, L.; Gratton, E. Conformational Dynamics of Bovine Cu, Zn Superoxide Dismutase Revealed by Time-Resolved Fluorescence Spectroscopy of the Single Tyrosine Residue. Biophys. J. 1994, 66 (4), 1185. https://doi.org/10.1016/S0006-

3495(94)80901-4. (6) Harries, W. E. C.; Akhavan, D.; Miercke, L. J. W.; Khademi, S.; Stroud, R. M. The Channel Architecture of Aquaporin 0 at a 2.2-Angstrom Resolution. Proc. Natl. Acad. Sci. U. S. A. 2004, 101 (39), 14045–14050. https://doi.org/DOI 10.1073/pnas.0405274101. (7) Domene, C.; Vemparala, S.; Klein, M. L.; Vénien-Bryan, C.; Doyle, D. A. Role of Aromatic Localization in the Gating Process of a Potassium Channel. Biophys. J. 2006, 90 (1). https://doi.org/10.1529/biophysj.105.072116. (8) Popp, B. V.; Ball, Z. T. Structure-Selective Modification of Aromatic Side Chains with Dirhodium Metallopeptide Catalysts. J. Am. Chem. Soc. 2010, 132 (19), 6660–6662. https://doi.org/10.1021/ja101456c. (9) Sul, S.; Feng, Y.; Le, U.; Tobias, D. J.; Ge, N. H. Interactions of Tyrosine in Leu-Enkephalin at a Membrane-Water Interface: An Ultrafast Two-Dimensional Infrared Study Combined with Density Functional Calculations and Molecular Dynamics Simulations. J. Phys. Chem. B 2010, 114 (2), 1180–1190. https://doi.org/10.1021/jp9105844. (10) Ghosh, A.; Tucker, M. J.; Gai, F. 2D IR Spectroscopy of Histidine: Probing Side-Chain Structure and Dynamics via Backbone Amide Vibrations. J. Phys. Chem. B 2014, 118 (28), 7799–7805. https://doi.org/10.1021/jp411901m. (11) Woys, A. M.; Almeida, A. M.; Wang, L.; Chiu, C.-C.; McGovern, M.; de Pablo, J. J.; Skinner, J. L.; Gellman, S. H.; Zanni, M. T. Parallel β-Sheet Vibrational Couplings Revealed by 2D IR Spectroscopy of an Isotopically Labeled Macrocycle: Quantitative Benchmark for the Interpretation of Amyloid and Protein Infrared Spectra. J. Am. Chem. Soc. 2012, 134 (46), 19118–19128. https://doi.org/10.1021/ja3074962. (12) Barth, A.; Zscherp, C. What Vibrations Tell about Proteins. Q. Rev. Biophys. 2002, 35 (4), 369–430. https://doi.org/10.1017/S0033583502003815. (13) Ding, B.; Laaser, J. E.; Liu, Y.; Wang, P.; Zanni, M. T.; Chen, Z. Site-Specific Orientation of an α-Helical Peptide Ovispirin-1 from Isotope-Labeled SFG Spectroscopy. J. Phys. Chem. B 2013, 117 (47), 14625–14634. https://doi.org/10.1021/jp408064b. (14) Marcus, Y. The Properties of Organic Liquids That Are Relevant to Their Use as Solvating Solvents. Chemical Society Reviews. 1993, pp 409–416. https://doi.org/10.1039/CS9932200409. (15) Kwon, Y. Handbook of Essential Pharmacokinetics, Pharmacodynamics and Drug Metabolism for Industrial Scientists; 2014. https://doi.org/10.1007/s13398-014-0173-7.2. (16) Egli, M.; Sarkhel, S. Lone Pair-Aromatic Interactions: To Stabilize or Not to Stabilize. Acc. Chem. Res. 2007, 40 (3), 197–205. https://doi.org/10.1021/ar068174u. (17) Jain, A.; Purohit, C. S.; Verma, S.; Sankararamakrishnan, R. Close Contacts between Carbonyl Oxygen Atoms and Aromatic Centers in Protein Structures: Π⋯π or Lone- Pair⋯π Interactions? J. Phys. Chem. B 2007, 111 (30), 8680–8683. https://doi.org/10.1021/jp072742l. (18) Dougherty, D. Cation-Pi Interactions Involving Aromatic Amino Acids. J. Nutr. 2007, 137 (6 Suppl 1), 1504–1508. https://doi.org/0022-3166/07. (19) Onsager, L. Electric Moments of Molecules in Liquids. J. Am. Chem. Soc. 1936, 58 (8), 1486–1493. https://doi.org/10.1021/ja01299a050. (20) Fried, S. D.; Bagchi, S.; Boxer, S. G. Measuring Electrostatic Fields in Both Hydrogen- Bonding and Non-Hydrogen-Bonding Environments Using Carbonyl Vibrational Probes. J. Am. Chem. Soc. 2013, 135 (30), 11181–11192. https://doi.org/10.1021/ja403917z. (21) Suydam, I. T.; Boxer, S. G. Vibrational Stark Effects Calibrate the Sensitivity of Vibrational

Probes for Electric Fields in Proteins. Biochemistry 2003, 42 (41), 12050–12055. https://doi.org/10.1021/bi0352926. (22) Pazos, I. M.; Ghosh, A.; Tucker, M. J.; Gai, F. Ester Carbonyl Vibration as a Sensitive Probe of Protein Local Electric Field. Angew. Chemie - Int. Ed. 2014, 53 (24), 6080–6084. https://doi.org/10.1002/anie.201402011. (23) Edington, S. C.; Flanagan, J. C.; Baiz, C. R. An Empirical IR Frequency Map for Ester C=O Stretching Vibrations. J. Phys. Chem. A 2016, 120 (22), 3888–3896. https://doi.org/10.1021/acs.jpca.6b02887. (24) Choi, J. H.; Cho, M. Vibrational Solvatochromism and Electrochromism of Infrared Probe Molecules Containing C≡O, C≡N, C=O, or C-F Vibrational Chromophore. J. Chem. Phys. 2011, 134 (15). https://doi.org/10.1063/1.3580776. (25) M. J. Frisch, G.; Trucks, W.; Schlegel, H. B. .; Scuseria, G. E. .; Robb, M. A. .; Cheeseman, J. R.; Scalmani, G.; Barone, V. .; Mennucci, B. .; Petersson, G. A.; et al. Gaussian 09, Revision E. 01; Gaussian; 2009. https://doi.org/111. (26) Phillips, J. C.; Braun, R.; Wang, W.; Gumbart, J.; Tajkhorshid, E.; Villa, E.; Chipot, C.; Skeel, R. D.; Kalé, L.; Schulten, K. Scalable Molecular Dynamics with NAMD. J. Comput. Chem. 2005, 26 (16), 1781–1802. https://doi.org/10.1002/jcc.20289. (27) Neidigh, J. W.; Fesinmeyer, R. M.; Andersen, N. H. Designing a 20-Residue Protein. Nat. Struct. Biol. 2002, 9 (6), 425–430. https://doi.org/10.1038/nsb798. (28) Culik, R. M.; Serrano, A. L.; Bunagan, M. R.; Gai, F. Achieving Secondary Structural Resolution in Kinetic Measurements of Protein Folding: A Case Study of the Folding Mechanism of Trp-Cage. Angew. Chemie-International Ed. 2011, 50 (46), 10884–10887. https://doi.org/10.1002/anie.201104085.

Figure 4.1 Depiction of the primarily ring breathing mode of the tyrosine ring found at 1517 cm-1 with the overall dipole moment direction shown by the yellow arrow. Created by F. Chalyavi. Used with permission.

Figure 4.2 The bandwidth of the peak diminishes with the strength of the solvent’s electric field.

Figure 4.3 Relationship between the vibrational bandwidth and the solvent electron donor ability (β).

Figure 4.4 The infrared mode as seen in DMSO and D2O. The center of the peak does not change substantially, but the change in bandwidth in notable. Taken from F. Chalyavi et al. J. Phys. Chem B, 121, 6380-6389. Used with permission.

Figure 4.5 Optimized configurations of solvent molecules show a strong dependence on the angle of approach. A DMSO molecule is in an optimized position at a distance of (a) 8.3 Å and (b) 2.4 Å from the center of the tyrosine ring. Taken from F. Chalyavi et al. J. Phys. Chem B, 121, 6380-6389. Used with permission.

Figure 4.6 Computer depiction of the position of the tryptophan residue in the hydrophobic pocket of folded Trp-cage. Created by M. Tucker. Used with permission.

Figure 4.7 a) Normalized infrared spectra of tyrosine in Trp-cage miniprotein folded in water (red) and under chemical denaturation conditions (green). b) The bandwidth of the Trp peak of the Trp- cage miniprotein shows experimental correlation to the electric field in the hydrophobic pocket folded in water (red) heated to 70C (blue) and chemically denatured (green). Taken from F. Chalyavi et al. J. Phys. Chem B, 121, 6380-6389. Used with permission.

Table 4.1 Solvatochromic Effect on Tyrosine Ring Breathing Vibrational Transition

Solvent Dielectric Electric Field Bandwidth (cm–1) Peak Position (cm–1) Constant (MV/cm)

Et2O 4.34 –9.11 10.2 1518.5 DMSO 46.7 –4.75 8.3 1516.7 MeOH 32.7 8.54 6.5 1518.9 ACN 37.5 15.72 6.7 1517.2

CHCl3 4.81 17.46 6.3 1518.9 TFE 27.68 26.45 5.6 1518.3

D2O 80.1 32.80 5.4 1516.5

Chapter 5: Fermi Resonance and Coriolis Coupling in p-azidobenzonitrile

5.1 Introduction

We have seen how the coupling between two vibrational modes and the vibrational Stark effect in solution can be calculated to extract information about local electric fields in proteins, and hence structural behavior. However, there are other conditions that can cause coupling from modes that otherwise would not affect each other: Fermi resonance (FR) is a phenomenon that introduces coupling into a system when dark modes are present. It appears in 2D IR as the coupling between two vibrational modes, but it is in fact the effect of a combination or hot band, causing complications in interpreting spectra.1–4

FR was observed in phenyl cyanate during studies evaluating its prospective use as a probe; the molecule was modified using isotopic 13C and 15N to eliminate the accidental resonance.5,6 While this may seem to be a simple, readily available solution, this phenomenon is an issue deserving consideration in a broader context. One issue here is the inherent difficulty of attributing a given signal to a given FR; in the OCN molecule the peaks were believed to be due to typical coupling between modes until detailed studies were performed. Evidence was amassed from spectra that showed that the suspect cross peaks appeared at atypically early waiting times, decayed along the diagonal, and showed similar ratios regardless of polarization. This was used to then rule out alternative explanations, such as motion in solvent, however there was no single test that could easily label a given mode as an FR. Secondly, DFT studies showed the existence of numerous dark states with frequencies that could couple to give accidental degeneracy, however even with complex quantum mechanical modeling it is time-consuming to find which vibrational modes contribute to a given FR and hence which modes must be modulated for it to be eliminated.

Other studies have shown this issue is by no means limited to cyanate alone. One such study shows that benzoylchloride exhibits a double peak structure in the CO peak, this has been attributed to FR between the carbonyl stretching band and an overtone of a low-frequency stretching mode of the aromatic ring. When dissolved in chloroform, the FR results in two peaks with about the same intensity, showing the frequency overlap is almost exact.1,7 Other studies involving acetonitrile demonstrated FR for numerous modes.2 However, studies on acetanilide showed that the double peak structure observed in the amide I band cannot be explained by FR alone, but via a combination of potential FR and vibrational self-trapping.7 Thus, even detailed studies can give contradictory evidence when multiple components play a factor.

The rationale for better understanding of FR is as follows. Phenyl cyanate is a small molecule, but rational probe design where elimination of unwanted resonance is not a matter of trial and error is still urgently needed as probe types become more numerous.8 Furthermore, when attached to a macromolecule, normal modes can be shifted in intensity and frequency, leading to suppression of some FRs and activation of others. Isolabelling adds costs and time to the creation of probe groups, and can limit the versatility of applications if isolabelling is employed as a site-specific label elsewhere in the molecule. It would be better to design probes to avoid accidental FRs as much as possible intrinsically, thus avoiding complex modulation.

Studies have shown the azido moiety to be an ideal probe for infrared experiments due to its small size, high extinction coefficient, and absorption away from many other molecular transitions. The information is available regarding FR and energy transfer between the fundamental asymmetric stretch of azide and carbon backmode modes in organic molecules, suggests the possibility of FR peaks similar to the phenomenon observed with phenyl cyanate.9,10 Therefore, further research

advancing the theory to predict FR and confirmation with experiment is needed before azide can be used with confidence as a probe.

5.2 Fermi Resonance Theory

Coupling is greatest when the coupled groups have approximately equal energies. This can mean two vibrational states that are close in energy. However, two modes coupled together can result in both being excited simultaneously. This can be an overtone or a combination band (v1+v2), as shown in Figures 1 and 2. When an overtone or combination band is close in energy to another mode, Fermi resonance can occur. The modes couple and mix together, causing the intensity and frequency to change. The lower modes becomes lower, and the higher frequency gets higher, resulting in greater splitting.

Fermi resonance theory is well understood, and the theoretical contributions to the band structure can be calculated simply through quantum mechanical equations.3,11 The band structure consists of excited states that have one-quantum and two-quantum contributions. First for the one-quantum states. Linear IR spectra can be computed as shown using Equation 1:

훾0푘 2 휋 ( ) 푆 = ∑ 휇0푘 2 2 1 (휔 − 휔0푘) + 훾0푘

휇0푘 is a vector that is the transition dipole from 0 to k, where k is a given state. If there is one vibrational mode, we can write this as 0 to 1. The second state is reached by going from 1 to 2, or

0 to 2 for an overtone excitation.

The probability that this happens is proportional to the transition dipole moment, written as 휇01, and by applying first order perturbation theory to this system we obtain equation (1) .

Equations (2) and (3) give the first and second level excitations, respectively, which correspond to the linear and 2D IR spectra, as shown in Figure 3.

|100⟩ |011⟩ 훼 ⟨100| 휔100 퐻1 = 2√2 (2) [ 훼 ] ⟨011| 휔011 2√2

|200⟩ |111⟩ |022⟩ 훼2 훼

⟨200| 휔200 2√2 2 퐻 = 훼2 훼 (3) 2 ⟨111| 휔111 2√2 √2 ⟨022| 훼 훼 휔022 [ 2 √2 ]

Third-order nonlinear responses are computed using (4) and (5) for rephasing and non-rephasing components respectively.

̂ ̂ ̂ ̂ ̂ ̂ ̂ ̂ 〈휇⃗0푘 ∙ 푘1휇⃗0푘′ ∙ 푘2휇⃗푘0 ∙ 푘3휇⃗푘′0 ∙ 푘푆〉 〈휇⃗0푘 ∙ 푘1휇⃗0푘′ ∙ 푘2휇⃗푘′퐾 ∙ 푘3휇⃗퐾푘 ∙ 푘푆〉 훽 = 4 ∑ − 2 ∑ (4) [−푖(휔푘0 − 휔휏) − 훾푘0][푖(휔푘′0 − 휔휏) + 훾푘′0] [−푖(휔푘0 − 휔휏) − 훾푘0][푖(휔퐾푘 − 휔휏) + 훾퐾푘] 푘,푘′ 푘,푘′,퐾

′ 〈휇⃗ ;∙ 푘̂ 휇⃗ ∙ 푘̂ 휇⃗ ′ ∙ 푘̂ 휇⃗ ∙ 푘̂ 〉 〈휇⃗ ∙ 푘̂ 휇⃗ ′ ∙ 푘̂ 휇⃗ ∙ 푘̂ 휇⃗ ′ ∙ 푘̂ 〉 훽 = ∑ 0푘 1 0푘 2 푘 0 3 푘0 푆 + ∑ 0푘 1 0푘 2 푘0 3 푘 0 푆 + [−푖(휔푘0 − 휔휏) − 훾푘0][푖(휔푘0 − 휔휏) + 훾푘0] [−푖(휔푘0 − 휔휏) − 훾푘0][푖(휔퐾푘 − 휔휏) + 훾퐾푘] 푘,푘′,퐾 푘,푘′,퐾

〈휇⃗ ∙ 푘̂ 휇⃗ ′ ∙ 푘̂ 휇⃗ ∙ 푘̂ 휇⃗ ′ ∙ 푘̂ 〉 〈휇⃗ ∙ 푘̂ 휇⃗ ′ ∙ 푘̂ 휇⃗ ∙ 푘̂ 휇⃗ ′ ∙ 푘̂ 〉 ∑ 0푘 1 0푘 2 푘0 3 푘 0 푆 − 2 ∑ 0푘 1 0푘 2 푘0 3 푘 0 푆 (5) [−푖(휔푘0 − 휔휏) − 훾푘0][푖(휔퐾푘 − 휔휏) + 훾퐾푘] [−푖(휔푘0 − 휔휏) − 훾푘0][푖(휔퐾푘 − 휔휏) + 훾퐾푘] 푘,푘′,퐾 푘,푘′,퐾

5.3 Calculation of Mode Frequencies for Anharmonic Coupling.

The goal of the method used is to obtain an accurate reproduction of 2D IR spectra using empirical parameters combined with theory. By doing this, we can explain the contributions that vibrational mode coupling and FR make towards the various observed cross-peaks. Past studies have shown

that DFT is inadequate to find true anharmonic terms due to electrostatic effects, thus it is prohibitively expensive to simulate the spectrum via pure theory alone.10,12 By obtaining the unmixed frequencies and coupling from the linear spectrum, we can an energy minimization algorithm to obtain accurate values for the non-linear anharmonic terms.

(all modeling of p-azidobenzonitrile was performed by Hari Pandey of the Leitner Group)

The initial geometry was constructed using the Avogadro visualization package and optimized using molecular mechanics (MM) with the General Amber Force Field (GAFF).13 The MM optimized geometry was introduced into a semi-empirical method (PM6) optimization, followed be a

Hartree-Fock level (HF) calculation with the 631G basis set. The implemented solvation model was the conducting polarizable continuum model (CPCM), applied for a self-consistent reaction field

(SCRF) water model, which has been shown to be adequate for reproduction of infrared spectra.14,15 The HF optimized geometry was taken as an initial structure for a DFT/B3LYP/6-31G- level calculation, followed by DFT/B3LYP/6-31+G** using the same water model for the solvent to obtain the optimized molecular geometry, Hessian, normal modes, frequencies, and anharmonic constants. An ultrafine integration grid for the two-electron integral calculation with accuracy 10−13 and very tight convergence criteria were applied throughout the electronic structure calculation process. All the DFT calculations were executed using the Gaussian 09 computational package.16

DFT was used with B3LYP/6-31+G** to calculate the cubic and quartic force constant terms. As these are minor relative to the harmonic force constants, ultrafine convergence was used. Quartic terms were found to be minor, as expected. As shown in Table 1, the third order resonance parameter TFRαβγ for the three frequency coupling modes () = (1154, 1154, 2317) is

largest and hence gives rise to the dominant anharmonic interaction. The mode with frequency

1154 cm-1 is localized to the ring and azide region of the molecule so the nitrile stretch couples mostly through the ring and a little to the azide portion of the molecule.

By fitting the linear IR spectra to our experimentally derived spectra (Figure 4) using the Nedler-

Mead algorithm, we obtained a coupling constant of 38 and uncoupled frequencies of 2160 and

2172 cm-1 respectively. Results showed that Gaussian substantially overestimates the frequencies involved. The FWHM of the two peaks located at 2104 and 2139 cm-1 are both approximately 19 cm-1. The linear IR spectrum of p-azidobenzonitrile in THF exhibits four distinct peaks at 2104 and

2139 cm-1 that correspond to the fundamental asymmetric stretch of the azide and the FR of the azide. In order to understand this effect, we developed code that simulates the spectra of coupling interactions between two modes. Using theoretical spectra from Gaussian with the B3LYP/6-31G basis set, we obtained uncoupled frequencies of 2234.82, 1366.26, and 843.19 cm-1 for the main mode and two combined modes, respectively. This shows good agreement with the experimental spectrum.

The 2D IR spectrum collected for p-azidobenzonitrile at T = 0 exhibits two peaks along the diagonal at 2105 and 2139 cm-1 giving excited states as 4196, 4235, 4273 cm-1 and distinct cross peaks with the negative regions shifted by 17 cm-1. By using the excited state frequencies as the target values and assuming the coupling constant remains the same, we can approximate the second-level anharmonicities, which do not follow linearly from the first-level anharmonicities, again using

Nedler Mead. Locating the resonances for the azide stretching mode via anharmonic interactions allows characterization of the resonant couplings. We have calculated the resonance distance,

훥휔, defined as 훥휔 = |휔푎 + 휔푏 − 휔𝑔|, as well as the third order resonance parameter, TFRαβγ

defined in equation (6). TFRαβγ is computed for a particular frequency coupling as a ratio between the anharmonic constant and resonance distance.17

Figure 5 shows the results of the experimental 2D IR spectrum with an uncorrected simulated spectrum, showing poor agreement. Thus, we conclude that the anharmonic components do play a significant role and must be added to obtain accurate representation. The next step is to add the parameters taken from the linear spectrum to the two-dimensional simulation method and move forward with matching theory with experiment. This can provide a basis to explain why

Fermi resonance occurs between the modes in our system.

References

(1) Bertran, J. F.; Ballester, L.; Dobrihalova, L.; Sánchez, N.; Arrieta, R. Study of Fermi Resonance by the Method of Solvent Variation. Spectrochim. Acta Part A Mol. Spectrosc. 1968, 24 (11), 1765–1776. https://doi.org/10.1016/0584-8539(68)80232-6. (2) Reimers, J. R.; Hall, L. E. The Solvation of Acetonitrile. J. Am. Chem. Soc. 1999, 121 (15), 3730–3744. https://doi.org/10.1021/ja983878n. (3) Kondratyuk, P. Analytical Formulas for Fermi Resonance Interactions in Continuous Distributions of States. Spectrochim. Acta - Part A Mol. Biomol. Spectrosc. 2005, 61 (4), 589–593. https://doi.org/10.1016/j.saa.2004.05.010. (4) Duncan, J. L. The Determination of Vibrational Anharmonicity in Molecules from Spectroscopic Observations. Spectrochim. Acta Part A Mol. Spectrosc. 1991, 47 (1), 1–27. https://doi.org/10.1016/0584-8539(91)80174-H. (5) Lipkin, J. S.; Song, R.; Fenlon, E. E.; Brewer, S. H. Modulating Accidental Fermi Resonance: What a Difference a Neutron Makes. J. Phys. Chem. Lett. 2011, 2 (14), 1672–1676. https://doi.org/10.1021/jz2006447. (6) Tucker, M. J.; Kim, Y. S.; Hochstrasser, R. M. 2D IR Photon Echo Study of the Anharmonic Coupling in the OCN Region of Phenyl Cyanate. Chem. Phys. Lett. 2009, 470 (1–3), 80–84. https://doi.org/10.1016/j.cplett.2009.01.025. (7) Edler, J.; Hamm, P. Two-Dimensional Vibrational Spectroscopy of the Amide I Band of Crystalline Acetanilide: Fermi Resonance, Conformational Substates, or Vibrational Self- Trapping? J. Chem. Phys. 2003, 119 (5), 2709–2715. https://doi.org/10.1063/1.1586694. (8) Nydegger, M. W.; Dutta, S.; Cheatum, C. M. Two-Dimensional Infrared Study of 3- Azidopyridine as a Potential Spectroscopic Reporter of Protonation State. J. Chem. Phys. 2010, 133 (13). https://doi.org/10.1063/1.3483688. (9) Lieber, E.; Rao, C.; Thomas, A. Infrared Spectra of Acid Azides, Carbamyl Azides and Other Azido Derivatives: Anomalous Splittings of the N3 Stretching Bands. Spectrochim. Acta 1963, 19, 1135–1144. (10) Zhang, J.; Wang, L.; Zhang, J.; Zhu, J.; Pan, X.; Cui, Z.; Wang, J.; Fang, W.; Li, Y. Identifying

and Modulating Accidental Fermi Resonance: 2D IR and DFT Study of 4-Azido-l- Phenylalanine. J. Phys. Chem. B 2018, 122 (34), 8122–8133. https://doi.org/10.1021/acs.jpcb.8b03887. (11) Wilson, E. B.; Decius, J. C.; Cross, P. C.; Sundheim, B. R. Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra. J. Electrochem. Soc. 2007, 102 (9), 235C. https://doi.org/10.1149/1.2430134. (12) Maj, M.; Ahn, C.; Błasiak, B.; Kwak, K.; Han, H.; Cho, M. Isonitrile as an Ultrasensitive Infrared Reporter of Hydrogen-Bonding Structure and Dynamics. J. Phys. Chem. B 2016, 120 (39), 10167–10180. https://doi.org/10.1021/acs.jpcb.6b04319. (13) Hanwell, M. D.; Curtis, D. E.; Lonie, D. C.; Vandermeersch, T.; Zurek, E.; Hutchison, G. R. Avogadro: An Advanced Semantic Chemical Editor, Visualization, and Analysis Platform. J. Cheminform. 2012, 4 (1), 17. https://doi.org/10.1186/1758-2946-4-17. (14) Barone, V. Anharmonic Vibrational Properties by a Fully Automated Second-Order Perturbative Approach. J. Chem. Phys. 2005, 122 (1). https://doi.org/10.1063/1.1824881. (15) Barone, V.; Biczysko, M.; Bloino, J. Fully Anharmonic IR and Raman Spectra of Medium- Size Molecular Systems: Accuracy and Interpretation. Physical Chemistry Chemical Physics. 2014, pp 1759–1787. https://doi.org/10.1039/c3cp53413h. (16) M. J. Frisch, G.; Trucks, W.; Schlegel, H. B. .; Scuseria, G. E. .; Robb, M. A. .; Cheeseman, J. R.; Scalmani, G.; Barone, V. .; Mennucci, B. .; Petersson, G. A.; et al. Gaussian 09, Revision E. 01; Gaussian; 2009. https://doi.org/111. (17) Zhang, Y.; Fujisaki, H.; Straub, J. E. Mode-Specific Vibrational Energy Relaxation of Amide I′ and II′ Modes in N-Methylacetamide/Water Clusters: Intra- and Intermolecular Energy Transfer Mechanisms. J. Phys. Chem. A 2009, 113 (13), 3051–3060. https://doi.org/10.1021/jp8109995.

Figure 5.1 a) Overtones do not exist in a pure harmonic system, as quantum selection rules forbid it. B) Introducing an anharmonic component allows for overtone states to occur, although they are not favorable, the probability of their occurence is still low.

Figure 5.2 (top) Fermi Resonance is quantum mechanical mixing due to accidental degeneracy leading to greater splitting of modes. Greater mixing leads to the modes being more equal in strength, and two strong transitions are observed. A combination band is two normal modes excited simultaneously, often due to one or both being dark states due to symmetry. Fermi resonance can occur between a fundamental and an overtone transition (bottom left) or between a fundamental and a combination transition (bottom right).

Figure 5.3 a) Components of a two state excitation leading to b) cross-coupling peaks observed in a theoretical spectrum, with the contribution designated by their appropriate colors.

Figure 5.4 The simulated linear infrared spectrum shows good agreement between experiment (red) and theory (blue).

Figure 5.5 The above simulation uses the cubic terms alone, and does not correctly reproduce the cross peaks seen in experiment. Work on accurate anharmonic constants is pending.

Table 5.1 Simulated resonance coupling contributing to Fermi Resonance.

Rate  Localization   Fabg TFRabg

1366.36 843.19 12.77 -49.14 25.26 1.945 (ring+azide, ring+azide)

The third order resonance parameter TFRαβγ is the largest for three frequency couplings

() = (1366, 843, 2234.82) from the simulation, hence it is the dominant anharmonic interaction. The modes with frequencies 843 cm-1 and 1366 cm-1 are localized to the ring and azide region of the molecule so the azide stretch couples mostly through the ring. All the units of the attributes given in the table are in cm-1.

Chapter 6: Future Directions in 2D IR

6.1 Phenyalanine Derivatives

The probe p-cyanophenyalanine (PheCN), was one of the very first side chain–based extrinsic IR probes to be exploited to study biological processes.1 Dynamics of the folded and unfolded villin headpiece (HP35) were measured with two CN-functionalized phenylalanines inserted into the hydrophobic core of the Chicken Villin peptide to measure solvent interactions compared to those of the chemically induced unfolded peptide.2 In another study PheCN was used as a vibrational probe of the Src homology 3 (SH3) domain, an archetypal models for the study of biological molecular recognition. Using amber codon suppression, PheCN was inserted at four locations and used to generate a site-specific picture of the changes in those sites upon binding.3 The frequency of the nitrile (C≡N) stretching vibration occurs in an isolated portion of the infrared spectrum

(2,100–2,400 cm-1). . The extinction coefficient of the C≡N stretching vibration of alkyl nitriles in water is relatively small (∼50 cm-1 M-1), compared to that of aromatic nitriles (~200 cm-1 M-1).

Finally, PheCN is also useable as a fluorescence probe, and this can be used in conjunction with IR data to confirm location specific interactions.

One significant constraint on the utility of nitrile-based probes such as PheCN is the short lifetime.

13 15 Isotopic substitution of Phe C N gives a CN lifetime of (8.7–10.5 ps) in H2O, giving a ∼2.0–2.5 fold longer lifetime than PheCN alone (4.0–4.6 ps), whereas those for Phe13CN (3.4–5.0 ps) and

PheC15N (0.90–2.2 ps) are roughly the same or even shorter.4–6 The increase in mass is not a significant enough effect to give a meaningful increase in lifetime of the PheCN probe, however the isotopic substitution shifts the CN frequency and results in tuning the CN frequency among the density spectrum. The causes the signal to overlap with a combination water mode band, limiting the concentrations of sample that generate useable signal. A search for an improved

PheCN has produced thiocyanate (SCN) and selenocyanate (SeCN). SeCN has a long lifetime but is difficult to incorporate into proteins due to oxidation and cannot be used with amber codon suppression.7–10 SCN is simple to incorporate into proteins via the chemical modification of solvent-exposed cysteine side chains but lacks the long vibrational life of SeCN (~300 ps).11–16 The asymmetric stretch of the azido group (R–N3) has a larger extinction coefficient than nitrile, making it more desirable when lower sample concentrations are required. The tradeoff is that the

Stark shift of the stretching frequency shows a lesser dependence on the environment than nitrile.

However, the 2D IR signal is still above the detection limit, allowing observation of the spectral diffusion processes occurring due to the surroundings. It should be noted that the aromatic azides tend to have added complexity to their vibrational band. Oftentimes features arise due to Fermi resonances that occur in the 2100 cm-1 and 2140 cm-1 region making observations difficult to interpret. Azide lifetimes are very short (∼1 ps), which limits their utility for studying dynamics.17–

20 Metal-carbonyl complexes, on the other hand, have longer lifetimes (∼20 ps) but are relatively large.21–23 PheCN has a moderate lifetime at (~4 ps).

6.2 Future Probe Design

The C–H stretching vibration has been widely used as a chromophore in the IR due to the high abundance of C–H groups in proteins and other biological molecules. However, for this same reason, this vibrational mode does not provide site-specific information for proteins and has a low molar extinction coefficient. By converting C–H group(s) of interest to C–D groups, a specific mode can be spectrally isolated. However, detecting this signal can require high concentrations as the signal strength is limited by the extinction coefficient, 5 to 30 M-1 cm-1. Pump-probe and dual- frequency 2D IR experiments have shown coupling between C–D and C=O modes in a sample formamide model.24,25

One relatively new amino acid to attach a nitrile probe on is tryptophan, but an analog to tryptophan is indole. Cyanoindole derivatives (4-cyanoindole, 5-cyanoindole, 6-cyanoindole, and

7-cyanoindole) are commercially available and act as analogs for the behavior of cyanotryptophan.26 Linear infrared spectra of these cyanoindole derivatives were collected and

Density Functional Theory (DFT) calculations were performed in order to determine extinction coefficients and transition dipole moment strengths. A distinct trend was observed between the placement of the nitrile and the extinction coefficient and transition dipole moment strengths.

Also, a Fermi resonance was observed in the spectrum of 4-cyanoindole. This Fermi resonance was reflected in the corresponding DFT calculation and it was determined that the Fermi resonance is the result of coupling between the fundamental nitrile frequency and a ring breathing mode of the indole rings.

The ideal infrared probe would excel in all of these criteria: (i) a physically small and chemically inert structure to avoid interference with the attached macromolecule’s activity, (ii) a strong transition dipole strength that gives a signal in an uncluttered region of the infrared spectrum where few other transitions occur, (iii) a group that can be easily incorporated into biomolecules at specific sites with reasonable accuracy. (iv) a probe with a wide variety of applications is preferable to one with a narrow scope, criteria (i)-(iii) being equal. No one molecular group can perform all three of these in arbitrary circumstances. Thus, based upon the circumstances of both the experiment and the target system being studied, we must decide which of these deserves greatest priority as shown in Figure 1.27

This is by no means a comprehensive list of all photoswitching or probe molecules available.

Photoswitches include compounds such as azobenzenes, stilbenes, hemithioindigos, thioxopeptides, selenoxopeptides, diarylethenes and acylhydrazones, and rhodopsin-like

molecules. Other photolinkers, such as esters of 3’,5’-dimethoxybenzoin have been shown to undergo efficient and clean photolysis under near-UV illumination.28 Linkers make the target peptides time-consuming to synthesize and add another layer of complexity to the dynamics of the molecule, as the behavior of the linker must now be considered as well. The target molecule’s potential energy surface may be sterically hindered or otherwise changed, and the peptide may not relax to its native conformation if a different energy minimum is more accessible.

6.3 Transient 2D IR Methods

With innovations in laser technology in the last two decades, the tremendous versatility and resolving power of 2D NMR multiple-pulsed methods is now accessible to the IR field, giving rise to a variety of powerful time-sensitive techniques for understanding molecular kinetics and structure. 2D IR has developed through the initial proof of concept phase, then to simple systems where 2D IR provide limited data that reinforces established methods, and now towards a fundamental method in its own right. Transient 2D IR extends this to the observation of information that is, at present, completely inaccessible to any other method. Advances in nonlinear optics have allowed for greater control over pulse trains, allowing more precise and rapid 2D IR experiments to be performed with greater resolution. The development of methods for analyzing 2D IR signals through use of molecular modeling and quantum mechanical simulations has becomes increasingly intertwined with experimental analysis in both the prediction and interpretation of spectra and the development of new probes and molecular tools for controlling fast reactions. The future development of 2D IR as an investigative method will open new horizons as more techniques become available. It is the goal of this section to give a overview of progress that has been made in this direction and areas that remain to be explored.

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The goal of transient 2D IR is to widen the range of the applications of 2D IR into the quasi- and non-equilibrium regime, to bring to fruition ultrafast snap shots of protein motion and chemical reactions on free energy surfaces on much longer timescales than allows for lifetime limited measurements. In these non-equilibrium experiments, the triggering of a structural change or an excited state by a UV/visible pulse is followed by a pulse sequence for a typical 2D IR experiment in most cases. By measuring a full heterodyned 2D IR signal following a given delay from the triggering pulse, it is possible to observe coupling and spectral diffusion at any point along the reaction pathway, limited only by the precision of our control of the structural change induced by the trigger.

As the number of pulses is increased, the number of factors that must be taken into account when selecting the optical triggering pulse grows. Polarization must be controlled to avoid artifacts from the measuring apparatus, while the possibility of signal drift and instability due to laser fluctuations and the energy of the pulse heating of the sample create additional complications for signal gathering. Polarization and signal drift can be accounted for through splitting the IR beam as for use as a reference, and the repeated measurement of multiple polarizations to allow for proper normalization. Increasing the beam size and making use of pulse stretching through the employment of glass prisms reduces the beam power while preserving coherence, however, whenever the pulse shape is affected, care must be taken not to inadvertently impart undesirable properties such as chirp to the excitation pulse. The work of Hamm et al. addresses this in their description of a sample transient 2D IR setup.29

6.4 Conclusions

Two-dimensional infrared (2D IR) spectroscopy has evolved from its theoretical underpinnings of multidimensional methods to provide a means of investigating detailed molecular structure on

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an ultrafast timescale. The advantage that multidimensional methods have is the ability to view coupling between multiple vibrating modes as these processes happen at the expense of increased cost and complexity of the laser setup. By doing so, relationships can be established such as between two parts of a helix, an active site with solvent, or binding between two peptides. Proteins involved in vision, light harvesting, and electron transfer already incorporate rapid photoinduced electronic processes to trigger conformational changes. It is only within recent years that research has the ability to use these properties to our own advantage, and follow these and other biological processes on their own time scale. The complexity of these issues suggest there is much left to learn, and that the further development of multidimensional spectroscopy is still sorely needed as the field continues to advance.

The development of methods for analyzing 2D IR signals through use of molecular modeling and quantum mechanical simulations has becomes increasingly intertwined with experimental analysis in both the prediction and interpretation of spectra and the development of new techniques. Rational design of phototriggered linker functional groups and would allow greater control of the potential energy surface, allowing the ability to induce both deliberate folding and misfolding to understand both how proteins work and how the folding process can go awry.

Simulated two dimensional infrared simulations will come into their own alongside other predictive models for spectroscopic methods, allowing rapid and accurate interpretation of data. Better and more versatile probe types will allow for greater ease and variety of systems to be studied. My work is based upon the power of two dimensional infrared spectroscopy and its applications to a wide variety of types of molecules.

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(15) Walker, D. M.; Hayes, E. C.; Webb, L. J. Vibrational Stark Effect Spectroscopy Reveals Complementary Electrostatic Fields Created by Protein–Protein Binding at the Interface of Ras and Ral. Phys. Chem. Chem. Phys. 2013, 15 (29), 12241. https://doi.org/10.1039/c3cp51284c. (16) Aaron T. Fafarman; Lauren J. Webb; Jessica I. Chuang, and; Boxer*, S. G. Site-Specific Conversion of Cysteine Thiols into Thiocyanate Creates an IR Probe for Electric Fields in Proteins. 2006. https://doi.org/10.1021/JA0650403. (17) Oh, K. I.; Lee, J. H.; Joo, C.; Han, H.; Cho, M. β-Azidoalanine as an IR Probe: Application to Amyloid Aβ(16-22) Aggregation. J. Phys. Chem. B 2008, 112 (33), 10352–10357. https://doi.org/10.1021/jp801558k. (18) Tucker, M. J.; Gai, X. S.; Fenlon, E. E.; Brewer, S. H.; Hochstrasser, R. M. 2D IR Photon Echo of Azido-Probes for Biomolecular Dynamics. Phys. Chem. Chem. Phys. 2011, 13 (6), 2237–2241. https://doi.org/10.1039/c0cp01625j. (19) Bazewicz, C. G.; Liskov, M. T.; Hines, K. J.; Brewer, S. H. Sensitive, Site-Specific, and Stable Vibrational Probe of Local Protein Environments: 4-Azidomethyl- l -Phenylalanine. J. Phys. Chem. B 2013, 117 (30), 8987–8993. https://doi.org/10.1021/jp4052598. (20) Taskent-Sezgin, H.; Chung, J.; Banerjee, P. S.; Nagarajan, S.; Dyer, R. B.; Carrico, I.; Raleigh, D. P. Azidohomoalanine: A Conformationally Sensitive IR Probe of Protein Folding Protein Structure and Electrostatics. Angew. Chemie - Int. Ed. 2010, 49 (41), 7473–7475. https://doi.org/10.1002/anie.201003325. (21) Moore, J. N.; Hansen, P. A.; Hochstrasser, R. M. Iron Carbonyl Bond Geometries of Carboxymyoglobin and Carboxyhemoglobin in Solution Determined by Picosecond Time- Resolved Infrared-Spectroscopy. Proc. Natl. Acad. Sci. U. S. A. 1988, 85 (14), 5062–5066. https://doi.org/10.1073/pnas.85.14.5062. (22) Baiz, C. R.; Nee, M. J.; McCanne, R.; Kubarych, K. J. Ultrafast Nonequilibrium Fourier- Transform Two-Dimensional Infrared Spectroscopy. Opt. Lett. 2008, 33 (21), 2533–2535. https://doi.org/10.1364/OL.33.002533. (23) Bredenbeck, J.; Helbing, J.; Kolano, C.; Hamm, P. Ultrafast 2D-IR Spectroscopy of Transient Species. ChemPhysChem 2007, 8 (12), 1747–1756. https://doi.org/10.1002/cphc.200700148. (24) Zimmermann, J.; Thielges, M. C.; Yu, W.; Dawson, P. E.; Romesberg, F. E. Carbon- Deuterium Bonds as Site-Specific and Nonperturbative Probes for Time-Resolved Studies of Protein Dynamics and Folding. J. Phys. Chem. Lett. 2011, 2 (5), 412–416. https://doi.org/10.1021/jz200012h. (25) Kumar, K.; Sinks, L. E.; Wang, J. P.; Kim, Y. S.; Hochstrasser, R. M. Coupling between C-D and C=O Motions Using Dual-Frequency 2D IR Photon Echo Spectroscopy. Chem. Phys. Lett. 2006, 432 (1–3), 122–127. https://doi.org/10.1016/j.cplett.2006.10.028. (26) Suydam, I. T.; Boxer, S. G. Vibrational Stark Effects Calibrate the Sensitivity of Vibrational Probes for Electric Fields in Proteins. Biochemistry 2003, 42 (41), 12050–12055. https://doi.org/10.1021/bi0352926. (27) Serrano, A. L.; Waegele, M. M.; Gai, F. Spectroscopic Studies of Protein Folding: Linear and Nonlinear Methods. Protein Sci. 2012, 21 (2), 157–170. https://doi.org/10.1002/pro.2006. (28) Hansen, K. C.; Rock, R. S.; Larsen, R. W.; Chan, S. I. A Method for Photoinitating Protein Folding in a Nondenaturing Environment. J. Am. Chem. Soc. 2000, 122 (46), 11567– 11568. https://doi.org/10.1021/Ja002949r.

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Figure 6.1 A flowchart depicting the situations where various infrared probe types are of use.

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Appendix A: Supplemental Information of the Theoretical Computation of Fermi Resonace

Part I: Nedler-Mead Algorithm

Nedler-Mead, developed in 1965, works by constructing an initial working simplex and terminating the process when the working simplex S becomes sufficiently small in some sense, or when the function values f are close enough in some sense (provided f is continuous). The

Nelder-Mead algorithm typically requires only one or two function evaluations at each step, while many other direct search methods use n or even more function evaluations. To minimize the function values at its vertices, the transformation is determined by computing one or more test points, together with their function values, and by comparison of these function values with those at the vertices. The simplex can change its size or its shape in one step.

The Nelder-Mead method frequently gives significant improvements in the first few iterations and quickly produces quite satisfactory results. Also, the method typically requires only one or two function evaluations per iteration, except in shrink transformations, which are extremely rare in practice. This is very important in applications where each function evaluation is very expensive or time-consuming. For such problems, the method is often faster than other methods, especially those that require at least n function evaluations per iteration.

Part II: Coupling Matrix

We write, as per Sakurai, the matrix as a function of the vibrational modes of the system. In isolation, modes 휀1, 휀2 푎푛푑 휀3 are three normal modes. If there was no anharmonic component to our system, the modes would not mix and there would be no fermi resonance.

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We write the state of our system as follows |휀1휀2휀3⟩ in 휀1 is the vibrational excitation of the fundamental and 휀2 and 휀3 are those of the mixed state. The first order matrix consists of the two possible excited states in the Fermi resonance picture: |100⟩ 푎푛푑 |011⟩

|100⟩ |011⟩ 퐻 = ⟨100| 휔 훽 (1) 1 [ 100 ] ⟨011| 훽 휔011

We wish to translate these into normal modes somehow. We construct an operator that gives

푚 푚 us 휀′푛 modes for each 휀푛 modes such that these modes form an orthonormal basis.

⟨100|푓(푄1, 푄2, 푄3)|011⟩ = ⟨1|푓(푄1)|0⟩⟨0|푓(푄2)|1⟩⟨0|푓(푄3)|1⟩ (2)

푣 푣 + 1 푣 + 1 ⟨1|푓(푄1)|0⟩ = √ ; ⟨0|푓(푄2)|1⟩ = √ ; ⟨0|푓(푄3)|1⟩ = √ (3) 2훾1 2훾2 2훾3

1 훼 1 훽 = = ; 훼 ≡ (4) √8훾1훾2훾3 2√2 √훾1훾2훾3

|100⟩ |011⟩ 훼 ⟨100| 휔100 퐻1 = 2√2 (5) [ 훼 ] ⟨011| 휔011 2√2

Doing the same with the second order excitations, we obtain:

|200⟩ |111⟩ |022⟩ 훼2 훼

⟨200| 휔200 2√2 2 퐻 = 훼2 훼 (5) 2 ⟨111| 휔111 2√2 √2 ⟨022| 훼 훼 휔022 [ 2 √2 ]

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Appendix B- Matlab Code function []= Linear_Fit(filnam,eval,sig1,sig2,gam1,gam2) close all; iter= 200; sigma= [sig1/2.355,sig2/2.355]; gamma= [gam1,gam2]; rho= [0.5,0.5];

norm_lin= load(filnam); norm_lin(:,2)= norm_lin(:,2)./max(norm_lin(:,2)); maxx= max(norm_lin,[],1); minn= min(norm_lin,[],1); stp= (maxx-minn)/(size(norm_lin,1)-1); x= minn(1):stp(1):maxx(1); x= (round(x*1000)/1000).';

% freq= [2160.82 1347.9 824.6]; % alpha= 30.14;

chi= [44.4, 44.1]; options=optimset('Display','iter','MaxFunEvals',eval,'tolfun',1e- 6); Est= fminsearch(@EstLin,[2160 2172 28],options,x,chi,norm_lin(:,2),sigma,gamma); freq(1)= Est(1); freq(2)= Est(2); freq(3)= 0; alpha= Est(3);

mat=[freq(1)-chi(1) alpha/(2*sqrt(2)); alpha/(2*sqrt(2)) freq(2)+freq(3)-chi(2)];

[eigv,eigl]= eig(mat); mu = [eigl(1,1) eigl(2,2)]; sig = [sigma(1)^2 (sigma(1)*sigma(2)*rho(2)); (sigma(1)*sigma(2)*rho(2)) sigma(2)^2]; Brd = mvnrnd(mu,sig,iter); linr_spec=0;

for r=1:iter, linr_spec=linr_spec+(eigv(1,1)).^2*((gamma(1)/pi)./(((x- Brd(r,1)).^2+gamma(1)^2))); linr_spec=linr_spec+(eigv(1,2)).^2*((gamma(2)/pi)./(((x- Brd(r,2)).^2+gamma(2)^2))); end norm_spec= linr_spec./max(linr_spec);

fprintf('Frequency: %f\n',freq); fprintf('Coupling Constant: %f\n',alpha); figure(1),plot(x,norm_spec); hold on; figure(1),plot(norm_lin(:,1),norm_lin(:,2),'r'); end

109 function []= 2D_Anharm(eval,sig1,sig2,gam1,gam2) close all; sig_const= 2.355; sigma= [sig1 sig2 30 20]; gamma= [gam1 gam2]; rho= [0.5 0.5 1 1];

sigma= sigma/sig_const; freq = [2162.165 2171.389 0.0]; chi = [40 50 -10]; twodr= [4138 4209 4095]; %Second harmonic; must be given. alpha= 37.994; options=optimset('Display','iter','MaxFunEvals',eval,'tolfun',1e- 6);

mat=[freq(1)-chi(1) alpha/(2*sqrt(2)); alpha/(2*sqrt(2)) freq(2)+freq(3)-chi(2)];

[eigv1,eigl1]= eig(mat); Est= fminsearch(@Est2D,chi,options,freq,alpha,twodr); chi= Est; mat= [2*freq(1)-chi(1) alpha^2/(2*sqrt(2)) alpha/2; alpha^2/(2*sqrt(2)) 2*(freq(2)+freq(3))-chi(2) alpha/sqrt(2); alpha/2 alpha/sqrt(2) freq(1)+freq(2)+freq(3)-chi(3)];

[eigv2,eigl2]= eig(mat);

FitCurve= comb_spec2(eigv,eigv2,sigma,gamma,rho); figure, contour(wt,wtau,FitCurve,60); line ([-4000 4000],[-4000 4000]) colorbar;axis equal; axis([2075 2160 2075 2160]) xlabel('\omega_{t} (cm^{-1})') ylabel('\omega_{\tau} (cm^{-1})') end

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%% PRogram to run combine rephasing and non-rephasing %rho=0.5;%0.89; %rho2=0.5;%0.89 %rho3=1; %rho4=1; %was 0.6

%gamma1=20/2;%20/2;18/2;9/2; %gamma2=10/2;%10/2;12/2;

%sigma1=1/2.355;%6/2.355;10/2.355; %sigma2=15/2.355;%10/2.355;20/2.355; %sigma3=30/2.355; %sigma4=20/2.355; function Fit_Curve= comb_spec2(eigval,eigvec,sigma,gamma,rho) mu = [eigval(1,1) eigval(1,1)]; SIGMA = [sigma(1)^2 (sigma(1)*sigma(1)*rho(2)); (sigma(1)*sigma(1)*rho(2)) sigma(1)^2]; R(1) = mvnrnd(mu,SIGMA,9050); %changed 150 to 250 mu = [eigval(2,2) eigval(2,2)]; SIGMA = [sigma(2)^2 (sigma(2)*sigma(2)*rho(2)); (sigma(2)*sigma(2)*rho(2)) sigma(2)^2]; R(2) = mvnrnd(mu,SIGMA,9050); %changed 150 to 250 mu = [eigval(1,1) eigval(2,2)]; SIGMA = [sigma(3)^2 (sigma(3)*sigma(4)*rho(3)); (sigma(3)*sigma(4)*rho(3)) sigma(4)^2]; %was [sigma1^2 (sigma1*sigma2*rho3); (sigma1*sigma2*rho3) sigma2^2] A = mvnrnd(mu,SIGMA,9050); %changed rho to rho3, 150 to 250 mu = [eigval(1,1) eigval(1,1)-eigval(1,1)]; SIGMA = [sigma(1)^2 (sigma(1)*sigma(1)*rho(2)); (sigma(1)*sigma(1)*rho(2)) sigma(1)^2]; C(1) = mvnrnd(mu,SIGMA,9050); %changed 150 to 250 mu = [eigval(1,1) eigval(2,2)-eigval(1,1)]; SIGMA = [sigma3^2 (sigma(3)*sigma(3)*rho(4)); (sigma(3)*sigma(3)*rho(4)) sigma(3)^2]; %was dependent on sigma1 C(2) = mvnrnd(mu,SIGMA,9050); %changed rho to rho4, 150 to 250 mu = [eigval(1,1) eigval(3,3)-eigval(1,1)]; SIGMA = [sigma(1)^2 (sigma(1)*sigma(1)*rho(1)); (sigma(1)*sigma(1)*rho(1)) sigma1^2]; C(3) = mvnrnd(mu,SIGMA,9050); %changed 150 to 250 mu = [eigval(2,2) eigval(1,1)-eigval(2,2)]; SIGMA = [sigma(2)^2 (sigma(2)*sigma(2)*rho(1)); (sigma(2)*sigma(2)*rho(1)) sigma(2)^2]; D(1) = mvnrnd(mu,SIGMA,9050); %changed 150 to 250 mu = [eigval(2,2) eigval(2,2)-Eigval1(2,2)];

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SIGMA = [sigma(2)^2 (sigma(2)*sigma(2)*rho(4)); (sigma(2)*sigma(2)*rho(4)) sigma(2)^2]; D(2) = mvnrnd(mu,SIGMA,9050); %changed rho2 to rho4, 150 to 250 mu = [eigval(2,2) eigval(3,3)-eigval(2,2)]; SIGMA = [sigma(2)^2 (sigma(2)*sigma(2)*rho(1)); (sigma(2)*sigma(2)*rho(1)) sigma(2)^2]; D(3) = mvnrnd(mu,SIGMA,9050); %changed 150 to 250 plotfreq_dom_RE5; plotfreq_dom_NONRE4; Fit_Curve=real(28000*total+28000*total2); end

%% Formula for the Rephasing part of the spectrum in the Frequency Domain %Tucker function total= NonRE4(gamma,C,A,R,D,eigvec) T=0; T1=2; total=0; [wt,wtau]=meshgrid(2050:1:2350,2050:1:2350);

for r=1:15, S1=(-(eigvec(1,1)).^4*(1./((-i*(R(1)*(r)-wtau)- gamma(1))))*(1./(i*(R(1)*(r)-wt)+gamma(1)))); S2=(-(eigvec(1,2)).^4*(1./((-i*(R(2)*(r)-wtau)- gamma(2))))*(1./(i*(R(2)*(r)-wt)+gamma(2)))); %S3=(-(Eigvec1(1,3)).^4*(1./((-i*(R3(r)-wtau)-gamma3)))*(1./(i*(R3(r)- wt)+gamma3))); S4=(-(eigvec(1,1))^2*(eigvec(1,2))^2*(1./((-i*(A(r,2)-wtau)- gamma(1)))*(1./(i*(A(r,1)-wt)+gamma(1))))); S5=(-(eigvec(1,1))^2*(eigvec(1,2))^2*(1./((-i*(A(r,1)-wtau)- gamma(2)))*(1./(i*(A(r,2)-wt)+gamma(2))))); S11=-2*(- (eigvec(1,2)^2*(eigvec(1,2)*eigvec(1,2)*sqrt(2)+eigvec(2,2)*eigvec(3,2) )^2))*((1./(-i*(D(2,r,1)-wtau)-gamma(2)))*(1./(i*((D(2,r,2))- wt)+gamma(2)))); %S11=-2*(- (Eigvec1(1,2)^2*(Eigvec1(1,2)*Eigvec2(1,2)*sqrt(2)+Eigvec1(2,2)*Eigvec2 (3,2)+Eigvec1(3,2)*Eigvec2(5,2))^2))*((1./(-i*(D2(r,1)-wtau)- gamma2))*(1./(i*((D2(r,2))-wt)+gamma2))); S12=-2*(- (eigvec(1,1)^2*(eigvec(1,1)*eigvec(1,2)*sqrt(2)+eigvec(2,1)*eigvec(3,2) )^2))*((1./(-i*(C(2,r,1)-wtau)-gamma(1)))*(1./(i*((C(2,r,2))- wt)+gamma(1)))); S25=-2*(- (eigvec(1,1).^2*(eigvec(1,1)*eigvec(1,1)*sqrt(2)+eigvec(2,1)*eigvec(3,1 )).^2))*((1./(-i*(C(1,r,1)-wtau)-gamma(1)))*(1./(i*((C(1,r,2))- wt)+gamma(1)))); %S25=-2*(- (Eigvec1(1,1).^2*(Eigvec1(1,1)*Eigvec2(1,1)*sqrt(2)+Eigvec1(2,1)*Eigvec 2(3,1)+Eigvec1(3,1)*Eigvec2(5,1)).^2))*((1./(-i*(C1(r,1)-wtau)- gamma1))*(1./(i*((C1(r,2))-wt)+gamma1)));

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S26=-2*(- (eigvec(1,2).^2*(Eigvec1(1,2)*Eigvec2(1,3)*sqrt(2)+eigvec1(2,2)*Eigvec2 (3,3)).^2))*((1./(-i*(D3(r,1)-wtau)-gamma2))*(1./(i*((D(3,r,2))- wt)+gamma(2)))); %S26=-2*(- (Eigvec1(1,2).^2*(Eigvec1(1,2)*Eigvec2(1,3)*sqrt(2)+Eigvec1(2,2)*Eigvec 2(3,3)+Eigvec1(3,2)*Eigvec2(5,3)).^2))*((1./(-i*(D3(r,1)-wtau)- gamma2))*(1./(i*((D3(r,2))-wt)+gamma2))); %S27=-2*(- (Eigvec1(1,3).^2*(Eigvec1(1,3)*Eigvec2(1,3)*sqrt(2)+Eigvec1(2,3)* %Eigvec2(3,3)+Eigvec1(3,3)*Eigvec2(5,3)).^2))*((1./(-i*(E3(r,1)-wtau)- gamma3))*(1./(i*((E3(r,2))-wt)+gamma3)));

Curve=(exp(-T/T1))*(S1+S2+S4+S5+S11+S12+S25+S26); clear S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S21 S22 S23 S24 S25 S26 S27; total= total+Curve; end end

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Appendix C- Fortran Code module FDM_Setup implicit none

! A data type that can perform the Finite Difference Method on a Gaussian file. ! We need the frequencies, the amount moved by each dipole, and the scaling factor type FDMObj double precision :: freq1,freq2,scal integer :: mode1,mode2 end type FDMObj

contains

subroutine FDMObj_Init(this) type(FDMObj) this

this%freq1= 0 this%freq2= 0 this%mode1= 0 this%mode2= 0 this%scal= -1 return end subroutine FDMObj_Init

! Sets frequencies subroutine FDMObj_SetFreq(this,freq1,freq2,scal) character(25), parameter :: prc= "FDMObj_SetFreq" character(100), parameter :: badval= & "Values cannot be negative"

type(FDMObj) this double precision :: scal,freq1,freq2

integer :: badfrq

badfrq= 1

if (freq1 <= 0) then badfrq= freq1 elseif (freq2 <= 0) then badfrq= freq2 elseif (scal <= 0) then badfrq= scal endif

if (badfrq == 1) then this%freq1= 4.105/sqrt(freq1) this%freq2= 4.105/sqrt(freq2) this%scal = scal else call ErrorWarn(prc,badval,badfrq) endif

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return end subroutine FDMObj_SetFreq

subroutine FDMObj_SetModes(this,md1,md2) character(25), parameter :: prc= "FDMObj_SetModes" character(100), parameter :: badnum= & "Mode number should be positive"

type(FDMObj) this integer :: md1,md2

integer :: badint

badint= 1 if (md1 < 1) then badint= md1 elseif (md2 < 1) then badint= md2 else this%mode1= md1 this%mode2= md2 endif

if (badint /= 1) call ErrorWarn(prc,badnum,badint) return end subroutine FDMObj_SetModes

subroutine FDMObj_GetShft(this,grp,shft1,shft2) character(25), parameter :: prc= "FDMObj_GetShft" character(100), parameter :: badmod= & "Mode not set" character(100), parameter :: badscal= & "Scale is not set"

type(FDMObj) this type(GGrp) grp double precision :: shft1,shft2

type(GOut) :: go type(NMode) dip type(GCoord), allocatable :: disp(:) character(100) :: err double precision :: vec(3) double precision :: totl integer :: i

! Error check time err= 'none'; shft1=0; shft2=0 if (this%mode1 == 0) then err= badmod elseif (this%scal == -1) then err= badscal endif

if (err .eq. 'none') then

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go= GGrp_GetOut(grp,'nmode')

! Mass weighted coordinates are normed dip= GOut_ToDipole(go,this%mode1) call NMode_GetAtms(dip,disp) call FDMObj_MakeShft(disp,this%scal)

totl= 0 do i=1,size(disp) vec= GCoord_GetCoor(disp(i)) totl=totl+SUM(vec ** 2) end do shft1= sqrt(totl)

dip= GOut_ToDipole(go,this%mode2) call NMode_GetAtms(dip,disp) call FDMObj_MakeShft(disp,this%scal)

totl= 0 do i=1,size(disp) vec= GCoord_GetCoor(disp(i)) totl=totl+SUM(vec ** 2) end do shft2= sqrt(totl) else call ErrorWarn(prc,err) endif return end subroutine FDMObj_GetShft

subroutine FDMObj_GetCoord(this,grp,newatms) character(25), parameter :: prc= "FDMObj_GetCoord" character(100), parameter :: badmod= & "Mode not set" character(100), parameter :: badscal= & "Scale is not set"

type(FDMObj) this type(GGrp) grp type(GCoord), allocatable :: newatms(:,:)

type(GOut) :: go(2) type(NMode) dip type(GCoord), allocatable :: disp1(:),disp2(:),old(:) character(100) :: err double precision :: vec1(3),vec2(3) integer :: i

err= 'none' if (allocated(newatms)) deallocate(newatms) if (this%mode1 == 0) then err= badmod elseif (this%scal == -1) then err=badscal endif

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if (err .eq. 'none') then go(1)= GGrp_GetOut(grp,'nmode') go(2)= GGrp_GetOut(grp,'coord')

dip= GOut_ToDipole(go(1),this%mode1) call NMode_GetAtms(dip,disp1) call FDMObj_MakeShft(disp1,this%scal)

dip= GOut_ToDipole(go(1),this%mode2) call NMode_GetAtms(dip,disp2) call FDMObj_MakeShft(disp2,this%scal)

call GOut_ToAtoms(go(2),old) allocate(newatms(size(disp1),4)) do i=1,size(disp1) vec1= GCoord_GetCoor(disp1(i)) vec2= GCoord_GetCoor(disp2(i)) newatms(i,1:4)= old(i)

call GCoord_Move(newatms(i,1),vec1) call GCoord_Move(newatms(i,2),vec1) call GCoord_Move(newatms(i,3),-1.0*vec1) call GCoord_Move(newatms(i,4),-1.0*vec1)

call GCoord_Move(newatms(i,1),vec2) call GCoord_Move(newatms(i,2),-1.0*vec2) call GCoord_Move(newatms(i,3),vec2) call GCoord_Move(newatms(i,4),-1.0*vec2) end do else call ErrorWarn(prc,err) endif return end subroutine FDMObj_GetCoord

! Norms the dipole and shifts it by a scaling factor. ! Puts everything back where it came from. subroutine FDMObj_MakeShft(atms,scal) type(GCoord) :: atms(:) double precision :: scal

type(GCoord), allocatable :: mv1(:),mv2(:) character(5) :: typ double precision :: mas,norm,vec(3) integer :: i

norm= Atom_MassSum(atms) do i=1,size(atms) mas= GCoord_GetMass(atms(i)) typ= GCoord_GetElem(atms(i)) vec= GCoord_GetCoor(atms(i)) vec= vec/norm*scal call GCoord_Init(atms(i)) call GCoord_Set(atms(i),typ,mas,vec(1),vec(2),vec(3))

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end do return end subroutine FDMObj_MakeShft

! Calculate the coupling constant for the two modes ! Use the frequency, the scaling factor, and the shifted amount function FDMObj_GetCoupl(this,grp,nrg) result(enary) character(25), parameter :: prc= "FDMObj_GetCoupl" character(100), parameter :: notset= "Frequencies not set"

type(FDMObj) this type(GGrp) grp double precision :: nrg double precision :: enary(4)

double precision :: shft1,shft2

enary= 0 if (this%scal > 0) then call FDMObj_GetShft(this,grp,shft1,shft2) enary(1)= nrg / (4 * this%scal ** 2) enary(2)= enary(1) * this%freq1 * this%freq2

enary(3)= nrg / (4 * shft1 * shft2) enary(4)= enary(3) * this%freq1 * this%freq2 else call ErrorWarn(prc,notset) endif return end function FDMObj_GetCoupl

! Mass weighted coordinate sum. function Atom_MassSum(atms) result(summ) type(GCoord) atms(:) double precision :: summ

double precision :: mas,vec(3) integer :: i

summ= 0 do i=1,size(atms) mas= GCoord_GetMass(atms(i)) vec= GCoord_GetCoor(atms(i)) vec= vec ** 2 * mas summ= summ+SUM(vec) end do summ= sqrt(summ) return end function Atom_MassSum end module FDM_Setup module FDMake use FDM_Setup implicit none

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contains

subroutine FDMake_WriteFile(npath,chk,ga) character(*) :: npath,chk type(GCoord) :: ga(:)

type(Perc) gp type(GInstr) gi type(GaussFile) gf character(50) :: charr(1) integer :: onum

onum= 200 open(file=npath, action='write',unit=onum)

call Gauss_Init(gf) call Perc_Init(gp) call GInstr_Init(gi)

call Perc_SetChk(gp,chk) call Perc_SetNProc(gp,3)

charr(1)= "" call GInstr_SetJob(gi,"","B3LYP/6-31G(d)") call GInstr_SetOther(gi,charr)

call Gauss_SetPercent(gf,gp) call Gauss_SetInstruct(gf,gi) call Gauss_SetTitle(gf,"Run FDM Calcs") call Gauss_SetCoor(gf,ga) call Gaussian_Print(gf,onum) close(onum) return end subroutine FDMake_WriteFile

subroutine FDMake_GetData(fgrp,gfile,mds) character(25), parameter :: prc= "FDMake_GetData" character(100), parameter :: badfil= & "File is not open!" character(100), parameter :: badsiz= & "There must be two modes"

type(GGrp) fgrp integer :: gfile integer :: mds(:)

type(GOut) :: go(3) logical :: isopn

inquire (unit=gfile, opened= isopn)

if (.NOT.(isopn)) then call ErrorWarn(prc,badfil) elseif (size(mds) /= 2) then call ErrorWarn(prc,badsiz)

119 else call GOut_Init(go(1)) call GOut_SetTask(go(1),'coord') call GOut_Init(go(2)) call GOut_SetTask(go(2),'nmode',mds) call GOut_Init(go(3)) call GOut_SetTask(go(3),'dipder',mds)

call GGrp_Dflt(fgrp) call GGrp_SetTasks(fgrp,go) call Basic_Setup(fgrp,gfile) call GGrp_SetValues(fgrp,gfile) endif return end subroutine FDMake_GetData subroutine FDMake_Main(fs,fname,modes,coord) type(FDMObj) fs type(GCoord), allocatable :: coord(:,:) character(*) :: fname integer :: modes(:) type(GGrp) fgrp integer :: fnum fnum= 255 open(unit=fnum,file=fname) call FDMake_GetData(fgrp,fnum,modes) call FDMObj_SetModes(fs,modes(1),modes(2)) call FDMObj_GetCoord(fs,fgrp,coord) close(fnum) return end subroutine FDMake_Main subroutine FDMake_Run(fpath,rlist,flist) character(*) :: rlist(:),flist(:) character(*) :: fpath type(FileP) fp type(Shell) rn character(100) :: rpath,mname,cmd,runfil,fdmfil integer :: i mname= trim(fpath) // '/MasterList.txt' open(unit=122,file=mname) write(122,'(A)') '#!/bin/bash' call FileP_Init(fp) call FileP_SetExt(fp,'sh') do i=1,4 runfil= trim(rlist(i)) // '.sh' fdmfil= trim(flist(i)) // '.com' write (122,'(2A)') 'qsub ',runfil call FileP_Set(fp,fpath,rlist(i)) rpath= FileP_Get(fp)

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call Shell_Init(rn) call Shell_Set(rn,rpath,3) call Shell_SetJob(rn,fdmfil) call Shell_MakeRun(rn) end do close(122) cmd= 'chmod 744 ' // mname call system(cmd) return end subroutine FDMake_Run subroutine FDMake_MakeFile(fpath,flist,crds) type(GCoord) crds(:,:) character(*) :: fpath,flist(:) type(FileP) fp character(100) :: chk,fname integer :: i call FileP_Init(fp) call FileP_SetExt(fp,'com') do i=1,4 call FileP_Set(fp,fpath,flist(i)) chk= flist(i) fname= FileP_Get(fp) call FDMake_WriteFile(fname,chk,crds(:,i)) end do return end subroutine FDMake_MakeFile subroutine FDM_Make(fs,fpath,fname,modes) type(FDMObj) fs character(*) :: fpath,fname integer :: modes(2) type(GCoord), allocatable :: crds(:,:) character(50) :: flist(4),rlist(4) character(100) :: cmd logical :: ext flist= (/'fdm1','fdm2','fdm3','fdm4'/) rlist= (/'run1','run2','run3','run4'/) call FDMake_Main(fs,fname,modes,crds) fname= trim(fpath) // '/m.txt' inquire(file= fname, exist= ext) if (.NOT.(ext)) then cmd= 'mkdir ' // fpath call system(cmd) open(111,file= fname) close(111) endif

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call FDMake_Run(fpath,rlist,flist) call FDMake_MakeFile(fpath,flist,crds) return end subroutine FDM_Make end module FDMake module TDC implicit none contains

! In case you are wondering, it is: ! TDC= [|U||V|-3*(U*n)(V*n)]/r^3 ! n is the norm of r ! r is the distance between the two dipoles (the center of mass of the relevant atoms) ! U,V are the transition dipole vectors

function DiplStrength(freq,intensiti) result(debbie) double precision :: freq,intensiti double precision :: debbie

debbie= sqrt(intensiti*100/4225.47) debbie= debbie*4.1058/sqrt(freq) return end function DiplStrength

function GetDiElec(solv) result(dilec) character(*) :: solv double precision :: dilec

select case(solv) case('gas') dilec= 1.0 case('water') dilec= 80.1 case('thf') dilec= 7.58 case default dilec= 0 end select return end function GetDiElec

function TLC_Coupl(dist,dip1,dip2,intens,dilec) result(coupl) double precision :: dist(:),dip1(:),dip2(:),intens(:) double precision :: dilec double precision :: coupl

double precision :: ndist(3) double precision :: const

ndist= dist call NormalVec(dip1) call NormalVec(dip2) call NormalVec(ndist)

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coupl= intens(1)*intens(2)*DotProd(dip1,dip2)- & 3*intens(1)*DotProd(dip1,dist)*intens(2)*DotProd(dip2,dist) & /Norm(dist)**2 coupl= 1/(4*dilec*pii*vacuum)*coupl/Norm(dist) ** 3 const= wavenum*(debbie ** 2)/(ang ** 3) coupl= const*coupl return end function TLC_Coupl

function TLC_Main(grp,irmod,solv) result(coupl) character(25), parameter :: prc= "TLC_Main" character(100), parameter :: badslv= & "Solvent not recognized"

type(IRMode) irmod(2) type(GGrp) grp character(*) :: solv double precision :: coupl

type(GOut) gut character(100) :: str(1),arry(10) double precision :: rad(3),dip1(3),dip2(3),dby(2) double precision :: dilec,freq,intens integer, allocatable :: atm1(:),atm2(:) integer :: mds(2) integer :: siz

siz= IRMode_GetSiz(irmod(1)) allocate(atm1(siz)) atm1= IRMode_GetAtoms(irmod(1)) siz= IRMode_GetSiz(irmod(2)) allocate(atm2(siz)) atm2= IRMode_GetAtoms(irmod(2)) mds(1)= IRMode_GetMode(irmod(1)) mds(2)= IRMode_GetMode(irmod(2)) freq= IRMode_GetFreq(irmod(1)) intens= IRMode_GetIntense(irmod(1)) dby(1)= DiplStrength(freq,intens) freq= IRMode_GetFreq(irmod(2)) intens= IRMode_GetIntense(irmod(2)) dby(2)= DiplStrength(freq,intens)

rad= Cake_Vector(grp,atm1,atm2) gut= GGrp_GetOut(grp,'dipder') str= GOut_GetValue(gut,mds(1)) call Chop(str(1),arry) dip1= MakeVec(arry) str= GOut_GetValue(gut,mds(2)) call Chop(str(1),arry) dip2= MakeVec(arry)

coupl= 0 dilec= GetDiElec(solv) if (dilec == 0) then

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call ErrorWarn(prc,badslv,solv) else coupl= TLC_Coupl(rad,dip1,dip2,dby,dilec) endif return end function TLC_Main end module TLC