QUANTITATIVE PREDICTION OF SPECTRAL PROPERTIES OF

POLARITY SENSITIVE DYES USING A MD/QM APPROACH

by

Swapnil Baral

A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics

Summer 2020

© 2020 Swapnil Baral All Rights Reserved QUANTITATIVE PREDICTION OF SPECTRAL PROPERTIES OF

POLARITY SENSITIVE DYES USING A MD/QM APPROACH

by

Swapnil Baral

Approved: Edmund Nowak, Ph.D. Chair of the Department of Physics and Astronomy

Approved: John Pelesko, Ph.D. Dean of the College of Arts and Sciences

Approved: Douglas J. Doren, Ph.D. Interim Vice Provost for Graduate and Professional Education and Dean of the Graduate College I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dis- sertation for the degree of Doctor of Philosophy.

Signed: Edward Lyman, Ph.D. Professor in charge of dissertation

I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dis- sertation for the degree of Doctor of Philosophy.

Signed: Björn Baumeier, Ph.D. Professor in charge of dissertation

I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dis- sertation for the degree of Doctor of Philosophy.

Signed: Sandeep Patel, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dis- sertation for the degree of Doctor of Philosophy.

Signed: Lars Gundlach, Ph.D. Member of dissertation committee

I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dis- sertation for the degree of Doctor of Philosophy.

Signed: James MacDonald, Ph.D. Member of dissertation committee ACKNOWLEDGEMENTS

I want to acknowledge the advice and support of many people and organiza- tions. This dissertation would not have been possible without my direct supervisor Dr. Edward R. Lyman, whose guidance and limitless patience have helped me de- velop professionally and become a critically thinking scientist. A huge thank you for mentoring and guiding me to complete this work. You have always encouraged me to focus on research and supported me to present my work at several conferences. I also want to thank all the group members (undergraduates, graduate students, and postdoctoral researchers) of our group (Lyman Group) from 2014 to 2020 for the engaging physics discussion on countless occasions. I want to thank Dr. Björn Baumeier (Eindhoven University of Technology (TU/e)), who taught me a lot about quantum mechanical calculations, especially the GW -BSE and its implementation in VOTCA-XTP. This work would not have been possible without the collaboration of Baumeier Research Group. I also thank you for generously covering a significant portion of my visit to TU/e. I want to express my gratitude towards Dr. Sandeep Patel for generously pro- viding about year-long financial support to me. You have also helped me understand the importance of Laurdan research and taught me how to get started with quantum calculations in Gaussian, especially during my early days in graduate school. I want to thank Dr. Lars Gundlach for raising profound discussions about my research and motivating me every time we met. Your group (Gundlach Group),

v thorough our collaboration, has played a crucial role in testing the results of my calculations. I want to thank Dr. James MacDonald for supporting me throughout my stay at UD and agreeing to serve on my committee. You have been extremely helpful even before I joined UD during the application process. I want to thank Dr. Jeffrey R. Comer (Kansas State University) for helping me compute the Adaptive Biasing Force calculation and Dr. Alex MacKerell and Dr. Fang-Yu Lin (University of Maryland, Baltimore) for teaching me protocols of CHARMM Force Field parametrization. This work was supported by a National Institutes of Health grant NIH R01GM116961. Computations were performed on the Stampede supercomputer at the Texas Advanced Computing Center of the University of Texas at Austin (sup- ported through the XSEDE program of the NSF (TG-MCB170146)) which is sup- ported by National Science Foundation grant number ACI-1548562. I want to express my gratitude towards all the faculty, staff, and members in the Department of Physics and Astronomy (DPA) at the University of Delaware for continuously supporting me in times of need. Thanks to Physics and Astronomy Graduate Student Association for allowing me to serve as a vice-president (2015-2016), Graduate Student Government for al- lowing me to serve as a senator representing DPA (2016-2017), and Nepalese Student Association for allowing me to serve as the president (2017-2018). I want to thank all my teachers and mentors, from my first schools, River Valley and Pushpa Sadan Boarding High School, to the current school, the University of Delaware. I also want to remember the physics department at the National School of Sciences (Lainchaur, Kathmandu), the University of Texas at Arlington, and the Georgia Institute of Technology.

vi I want to express my love to my mother, Dr. Sushma Banskota Baral, who was the first person to support me when I decided to study Physics in the United States. I want to thank my father, Dr. Krishnahari Baral, the hero of my life, for helping me pursue a career in Physics. I want to dedicate this work to both of you. I want to remember my siblings (Smriti Baral and Swikriti Baral), relatives, and friends who always stand by my side and continuously support me. I express my love to my wife, Aastha Dahal, for supporting me on my journey. Without your colorful smile, everything I achieve becomes colorless.

vii TABLE OF CONTENTS

LIST OF TABLES ...... xiii LIST OF FIGURES ...... xv ABSTRACT ...... xxii

Chapter

1 BILAYERS AND CELL MEMBRANES: COMPOSITION, PHASES, AND PACKING ...... 1

1.1 Introduction ...... 1 1.2 Basic lipid chemistry ...... 3

1.2.1 Membrane heterogeneity and lipid phases ...... 6

2 INTRODUCTION TO FLUORESCENCE ...... 9

2.1 Introduction ...... 9 2.2 Jablonski Diagram ...... 9 2.3 Stokes Shift ...... 12 2.4 Quantum Yield and Lifetime ...... 12 2.5 Solvent effects and Polarity Sensitive Dyes ...... 13

2.5.1 Solvent relaxation ...... 13

3 SPECTRAL PROPERTIES OF PRODAN AND LAURDAN .. 16

3.1 Introduction ...... 16 3.2 Prodan Structure ...... 16

viii 3.3 Nature of Prodan Emitting State ...... 17 3.4 Prodan Dipole Moment ...... 20 3.5 Prodan Absorption and Emission Spectra in Bulk Solvents ..... 26

4 THEORETICAL BACKGROUND ...... 28

4.1 Introduction ...... 28 4.2 Many-electron quantum mechanics ...... 29 4.3 Wave-function ...... 30 4.4 Born-Oppenheimer approximation ...... 32 4.5 Classical molecular dynamics simulation ...... 34

4.5.1 Force-Fields and potential energy function ...... 35

4.5.1.1 Bonded terms ...... 36 4.5.1.2 Nonbonded terms ...... 36

4.5.1.2.1 Lennard-Jones ...... 36 4.5.1.2.2 Electrostatics ...... 37

4.6 Solving electronic Schrödinger equation ...... 38 4.7 Density Functional Theory ...... 42

4.7.1 Exchange-Correlation Functional ...... 45 4.7.2 Basis sets ...... 46

4.8 Electronic excitations ...... 47

4.8.1 General information about excited state calculations .... 48

4.9 Electronically excited states calculation using Many-body Green’s Function: GW -BSE ...... 51

4.9.1 One particle excitation ...... 51 4.9.2 Two-particle excitations ...... 59

4.10 GW -BSE/MM ...... 65

ix 5 CHARMM GENERAL FORCE FIELD PARTIAL CHARGE PARAMETRIZATION OF PRODAN/LAURDAN ...... 67

5.1 Introduction ...... 67 5.2 Method ...... 67

5.2.1 Results ...... 72

6 MOLECULAR DYNAMICS SIMULATION OF LAURDAN IN LIPID MEMBRANES ...... 77

6.1 Introduction ...... 77 6.2 MD Simulation ...... 78 6.3 Simulation Results ...... 79 6.4 Discussion ...... 84

7 MD/QM CALCULATION OF PRODAN IN BULK SOLVENTS 86

7.1 Background Information ...... 87 7.2 Theory ...... 89

7.2.1 Quasi-Particle Excitation ...... 89 7.2.2 Coupled Electron-Hole Excitations ...... 90 7.2.3 Electrostatic embedding ...... 92 7.2.4 Transition probability and oscillator strength ...... 92

7.3 MD/GW -BSE: Excited state calculation and Iterative solvent relaxation ...... 93

7.3.1 Preparing MD trajectory for QM calculation ...... 93 7.3.2 MD/GW -BSE iterative protocol ...... 94

7.4 Prodan MD/GW -BSE calculation in bulk solvents ...... 95

7.4.1 System setup and sampling heterogeneity using classical MD simulation ...... 95

x 7.4.2 Determining the size of MM region for MD/GW -BSE calculation ...... 96 7.4.3 GW -BSE: absorption energy calculation and hydrogen bonding ...... 98 7.4.4 Simulation Results: Prodan emission and dipolar evolution in six different solvents ...... 99 7.4.5 Experimental Measurement: Ultrafast Spectroscopy ..... 106 7.4.6 Solvent relaxation dynamics: Comparison of ultrafast spectroscopy to MM/GW-BSE approach...... 107

7.5 Discussion ...... 111 7.6 Conclusion ...... 113

8 MD/QM CALCULATION OF LAURDAN IN SIMPLE LIPID BILAYERS ...... 114

8.1 Laurdan experimental emission spectrum ...... 114 8.2 Laurdan MD/GW -BSE calculation in lipid bilayers ...... 115

8.2.1 System Setup and sampling heterogeneity using classical MD simulation ...... 115 8.2.2 Determining size of MM region for MD/GW -BSE calculation 115 8.2.3 Simulation Results: Laurdan emission and dipolar evolution in lipid bilayers ...... 116

9 PARTITIONING OF PRODAN AT AN OIL-WATER INTERFACE ...... 122

9.1 Free energy calculation ...... 122

9.1.1 Thermodynamic Integration to compute free energy differences ...... 124

9.1.1.1 Adaptive Biasing Force ...... 126

9.2 ABF calculation of Prodan in hexadecane-water slab ...... 128

9.2.1 Method ...... 129

xi 9.2.2 Results ...... 132

10 CONCLUSION AND OUTLOOK ...... 137

BIBLIOGRAPHY ...... 141

Appendix

A SAMPLE MDP FILE USED IN MD SIMULATION ...... 163 B PERMISSIONS ...... 166

xii LIST OF TABLES

1.1 Names of lipids acronyms and melting temperature for 5 different lipid types...... 4

3.1 Spectral Properties of Prodan Solutions. Reprinted (adapted) with permission from Weber, G., Farris, F. J. (1979). Synthesis and spectral properties of a hydrophobic fluorescent probe: 6-propionyl-2-(dimethylamino) naphthalene. Biochemistry, 18(14), 3075–3078. https://doi.org/10.1021/bi00581a025. Copyright 1979 American Chemical Society [1] ...... 22

3.2 Prodan dipole moments reported in the literature. [2, 3, 4, 5, 6, 7, 8, 9] µG and µE are ground state and excited state dipole moment of Prodan measured in Debye (D)...... 26

5.1 The components of dipole moment of the model compound calculated by using set of initial partial charges and final charges compared to QM target value calculated with HF/6-31G(d) level of theory. The unit of dipole moment is in Debye (D) ...... 69

kcal 5.2 The QM and MM interaction energies ( mol ) between various atoms in model compound and a water molecule. The MM results are presented for the initial (IC) and final (FC) set of charges. The difference between MM and QM energies are also presented for both cases. The atom names correspond to Fig. 5.1 and 5.2 ... 71

5.3 The QM and MM interaction distance() between various atoms in model compound and a water molecule. The MM results are presented for the initial (IC) and final (FC) set of charges. The difference between MM and QM distances are also presented for both cases. The atom names correspond to Fig. 5.1 and 5.2 ... 73

xiii 5.4 The atom name, atom type, and the partial charges (both initial and final or optimized) in the unit of elementary charge (e). Allthe hydrogen connected to C1, C2, and C14 have the default charge of +0.09e. The atom names in the able correspond to the atom names in Fig. 5.1 and 5.2 ...... 75

6.1 The composition of simulation systems...... 78

7.1 Structure of Prodan in S0 and S1 states. The names of atom correspond to Fig. 5.1 ...... 95

7.2 The emission energies of Prodan in water for the first four excited states and the number of water molecules (MM region) for different values of the cutoff distance...... 97

7.3 Prodan dipole moments reported in the literature. [2, 3, 4, 5, 6, 7, 8, 9] µG and µE are ground state and excited state dipole moment of Prodan measured in Debye (D)...... 104

7.4 Prodan S1 state relaxation time in simulation and experiment obtained from single and double exponential fitting function, − t − t F (t) = ae τ1 + be τ2 + c...... 110

7.5 Stokes shift of Prodan spectra from MD/GW -BSE calculated absorption and emission energies compared to experimental measurements from literature [10] and Fig. 7.11...... 110

8.1 The first three excited state energies of Laurdan inDOPC membrane for different values of the cutoff distance...... 116

xiv LIST OF FIGURES

1.1 Chemical structure of 2–dimethylamino–6–propionylnaphthalene (Prodan) and 2 –dimethylamino 6 –lauroylnaphthalene (Laurdan). 2

1.2 Schematic diagram of cell membrane and lipid bilayer drawn in ChemDraw version 19.0.1.32...... 4

1.3 Chemical structure of some lipids used in this work drawn using ChemDraw 19.0.1.32. The name of acronyms are given in Table 1.1. 5

1.4 Observed phase diagram showing liquid immiscibility region in GUVs at 30°C. Compositions of vesicles in micrographs 1–8 are as follows: 1), 1:1 DOPC/DPPC + 5% Chol; 2), 2:1 DOPC/DPPC + 45% Chol; 3), DPPC + 40% Chol; 4), 2:1 DOPC/DPPC + 20% Chol; 5), 1:1 DOPC/DPPC + 30% Chol; 6), 1:2 DOPC/DPPC + 20% Chol; 7), 1:2 DOPC/DPPC + 40% Chol; and 8), 1:9 DOPC/DPPC + 30% Chol. All scale bars are 20￿m. Vesicles 4–8 were imaged at 30 ± 1°C, and domains are not at equilibrium sizes. Reprinted (adapted) with permission from Sarah L. Veatch and Sarah L. Keller (2003). Separation of Liquid Phases in Giant Vesicles of Ternary Mixtures of Phospholipids and Cholesterol. Biophysical Journal, Copyright 2003 Elsevier [11]...... 7

2.1 Jablonski diagram: IC: Internal Conversion , ISC: Intersystem crossing, VR: Vibrational Relaxation; Adapted from [12]...... 11

2.2 Jablonski diagram with solvent relaxation...... 13

xv 2.3 Prodan emission spectra from left to right in cyclohexane, chlorobenzene, dimethylformamide, ethanol, and water. Reprinted (adapted) with permission from Weber, G., Farris, F. J. (1979). Synthesis and spectral properties of a hydrophobic fluorescent probe: 6-propionyl-2-(dimethylamino) naphthalene. Biochemistry, 18(14), 3075–3078. https://doi.org/10.1021/bi00581a025. Copyright 1979 American Chemical Society [1]...... 14

3.1 Laurdan emission spectra in gel phase DPPC at 35◦C (red) and liquid phase DPPC at 50◦C (blue). Reprinted/adapted from FORMATEX [13]...... 17

3.2 Chemical structure of 2–dimethylamino–6–propionylnaphthalene (Prodan) and 2 –dimethylamino 6 –lauroylnaphthalene (Laurdan) 18

3.3 Mechanism of intramolecular charge transfer(ICT) in lowest excited state of Prodan upon excitation. Planar-ICT(PICT) and Twisted-ICT(TICT) is defined based on angle between the planes consisting donor and acceptor groups (ϕ = 0◦ correspond to PICT and ϕ = 90◦ correspond to TICT) ...... 19

3.4 Prodan and its derivative with the planar amino group ...... 21

3.5 Prodan ground state electric dipole moment calculated using QMMM approach. Reprinted (adapted) with permission from Cintia Vequi-Suplicy, Kaline Coutinho, and M Teresa Lamy (2014). Electric dipolemoments of the fluorescent probes prodan and laurdan: Experimental and theoretical evaluations. Biophysical Reviews, 6:63–74, 03. Copyright 2014 SPRINGER NATURE [14]. 24

3.6 Prodan(black) and Laurdan (red) absorption spectra in water(a), methanol(b), aceto-nitrile(c), dichloromethane(d), chloroform(e), and cyclohexane(f). Reprinted (adapted) with permission from Cintia Vequi-Suplicy, Kaline Coutinho, and M Teresa Lamy (2014). Electric dipolemoments of the fluorescent probes prodan and laurdan: Experimental and theoretical evaluations. Biophysical Reviews, 6:63–74, 03. Copyright 2014 SPRINGER NATURE [14]. 25

xvi 3.7 Prodan(black) and Laurdan (red) emission spectra in water(a), methanol(b), aceto-nitrile(c), dichloromethane(d), chloroform(e), and cyclohexane(f). Reprinted (adapted) with permission from Cintia Vequi-Suplicy, Kaline Coutinho, and M Teresa Lamy (2014). Electric dipolemoments of the fluorescent probes prodan and laurdan: Experimental and theoretical evaluations. Biophysical Reviews, 6:63–74, 03. Copyright 2014 SPRINGER NATURE [14]. 25

4.1 Schematic representation of basic molecular dynamics workflow . 35

4.2 Schematic representation of bonded and nonbonded terms in a typical force-field ...... 37

4.3 Schematic representation of self-consistent DFT calculation ... 45

4.4 Schematic representation of a) photoemission b) inverse photoemission, and c) absorption in a molecule. Adapted from [15] 49

4.5 Schematic representation Hedin’s equations ...... 55

4.6 The system of interacting particles (a) when approximated by quasi-particle approximation (b) interact via screened Coulomb interaction,W instead of bare Coulomb, v. Adapted from [15]. .. 57

4.7 Spectral function of an infinite system within the quasi-particle approximation. Adapted from [15]...... 58

4.8 Schematic diagram showing Prodan in QM region and water molecules in MM regions for QM/MM calculation ...... 66

5.1 Chemical structure of 6-Dodecanoyl-N,N-dimethyl-2-naphthylamine, commonly known as Laurdan (left), 2-dimethylamino-6-propionylnaphthalene, commonly known as Prodan (middle) and model compound used in partial charge parametrization (right)...... 68

xvii 5.2 The schematic diagram showing interaction orientations of the model compound and water molecules used for charge optimization. Only one water molecule interacts with the model compound during each calculation. All water molecules are shown in this figure for convenience only. Only the cases represented by the dashed blue lines are used for the charge optimization to avoid the effect of secondary interaction from other atoms of the model compound...... 70

5.3 Model compound-water interaction energies obtained from HF/6-31G(d)(in red), CGenFF with initial set of partial charges (in cyan) and final set of partial charges (in blue). * and ** correspond to two different geometric orientations0 ( ◦ and 90◦ respectively) of a water oxygen next to the specified atom in the model compound. 73

5.4 Prodan ground state partial charge values before (left) and after parametrization (right) in the unit of elementary charge. The red, cyan, and blue arrows represent twice the dipole moment vector obtained by MP2/6-31G(d), CGENFF with initial and final set of partial charges respectively...... 74

6.1 A MD simulation snapshot showing liquid ordered (Lo) phase (A) and liquid disordered (Ld) phase (B)...... 79

6.2 Time series showing location of the Laurdan relative to the phosphates of each leaflet for four systems: Panel A DPPCLo B PSM Lo C DPPC Ld D PSM Ld...... 80

6.3 Density profile of the substructures of Laurdan for different systems: Panel A: DPPC Lo phase. B: PSM Lo phase C: DPPC Ld phase. D: PSM Ld phase. Black, water; Red, Laurdan head group, blue, Laurdan ring, green, Laurdan tail...... 81

6.4 Radial density function and its integral of C=O and N in Lo and Ld systems: Panel A: DPPC C=O. B: DPPC Ld N. C: PSM C=O. PSM N. Red, Ld. Blue, Lo...... 82

xviii 6.5 Tilt angle of dipole moment vector of Laurdan with respect to bilayer normal in different systems: Panel A: DPPC Lo, B: PSM Lo, C: DPPC Ld, D:PSM Ld ...... 83

6.6 H bonding water occupancy percentage of Laurdan oxygen in Lo and Ld bilayers...... 84

7.1 (a) Six different rigid fragments in Prodan: nitrogen atom, naphthalene core, the carbonyl and methylene groups (b) Visualization of mapping QM optimized copies into MD structure. 94

7.2 The schematic diagram showing Prodan (in QM region) surrounded by water molecules (in MM region)...... 96

7.3 The schematic diagram representing GW -BSE absorption energy calculation of Prodan with and without hydrogen bonded water and presence of water in MM region. See section 7.4.3 for the discussion...... 97

7.4 The effect of inclusion of water in MM region and hydrogen bonded water molecule in QM region on absorption spectra of Prodan. .. 98

7.5 Prodan emission energy from S1 to S0 transition averaged over 2, 5, 15, 20, 24, and 25 MD configurations from fainter to darker curves. 99

7.6 Prodan excitation and solvent relaxation workflow ...... 100

QP 7.7 Quasiparticle energy levels (εi ) and contributions of interlevel transitions to the electron-hole wavefunctions for (a) absorption and (b) emission in vaccum on optimized geometries, as well as sample MD/GW -BSE relaxed geometries in (c) hexane, and (d) water solvent. Isosurfaces of selected quasiparticle orbitals (isovalue 10−2, red/blue) and difference densities of the excited state (isovalue 5 · 10−4 e, orange/green) are shown as insets...... 101

xix 7.8 (a)/(c) Prodan S1 state dipole moment and (b)/(d) S1 → S0 emission energy in hexane (red), octanol (black), acetone (cyan), ethanol (magenta), methanol (green), and water (blue). Error bars and background shading indicate the standard deviation among 25 different solvent configurations at each time point. The leftand right panels correspond to 1 ps and 20 ps of MD relaxation respectively...... 103

7.9 Normalized absorption (dashed lines), emission immediately after absorption (dashed-dotted lines) and after 49 ps MD/GW -BSE relaxation (solid lines) spectra of Prodan in different solvents. .. 105

7.10 Prodan TA maps in acetone (a), ethanol (b), and methanol (c). . 106

7.11 Prodan absorption (left) and emission (right) spectra in acetone, ethanol, and methanol from simulation (solid lines) and experimental measurements (dashed lines)...... 108

7.12 Prodan time resolved emission spectra (black) compared with simulation in acetone (cyan), ethanol (magenta), and methanol (green)...... 109

7.13 Comparison of Stokes shift obtained in simulation with experimental measurements from the literature [10] and Fig. 7.11. 111

7.14 Panel (a) and (b): Prodan time resolved emission spectra in acetone (cyan) and methanol (green) from simulation for 20 ps and 1 ps MD solvent relaxation between consecutive emission energy calculation compared with experiment(dashed lines). Panel (c): Prodan time resolved emission spectra in acetone (cyan) and methanol (green) from simulation for 20 ps and 1 ps MD solvent relaxation between consecutive emission energy calculation compared with experiment (dashed lines)...... 112

8.1 Panel a) shows molecules included in GW -BSE calculation: Laurdan(blue; in QM region), DOPC, and water molecules in MM region. Panel b) shows the chromophore included in QM region and the excluded hydrocarbon tail (blue)...... 117

xx 8.2 Laurdan excitation and solvent relaxation workflow ...... 118

8.3 (a) Laurdan S1 state dipole moment and (b) S1 → S0 emission energy in DOPC bilayer (blue) and DOPC/CHOL (red). Error bars and background shading indicate the standard deviation among 45 different solvent configurations at each time point. ... 119

8.4 (a) S1 → S0 emission energy and (b) Laurdan S1 state dipole moment in DOPC bilayer (blue) and DOPC/CHOL (red). Each iteration step in this figure, unlike Fig. 8.3, correspond to 20 ps classical MD...... 120

9.1 The schematic diagram showing ABF calculation setup for Prodan in hexadecane-water slabs. Prodan molecule moves along the direction of transition coordinate (ξ) as shown by a dotted line which is stratified into several bins...... 129

9.2 Potential of mean force (PMF) of six replicas along the transition coordinate...... 132

9.3 Time series obtained from 475 ns simulation showing the location of center of mass of Prodan moving along the direction of transition coordinate...... 133

9.4 Gradient(top) and potential of mean force (bottom) of six replicas along the transition coordinate. The black curve is the error in measurement. The vertical lines represent the hexadecane-water interfaces...... 134

9.5 Symmetrized version of Free-energy profile (PMF) from Fig. 9.4b. 135

B.1 Permission statement to use Fig. 5.3, 5.4, 7.6, 7.7, 7.8, 7.9, 7.10, 7.11, 7.12, 7.13, and 7.14 and Table 7.1, 7.4, and 7.5...... 166

xxi ABSTRACT

The cell membrane is spatially heterogeneous, mostly in a fluid state, and composed of hundreds of different lipid types and proteins. Based on the composition and temperature, the lipids on cell membranes are packed into different kinds of phases, namely the gel phase, liquid-ordered phase, and liquid disordered phase. Polarity sensitive dyes such as Laurdan and Prodan, interrogate different lipid phases as the emission spectrum is red-shifted in the less dense liquid disordered phase compared to more dense gel or liquid-ordered phase. The shift in emission occurs because of the dipolar relaxation of solvents molecules present in the vicinity of the chromophore. An MD/QM approach is developed in which (i) the local environment is sampled by classical molecular dynamics (MD) simulation of the dye, (ii) the elec- tronically excited state of Laurdan/Prodan is predicted using numerical quantum mechanics (QM), (iii) the environment around the excited dye is iteratively relaxed by MD coupled with the evolution of the excited state, and (iv) the emission proper- ties are predicted by QM. The QM calculation is performed using many-body Green’s function approach within GW approximation. The Bethe-Salpeter equation is solved to obtain coupled electron-hole excitation, and the environment is modeled as fixed point charges sampled from MD simulation. This combination of methods is referred to below as MD/GW-BSE.

xxii The simulation results of MD/GW -BSE calculation explained above per- formed on Prodan molecule in bulk solvents of various polarity agrees with the ex- perimental measurements of the Stokes shift. The dynamics of the coupled solvent relaxation and evolution of the excited state is in excellent agreement with ultrafast transient absorption spectroscopy performed in the Gundlach lab. The MD/GW -BSE calculation is repeated in a more complex system con- sisting of Laurdan in DOPC and 1:1 mixture of DOPC/CHOL bilayer. The result of Laurdan simulation does not agree with experimental measurement but provides valuable insights regarding the need to improve Laurdan Force Field parameters. In particular, the results suggest that the MD model for Laurdan is not partitioning correctly at the lipid/water interface. In order to assess the partitioning of the Laurdan chromophore at the oil/water interface, Adaptive Biasing Force (ABF) Free energy calculation was per- formed on Prodan in the water-hexadecane slab system. The free energy profile (PMF) of Prodan in the direction perpendicular to the water hexadecane interface was analyzed to obtain the preferred location of Prodan in the water-hexadecane slab, and to determine the bulk partitioning of the Prodan model between water and hexadecane.

xxiii Chapter 1

BILAYERS AND CELL MEMBRANES: COMPOSITION, PHASES, AND PACKING

1.1 Introduction Animal cells are enclosed by cell membranes that separate the inside of the cell from the outside environment (Fig 1.2)[16]. The cell membrane is a thin 5 nm layer of lipids and protein molecules and plays an important role in various cellular functions related to cellular communication [17, 18, 19]. Intracellular organelles such as endoplasmic reticulum, Golgi apparatus, mitochondria, cytoplasm are also enclosed by membranes. Most cell membranes are in a fluid state, which is essential to their function as platforms for signaling and interfacial chemistry [20]. Membrane components that

2 freely diffuse do so with diffusion constants ranging from 0.01to1 um /sec [21, 22, 23]. Although the composition of the membrane is complex and spatially heterogeneous (more on this below), the fluidity of membranes is carefully maintained by cells by varying lipid content. This was first shown by Sinensky, who discovered that bacteria grown at different temperatures vary the lipid content of their membranes in order to maintain a fluid state, a property he named “homeoviscous adaptation.”[24] Thus, experimental probes which report on fluidity have become important tools in membrane biophysics [25]. Several small-molecule chromophores have been developed, the spectral prop- erties of which depend on the local membrane environment. Polarity sensitive dyes,

1 N O

Prodan

N O

C10H21

Laurdan Figure 1.1: Chemical structure of 2–dimethylamino–6–propionylnaphthalene (Pro- dan) and 2 –dimethylamino 6 –lauroylnaphthalene (Laurdan). such as Laurdan and Prodan (Fig 1.1) are widely used in the study of membrane structure as the spectral properties of such fluorophores report lipid packing order [26, 27]. These fluorophores are lipophilic (consisting both hydrophobic fatty acid chain and hydrophilic head group) and therefore partition to the membrane. The emission spectra of these fluorophores undergo a dramatic change in energy cor- responding to 50 nm shift in less ordered, more fluid membranes when compared between the more ordered membranes (more on this below)[26]. Recent advances in mass spectrometry for lipids have revealed the membrane to be a complex mixture of hundreds of different lipid types, which differ in hydrocar- bon chain length and unsaturation, in the chemistry of the amphiphilic headgroup, and in the linkage that connects the headgroup and the hydrocarbon tails [28]. The details of this composition vary greatly across the kingdoms of life and even across different cell types in the human body. Despite this complexity, within mammals

2 between 20 and 40 mol % of the lipid fraction is cholesterol. In order to make the complexity of the membrane experimentally tractable, simpler binary and ternary mixtures (usually containing cholesterol as one of the components) are used as model systems. The role of cholesterol and the properties of these model systems are dis- cussed next.

1.2 Basic lipid chemistry Lipid molecules are amphiphilic, meaning they consist of a polar head group and non-polar tail in the same molecule. The most abundant lipids in the cell membrane are phospholipids (Fig 1.3), which consists of a phosphate containing zwitterionic or negatively charged headgroup and hydrocarbons in the fatty acid tail [16]. In the presence of water, the lipid molecules self-assemble to exclude water from hydrophobic tails and form two monolayers (also known as two leaflets) with the head group exposed towards water molecules and hydrophobic fatty acid tails hidden. Since the thickness of the bilayer is about 5 nm, the membrane can be treated as a two-dimensional fluid with the lipids free to diffuse in the plane of the membrane.A hydrocarbon tail without double bonds between carbon atoms is called “saturated” as each carbon atom contains the maximum allowed number of hydrogen atoms. A hydrocarbon tail that contains at least one double bond is known as “unsaturated”. The presence of a double bond provides kink in the fatty acid tails, lowering the melting temperature compared to the lipid membranes with saturated lipids. The lipid membranes studied in this work are mixtures of DOPC, DPPC, POPC, PSM, and CHOL. The acronyms are defined and their respective melting temperatures are listed in Table 1.1, and their structures are provided in Figure 1.3

3 Figure 1.2: Schematic diagram of cell membrane and lipid bilayer drawn in Chem- Draw version 19.0.1.32.

Table 1.1: Names of lipids acronyms and melting temperature for 5 different lipid types.

Abbreviation Chemical Name Melting Temperature (K) DOPC 1,2-Dipalmitoyl-sn-glycero-3-phosphocholine 256 DPPC 1,2-Dioleoyl-sn-glycero-3-phosphocholine 314 POPC 1-Palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine 271 PSM Palmitoyl sphingomyelin 314 CHOL Cholesterol N/A

4 Figure 1.3: Chemical structure of some lipids used in this work drawn using Chem- Draw 19.0.1.32. The name of acronyms are given in Table 1.1.

5 1.2.1 Membrane heterogeneity and lipid phases The simplest membrane system, which consists of one type of lipid species, is known as the single component bilayer. The fluidity of the single component phospholipid bilayer increases with the increase in temperature. At the melting tem- perature, the system transitions from a more ordered solid phase (called gel phase), where hydrocarbon chains are fully extended and lipids pack tightly within each leaflet to a more disordered fluid phase (called Ld or Lα), where hydrocarbon tails are more disordered, the area per lipid is increased relative to the gel, and the mem- brane is in a fluid state. The unsaturation of hydrocarbons in the lipid tail lowers the melting temperature of the bilayer. For example, DOPC contains two singly un- saturated hydrocarbon tail and lower melting temperature (256 K) compared to the melting temperature of DPPC (314 K), which contains two saturated hydrocarbon tail. Increasing the concentration of cholesterol (see Fig. 1.3) has both ordering and fluidizing effects on a lipid membrane. The presence of the cholesterol molecules melts the gel phase but orders the hydrocarbon chains of the liquid disordered phase, hence introducing a new phase which is fluid but with more ordered hydrocarbon chains, called the liquid-ordered (L0) phase [29]. The ordering effect of the cholesterol, with hydroxyl group followed by four fused rings and a short hydrophobic tail, intercalates between the lipid molecules filling in space. Combining an unsaturated and a saturated phospholipid with cholesterol makes a ternary mixture which approximates the complexity of a cell membrane, which is a mixture of cholesterol and saturated and unsaturated phospholipid. The phase diagram (Fig. 1.4) for such a mixture is portrayed by an equilateral trian- gle with a concentration of each lipid species being maximum on the vertex. The

6 Figure 1.4: Observed phase diagram showing liquid immiscibility region in GUVs at 30°C. Compositions of vesicles in micrographs 1–8 are as follows: 1), 1:1 DOPC/DPPC + 5% Chol; 2), 2:1 DOPC/DPPC + 45% Chol; 3), DPPC + 40% Chol; 4), 2:1 DOPC/DPPC + 20% Chol; 5), 1:1 DOPC/DPPC + 30% Chol; 6), 1:2 DOPC/DPPC + 20% Chol; 7), 1:2 DOPC/DPPC + 40% Chol; and 8), 1:9 DOPC/DPPC + 30% Chol. All scale bars are 20￿m. Vesicles 4–8 were imaged at 30 ± 1°C, and domains are not at equilibrium sizes. Reprinted (adapted) with permis- sion from Sarah L. Veatch and Sarah L. Keller (2003). Separation of Liquid Phases in Giant Vesicles of Ternary Mixtures of Phospholipids and Cholesterol. Biophysical Journal, Copyright 2003 Elsevier [11].

shaded region in the diagram (Fig. 1.4) is the region of coexistence for L0 and Ld phase. Fig 1.4 shows the phase diagram of ternary mixture of DOPC/DPPC/Chol at 30◦C. To the right of the phase diagram, the figure shows the fluorescence image showing two kinds of phases (Lo in darker areas and Ld in lighter areas) for different composition mixture represented by the different points labeled in the diagram. The measurement is performed in the Giant Unilamellar Vesicle (GUV) that are 10-50 µm diameter large spherical models for bio-membranes. This work focuses on predicting the spectral properties of chromophores used

7 to study membrane packing and fluidity. Chapter 2 gives a brief background on fluorescence. Chapter 3 consists of the literature review of Prodan and Laurdan. In chapter 4, the theoretical framework of molecular dynamics (MD) simulation and MD/QM calculation used in this work is provided. In chapter 5, the result of Prodan partial charge optimization is provided using which the MD simulation of Prodan and Laurdan in bulk solvents and lipid bilayer is run. The result of MD simulation of Laurdan is provided in chapter 6. MD/QM result of Prodan in bulk solvent is provided in chapter 7 and of Laurdan in lipid bilayer is provided in chapter 8. Chapter 9 consist result of free energy calculation of Laurdan in membrane environment, in order to improve the parametrization of the Laurdan model. Finally, the conclusion and the future outlook is provided in chapter 10.

8 Chapter 2

INTRODUCTION TO FLUORESCENCE

2.1 Introduction When a molecule absorbs light, electrons move to an excited state, which may either be a singlet or triplet state. A singlet state has a total spin of zero and is 1 |↓↑⟩−|↑↓⟩ represented by 2 ( ). In this notation, the first arrow represents the spin ofa promoted electron and the second arrow represents the spin of the hole (inverse of the electron spin) in the lower state left behind due to the transition of the electron. The |↑↑⟩ |↓↓⟩ 1 |↓↑⟩ |↑↓⟩ triplet spin has the total spin of 1 and is represented by , , and 2 ( + . An electron excited to singlet state has the opposite spin as compared to the other electron in the ground state, whereas an electron excited to triplet state has the same spin. An electron in a singlet excited state returns to the ground state without having to flip the spin, a process known as fluorescence. In contrast, the transition from the triplet excited state requires the change of spin orientation and is known as phosphorescence. Fluorescence is the quantum mechanically spin allowed transition and typically occur with emissive rates of 108 s−1 and time scale of 1-100 ns [30].

2.2 Jablonski Diagram The Jablonski diagram [31] (see Fig. 2.1) illustrates the absorption of an elec- tron from a ground state to singlet excited states and several mechanisms that the electron can take to return to the ground state. The ground state and singlet excited

9 states are labeled as S0 and S1 respectively, whereas triplet excited state is labeled as

T1. The vibrational levels, corresponding to the nuclear degree of freedom, for each electronic state are labeled as horizontal lines. Mechanisms of both radiative (Flu- orescence and Phosphorescence) and non-radiative (Internal conversion, Vibrational relaxation, and inter-system crossing ) transfer pathways are demonstrated. At room temperature, most of the molecules will reside only in the lowest vibrational state, as described by Boltzmann distribution. Since the energy differ- ence between two electronic states is much larger than two vibrational states, it is not possible to thermally excite a molecule to excited states for the molecules and temperatures considered here. As a molecule absorbs light, an electron is typically excited to higher singlet electronic state. Because a nucleus is approximately 1800 times heavier than an electron, the wave function can be separated into the electronic wave function and nuclear wave function, an approximation known as Born-Oppenheimer [32] (More about this on Chapter 4). The Frank-Condon principle, which is a direct conse- quence of Born-Oppenheimer approximation, states that the nucleus cannot move immediately during electronic excitation. Therefore, when an electron residing on a lower vibrational level of the ground state gets excited, it ends up in a higher vibrational level of the excited electronic state. In most cases, an electron in higher vibrational level releases energy non-radiatively, transferring to the lowest vibration level of the excited state. The time scale of this process is of order 0.1 ns which is much faster than the timescale of fluorescence, often permitting the two processes to be considered separately in a theoretical treatment. Also, in most cases, electrons in second excited or higher singlet states, trans- fer to first excited state non-radiatively as a consequence of Kasha’s rule[33]. There- fore, the emission wavelength does not depend on the excitation wavelength because

10 S2 IC VR VR

S1 IC ISC T VR 1 ISC VR Absorption Fluorescence Phosphorescence

Ground state S0

Figure 2.1: Jablonski diagram: IC: Internal Conversion , ISC: Intersystem crossing, VR: Vibrational Relaxation; Adapted from [12].

11 the electron moves to the lowest excited state from higher excited states as a re- sult of the strong overlapping of numerous vibrational levels of comparable energies. These transfers are shown in Fig. 2.1 as the internal conversion with the time scale of a picosecond. The electron in the first excited singlet state, in most cases, may transfer to ground state (Fluorescence; time scale 1-100 ns) or travel to triplet state via non-radiative transfer known as intersystem crossing before returning to ground state (Phosphorescence; time scale ms).

2.3 Stokes Shift The difference in energy between absorption and emission peaks is knownas Stokes shift. The loss of energy is due to different non-radiative energy losses during internal conversion and intersystem crossing. During the process of fluorescence, an electron may also end up in a higher vibrational level of ground state from which it may lose further energy via the non-radiative mechanism. Solvent effects may also influence the Stokes shift. As a consequence of similar vibrational levels involved in excitation and emis- sion, the absorption and emission spectrum sometimes appears as the mirror image of each other. The exception to the mirror image rule suggests different geometry of nuclei in an excited state as compared to the ground state.

2.4 Quantum Yield and Lifetime The ratio of the number of photons emitted to the absorbed is known as quantum yield, and the average time a molecule spends in an excited state is known as the fluorescence lifetimes (about 10 ns for the molecules considered here).

12 Figure 2.2: Jablonski diagram with solvent relaxation.

2.5 Solvent effects and Polarity Sensitive Dyes 2.5.1 Solvent relaxation Since the absorption process occurs on a much shorter time scale (Frank- Condon principle) as compared to emission, the absorption process is not very sen- sitive to the dynamics of solvent. The emission spectrum, on the other hand, can be sensitive to various solvent properties such as solvent polarity and viscosity, inter- nal charge transfer because the lifetime of the fluorophore is longer than timescales typical of solvent reorganization.

A typical fluorescent dye has a larger dipole moment in an excited state, µE, compared to ground state dipole moment, µG. The change in dipole moment during the excitation process reorients solvent dipole moments, a process known as solvent relaxation, which in turn lowers the energy of the excited state of the fluorophore. This effect is more pronounced in more polar solvents, resulting in a larger decrease

13 Figure 2.3: Prodan emission spectra from left to right in cyclohexane, chlorobenzene, dimethylformamide, ethanol, and water. Reprinted (adapted) with permission from Weber, G., Farris, F. J. (1979). Synthesis and spectral properties of a hydropho- bic fluorescent probe: 6-propionyl-2-(dimethylamino) naphthalene. Biochemistry, 18(14), 3075–3078. https://doi.org/10.1021/bi00581a025. Copyright 1979 American Chemical Society [1].

14 in the emission energy maxima, as shown in Fig. 2.2. Fluorescent molecules that are sensitive to the solvent polarity of the micro-environment are called polarity sensitive. Figure 2.3 [1] shows the emission spectra of Prodan in cyclohexane, chlorobenzene, dimethylformamide, ethanol, and water clearly demonstrating a sensitivity to the polarity of the environment. The time scale of solvent relaxation, 10-100 ps, is shorter compared to fluores- cence lifetimes 1-100 ns and this is the reason that the steady-state emission spectra such as Fig. 2.3 usually represents solvent relaxed behavior. In the next chapter, we will discuss some properties of a family of polarity sensitive dyes, especially Laurdan and Prodan, that are widely used in the study of bio-membranes.

15 Chapter 3

SPECTRAL PROPERTIES OF PRODAN AND LAURDAN

3.1 Introduction Dyes such as 6-propionyl-2-(dimethylamino)naphthalene and 2 –dimethylamino 6 –lauroylnaphthalene commonly known respectively as Prodan and Laurdan (Fig. 3.2) are sensitive to the polarity of the environment because of the mechanisms de- scribed in the earlier chapter. These lipophilic dyes are widely used to study lipid structure in biological membranes because the emission wavelength shifts by as much as 50 nm depending on the phase and composition of the membrane (Fig. 3.1), pre- sumably due to changes in the local solvation of the chromophore [34, 35, 27, 36, 37]. In this chapter, Prodan structure, nature of the emitting state, and dipole moment are reviewed.

3.2 Prodan Structure Weber and Farris [1] introduced the polarity sensitive dye Prodan in 1979; since that time it has found wide use, with over 50,000 articles reported by Google Scholar. The presence of a good electron donor (amino) and a good electron acceptor (C=O) groups at the two ends of the naphthalene ring lends a charge transfer (CT) character to the lowest excited state (see Fig. 3.3). This effect becomes the most pro- nounced when the physical separation between the donor and the acceptor is largest. In the case of naphthalene derivative, this takes place when donor and acceptor are

16 Figure 3.1: Laurdan emission spectra in gel phase DPPC at 35◦C (red) and liquid phase DPPC at 50◦C (blue). Reprinted/adapted from FORMATEX [13]. fixed to the second and sixth position of the rings[1]. With this configuration, the electronic jump from the ground state to locally excited (LE) state evolves into CT excited state becomes very sensitive to the polarity of the surrounding environment [38].

3.3 Nature of Prodan Emitting State The excited-state properties of a chromophore can be affected by the relax- ation of its geometry. In the case of the chromophores that are composed of a donor (D) and an acceptor (A), one of the most important geometrical changes is the rota- tion around the bond connecting the D-A group as it can have significant effects on excited-state properties. Two models that describe the orientation of D-A bond in intramolecular charge transfer (ICT) state are twisted-ICT (TICT) and planar-ICT (PICT)(see Fig. 3.3).

17 N O

Prodan

N O

C10H21

Laurdan Figure 3.2: Chemical structure of 2–dimethylamino–6–propionylnaphthalene (Pro- dan) and 2 –dimethylamino 6 –lauroylnaphthalene (Laurdan)

In TICT state, proposed by Grabowski et al. [39], D and A group twist to adopt a perpendicular orientation in the excited state, and this results in orbital decoupling leading to complete charge transfer character. This state is also associated with an increased excited state dipole moment and lowered emission energy of a molecule as compared to PICT state. The increased dipole moment is reported to be stabilized by polar solvents resulting in so called La type emission band from the charge transfer state, whereas a Lb type emission band is the result of a locally excited (LE) (also known as Frenkel) state [40, 41]. In a PICT state, proposed by Zachariasse et al. [42], the D-A groups remain planar, which tends to conjugate them in the excited state. Because of this, the excited state bond becomes slightly shorter, the dipole moment becomes smaller, fluorescence becomes more pronounced, and the Stokes shift becomes more modest

18 Figure 3.3: Mechanism of intramolecular charge transfer(ICT) in lowest excited state of Prodan upon excitation. Planar-ICT(PICT) and Twisted-ICT(TICT) is defined based on angle between the planes consisting donor and acceptor groups (ϕ = 0◦ correspond to PICT and ϕ = 90◦ correspond to TICT) as compared to TICT state. In this alternative explanation to TICT, the solvent- induced vibronic coupling between locally excited (LE) states and CT states is ex- plained to take place because of the small energy gap between the two states. This, along with the wagging motion of the amino group, is said to cause splitting of quasi-degenerate levels that stabilize one state considerably, and this represents the charge transfer state leading to second emission band [43]. The transitions involving in TICT and PICT states are reported to be n –> Π∗ and Π –> Π∗ respectively [44]. Early theoretical studies (semi-empirical ground-state calculations using Mod- ified Neglect of Diatomic Overlap (MNDO/3, MNDO), and Austin Model 1(AM1) and electronic transitions using Intermediate Neglect of Differential Overlap (INDO/S))

19 in vacuum reported that Prodan has planar ground state geometry, but the fluores- cence is described to occur from a highly polar excited state with twisted geome- try [44]. Later work, using higher level theory (Configuration Interactions Singles (CIS) and Density functional theory (DFT/SCI) suggested that the emission from the TICT is stabilized by the polar solvents by using self-consistent reaction field (SCRF) approach describing these in terms of Lippert-Mataga plot of the Stokes shift under several assumptions: i) chromophore is spherical, ii) it does not interact with solvent (neglecting the polarizability of the chromophore) iii) excited state and ground state dipole moment vectors are parallel [45][46][47]. By comparing the experimentally measured fluorescence spectra of two sys- tems similar to Prodan in which the amino group is forced to be planer (Fig. 3.4) to that of Prodan, Abelt, et al. [48][49] concluded that Prodan emits from planar excited state geometry. These authors also discuss that the theoretical studies in- dicating emission from a TICT state were dependent on the choice of the Onsager radius. A radius of 4.6 Å favored a TICT state while a radius of 5.6 Å favored the PICT state [49]. Mennucci, et al. [50] then showed that the lower energy TICT state is an artifact of TDB3LYP with approximate long-range functional that does not describe the n-> π∗ transition correctly. The authors further showed that a more accurate calculation based on CI yields lower energy for PICT state. Based on present knowledge, the TICT model is considered to apply to less ex- tended molecular systems with only one aromatic ring, whereas Prodan is considered to emit from PICT state.

3.4 Prodan Dipole Moment In 1979, based on the data from the Table 3.1 and equations 3.1 and 3.2, Weber and Farris [1], within the framework of dipole interaction theory by Lipert (1957),

20 (a) Prodan (b) Modified Prodan (pla- nar amino group)

Figure 3.4: Prodan and its derivative with the planar amino group estimated a 20 Debye increase in dipole moment upon excitation by considering the 4338 cm−1 Stokes shift from cyclohexane to water that correspond to ∆f = 0.32. The dipole interaction framework assumes the fluorophore to be a dipole in a continuous environment with the uniform dielectric constant and therefore does not incorporate solvent-fluorophore interaction30 [ ]. The fluorescence polarization measurement data in Table 3.1 is measured using a photometer, and the lifetime data is measured using the cross-correlation method of Spencer and Weber [51]. In equations 3.1 and 3.2, n is the refractive index, ε is dielectric constant, and a is the radius of cavity scooped in the solvent to place the dye estimated from the crystallographic data on naphthalene derivatives. N to O distance is 8.4 Å), and ∆ν and ∆f are Stokes shift change and change in polarizability respectively,

n2 − 1 ε − 1 ∆f = − (3.1) 2n2 + 1 2ε + 1

21 Table 3.1: Spectral Properties of Prodan Solutions. Reprinted (adapted) with per- mission from Weber, G., Farris, F. J. (1979). Synthesis and spectral properties of a hydrophobic fluorescent probe: 6-propionyl-2-(dimethylamino) naphthalene. Biochemistry, 18(14), 3075–3078. https://doi.org/10.1021/bi00581a025. Copyright 1979 American Chemical Society [1]

Solvent Abs max (eV) Ems max (eV) Stokes shift (eV) (eV) f (ns) Cyclohexane 3.63 3.09 0.53 0.27 0.001 1.6,1.8 Acetone 3.54 2.74 0.8 0.37 0.287 Acetonitrile 3.54 2.68 0.86 0.36 0.304 Methanol 3.42 2.46 1.02 0.35 0.308 Water 3.41 2.33 1.07 0.32 0.32 2.1, 2.4

( ) 2a3 µ∗ − µ = hc∆ν (3.2) ∆f Weber and Farris [1] also estimated the minimum increase in dipole moment upon excitation by comparing the fluorescence spectral displacement in a solvent at two temperatures. Arguing that larger viscosity of the solvent at lower temperature resists the solvent relaxation during the lifetime of the dye and lower viscosity at higher temperature permits the relaxation from Frank-Condon state, the authors considered the 2226 cm−1 shift of emission maxima of Prodan in Propylene glycol at -50 and 20 °C to show that the change in dipole moment upon excitation becomes 14.8 D. The difference of this value with 20 D (obtained by using emission incyclo- hexane to water) is described to have arisen from the residual orientation effects of Prodan at low temperature due to its significant dipole moment in the ground state. In 1987, Balter et al. [52] pointed that the factor of 2 in the lipert equation (equation 3.2) was placed incorrectly in numerator instead of denominator which yields a change in Prodan dipole moment of 10 D instead of 20 D. They further mentioned that this value is obtained using two points only (water and cyclohexane) in the Lipart-Mataga plot and that Weber and coworkers used a protic solvent, which

22 should generally be excluded for solvatochromic dipole moment estimation, arguing that the 10 D change in dipole moment was also an overestimation. By reanalyzing the spectral data from [1], Balter et al.[52] used the seven aprotic solvents to estimate the ground state, excited state, and change in dipole moment as 2.9 D, 10.9 D, and 8.0 D respectively. During this analysis, cyclohexane was excluded because the 30 times lower quantum yield and a shorter life time in cyclohexane implied inter-system crossing into triplet excited state, and the change in state type is not considered within the framework of Lippert’s thoery. By adding a solvent acidity term in the Lippert equation, in 1991, Catalan et al. [38] estimated the ground state, excited state, and increase in dipole moment upon excitation to be 4.7 D, 11.7 D, and 7.0 D respectively. Unlike Weber and Farris [1], Catalan et al. [38] pointed out the necessity to reconsider the value of µ∗/µ as they estimated it to be 2.5 instead of 8. In 1999, Kawski [53] developed an analysis based on the solvent perturba- tion method [54, 55] to estimate the ground state and excited state dipole moments as 2.1 D and 6.4 D for a = 4.2 Å, and 2.5 D and 7.4 D for a = 4.6 Å respectively. Key parameters for this solvatochromic perturbation calculations were determined from the absorption and fluorescence band shifts of various solvents. Their calcu- lations suggest that that the excited state dipole moment for Prodan is only three times the value of ground-state dipole moment. In 2000, Kawski [56] showed that the thermochromic shift method also yielded similar dipole moment for the excited state (6.65/7.6 D for 4.2/4.6 Å cavity respectively) and slightly different for ground state (2.45/2.8 D for 4.2/4.6 Å cavity) when compared to the results obtained using solvatochromic method. In 2000, Samanta and Fessenden [57] employed time-resolved changes in mi- crowave dielectric absorption to determine the dipole moments in the ground state

23 Figure 3.5: Prodan ground state electric dipole moment calculated using QMMM approach. Reprinted (adapted) with permission from Cintia Vequi-Suplicy, Kaline Coutinho, and M Teresa Lamy (2014). Electric dipolemoments of the fluorescent probes prodan and laurdan: Experimental and theoretical evaluations. Biophysical Reviews, 6:63–74, 03. Copyright 2014 SPRINGER NATURE [14]. and the excited state as 5.2 D and 10.2 D, respectively. They also pointed out that the prior mentioned change in the dipole moment of 20 D as the overestimation and that the dye does not emit from the TICT state. They also argued that the change in dipole moment could not be the sole reason for the shift of the emission band, and other factors such as hydrogen bonding can contribute. In 2014, Cintia et al. [14], based on the solvatochromic method, estimated the ground state and excited state dipole moment of Prodan to be 5.5 D and 20 D respectively for the spherical cavity of 6.3 Å. The authors also showed the convergence of Prodan ground state electric dipole moment during the polarization process in several solvents Fig. 3.5 via sequential QM/MM (MP2 for QM and Monte Carlo for MM) approach. The Prodan dipole moments discussed in this chapter are summarized in Table 7.3 from which one can conclude that there seems to be ambiguity about the Prodan

24 Figure 3.6: Prodan(black) and Laurdan (red) absorption spectra in water(a), methanol(b), aceto-nitrile(c), dichloromethane(d), chloroform(e), and cyclohex- ane(f). Reprinted (adapted) with permission from Cintia Vequi-Suplicy, Kaline Coutinho, and M Teresa Lamy (2014). Electric dipolemoments of the fluorescent probes prodan and laurdan: Experimental and theoretical evaluations. Biophysical Reviews, 6:63–74, 03. Copyright 2014 SPRINGER NATURE [14].

Figure 3.7: Prodan(black) and Laurdan (red) emission spectra in water(a), methanol(b), aceto-nitrile(c), dichloromethane(d), chloroform(e), and cyclohex- ane(f). Reprinted (adapted) with permission from Cintia Vequi-Suplicy, Kaline Coutinho, and M Teresa Lamy (2014). Electric dipolemoments of the fluorescent probes prodan and laurdan: Experimental and theoretical evaluations. Biophysical Reviews, 6:63–74, 03. Copyright 2014 SPRINGER NATURE [14].

25 Table 3.2: Prodan dipole moments reported in the literature. [2, 3, 4, 5, 6, 7, 8, 9] µG and µE are ground state and excited state dipole moment of Prodan measured in Debye (D).

Reference µG (D) µE(D) Balter et al. [2] 2.9 10.9 Catalan et al. [3] 4.7 11.7 Bunker et al. [4] 2.85 9.8 Kawski [5] 2.1/2.5 6.4/7.4 Kawski [9] 2.14/2.46 6.46/7.37 Kawski et al. [8] 2.45/2.80 6.65/7.6 Samanta, Fessenden [6] 5.2 10.2 Cintia et al. [7] 5.5 20.0 ground state and excited state dipole moment in the literature.

3.5 Prodan Absorption and Emission Spectra in Bulk Solvents In 1979, Weber and Farris [1] showed the polarity- dependent shift of absorp- tion and emission peak of Prodan in solvents of varying polarity (Some listed in Table 3.1). In 2006, Moyona et al. [58] showed that the absorption and emission spectra not only depend on the polarity of the solvent but also on the hydrogen bonding ability of the solvent. In 2014, Cintia et al. [14] showed that the Prodan and Laurdan yield similar optical absorption and emission spectra Fig. 3.6 and 3.7. To conclude, it seems necessary to study more about the nature of Prodan emitting state and dipole moment and the effect of solvent parameters such as polar- ity and hydrogen bonding to the Prodan excited state. More complete understanding of the mechanism of polarity sensitivity would aid in the interpretation of experi- mental spectra, as well as perhaps guide the design of choromophores with improved characteristics for membrane imaging.

26 The next chapter consists of a brief introduction of molecular dynamics sim- ulation and different kinds of quantum mechanical calculations used to solve many- body Schrodinger equation.

27 Chapter 4

THEORETICAL BACKGROUND

4.1 Introduction To study the stokes shift of the emission spectrum of Laurdan and Prodan in polar environments, it is necessary to perform a numerical quantum calculation to obtain the excited state energies and their evolution. The timescale of the solvent relaxation process, which is approximately a few hundreds of picosecond long, re- quires the use of classical molecular dynamics for the study of the solvent dynamics. This chapter consists of an overview of molecular dynamics simulation, electronic structure theory, and GW -BSE calculation using the method of Green’s function. Quantum calculations comes in two variations: wave-function based and den- sity based approaches. The wave function based approach scales O(N 5−7) whereas GW -BSE, which is built on the top of density functional theory (DFT) scales O(N 3−4) where N is the number of electrons [59]. In either case, the computational effort is too high to treat a complete solute-solvent system using quantum calculation. We therefore developed a mixed approach in which the solvent is relaxed using classical molecular dynamics and the electronic degrees of freedom relevant to the fluorescence are treated quantum mechanically. First the QM method is discussed, some aspects of MD are reviewed, and finally the mixed QM/MD approach is described.

28 4.2 Many-electron quantum mechanics

A molecule consisting N electrons at position r1, r2, .., rN and M nuclei at

R1, R2, .., RM respectively described by non-relativistic Hamiltonian,

ˆ ˆ ˆ ˆ ˆ ˆ H = Tel + Tnuc + Vnuc-nuc + Vnuc-el + Vel-el (4.1) where Tˆ represents kinetic energy operators, Vˆ represents potential energy opera- tor of an electron (el) and a nucleus (nuc). The Hamiltonian consists of following interactions.

∑ pˆ2 Tˆ = i (4.2) el 2m i e

∑ 2 ˆ 1 e Vel-el = (4.3) 2 | ri − rj | i≠ j

∑ 2 ˆ pˆK Tnuc = (4.4) 2mK K

∑ 2 ˆ ZK e Vel-nuc = − (4.5) | ri − RK | i,K

∑ 2 ˆ 1 ZJ ZK e Vnuc-nuc = (4.6) 2 | rJ − rK | K≠ J In above equations, pˆ is the momentum operator, and Z is the atomic number. The time evolution of a many body quantum state ψ is described by the time dependent Schrödinger equation,

∂Ψ(r, R, σ, t) Hˆ (r, R, σ, t)Ψ(r, R, σ, t) = iℏ (4.7) ∂t

29 In the above expression, σ is spin variable, which is needed to obtain cor- rect particle statistics, included only for completeness and will be neglected in the discussion below.

4.3 Wave-function ∫ b 2 According to Born’s statistical interpretation of wave function, a ψ(x) dx gives the probability of finding a particle between a and b. Along with ψ(x), Aψ(x) is also the solution of the Schrödinger equation. The wave function is a complex quantity and A is the complex constant which is determined by normalizing the wave function. The normalization is given by,

∫ +∞ ψ(x) 2dx = 1 (4.8) −∞ Knowing the wave function allows one to obtain the expectation value of any operator of interest. The operator act on wave function, represent the observables and can be thought of as the instruction provided to perform an operation of interest to the wave function [60]. For example, position operator, xˆ, is an instruction to multiply the function by x. The expectation value of the operator is calculated by a) sandwiching the operator between the complex conjugate of the wave function and the wave function and b) integrating. For example, because the Hamiltonian operator is related to the energy of the system and the expectation value of the energy can be found as,

∫ ⟨H⟩ = ψ∗Hψdxˆ (4.9)

30 In the notation of linear algebra, the wave function ψ(x) can also be repre- sented as vectors, ∑n |ψ⟩ = ai |ai⟩ (4.10) i=1 where |ψ⟩ is a sate vector also known a ”ket” (according to Dirac’s bra-ket notation), ai and |ai⟩ are expansion coefficients and the basis vectors respectively. The setof expansion coefficients can be written as a column vector and the adjoint whichis represented by row vector (also known as ”bra”). In this notation, the inner product of two functions, ϕ(x) and ψ(x) are is given by,

∫ b ⟨ϕ| ψ⟩ = ϕ(x)∗ψ(x)dx (4.11) a In Dirac’s notation, an expectation value of an operator, oˆ, is given by,

⟨o⟩ = ⟨ϕ| oψˆ ⟩ (4.12)

The wave function could be symmetric or anti-symmetric with respect to exchange. The particle with symmetric wave functions have integral multiple of spin (example bosons) and anti-symmetric wave function have half-integral multiple of spin (example fermions).

When the exact wave function is unknown, the trial wave function ψtrial can be used as an initial guess to calculate ground state energy. The energy calculated by using the trial wave function is always larger than the actual ground state energy we would get if we knew the true wave function ψ0 as follows,

ˆ ⟨ψtrial|H|ψtrial⟩ > E0 (4.13)

This is known as the variational principle of quantum mechanics. With the better trial wave function, one can get closer to the real ground state energy, E0

31 Equation 4.7 can only be solved analytically for few special cases such as for N = 1 and M = 1 (hydrogen atom) because it is possible in hydrogen to a) separate the center of mass motions reducing the number of dimensions and b) separate the electronic and nuclear motion. The larger systems that we are interested in contains much more electrons than the nuclei and the gain of reducing the dimensions is not as pronounced. These equations are very difficult to solve even with modern computers so to simplify the computation, Born-Oppenheimer(BO) approximation can be used which we describe in the next section.

4.4 Born-Oppenheimer approximation Nuclear and electronic dynamics can be separated as a nucleus is approxi- mately 1800 times heavier than an electron. The separation is possible because the electron can respond almost instantaneously to the movement of heavier nuclei so it is possible to approximate nuclei as fixed and this allows the system to begov- erned by electronic Schrödinger equation in which nuclear coordinate, R, enters the electronic part of Hamiltonian only as a parameter. The total wave function containing nuclear and electronic wave function is written as,

∑ ψ(r, R) = ψa(r; R)ϕa(R) (4.14) a In equation 4.14, the total wave function is expanded in terms of electronic wave function, ψa(r; R) which is found by solving electronic Schrödinger equation. Because of the mathematical consequence of the physical argument within BO ap- proximation, an electronic Schrödinger equation for the system with fixed nuclear

32 configuration is given by,

ˆ Helψa(r; R) = Ea(R)ψa(r; R) (4.15)

In equation 4.15, Ea and ψa refer to adiabatic energies and wave functions ˆ ˆ ˆ ˆ ˆ respectively and Hel = Tel + Vel-el + Vnuc-el. Inter nuclei interaction (Vnuc-nuc) can only shift the eigenvalue by a constant amount and is often neglected as R enters the Hamiltonian as a parameter rather than an operator.

The equation for ϕa(R) can be obtained by inserting equation 4.14 into equa- tion 4.7 and using equation 4.15.

∑ ∂ϕ (R) Hˆ a (R)ϕ (R) + Θˆ abϕ (R) = iℏ a (4.16) nuc a b ∂t b≠ a The nuclear Hamiltonian is given by,

ˆ a ˆ ˆ ˆ aa Hnuc = Tnuc + V| nuc-nuc + E{za(R) + Θ } (4.17) U a(R) In equation 4.16, Θˆ ab is the non-adiabacity operator that couples the equation of motion for different nuclear states which is given by,

∫ ∑M ∫ ˆ ab ˆ 1 ˆ ˆ Θ = drψa(r; R)Tnucψb(r; R) + drψa(r; R)PK ψb(r; R)PK (4.18) MK K=1 ˆ In equation 4.18, PK = −iℏ∇K Hamiltonian, eigenstates, and eigenvalues of equation 4.15 depend on the nuclear configuration. Basically, BO approximation allows one to compute electronic structure of a molecule without having to worry about the quantum mechanics of the nuclei. Equation 4.15 can be thought of as the Schrödinger equation for ”fixed”

33 nuclei and this equation has to be solved for the range of {R} to obtain potential energy surface (PES) (U a(R)) in equation 4.17) and it can be assumed that the nuclei moves within this surface. Computational cost of determining the PES can be highly reduced by using the classical approximation of many body quantum potential energy surface (QM- PES). In a next section, we will shift the focus and first describe Molecular Dynamic (MD) simulation and the classical Force Field. Then, we will come back to this discussion and explain various ways of solving electronic Schrödinger equation.

4.5 Classical molecular dynamics simulation Classical molecular dynamics simulation is simply the integration of Newton’s equation of motion for a system of classical particles. An initial configuration is generated for the system of interest, specifying the position of every atom. Then each particle in the system is initialized with velocities obtained from the Maxwell- Boltzmann distribution for the desired temperature. The positions and velocities are updated at discrete time steps using an integration scheme such as leapfrog or the Verlet integrator [61]. At each integration step the forces are obtained from the gradient of the potential with the potential defined by a set of parameters called a ”force-field,” described in the next section. This process is repeated for the desired number of steps. To make the simulation relevant to the bulk phase, avoid boundary effect problems caused by the finite size effect and make the system more like infinite one, the periodic boundary condition is used.

34 Figure 4.1: Schematic representation of basic molecular dynamics workflow

4.5.1 Force-Fields and potential energy function The force-field consists of the set of parameters that describe the many-body PES by using intra-molecular (bonded) and inter-molecular (non-bonded) terms (see Fig. 4.2) as in equation 4.19 and 4.20, The potential energy function which is used to compute forces during molec- ular dynamic simulation is given by.

∑ ∑ ∑ 2 2 Vbonded = kb(b − b0) + kθ(θ − θ0) + kϕ[1 + cos(nϕ − δ)]+ bonds angles dihedrals ∑ (4.19) 2 kω(ω − ω0) ha impropers

     12 6 ∑ Rminij  Rminij   qiqj Vnonbonded = ε − 2 + (4.20) rij rij ϵrij nonbonded

35 The parameters that define the force field are determined by a mixture ofQM calculations on small, representative molecular fragments and empirical data, such as solvation free energies and partition coefficients.

4.5.1.1 Bonded terms The first term in equation 4.19 represents stretching of the bond described by a harmonic potential with bond force constant kb and equilibrium length b0. The second term represents angle stretching, which is also described by a harmonic potential with angle force constant kθ and equilibrium angle θ0. The third term is a dihedral angle or torsion angle, which describes the set of four atoms (a,b,c, and d) with the angle between two planes that consist of atoms a, b, c, and b, c, d. kϕ is the dihedral force constant, and n is the multiplicity. The cosine function describes the periodicity of dihedral energy , capturing the symmetry of rotations about rotateable bonds. The last term is the improper term, which ensures the planarity of the atoms in chemical groups and prevents an atom from flipping into its mirror image through a harmonic potential described by the force constant kω and equilibrium out of plane angle ω0.

4.5.1.2 Nonbonded terms The nonbonded potential energy consists of the sum of Lennard-Jones and electrostatic interactions.

4.5.1.2.1 Lennard-Jones The dispersion and exchange interaction is approximated by a standard 12-6

Lennard-Jones (LJ) potential. In equation 4.20, ϵij is well depth, Rmin,ij is radius and rij is distance between i and j atoms.

36 Figure 4.2: Schematic representation of bonded and nonbonded terms in a typical force-field

1 The interaction parameters, ϵ and σ where Rmin,ij = 2 6 × σ are determined for pair of atom types (labelled by indices i and j in equation 4.21 and 4.22) using mixing rule. These parameters are determined in the CHARMM force-field by using Lorentz-Berthelot mixing rule as,

( ) 1 2 ϵij = ϵiϵj (4.21)

( ) σ + σ i j (4.22) σ = ij 2 4.5.1.2.2 Electrostatics The electrostatic interaction is described by the Coulomb potential. In equa- tion 4.20, qi and qj are partial atomic charges of atom i and j respectively. Unlike

37 LJ potential, Coulomb potential is long-range and cannot be simply cutoff as this can lead to serious artifacts [62]. Ewald sum method minimizes the summation by decomposing the interaction into a short-range component (typically 1 nm or so), which is treated with a cutoff in real space and long-range component, which is summed in reciprocal space. In modern MD software the long range electrostatic interactions are computed on a grid and interpolated, an algorithm called “particle mesh Ewald” [63].

4.6 Solving electronic Schrödinger equation In this section we focus on solving electronic Schrödinger equation (equation 4.15), since this is relevant for both determining the parameters for classical force fields, as well as determining the optical properties of chromophores such asProdan. The electronic part of the Schrödinger equation is given by

∑ 2 ∑ 2 ∑ 2 ˆ pˆi 1 e ZK e Hel = + − (4.23) 2me 2 | ri − rj | | ri − RK | i i≠ j i,K Among all of the above interactions, the electron electron interaction is the most complicated because the motion of each electron is influenced by other electrons and consideration of these interactions make the computation difficult. Assuming that the electrons do not interact, Hamiltonian can be written as the sum of single- particle Hamiltonians as follows,   ∑ 2 ∑ 2 ∑ ˆ  pˆ Zke  ˆ Hel = − = hi (4.24) 2me | ri − RK | i K i

38 Now, the single particle problem can be solved to obtain single particle ener- gies εi and single particle wave function ϕi,

ˆ hiϕi = εiϕi (4.25)

The single particle Hamiltonian does not include an electron-electron interac- tion. The exact wave function for non-interacting Hamiltonian is given by product of single-particle wave functions,

∏ Hartree ψi (r) = ϕi(ri) (4.26) i Since the Hartree wave function lacks the anti symmetric property of fermionic wave function, Hartree-Fock defines the wave function as Slater determinant asfol- lows,   1 ∑ ψHF(r) = √ (−1)pP ψ{Hartree} (r) (4.27) i N! perm where P is permutation operator and the counter p is increased for each exchange of electrons. Next, we add electron-electron interaction by using Hartree-Fock wave func- tion as a guess and inserting it into the full Hamiltonian Hel. Also, we drop HF label from ψ(r) and complex conjugation (as we choose real ψ(r) for the finite system of interest).

∫ ˆ 3 ˆ ⟨ψ(r)|Hel|ψ(r)⟩ = d rψ(r)Helψ(r) (4.28)

39 Using equation 4.27 and 4.23, we get

∑N 1 ∑N ⟨ψ(r)|Hˆ |ψ(r)⟩ = ⟨ϕ |hˆ |ϕ ⟩ + (ϕ ϕ |ϕ ϕ ) − (ϕ ϕ |ϕ ϕ ) (4.29) el i i i 2 i i j j i j j i i i,j In equation 4.29,

∫ ∫ e2 (ϕ ϕ |ϕ ϕ ) = d3rd3r′ | ϕ (r) |2 | ϕ (r′) |2 (4.30) i i j j i | r − r′ | j ∫ ∫ e2 (ϕ ϕ |ϕ ϕ ) = d3rd3r′ϕ (r)ϕ (r′) ϕ (r)ϕ (r′) (4.31) i j j i i j | r − r′ | j i The first term in equation 4.29 is single particle energy contribution, the second term is classical electrostatic energy usually known as coulomb or Hartree energy and the last term is called exchange energy. When i = j, the second and third terms cancels each other forbidding the single particle to interact with itself. ˆ The variation of ⟨ψ(r)|Hel|ψ(r)⟩ with respect to single particle wave function

ϕi,

δ⟨ψ(r)|Hˆ |ψ(r)⟩ el = 0 (4.32) δϕi

Imposing the ortho-normality on ϕi as, ∫ 3 d rϕi(r)ϕj(r) = δij (4.33) where δij is Kronecker delta symbol. Above functional derivative yields system of single particle equations known as Hartree-Fock equations as   ∑  ˆ ˆ  HF hi + VH + Vx ϕi = εi ϕi (4.34) i

40 where

∫ ∑N e2 V = d3r′ | ϕ (r′) |2 (4.35) H j | r − r′ | j ∫ ∑N e2 V ϕ (r) = d3r′ϕ (r′) ϕ (r′)ϕ (r) (4.36) x i j | r − r′ | i j j ˆ ˆ VH and Vx are mean-field potential as an individual electron is not affected by all other electrons but only by the averaged potential. Few points about solving Hartree-Fock equations: 1. Equation 4.34 has to be solved self-consistently.

2. The exchange potential Vx (equation 4.36) being non-local acts on ϕi requiring the value of ϕi at every point in space. ˆ ˆ 3. Each electron experiences average potential as VH and Vx sum over all electrons. Hartree-Fock approximation is therefore a mean-field theory. 4. Hartree-Fock wave function is single determinant wave function. As it does not contain linear combination of determinants, it lacks correlation between electrons. Using linear combination of determinants introduces the correlation effect between electrons , ∑ ∑ HF ψ = ψ + ψi + ψij..., (4.37) i i≠ j In equation 4.37, the first term is Hartree-Fock wave function. For system with

N electrons, the ground state determinant is given by |ϕ1(r1)ϕ2(r2)....ϕi(ri)....ϕN (rN)|. For excited state j > N, a singly excited determinant has a single particle excited state of electron i resulting the determinant of |ϕ1(r1)ϕ2(r2)....ϕj(ri)....ϕN (rN)|. The second term and the third term consists of all singly excited and dou- bly excited Slater determinants respectively. The inclusion of the correlation effects

41 makes the computation more costly and different methods such as Configuration interaction, Coupled cluster, Moller-Plesset pertubation theory approximate the ex- pansion to minimize the cost. The correlation energy is defined as the difference in exact value of energy and the best result that can be obtained using HF calculation. This is because HF does not contain correlation effect. In the next section, we discuss another approach that can deal with correlation effects in reasonable computational cost.

4.7 Density Functional Theory Like, Hartree Fock, Density Functional Theory is also a single particle theory. It is not necessary to start with wave function based method because electron density is enough to obtain observable of interest. Unlike wave function, electron density, n, depends only on the position. The total energy of the system is a functional of electron density. The electron density is written in terms of wave function ψ as,

∫ 3 3 2 n(r1) = N d r2..d rN | ψ(r1, ..rN ) | (4.38)

Hohenberg and Kohn have put two simple theorems: i) For any system of interacting particles in an external potential, the density is uniquely determined. ii) The universal functional for the energy E[n] can be defined in terms of the density, and the exact ground state is the global minimum of this functional.

ETot[n] ≥ ETot[n0] ≡ E0 (4.39)

42 where E0 is ground state energy and n0 is ground state density and the functional derivative of E

∂ETot[n] = 0 (4.40) ∂n n=n0

The explicit form of ETot may be written as,

ETot[n] = Eel-nuc[n] + Tel[n] + Eel-el[n] + EXC[n] (4.41)

In above equation (equation 4.41) the exact form of Tel[n] and EXC[n] is not known but the classical contributions of other two terms are given by,

∫ ∫ 1 n(r)n(r′) E [n] = d3(r)d3(r′) (4.42) el-el 2 | r − r′ | ∫ 3 ˆ Enuc-el = d (r)Vnuc-el(r)n(r) (4.43) where Vˆ is given in Eqn 4.5. Since the actual form of kinetic energy is unknown, Kohn and Sham postu-

KS lated the use of fictitious wave function, ϕi in place of real wave function given that the electron density remains unchanged. For the non-interacting fermions, density can be written in terms of non-interacting single particle Kohn-Sham wave functions

KS (ϕi ),

∑N KS 2 n(r) = ϕi (r) (4.44) i=1 Based on single particle formalism, now we can write total energy as,

∫ ∫ ∫ 1 n(r)n(r′) E [n] = T KS[n] + d3rV (r) + d3rd3r′ + E˜ (4.45) Tot el nuc-el 2 | r − r′ | XC

43 ˜ where EXC consists of the exchange-correlation effect and the unknown kinetic energy KS term since Tel [n] is not the true kinetic energy. By ignoring the exchange-correlation for now, we can use variational method to find the stationary point of the energy with respect to change in density as,

∂E Tot = 0 (4.46) ∂ϕi

Imposing the ortho-normality on ϕi as, ∫ 3 d rϕi(r)ϕj(r) = δij (4.47) where δij is Kronecker delta symbol. Now we use Lagrange multiplier εi to enforce orthonormality constrain as,    ∑ ∫ ∂   3  ETot[n] − εj d rϕkϕj − δjk = 0 (4.48) ∂ϕi j,k ∫ ′ 3 ′ n(r ) Substituting Eq 4.45 on above expression (Eq 4.48) and defining V = d r ′ H |r−r | ∂EXC[n] and VXC = ∂n , we can obtain Kohn-sham equation,   ∇2  − ˆ ˆ ˆ  KS KS KS + Vel-nuc + VH[n] + VXC[n] ϕi (r) = εi ϕi (r) (4.49) | 2me {z } Hˆ DFT The above expression has to be calculated self-consistently (Fig 4.3) because

KS KS the density n depends on ϕi and ϕi again depends on n. Also, the above equation describes the fictitious system of non-interacting particles whose density is identical

KS to that of the real system, and εi are Legendre parameters and not the physical energies.

44 Figure 4.3: Schematic representation of self-consistent DFT calculation

4.7.1 Exchange-Correlation Functional The exact form of the exchange-correlation functional is unknown, and there- fore it has to be approximated. The simplest approximation uses the exchange and correlation from the homogeneous electron gas, and VXC in this approximation only depends on the local density of electron. This approximation is known as the Local Density Approximation (LDA). This can be further improved by considering de- pendence of VXC to both electron density n(r) and density gradient | ∇n(r) |. This approximation is known as Generalized Gradient Approximation (GGA). These func- tionals are semilocal as the inclusion of density gradient takes the space around r into consideration. An example of such semi-local functional is PBE0 [64].

HF Some functionals mix the GGAs with Hartree-Fock exchange term Vx [65]

45 as,

GGA m HF − m GGA VXC = Vc + α Vx + (1 α )Vx (4.50) where αm is 0.2 for B3LYP and 0.25 for PBE0 [66]. The long-range behavior of exchange-correlation functional does not show

1 correct r behavior because, in the long-range regime, GGAs falls off proportional to − αm exponential decay exp( r) and hybrid functionals fall off as r . This can be prob- lematic for system with charge-transfer character [67]. The range-separated func- tionals improve this by using DFT exchange at short range and Hartree-Fock exact exchange for long-range and by replacing αm by range dependent cutoff function c(r) = erf(µ | r |). However, the use of these functionals requires system-specific tuning.

4.7.2 Basis sets

The single-particle wave function, ϕ(r)i , consists of infinitely many points so representing it exactly is impossible. The wave function ϕi is therefore expanded in some finite basis with M basis functions χm(r) and expansion coefficients Xij as

∑M ϕi(r) = Xijχj(r) (4.51) j=1 Inserting equation 4.51 into equation 4.55, we obtain,

∑M ∑M ⟨χ |Hˆ DFT|χ ⟩ X = ⟨χ |χ ⟩ X ϵ (4.52) | k {z j} ij | k{z j} ij i j=1 j=1 Fkj Skj In equation 4.52 F , called Fock-Matrix, consist of the expectation value of DFT Hamiltonian and basis functions and S, called overlap matrix, consists of the

46 overlap between basis functions. In matrix notation, the above equation can be rewritten as,

FXi = ϵiSXi (4.53)

The electron density can also be rewritten as,

∑M ∑N n(r) = XikXij χj(r)χk(r) (4.54) jk |i=1 {z }

Dkj where Dkj is the density matrix. Slater Type Orbitals (STO) which has the functional form of χ(r) = l Nl,αYlm|r| exp(−α|r|) are useful in studying the molecular system using linear combi- nation of solutions to the hydrogen atom. STO decay as exp(−|r|) and also consists cusps at atomic nuclei. The exponential form makes evaluation of integrals difficult. Furthermore, STOs do not form orthogonal basis sets which does not gurantee con- vergence when more basis functions are added. Therefore, the sum of gaussian type

l 2 orbitals (GTOs) with the functional form of χ(r) = Nl,αYlmr exp(−αr ) are often used to approximate the slater type orbitals (STOs) because of the computational efficiency to integrate gaussian functions.

4.8 Electronic excitations The closed-shell ground state of the N electron system is described by the

N occupation of the 2 lowest energy single-particle states, also known as orbitals. With the effect of external perturbation such as light, the electron density ofthe system gets rearranged, and an electron or more can occupy the energy level above the Fermi energy, leaving a hole(s) behind in the energy level where it earlier existed. The excited electron interacts with the hole via Coulomb interaction (more precisely

47 via screened Coulomb interaction if the hole is screened by other electrons). The electron-hole pair is known as an exciton. The energy of a bound exciton is less than that of an electron and hole independently because of the interaction between the two. The excitons that is localized within a molecule are known as Frenkel excitons. The states in which electron and holes reside on two molecules or within a different region of a molecule is known as the charged-transfer (CT) states. A closed shell system in a ground state has all the energy levels up to the highest occupied molecular orbital (HOMO) completely filled. All the electrons in this system are paired up, and the total spin of the system is 0. When an exciton gets excited, it can be in a singlet state or a triplet state. Only the singlet state can be filled by an excitation due to the absorption of light because as discussed in Chapter 1, the electron spin in triplet excited state is parallel to that of singlet ground state and the change in multiplicity makes this a forbidden transition.

4.8.1 General information about excited state calculations In spectroscopic experiments, a system responds to an external perturbation (due to incoming photons, electrons, and so on) by changing its electronic configura- tion such that the ground state electron can be promoted to excited states. This could result in direct photoemission, inverse photoemission, and absorption, as sketched in Fig. 4.4 [68]. In the case of the photoemission, an electron gets ejected with some kinetic energy after the system absorbs a photon of energy hν. At a significant distance where the electron gets decoupled from the sample, one can calculate the change in the total energy of the system based on the momentum and energy conservation. The change in energy is the energy of the hole, which is resulted due to the ejected photo- electron. During photoemission, the density of occupied states is measured, and the

48 Figure 4.4: Schematic representation of a) photoemission b) inverse photoemission, and c) absorption in a molecule. Adapted from [15] single electron levels are renormalized to include the response of other electrons that can relax after the emission of the photo-electron and other quantum-mechanical contributions. Likewise, in the case of the inverse photoemission, the density of unoccupied states is measured, which is also renormalized. One can still use one particle energy level as long as quasi-electrons and quasi-holes are used, which con- tain the effects of all the particles. Thus, in the case of photoemission and inverse photoemission, the quasi-particle (quasi-hole or quasielectron) is considered, and the solution of one-quasi-particle Schrödinger equation is adequate [68]. In contrast, in the process of absorption, the excited electron remains coupled to the hole it leaves behind in the ground state. Therefore, the joint density of occupied and unoccupied states must be taken into account as one cannot treat absorption as the sum process of photoemission and inverse photoemission. A small change in excitation energy can bring a significant change in transition probability.

49 One-quasi-particle Schrödinger equation is inadequate as the quasi-hole and quasi- electron have to be simultaneously considered. Therefore, the effective two-particle equation is required [68]. In the case of absorption, the following methods are used in calculating excitation energies.

• Many-Body Perturbation Theory (used in the present work and discussed be- low),

• Time-Dependent Density Functional Theory (TD-DFT): Less cumbersome but Hamiltonian is not well established. Because of approximate exchange- correlation functional and adiabatic approximation, in an extended system, TD-DFT can suffer from the following self-interaction errors, especially forthe Charge Transfer (CT) excitations.

1 – Incorrect r potential (that dies too rapidly) where r is distance between electron and hole, – High exchange correlation potential in inner region resulting in unbounded virtual orbitals which can lead to substantial errors for excitation energies in Rydberg states where electron travels far away from nucleus and – Incorrect interaction (unscreened or bare) between electron and hole

• Configuration Interaction: Good but computational cost is very high[69]

In this work, GW -BSE calculation is performed for excited-state calculations, and the Kohn-Sham energy levels from Density functional theory (DFT) are used as the starting point. The Kohn-Sham (KS) equation (equation 4.49) is solved to find the ground ˆ ˆ state, |N,0⟩ energy levels in which VH is hartree potential, VXC is exchange-correlation ˆ potential, and Vext is external potential.   ∇2  − ˆ ˆ ˆ  KS KS KS + Vext + VH[n] + VXC[n] ϕi (r) = εi ϕi (r) (4.55) | 2me {z } Hˆ DFT

50 In the next section, excited states calculation using Many-body Green’s Func- tion is discussed.

4.9 Electronically excited states calculation using Many-body Green’s Function: GW -BSE 4.9.1 One particle excitation The one particle Green’s function (equation 4.56) describes the addition (and removal) of electron to (and from) the system in ground state, |N, 0⟩. In equation 4.56, Tˆ is the fermionic time ordering operator (equation 4.59) and ψ and ψ† are the annihilation and the creation electron field operator (equation 4.57 and 4.58), respectively in Heisenberg picture as described in equation 4.60 . Also, the space-time variables are contracted into single variable; for instance 1 ≡ r1, t1.

ˆ † G1(1, 2) = −i⟨N, 0|T ψ(1)ψ (2)|N, 0⟩ (4.56)

∑ ˆ† † ψ (r) = ψi(r)ˆa (4.57) i ∑ ˆ ψ(r) = ψi(r)ˆa (4.58) i [ ] Tˆ ψˆ(1)ψˆ†(1′) = Θ(t − t′)ψˆ(1)ψˆ†(1′) − Θ(t′ − t)ψˆ(1′)ψˆ†(1) (4.59)

ψˆ(r, t) = exp(iHtˆ )ψˆ(r)exp(−iHtˆ ) (4.60)

51 Creation operator can be used to calculate excitation energy as shown below [15]:

⟨n, 0|ψˆ(r, t)|n + 1, s⟩ = ⟨n, 0|exp(iHtˆ )ψˆ(r)exp(−iHtˆ )|n + 1, s⟩ ˆ = ⟨n, 0|exp(iEn,0t)ψ(r)exp(−iEn+1,st)|n + 1, s⟩   (4.61) ˆ   = ⟨n, 0|ψ(r)|n + 1, s⟩exp − i(En+1,s − En,0)t

ˆ where ⟨n, 0|ϕ(r)|n + 1, s⟩ = fs(r) and (En+1,s − En,0) = εs ′ Assuming t greater than t , G1 can be written in frequency domain as,

∑ f (r)f ∗(r′) G (r, r′, ω) = s s (4.62) 1 ω − ε − iη s s

Equation of motion for one particle Green’s function, G1 (equation 4.64), can be found by using equation of motion for field operator in Heisenberg picture (equation 4.63). In equation 4.64, h can be non-interacting, Hartree, Hartree-Fock, or Kohn-Sham hamiltonian and v is bare Coulomb potential. G2 is 2-particle Green’s function which is defined in equation 4.65 by using contracted variable (r1, t1 ≡ 1)   ∂ψˆ(r, t) = ψˆ(r, t), Hˆ  (4.63) ∂t

  ∫ ∂  −hG (rt, r′t′)+i d3r′′v(r, r′′)G (r′′, t, r′, t′, r′, t′, r, t) = δ(r−r′)δ(t−t′) (4.64) ∂t 1 2

2 ˆ † † G2(1, 2, 3, 4) = i ⟨N, 0|T ψ(1)ψ(3)ψ (4)ψ (2)|N, 0⟩ (4.65)

52 The equation of motion for 1-particle Green’s function 4.64 contains 2-particle Green’s function and this continues for higher order Green’s functions. Therefore, one arrives at infinite set of equations and to get around this problem, we introduce the self-energy operator, Σ(r, t, r′, t′) as follows:

∫ ∫ 3 ′′ ′′ ′′ ′ ′ ′ ′ 3 ′′ ′′ ′′ ′′ ′′ ′′ i d r v(r, r )G2(r , t, r , t , r , t , r, t) = − d r dt Σ(r, t, r , t )G1(r , t , r, t) (4.66) Now that 2-particle Green’s function is expressed in terms of the product of self-energy operator and 1-particle Green’s function, we will focus on how to find the self-energy operator. The self-energy operator can be obtained from the following closed set of coupled equations: [70]

∫

G1(1, 2) = G0(1, 2) + d34G0(1, 2)Σ(3, 4)G1(4, 2) (4.67)

∫

Σ(1, 2) = i d34G1(1, 3)W (1, 4)Γ(4, 2, 3) (4.68)

∫ ∂Σ(1, 2) Γ(1, 2, 3) = δ(1, 2)δ(1, 3) + d4567 G1(4, 6)G1(5, 7)Γ(6, 7, 3) (4.69) ∂G1(4, 5)

∫

P (1, 2) = −i d34G1(1, 3)G1(4, 1)Γ(3, 4, 2) (4.70)

∫ W (1, 2) = v(1, 2) + d34v(1, 3)P (3, 4)W (4, 2) (4.71)

53 In the above set of equations (4.67 through 4.71), G0 is non-interacting Green’s function, Γ is vertex correction, P is polarizability, W is screened Coulomb interac- tion, and v is the Coulomb interaction. The macroscopic dielectric function (ϵ) is also related to polarizability and screened Coulomb interaction (equation 4.73 and 4.74). The screened Coulomb interaction, W is evaluated by first calculating the polariz- ability, P is evaluated within the random-phase approximation (equation 4.72)[71].

∑occ ∑vir ∗ ′ ∗ ′ ψi(r)ψ (r)ψj(r )ψ (r ) P (r,r′, ω) = (f − f ) j i (4.72) i j ω − (E − E ) + iη i j j i ∫ ϵ(1, 2) = δ(1, 2) − d3v(1, 3)P (3, 2) (4.73)

∫ W (1, 2) = d3ϵ−1(1, 3)v(3, 2) (4.74)

∫ d2ϵ(1, 3)ϵ−1(2, 3) = δ(1, 3) (4.75)

Since solving the derivative of the self-energy with respect to Green’s function in equation 4.69 is tedious, one reduces the vertex correction to an identity within the framework of GW approximation as follows:

Γ(1, 2, 3) = δ(1, 2)δ(1, 3) (4.76)

This greatly simplifies equation 4.68 through 4.71 as follows (schematic diagram in Fig 4.5):

Σ(1, 2) = iG1(1, 2)W (1, 2) (4.77)

∫ ϵ(1, 2) = δ(1, 2) − d3v(1, 3)P (3, 2) (4.78)

54 Figure 4.5: Schematic representation Hedin’s equations

P (1, 2) = −iG1(1, 2)G1(2, 1) (4.79)

∫ W (1, 2) = d3ϵ−1(1, 3)v(3, 2) (4.80)

Even though the GW approximation is not a formal perturbation, it is a good idea to expand self-energy in terms of screened Coulomb interaction instead of bare Coulomb interaction because writing self-energy as power series of bare Coulomb ei- ther diverges quickly or converges very poorly with increased polarizability. Since the screened Coulomb interaction is docile in long-range, the dielectric function reduces to Random Phase Approximation (RPA) screening [70]. Expressing 1-particle Green’s function in spectral representation by consider- ing complex frequency (expressing ω to z ∈ C), one gets equation 4.81 and combining

55 this to the equation of motion we obtained in equation 4.64 and 4.66, we get equation 4.82.

∑ ψ (r, z)ψ¯ (r′, z) G (r, r′, z) = i i (4.81) 1 z − E (z) i i ∫ ˆ 3 ′ ′ ′ h0ψi(r, z) + d r Σ(r, r , z)ψi(r , z) = Ei(z)ψi(r, z) (4.82)

Since it is difficult to evaluate self-energy Σ for all frequencies, we further assume that the low lying frequencies can be described by the quasi-particle approx- imation where quasi-particle energies εQP are calculated within GW approximation (Σ = iGW ). The poles of the 1-particle Green’s function yield the energies of a particle-like excitations as follows:

QP QP εi = Ei(εi ) (4.83)

Inserting equation 4.83 into equation 28, we get the following in Dirac nota- tion:  

ˆ QP | QP⟩ QP| QP⟩ h0 + Σ(εi ) ϕi = εi ϕi (4.84)

Quasi-particles are mathematical constructs but as long as their lifetime is long (ie, Im(Σ) and correlation are small), one can think of these quasi-particles as particles that interact with each-other via screened Coulomb interaction (W) instead of bare Coulomb interaction (v) as shown in Fig. 4.6 [15]. Spectral function A(ω) (equation 4.85 where µ is chemical potential and sgn is signum function) can be found by taking the imaginary part of the 1-particle Green’s function. The coherent part of the spectrum (Fig. 4.7) is centered around εGW ; shift

56 Figure 4.6: The system of interacting particles (a) when approximated by quasi- particle approximation (b) interact via screened Coulomb interaction,W instead of bare Coulomb, v. Adapted from [15]. of the energy compared to non-interacting energy is given by real part and the full half width maxima of the Lorentzian is given by the imaginary part of the self-energy (Σ(εGW )). The incoherent part of the spectrum is neglected within Quasi-particle approximation [15].

1 A(r, r′, ω) = − Im(G (r, r ′, ω)sgn(ω − µ) (4.85) π 1 KS In practice, KS equation (equation 4.49) is solved to find eigenvalues (εi ) and KS eigenfunctions (ϕi (r)). These eigenvalues and eigenfunctions are used to construct non-interacting 1-particle Green’s function(G0) by using equation 4.86.

∑ ϕKS(r)ϕKS(r′) G (r, r′, ω) = i i (4.86) 0 ω − εKS  iη i i

57 Figure 4.7: Spectral function of an infinite system within the quasi-particle approx- imation. Adapted from [15].

ˆDFT Next step is to solve equation 4.87 where h0 is DFT Hamiltonian in self- consistent approach.  

ˆDFT QP − ˆ | QP⟩ QP| QP⟩ h0 + Σ(εi ) VXC ϕi = εi ϕi (4.87)

In simpler case, first, Kohn-Sham states can be approximated as GW quasi- particle states,

| QP⟩ ≈ | KS⟩ ϕi ϕi (4.88)

and then Kohn-Sham energies can be corrected to get GW corrected KS-state energies as follow:

GW − KS ⟨ KS| GW − ˆ | KS⟩ εi εi = ϕi Σ(εi ) VXC ϕi (4.89)

If G and W are evaluated based on KS levels, the approach is known as

GW G0W0. Substituting energy output from the G0W0 calculation back into Σ until εi

58 converges is represented as GW0. This energy can be feed into the self-consistent calculation of W again and this method is named as evGW.[15] If the off-diagonal elements of the self-energy operator is not negligible, equa- tion 4.88 does not hold and hence quasi-particle state has to be constructed by using Kohn-Sham states in following way: ∑ | QP⟩ | KS⟩ ϕi = Cij ϕj (4.90) j

Since the self-energy operator is non-hermitian, equation 4.84 has to be solved approximately [72],   1 ⟨ϕQP|Σ|ϕQP⟩ = ⟨ϕQP|Σ(εQP)|ϕQP⟩ + ⟨ϕQP|Σ(εQP)|ϕQP⟩ (4.91) i j 2 i i j j j i

Even though this method is very good for single-particle excitations (such as photoemission and inverse photoemission), it is not adequate for absorption (need to consider quasi-particle quasi-particle interaction). As the vertex correction is neglected in this method, the GW approximation does not incorporate quasi-particle quasi-particle interaction, and hence the 2-particle excitations have to be taken into account. As we have calculated dielectric-matrix (ϵ(ω)) within GW approximation (neglecting vertex correction Γ), the screening becomes an independent particle. Including the vertex correction (Γ) is also not practical, as this makes numerical calculation very difficult. Therefore, we will proceed with the 2-particle excitation in the next section.

4.9.2 Two-particle excitations The neutral excitations, unlike single-particle excitations, cannot be described by 1-particle Green’s functions. Since the polarizability was calculated by neglecting

59 vertex correction, Γ, it reduces to RPA (equation 4.72), and therefore the dielectric matrix, ϵ(ω) and screened Coulomb interaction, W are also calculated within RPA. On the other hand, the inclusion of vertex correction term, Γ poses numerical dif- ficulty as the expansion of polarizability would depend on higher-order polarization functions [15]. Therefore, we will expand two-point Green’s function to four-point Green’s function. First, we define two-particle correlation function as,

′ ′ ′ ′ ′ ′ L(12, 1 2 ) = −G2(12, 1 2 ) + G1(12)G1(1 2 ) (4.92)

The first term of the right-hand side of equation 4.92 is the two-particle Green’s function (correlation of electron and hole), and the second term is the prod- uct of two one-particle Green’s function (uncorrelated motion of electron and hole) [15]. Next task is to introduce the Dyson-like equation for L: the Bethe-Salpeter- Equation(BSE),

∫ ′ ′ ′ ′ ′ ′ L(12, 1 2 ) = L0(12, 1 2 ) + d3456L0(14, 1 3)K(35, 46)L(62, 52 ) (4.93)

′ ′ ′ ′ Equation 4.93 (where L0(12, 1 2 ) = G1(1, 1 )G1(2, 2 )) can be simplified by as- suming simultaneous creation and annihilation of quasi-particles during optical exci- tations. Considering Hamiltonian that does not explicitly depend on time, L(12, 1′2′) simplifies to L(12, 1′2′, ω) where we omit time variables. The Kernel (in equation 4.93) is given by [15],     δ Σ(3, 4) + VH K(35, 46) = (4.94) δG1(6, 5)

60 Using GW approximation (Σ = iGW) and considering that the screening is unaffected by the excitation ( ∂W ≈ 0), we can write the interaction Kernel as, ∂G1

K(35, 46) = −iδ(3, 4)δ(5, 6)ν(3, 6) + iδ(3, 6)δ(4, 5)W (3, 4) (4.95) = kx(35, 46) + kd(35, 46)

In equation 4.95, kd is direct interaction which binds electron-hole pair via screened interaction, W and kx is exchange interaction which is responsible for singlet-triplet splitting and this acts through unscreened interaction, v [73].

  ∑ ϕ (r )ϕ (r )ϕ∗(r ′)ϕ∗(r ′) ϕ (r )ϕ (r )ϕ∗(r ′)ϕ∗(r ′) L (r , r , r ′, r ′, ω) = i  c 1 v 2 v 1 c 2 − v 1 c 2 c 1 v 2  0 1 2 1 2 ω − (ε − ε ) + iη ω − (ε − ε ) − iη νc c v c v (4.96)

Equation 4.96 is the position representation of L0 as the sum of independent ex- citations in which ν runs over occupied holes and c runs over unoccupied electron states[73]. Electron-hole amplitude is defined in terms of product state of independent excitations (in a different basis: n1, n2..) to get single-particle wave function,

∑ ex ′ ∗ χ (r, r ) = ϕn1 (r1)ϕn2 (r2) (4.97) n1,n2 ′ ′ We can transfer equation 4.96 from (r1,r2,r1 ,r2 ,ω) basis to (n1, n2, n3, n4, ω) basis,

  ∑ δ δ δ δ δ δ δ δ −iLn1,n2,n3,n4 (ω) =  n1c n2v n3c n4v − n1c n2v n3c n4v  (4.98) 0 ω + (ε − ε ) − iη ω − (ε − ε ) + iη νc c v v c

61 Equation 4.98 becomes diagonal when n1=n3 and n2=n4 and hence occupied to unoccupied (and vice-versa) transitions take place. Consider the operator in matrix notation,  

 (n3n4)   | {z }      (|n{z1n2}) vv cc vc cv       vv 0 0 0 0  −iL0(ω) =   (4.99)    cc 0 0 0 0     −1   vc 0 0 ε −ε −ω 0   n2 n1  1 cv 0 0 0 − − εn2 εn1 ω Considering the subspace of matrix 4.99 in occupied to unoccupied transitions and vice-versa, we define matrix F (writing the 2-particle non-interacting Green’s function in occupation representation, fi where fν = 1 and fc = 0 and defining occupation matrix) as,   f − f 0  |2 {z }1     −   1  F =   (4.100)  −   |f2 {z f}1   0 1

and using L0 = L0F we can write Bethe-Salpeter equation in matrix notation as, 1 − −1 iL (ω) = −1 (4.101) −iL0 (ω) − iFK(ω)

62 The first term in the right hand side of equation 4.101 is diagonal and can be inverted,

     

εn − εn − ω 0  εn − εn 0  ω 0 −iL−1(ω) =  2 1  =  2 1  −   0 − − − 0 εn2 εn1 ω 0 εn2 εn1 0 ω (4.102) Now, we will write BSE Hamiltonian as,  

εn − εn 0  HBSE =  2 1  − iFK(ω) − 0 εn2 εn1     (4.103)

εn − εn 0   Kvc,v′c′ Kvc,c′v′  =  2 1  + i   − − ′ ′ − ′ ′ 0 εn2 εn1 Kcv,c v Kcv,c v

More generally BSE Hamiltonian is written as,   Hres K  HBSE =   (4.104) −K −Hres

In equation 4.104, resonant part, Hres and anti-resonant part, −Hres describe transitions from occupied to unoccupied states (positive frequency) and unoccupied to occupied states (negative frequency) respectively. The off-diagonal term cou- ples these transitions. When the off-diagonal terms are neglected, it is known as Tamm-Dancoff approximation (TDA) which reduces the size of Hamiltonian byhalf. The full BSE calculation contains both the diagonal and off-diagonal terms and is computationally more rigorous.

63 Now the task is to solve the following eigenvalue problem using the non- Hermitian operator,

BSE H |χS⟩ = ΩS|χS⟩ (4.105)

In matrix notation we get,       res S S BSE H K  A  A  H =     = ΩS   (4.106) −K −Hres BS BS

⟨ | ⟩ S ∗ S ∗ where r1, r2 χS = χS(r1, r2) = Avcϕc(r1)ϕv(r2) + Bvcϕv(r1)ϕc (r2) The diagonal and off-diagonal terms of the Hamiltonian in equation 4.106 is given by,

res x d ′ ′ Hvc,v′c′ (ω) = Dvc,v′c′ + Hvc,v′c′ + H vc, v c (4.107) x d Kcv,v′c′ (ω) = Hcv,v′c′ + Hcv,v′c′

In non-interacting quasi-particle basis, the matrix elements in equation 4.107 become:

Dvc,v′c′ = (εc − εv)δvv′ δcc′ ∫ ′ ∗ ′ ′ ∗ ′ x 3 3 ′ Hvc,v′c′ = d r d r ϕc (r)ϕv(r)ν(r, r )ϕc (r )ϕv′ (r ) (4.108) ∫ ′ ∗ ′ ∗ ′ ′ d 3 3 ′ Hvc,v′c′ = d r d r ϕc (r)ϕc (r)ϕv(r )ϕv′ (r )W (r, r , ω = 0)

In equation 4.107, we use static approximation as dynamical properties of screened Coulomb interaction, W, is neglected which is valid if ΩS − (εc − εv) << ωl where ωl is plasmon frequency [15]. In everything we have done until now we have assumed: a) Born-Oppenheimer approximation, b) the system is closed shell and c) the net spin is zero. Assuming

64 that the spin-orbit coupling is negligible, the following basis span the Hilbert space of the spin singlet ground state,

|↑↑⟩, |↑↓⟩, |↓↑⟩, |↓↓⟩ (4.109)

The first arrow represents spin of the electron and the second arrow represents the spin of the hole. The singlet and triplet states can be calculated separately by using following BSE Hamiltonian:

BSE d x Hsinglet = D + H + 2H (4.110) BSE d Hsinglet = D + H However, this only works as long as the spin-orbit coupling is small. If not, the full spin structure must be considered [15]. In the next section, GW -BSE/MM will be briefly discussed.

4.10 GW -BSE/MM The system of a dye in bulk solvent or bio-membranes is divided into two regions: quantum (QM) region and the classical (MM) region, as shown in Fig. 4.8. Prodan or Laurdan exist in QM region, whereas the solvents and lipids are present in the MM region. The mixed approach is used, in which the excited state of the dye is cal- culated within GW -BSE embedded in the electrostatic background of the solvent sampled by classical molecular dynamics. The coupling of the solvent and the solute in the evaluation of the excitations within GW -BSE is performed using a molecular mechanics (MM) representation of the electrostatic potential. Specifically, the sol- vent molecules in the MM region are represented by the static atomic point charges

65 Figure 4.8: Schematic diagram showing Prodan in QM region and water molecules in MM regions for QM/MM calculation

a Q at positions Ra where a indicates the atoms in the MM region. The classical electrostatic potential

∑ Qa VMM(r) = (4.111) |r − Ra| a∈MM ˆ ˆ from the MM region is added to Vext in h0 in Eqs. (4.55) and (4.84), respec- tively.

66 Chapter 5

CHARMM GENERAL FORCE FIELD PARTIAL CHARGE PARAMETRIZATION OF PRODAN/LAURDAN

5.1 Introduction The quality of the force-field parameters determines the quality of the result of MD simulation. Of all the interactions present in the CHARMM Force Field, electrostatic interaction plays a vital role in governing the solvent relaxation process explained in Chapter 2. In equation 4.20, the correctness of the Coulomb interaction between a fluorescent molecule (Prodan or Laurdan) and other molecules (solvent and lipids) is based on the accuracy of the partial charges of both fluorophore and its surrounding molecules. In CHARMM, both lipid and water parameters have been parametrized, widely used, and improved when necessary. However, the Prodan or Laurdan pa- rameters have not been as widely used and as systematically verified. Thus, these parameters need to be reviewed and improved, if necessary. In this chapter, the CGenFF partial charge parametrization protocol is ex- plained, and the result of Prodan/Laurdan partial charge parametrization is pre- sented.

5.2 Method The appropriate model compound is identified, as shown in Fig. 5.1, and used for charge optimization. The model compound excludes the hydrocarbon tail

67 Figure 5.1: Chemical structure of 6-Dodecanoyl-N,N-dimethyl-2-naphthylamine, commonly known as Laurdan (left), 2-dimethylamino-6-propionylnaphthalene, com- monly known as Prodan (middle) and model compound used in partial charge parametrization (right). of Laurdan, which is justified because the hydrocarbon tail, also present in lipids, has already been parametrized by CHARMM force field developers. Furthermore, excluding the hydrocarbon tail decreases the number of particles in the numerical quantum calculation and minimizes the computational cost. The structure of the model compound is constructed using Molden [74] and the mol2 file is uploaded to the Paramchem web server [75, 76] to obtain the initial guess of the partial charges from the CGenFF [77]. The Paramchem webserver outputs a stream file that consists of all the bonded and non-bonded parameters of the model compound. These parameters (partial charges in our case) is used as the initial guess. The initial charges did not yield a ground-state dipole moment in agreement with the available data (see Table 5.1), so it was necessary to optimize

68 Table 5.1: The components of dipole moment of the model compound calculated by using set of initial partial charges and final charges compared to QM target value calculated with HF/6-31G(d) level of theory. The unit of dipole moment is in Debye (D)

Dipole moments µx(D) µy(D) µz(D) HF/6-31G(d) and MP2/6-31G(d) -3.68 2.34 -0.94 Initial Charges -3.8 -0.24 -1.27 Final Charges -3.06 2.49 -0.75 the partial charges. Partial charges were optimized using the standard procedure provided by the CHARMM family of force fields [77, 78]. First, the geometric optimization of the model compound was performed at MP2/6-31G(d) model chemistry using Gaussian09 [79]. CHARMM minimized structure was taken as the starting structure for this optimization. The model compound-water complex is constructed by placing water molecules next to atoms of the model compound (see Fig. 5.2) at the distance of 2.1 Åat specified geometry. The angles made by all water (TIP3P geometry [80]) with respect to an atom of interest from the model compound (MP2/6-31G(d) optimized geometry) are provided in Table 5.2 in which the angle between an atom X in the model compound and atom O/H of water is defined as X–OHH/X–HOH. In most of ◦ the cases, only one geometry is considered while for C14 and O two geometries (0 and 90◦) are considered (Fig. 5.2). Model compound-water interaction energies are computed using Gaussian09 [79] using HF/6-31G(d) model chemistry. The model compound-water interaction distance is optimized at HF/6-31G(d) by keeping all other degrees of freedom fixed. The optimized distance is often different from 2.1 Å. The QM single-point energies are then calculated for the model compound-water complex. All the QM calculations

69 Figure 5.2: The schematic diagram showing interaction orientations of the model compound and water molecules used for charge optimization. Only one water molecule interacts with the model compound during each calculation. All water molecules are shown in this figure for convenience only. Only the cases represented by the dashed blue lines are used for the charge optimization to avoid the effect of secondary interaction from other atoms of the model compound.

70 kcal Table 5.2: The QM and MM interaction energies ( mol ) between various atoms in model compound and a water molecule. The MM results are presented for the initial (IC) and final (FC) set of charges. The difference between MM and QM energies are also presented for both cases. The atom names correspond to Fig. 5.1 and 5.2

IC IC FC FC Interacting atoms EQM EMM ∆E EMM ∆E N-HOH 0◦ -4.51 -5.23 -0.73 -4.36 0.14 ◦ C1H–OHH 90 -0.14 -0.6 -0.46 -0.19 -0.05 ◦ C2H–OHH 90 -0.07 -0.55 -0.48 -0.11 -0.04 ◦ C7H–OHH 90 -1.71 -1.41 0.30 -1.21 0.5 ◦ C8H–OHH 90 -0.81 -1.02 -0.20 -0.82 -0.01 ◦ C9H–OHH 90 -1.11 -1.98 -0.87 -0.80 0.31 ◦ C10H–OHH 90 -2.47 -2.9 -0.43 -3.0 -0.53 ◦ C14H–OHH 90 -0.45 -2.0 -1.56 -0.87 -0.42 ◦ C14H–OHH 90 -0.44 -1.99 -1.55 -0.88 -0.43 O–HOH 0◦ -7.2 -7.09 0.11 -6.08 1.11 O–HOH 90◦ -5.74 -6.41 -0.67 -6.37 -066 are performed using Gaussian09 [79]. The water molecule may not be in the same position after distance optimiza- tion; it may slightly move closer towards other atoms (other than the atom initially linked to) in the model compound, which can cause the possibility of the unwanted secondary interaction. Therefore, the cases corresponding with the possibility of the secondary interactions are removed from the parametrization process. As shown in Fig. 5.2, only the subset of cases represented by blue dashed lines is used for charge optimization. The calculation mentioned above was also repeated for the model compound and a water molecule separately. The water interaction energies are obtained by taking the difference of the single point energy of the model compound(mc)-water complex and the sum of the single point energies of the water molecule and the model compound separately, as shown in equation 5.1. Water interaction energies

71 are scaled by the factor of 1.16 to be relevant for the bulk phase [77].

EInt = Emc-water − Emc − Ewater (5.1)

The level of theory and the scaling of energy is done to maintain the compat- ibility with CHARMM additive force fields. Scaled HF/6-31G(d) model compound- water interaction energies, dipole moment, and distance between the model com- pound and individual water molecule are used as the target data. The water interaction calculation mentioned above was also performed using the CHARMM force field (Molecular Mechanics (MM)) for the geometries corre- sponding to QM calculation with an initial and adjusted set of partial charges and compared to QM calculated energies. Partial charges are adjusted to obtain the best fit between QM and MM interaction energies and dipole moments. Also, the dipole moment of the model compound changes when the partial charges are adjusted, so the dipole moment has to be iteratively checked. As the charges are adjusted, the charges of aliphatic hydrogen atoms are unchanged, and the default value of +0.09e is used.

5.2.1 Results Figure 5.3 shows the HF/6-31G(d) model compound-water interaction ener- gies calculated using the HF/6-31G(d) level theory, and MM with both initial and the final set of charges. The atom names of the model compound correspond tothe names provided in Fig. 5.1. For most of the cases, the interaction energy of the final set of parametrized charges agrees better with target data. The water inter- action energies calculated using QM, MM, and the difference between the two are presented in Table 5.2, and the water interaction distances are shown in Table 5.3 for both initial and final set of partial charges.

72 Figure 5.3: Model compound-water interaction energies obtained from HF/6- 31G(d)(in red), CGenFF with initial set of partial charges (in cyan) and final set of partial charges (in blue). * and ** correspond to two different geometric orientations (0◦ and 90◦ respectively) of a water oxygen next to the specified atom in the model compound.

Table 5.3: The QM and MM interaction distance() between various atoms in model compound and a water molecule. The MM results are presented for the initial (IC) and final (FC) set of charges. The difference between MM and QM distances are also presented for both cases. The atom names correspond to Fig. 5.1 and 5.2

IC IC FC FC Interacting atoms dQM dMM ∆d dMM ∆d N-HOH 0◦ 2.20 -5.23 -0.73 2.04 -0.16 ◦ C1H–OHH 90 2.66 -0.6 -0.46 2.89 0.23 ◦ C2H–OHH 90 2.66 -0.55 -0.48 2.92 0.26 ◦ C7H–OHH 90 2.44 -1.41 0.30 2.74 0.30 ◦ C8H–OHH 90 2.59 -1.02 -0.20 2.78 0.19 ◦ C9H–OHH 90 2.54 -1.98 -0.87 2.97 0.43 ◦ C10H–OHH 90 2.78 -2.9 -0.43 2.57 -0.21 ◦ C14cH–OHH 90 2.59 -2.0 -1.56 2.8 0.21 ◦ C14cH–OHH 90 2.59 -1.99 -1.55 2.8 0.21 O–HOH 0◦ 2.01 -7.09 0.11 1.75 -0.26 O–HOH 90◦ 2.05 -6.41 -0.67 1.7 -0.35

73 Figure 5.4: Prodan ground state partial charge values before (left) and after parametrization (right) in the unit of elementary charge. The red, cyan, and blue ar- rows represent twice the dipole moment vector obtained by MP2/6-31G(d), CGENFF with initial and final set of partial charges respectively.

The dipole moment of the model compound is calculated using MP2/6- 31G(d), HF/6-31G(d), and CGenFF with both initial and final set of partial charges are shown in Table 5.1. The y-component of the dipole moment calculated using an initial set of partial charges are dramatically improved by the parametrization process and compared much better to target data. Although not as dramatic, the same is also true for the z-component of the dipole moment vector. Both initial and optimized charges and the atom types are shown in Table 5.4. Figure 5.4 shows the initial and final sets of partial charges along with the comparison of initial and final dipole moment vectors to the QM calculated dipole moment. All of the dipole moment vectors in Fig. 5.4 are scaled by the factor of two.

74 Table 5.4: The atom name, atom type, and the partial charges (both initial and final or optimized) in the unit of elementary charge (e). All the hydrogen connected to C1, C2, and C14 have the default charge of +0.09e. The atom names in the able correspond to the atom names in Fig. 5.1 and 5.2

Atom Type Final charge Initial charge N NG301 -0.420 -0.400 C1 CG331 -0.225 -0.149 C2 CG331 -0.225 -0.149 C3 CG2R61 0.350 0.155 C4 CG2R61 -0.008 -0.119 H4 HGR61 0.058 0.115 C5 CG2R61 0.297 0.006 C6 CG2R61 0.327 0.005 C7 CG2R61 -0.153 -0.114 H7 HGR61 0.036 0.115 C8 CG2R61 -0.200 -0.119 H8 HGR61 0.086 0.115 C9 CG2R61 -0.320 -0.115 H9 HGR61 0.020 0.115 O OG2D3 -0.490 -0.470 C10 CG2R61 -0.500 -0.118 H10 HGR61 0.321 0.115 C11 CG2R61 0.569 0.071 C12 CG2R61 -0.198 -0.118 H12 HGR61 0.015 0.115 C13 CG205 0.429 0.359 C14 CG331 -0.580 -0.229

75 The optimized partial charges are used in the Prodan and Laurdan simula- tions. The model compound used during the charge optimization does not contain the hydrocarbon tail of Laurdan (C11H23) and CH2 of Prodan. As mentioned in the introduction, the parameters of hydrogen and carbons of the fatty acid tail are well established and verified. Therefore, the partial charges of these atoms (+0.09e for hydrogen and -0.180e for carbons) are directly used from the CHARMM Force Field. The linking hydrogen atom has to be removed when connecting the model compound to the excluded fragment of Prodan or Laurdan, which requires the adjustment of the charge of removed hydrogen (0.09e). The results of QM/MM simulation of Prodan in solvents of various polarities are presented in the next chapter.

76 Chapter 6

MOLECULAR DYNAMICS SIMULATION OF LAURDAN IN LIPID MEMBRANES

6.1 Introduction The plasma membrane in the animal cell is a complex mixture of hundreds of spatially heterogeneous lipid species and proteins that is asymmetric with respect to two leaflets. The local structural properties of the membranes can be studied using polarity sensitive dyes such as Prodan, Laurdan, and C-Laurdan. The fluorescent emission of these dyes is sensitive to membrane properties like membrane order and packing because the emission spectra are about 50 nm blue-shifted going from liquid disordered (Ld) phase to gel phase or cholesterol-rich liquid-ordered (Lo) phase. The standard model used to explain the spectral shift is based on the polar- ity of the environment surrounding the chromophore. Upon excitation, the dipole moment of the Prodan-class of dyes increases from (2.1-5.5) Debye to about (6.4- 20.0) Debye, as shown in Table 7.3. A polar environment will relax in response to this increase in dipole, reducing the energy of the excited state relative to a less polar environment, which in turn reduces the energy of the emission; thus the less polar environment is blue-shifted (higher energy) relative to the more polar environment (see Fig. 2.2). The solvent relaxation mechanism implies that the chromophore is better sol- vated in disordered lipid environments than ordered lipid environments, by a deeper penetration of water into the headgroup-hydrocarbon interface. In this chapter, the

77 Table 6.1: The composition of simulation systems.

System mol % Membrane Area (nm2) DPPC/DOPC/CHOL(Ld) 29/60/11 120.14 DPPC/DOPC/CHOL(Lo) 55/15/30 108.95 POPC/PSM/CHOL(Ld) 69/23/8 165.09 POPC/PSM/CHOL(Lo) 8/61/31 117.01 evidence is presented for a more complex picture, in which the chromophore solvation is insufficient to explain the shift in emission. Instead, a more complex pictureis proposed, in which the orientation of the chromophore changes, being more parallel to the bilayer normal in ordered lipid phases. Additionally, local molecular inter- actions appear to play a significant role, in particular, hydrogen bonding with the carboxyl moiety of Laurdan [81]. The remainder of the chapter is organized as follows. First, it is shown that bulk solvation of the chromophore is not increased in simulations of Ld phases relative to Lo phases, but that the naphthalene moiety is significantly tilted in Ld, while in

Lo the long axis of the ring remains more parallel to the bilayer normal. As a result of this reorientation in Ld, the carboxyl oxygen is presented to the water interface, and consequently, the likelihood of forming a hydrogen bond is greatly increased.

6.2 MD Simulation

Two Lo and two Ld simulation systems shown in Table 6.1 were built in which one Laurdan molecule was inserted on each leaflet by keeping the Laurdan head group in the level of lipid glycerol as described in literature [35]. MD simulation was run using Gromacs 5.1.1 [82] with CHARMM36 force field [77, 78] and TIP3P water model [80]. Steepest descent energy minimization was run followed by the 30 ns in the NVT ensemble where N is the number of particles,

78 Figure 6.1: A MD simulation snapshot showing liquid ordered (Lo) phase (A) and liquid disordered (Ld) phase (B).

V is the volume of the simulation box, and T is the temperature. Nose-Hoover [83] thermostat was used for temperature coupling at 298 K. NPT (where P is the pressure) equilibration was ran for 30 ns using V-rescale thermostat [84] (298 K) and semi-isotropic Parrinello-Rahman barostat [85](1 atm). The 500 ns production simulation was performed using Nose-Hoover [83] thermostat and Parrinello-Rahman [85] barostat. Lennard-Jones interactions were cut off using a switching function between 8.0 and 10.0 Å. Particle mesh Ewald [63] method with 10.0 Å cutoff radius was used for long-range electrostatics. The Lo and Ld simulation snapshots showing Laurdan molecules on each leaflet of the DPPC/DOPC/CHOL ternary mixture is shown in Fig. 6.1.

6.3 Simulation Results Figure 6.2 shows the location of the Laurdan chromophore (red) relative to the phosphates (blue) of each leaflet for both Lo and Ld phases. In Fig. 6.2 and in the following, two different ternary mixtures are shown for both phases — DOPC/DPPC/CHOL and POPC/PSM/CHOL — to determine whether results are particular to the lipid mixture, or are instead a property of the phase. Figure 6.2

79 Figure 6.2: Time series showing location of the Laurdan relative to the phosphates of each leaflet for four systems: Panel A DPPCLo B PSM Lo C DPPC Ld D PSM Ld.

shows that in each case the Lo phase is ca. 0.6 Å thicker than the corresponding

Ld phase, as measured by the phosphate-phosphate distance. As reported by the position of the N of the dimethylamine relative to the average phosphate position, the position of the chromophore, somewhat unexpectedly, appears to be a bit deeper in the bilayer in the Ld phase. This is unexpected because the conventional model for the shift in emission to lower energy in the Ld phase compared to Lo is that the chromophore is better solvated in Ld than in Lo but in Fig. 6.2 it appears that the chromophore is deeper in the hydrocarbon region in Ld than in Lo. More polar environments will respond to the dipole of the excited state, with the relaxation of the environment lowering the energy of the excited state relative to less polar environments.

Although Fig. 6.2 indicates a deeper chromophore in Ld, it does not show the position of the chromophore relative to the water hydrating the interface. Figure 6.3 therefore, reports the densities of chemical groups along the z-coordinate and

80 Figure 6.3: Density profile of the substructures of Laurdan for different systems: Panel A: DPPC Lo phase. B: PSM Lo phase C: DPPC Ld phase. D: PSM Ld phase. Black, water; Red, Laurdan head group, blue, Laurdan ring, green, Laurdan tail. averaged over the membrane plane. Comparing the Lo phases to their corresponding

Ld phases (i.e., comparing the top panel in Fig. 6.3 to the bottom panels), it is apparent that the Laurdan indeed moves deeper into the bilayer. In the Lo phases, the rings sit just inside the 50% solvation mark (the point at which the water density

1 falls to 2 it’s bulk value), while the headgroup sits just on or outside this line. In both Ld phases, both the ring and the headgroup sit inside this line. Note also that the Laurdan in the Lo phase seems to sample two different orientations, evidenced by the bimodal distribution of the headgroup location. This behavior is also evident in the time series in Fig. 6.2. Note that relatively long simulations are required to observe the range of conformations adopted by Laurdan in the membrane; other recent MD simulations of the dye have relied on significantly shorter trajectories [86].

Although the data in Fig. 6.3 suggest a less solvated chromophore in Ld

81 Figure 6.4: Radial density function and its integral of C=O and N in Lo and Ld systems: Panel A: DPPC C=O. B: DPPC Ld N. C: PSM C=O. PSM N. Red, Ld. Blue, Lo.

than in Lo, examination of the local water environment in each case reveals a more nuanced picture. Comparing the integrated radial distribution functions (dashed lines, Fig. 6.4B and 6.4D) shows that the methylamine headgroup of Laurdan is more solvated in Lo than in Ld, consistent with the location of this chemical group being deeper in the membrane as indicated in Fig. 6.3. However, the situation for the carboxyl hydrogen (located between the naphthalene rings and the hydrocarbon tail) is reversed — it is better solvated in Ld than in Lo in both cases. Note that the carboxyl is the only group in Laurdan capable of participating in a hydrogen bond, a point which will be revisited later. The data in Figs. 6.2-6.4 are explained by a reorientation of the chromophore relative to the bilayer normal in Lo as compared to Ld. In Lo, the chromophore tends to remain oriented with its long axis parallel to the bilayer normal, as it is typically drawn in review articles. In Ld, however, the rigid ring system adopts an orientation that are significantly tilted, as shown in Fig. 6.5, which plots the orientation of

82 Figure 6.5: Tilt angle of dipole moment vector of Laurdan with respect to bilayer normal in different systems: Panel A: DPPC Lo, B: PSM Lo, C: DPPC Ld, D:PSM Ld the chromophore dipole moment relative to the bilayer normal. In the bottom row, showing the orientation in Ld, the distribution is shifted toward a ring which lies parallel to the interface as compared to the Lo phases in the top row. Figure 6.5 also reveals a difference between the two Lo phases: Although both distributions are bimodal, the DPPC Lo phase favors orientations with the rings parallel to the normal. While in PSM the population of tilted chromophores is the larger mode. In other words, in DPPC the distributions of orientations in Lo and

Ld are more different from one another than in PSM. The increase in tilt anglein disordered phase is consistent with simulation result [87, 88] and microscopy images using photoselection effect89 [ ] available in the literature.

The re-orientation of the chromophore in Ld brings the carboxyl closer to the interface, where it is better solvated in the Ld phases than in Lo. As a result, the carboxyl in Ld is more often hydrogen bonded to water than in Ld, as shown in Fig.

6.6. When in the Ld phase, the carboxyl is hydrogen bonded to a water ca. 60% of

83 Figure 6.6: H bonding water occupancy percentage of Laurdan oxygen in Lo and Ld bilayers.

the time in both mixtures. In the PSM Lo phase, this only drops to 55%, consistent with the large population of titled chromophores in Fig. 6.5. In contrast, in the DPPC Lo phase a water is hydrogen bonded to the carboxyl less than 35% of the time.

6.4 Discussion The simulation results presented here are apparently inconsistent with the accepted explanation for the dependence of Laurdan’s Stokes shift in ordered vs. disordered membrane environments. The conventional view holds that the chro- mophore is better solvated in the Ld phase than in the Lo phase, while the present results show the reverse behavior. There is evidence that hydrogen bonding can contribute to the Stokes shift at a level that is comparable to the polarity of the solvent. Moyano, et al [90] compared the emission of Prodan in a series of solvents which vary in both polarity and hydro- gen bonding capacity. Comparison of solvents with similar polarity (methanol and

84 acetone) but very different hydrogen bonding capacities shows that hydrogen bond- ing can red shift emission by 57.06 nm/0.32 eV. In the simulation data presented above, the tilt of the chromophore in Ld results in more hydrogen bonding between the carboxyl and water than in the Lo phase. However, it might also be the case that the parametrization of partial charges described in Chapter 5 needs further optimization. Slight differences in partial charges can change the partitioning of the chromophore at the interface. If the parametrization yields a model which is too hydrophobic, the chromophore might partition too deeply in the membrane in the Ld phase where there is more free volume than in Lo. These questions are quantitatively assessed in the upcoming chapters.

85 Chapter 7

MD/QM CALCULATION OF PRODAN IN BULK SOLVENTS

In this chapter, an iterative molecular dynamics/GW -BSE scheme used to compute spectral properties and excited state of Prodan and Laurdan in bulk solvents and simple lipid bilayer systems is explained. The simulation results are presented and compared to ultrafast spectroscopic measurements. The content of the chapter is organized in the following order:

• Background information

• Theory (Detailed discussion in chapter 4)

– Quasi-particle excitation – Coupled electron-hole excitations – Electrostatic embedding

• GW -BSE: effect of hydrogen bonding in absorption energy

• MD/GW -BSE: Excited state calculation and iterative solvent relaxation

• Prodan MD/GW -BSE calculation in bulk solvents

– Simulation Results: Prodan emission and dipolar evolution in six different solvents – Experimental measurement: ultrafast spectroscopy – Solvent relaxation dynamics: Comparison of ultrafast spectroscopy to MM/GW-BSE approach – Discussion

• Conclusion

86 7.1 Background Information Polarity-sensitive dyes, 4–(2–(6–(dibutylamino)–2–naphthalenyl)ethenyl)– 1 –(3–sulfopropyl) –hydroxide (di–4–ANEPPS), and the naphthalene deriva- tives 2–dimethylamino–6–propionyl-naphthalene (Prodan), 2 –dimethylamino 6 – lauroylnaphthalene (Laurdan), and C–Laurdan are widely used as reporters of mem- brane structure in both model and cell membranes [91, 27]. All are lipophilic, so that they partition to the membrane, and have spectral properties that depend on mem- brane structure, including lipid packing, membrane thickness, and chain ordering. The sensitivity of the Stokes shift of the Prodan family is especially dramatic, with a difference of 50 nm when comparing emission in a highly ordered and tightly-packed gel-phase membrane to a less ordered fluid membrane [91]. This strong signal, to- gether with tolerable toxicity underlie an enormous literature using these dyes to study membrane structure over the past 40 years [91, 27, 92]. The pronounced Stokes shift of the Prodan-derived dyes is due to a transition from an excitonic state to a charge transfer (CT) state that is extremely sensitive to the electronic environment [93]. Applications to biomembranes have traditionally relied on a comparison of the relative fluorescence intensity at two different wave- lengths (termed ”generalized polarization”, or GP) because the implementation is straightforward and is easily combined with confocal microscopy [91, 37]. More re- cently, methods have been introduced that make use of the full fluorescence spectrum [94] and of the fluorescence lifetime (phasor methods) [36]. There have been some attempts to elucidate the excited state dynamics employing ultrafast spectroscopic methods [95]. The stabilization of the charge transfer state may include changes in both hydrogen bonding [90] and local polarization [35] due to solvent reorientation, complicating an unambiguous interpretation of the experimental results.

87 Prior numerical work on the spectral properties of Prodan (Fig. 5.1) and related dyes have in a few cases used wave-function based methods [96, 97] but most studies have been performed using time-dependent density functional theory (TD-DFT) [96, 98, 99, 100, 101, 102, 103]. While some attempts have been made to include effects of one [99] or many [102, 101] solvent environments explicitly, a majority of studies reply on implicit approaches such as polarizable continuum models (PCM) [96, 98, 97, 100, 103]. Although most of these studies give results in qualitative agreement with polarity-dependent spectroscopies, they fall short in two important respects when aiming at the simulation of spectra in membrane structures. First, conventional DFT functionals are not well-suited to CT states due to the lack of proper long-range interactions [104]. This can be overcome when special system- tuned corrections, e.g., CAM-B3LYP as in Ref. [103], are taken into account [105] but in particular for the emission spectra, the tuning may also depend on the specifics of the environment before, during, and after relaxation of the excited state. Second, even though a continuum treatment of the environment can be tuned to represent the effects of a molecular embedding, it will not capture the relaxation dynamics and coupled stabilization of the CT excited state in a complex (e.g., fluid membrane vs. gel phase), dynamic, atomistic environment. For a reliable prediction of the absorption and time-dependent emission spectra, it is therefore desirable to choose a simulation approach that combines an accurate quantum-mechanical method for the calculation of excited states and an explicit, atomistic environment model without the necessity of tuning either of those. In this chapter, ultrafast spectroscopy of Prodan is combined with state-of- the-art molecular dynamics/quantum mechanics simulations in order to resolve the time-evolution of the excited state, overcoming the limitations of previous work.

88 Femtosecond transient absorption of Prodan in different solvents is applied to mea- sure the solvent-dependent sub-picosecond formation of the CT state. The opti- cal excitations of the dye are treated using Many-Body Green’s Functions [106] within the GW approximation [106] and the Bethe-Salpeter Equation [107](GW - BSE), which yields accurate descriptions of localized excitonic and CT excitations on an equal footing, without the need for system- and environment-specific adjust- ments [108, 109, 110].

7.2 Theory This section consist a summary of what has been laid out in detail in chapter 4. Electronic excitations of a closed-shell system with net spin of zero are calcu- lated using GW -BSE. This approach builds upon a DFT [111] ground state calcula- | ⟩ tion in which Kohn–Sham (KS) energy levels εi and wave functions ϕi are obtained as solutions to ˆ ˆ | ⟩ | ⟩ [h0 + VXC] ϕi = εi ϕi , (7.1) ˆ ˆ ˆ ˆ ˆ ˆ where h0 = T0 + Vext + VH. T0 is the kinetic energy operator, Vext the external ˆ ˆ potential, VH the Hartree potential, and VXC the exchange-correlation potential [112].

7.2.1 Quasi-Particle Excitation First, quasi-particle (QP) states representing independent single-particle (electron or hole) excitations are calculated within the GW approximation [106] from   ˆ ˆ QP  | ⟩ QP | ⟩ h0 + Σ(ϵi ) ϕi = εi ϕi , (7.2)

89 ⟩

QP QP where ϕi are QP wave functions and εi are the excitation energies of quasi- electron and quasi-hole states. In Eq. (7.2), the approximate exchange-correlation potential of KS-DFT as in Eq. (7.1) has been replaced by the self-energy Σ(ˆ E) which is a non-local, energy-dependent operator. Within the GW approximation it is calculated as Σˆ = iGW, (7.3) which is a convolution of the single-particle Green’s function G and the screened

−1 ′ −1 Coulomb interaction W = ϵ vc, where vc = |r − r | is the bare Coulomb in- teraction and ϵ(r, r′, ω) is the microscopic dielectric function calculated within the random-phase approximation. These quasi-electron and quasi-hole states are typi- cally expanded in terms of KS states as ⟩ ⟩ ∑ QP KS ϕi = Cij ϕj . (7.4) j Assuming that the Kohn–Sham states approximate the GW quasi-particle ⟩ ⟩ QP ≈ KS states, i.e., ϕi ϕi , the quasi-particle energies can be obtained perturbatively [110] according to

εQP = εKS + ∆εQP i i i ⟩ (7.5) KS ⟨ | ˆ QP − ˆ KS = εi + ϕi Σ(εi ) VXC ϕi .

QP ˆ In equation 7.5, both the correction term ∆εi and Σ (via G and W ) depend on QP QP εi , requiring an iterative procedure until self-consistency in the eigenvalues εi is reached.

7.2.2 Coupled Electron-Hole Excitations Quasi-particle excitations as above correspond to charged excitations of the system. Neutral excitations which involve a coupled electron-hole pair, e.g., after the

90 absorption of a photon, are not included in this framework and need to be described using a two-particle wavefunction [110, 59]. The S-th excitation, χS, can be written as the linear combination of quasi-particle product states as

S ∗ S ∗ χS(rh, re) = Avcϕv(rh)ϕc (re) + Bvcϕc(rh)ϕv(re), (7.6) where re(rh) are the electron (hole) coordinates, v and c run over occupied and S S unoccupied single particle states respectively, and Avc (Bvc) are the (anti-)resonant electron-hole amplitudes. These amplitudes and the associated excitation energy ΩS are obtained by solving the Bethe-Salpeter equation, which can be cast in the form of a non-Hermitian eigenvalue problem

ˆ BSE H |χS⟩ = ΩS |χS⟩ (7.7) or in matrix form       Hres K  AS  AS      = ΩS   . (7.8) −K −Hres BS BS

For spin-singlet excitations as considered in this work, the diagonal and off-diagonal blocks of the Hamiltonian in Eq. (7.8) are determined in the non-interacting quasi- particle basis according to

res x d Hvc,v′c′ = Dvc,v′c′ + 2Hvc,v′c′ + Hvc,v′c′ (7.9) x d Kcv,v′c′ = 2Hcv,v′c′ + Hcv,v′c′ . with

D ′ ′ = (ε − ε )δ ′ δ ′ vc,v c ∫ c v v,v c,c x 3 3 ∗ ∗ Hvc,v′c′ = d rhd reϕc (re)ϕv(re)vC(re, rh)ϕc′ (rh)ϕv′ (rh) ∫ d 3 3 ∗ − ′ Hvc,v′c′ = d red rhϕc (re)ϕc (re)W (re, rh, ω = 0)

× ∗ ϕv(rh)ϕv′ (rh).

91 The contribution Hx is the repulsive exchange interaction originating from the

d unscreened interaction vc, while the direct interaction H contains the attractive, but screened (W ), interaction between electron and hole and causes the binding of the electron hole pair. Furthermore, it is assumed here that the dynamic properties of W (ω) are negligible and the computationally less demanding static approximation ω = 0 is sufficient.

7.2.3 Electrostatic embedding As discussed in section 4.5.1.2.2 of chapter 4, the GW -BSE is performed using a molecular mechanics (MM) representation of the electrostatic potential. Solvent molecules in the MM region are represented by the static atomic point charges Qa at positions Ra where a indicates the atoms in the MM region. The classical elec- trostatic potential

∑ Qa VMM(r) = (7.10) |r − Ra| a∈MM ˆ ˆ from the MM region is added to Vext in h0 in Eqs. (7.1) and (7.2), respectively. Relaxation of the environment and corresponding evolution of the excited state is handled by an iterative process described in the next section.

7.2.4 Transition probability and oscillator strength

The probability, PT, of transition between the ground state |0⟩ and excited state |S⟩ is proportional to the square of transition dipole moment, µT,

− |⟨ | | ⟩|2 2 PT = e 0 rˆ S = µT (7.11)

92 The oscillator strength can by obtained by normalizing the transition dipole by the excitation energy.

2 f = µ2 Ω (7.12) 3 T S

7.3 MD/GW -BSE: Excited state calculation and Iterative solvent relax- ation In this section, MD/GW -BSE excited state calculation is described and the results of excited state calculation of Prodan in bulk solvents and Laurdan in simple lipid membranes are presented.

7.3.1 Preparing MD trajectory for QM calculation GW -BSE calculations for singlet excited states are run using VOTCA-XTP [110]. First, the VOTCA-XTP code reads the atomistic trajectory of the MD simula- tion. Before performing the GW -BSE calculations, special care needs to be taken to avoid spurious effects arising from inconsistencies of the classical and quantum molec- ular potential energy surface. Deviations in equilibrium bond lengths can change ex- citation energies by several tenths of an eV. VOTCA-XTP therefore post-processes the structures read from the MD trajectory and replaces the rigid parts of Prodan and Laurdan (see Fig. 7.1), i.e., its naphthalene core, the carbonyl, nitrogen atom, and methylene groups, with copies optimized on PBE0/cc-pVTZ level for the DFT

S0 ground state (for absorption) and GW -BSE S1 excited state (for emission). The MD structure and mapped QM structure can be visualized together using Visual Molecular Dynamics (VMD) (Fig. 7.1) VOTCA-XTP provides the interface to run Gaussian09 for the preceding ground state DFT calculation. DFT calculations are computed using PBE0 hybrid

93 Figure 7.1: (a) Six different rigid fragments in Prodan: nitrogen atom, naphthalene core, the carbonyl and methylene groups (b) Visualization of mapping QM optimized copies into MD structure. functional [113] and the cc-pVTZ basis set and its optimized auxiliary basis [114] to express two-electron integrals within the resolution-of-identity technique. Seven hundred and forty eight KS orbitals are used within the RPA, and product functions for the BSE are formed using 61 (687) occupied (unoccupied) orbitals. The fre- quency dependence of the self-energy is evaluated using a generalized plasmon-pole model [115]. Quasi-particle energies are determined self-consistently, as discussed above, until changes are smaller than 10−5 Hartree.

7.3.2 MD/GW -BSE iterative protocol The MD/GW -BSE absorption and emission calculations are performed as follows:

• Several configurations are randomly sampled from classical NPT MD simula- tion of Prodan (Laurdan) in bulk solvents (lipid membranes).

• Absorption energy from S0 → S1 state is computed.

• Initial calculation of the S1 → S0 emission energy is performed for the same configurations for Prodan (Laurdan).

94 • For each, the partial charges of the S1 state are determined with the CHELPG method by constraining the partial charges of terminal hydrogens to 0.09e where e is the elementary charge. • The excited state partial charges are used to continue classical MD for another time (t) to relax the solvent environment in the presence of the changed dipole moment of the dye. • The GW -BSE emission calculation is performed again in the new solvent back- ground, and the procedure is repeated for n iterations.

Table 7.1: Structure of Prodan in S0 and S1 states. The names of atom correspond to Fig. 5.1

Bond/Angle S0 S1 NO distance(Å) 8.3732 8.3739

NC3 distance(Å) 1.3699 1.3667 OC13 distance(Å) 1.2137 1.2219 ◦ NC1C2 angle( ) 1.2137 1.2219

7.4 Prodan MD/GW -BSE calculation in bulk solvents 7.4.1 System setup and sampling heterogeneity using classical MD sim- ulation Six different Prodan solvent systems are built with Packmol [116] each of which consists of one Prodan molecule in 7200 TIP3P water, 2225 ethanol, 1770 acetone, 826 octanol, 3212 methanol, or 990 hexane. The MD simulations are run with GROMACS 5.1.1 [82] using the CHARMM36 force field77 [ , 78] and TIP3P water model [80]. Steepest descent energy minimization was followed by 20 ns NVT (Nose-Hoover [83] for temperature coupling at 298 K), and then 30 ns NPT equlibration (Berendsen [117] for temper- ature and isotropic Parrinello-Rahman [85] pressure coupling for 298 K and 1 atm pressure respectively), then 300 ns production simulation (Nose-Hoover temperature

95 Figure 7.2: The schematic diagram showing Prodan (in QM region) surrounded by water molecules (in MM region). coupling [83] and Parrinello-Rahman pressure coupling [85]). Lennard-Jones [118] interactions are cut off using a switching function between 10 Å and 12 Å and particle mesh Ewald [63] method with 12 Å cutoff radius was used for long range electrostat- ics.

7.4.2 Determining the size of MM region for MD/GW -BSE calculation The MD/GW -BSE calculation consists of the QM region and the MM region as shown in Fig. 4.8 and 7.2. An appropriate cutoff distance defined as the distance from the Prodan or Laurdan molecule (in QM region) to the outer MM boundary is chosen to include sufficient number of solvent molecules in the MM region. The2nm cutoff is carefully chosen after testing the effect of cutoff distance on absorption and emission energies. The emission energies of Prodan in water from first four excited states are given in Table 7.2.

96 Table 7.2: The emission energies of Prodan in water for the first four excited states and the number of water molecules (MM region) for different values of the cutoff distance.

Cutoff dist(nm) S1(eV) S2(eV) S3(eV) N 0.4 3.33 3.44 3.59 3 0.5 3.21 4.43 3.58 9 0.8 3.05 3.44 3.57 58 0.9 3.05 3.44 3.57 58 1.0 3.08 3.49 3.54 92 2.0 2.97 3.48 3.55 1097 2.5 2.97 3.48 3.55 2170

Figure 7.3: The schematic diagram representing GW -BSE absorption energy calcu- lation of Prodan with and without hydrogen bonded water and presence of water in MM region. See section 7.4.3 for the discussion.

97 Figure 7.4: The effect of inclusion of water in MM region and hydrogen bonded water molecule in QM region on absorption spectra of Prodan.

7.4.3 GW -BSE: absorption energy calculation and hydrogen bonding In order to test the effect of inclusion of MM solvents and also the presence ofa MM solvent in QM region alongside Prodan, GW -BSE absorption energies computed in following systems (Fig. 7.3): a) 1 Prodan (QM region), b) 1 Prodan (QM region) + Water molecules (MM region of 2 nm radii), c) 1 Prodan + 1 water (QM region), and d) 1 Prodan + 1 water (QM region) + Water molecules (MM region of 2 nm radii). The result is shown in Fig. 7.4 and as expected the difference in low energy region is minimal. After testing the effect of hydrogen bonding and the presence of MM solvents on absorption spectra and before showing the result of emission spectra of Prodan, we first describe the iterative protocol developed to iteratively calculate emission energies evolving the excited state and relaxing the system via MD simulation.

98 Figure 7.5: Prodan emission energy from S1 to S0 transition averaged over 2, 5, 15, 20, 24, and 25 MD configurations from fainter to darker curves.

The MD/GW -BSE calculation as described in the workflow of section 7.3.2 is performed in twenty five configurations for Prodan in bulk solvents with two duration of MD relaxation time/number of iterations (t=1 ps/n=49 and t=20 ps/n=9) (see Fig. 7.6). The choice of twenty five configurations for Prodan calculation is based on the following: when absorption or emission energies from different configurations are averaged, the spectra start to converge as shown in Fig. 7.5 and the result does not change beyond 25 configurations.

7.4.4 Simulation Results: Prodan emission and dipolar evolution in six different solvents We begin with a short discussion about the composition of the excited states in vacuum calculations. As shown in Fig. 7.7(a), the S1 absorption excitation as obtained from a GW -BSE calculation in the ground state geometry is formed mainly

99 Classical MD Simulation (300 ns)

Sample 25 configurations

GW -BSE: S0 → S1 Absorption Energy Calculation

GW -BSE: S1 → S0 Emission Energy Calculation

49(9) Prodan S1 state par- tial charges extraction iterations

Classical MD Solvent Relax- ation (1 ps (20 ps)) using Prodan S1 state partial charges

Figure 7.6: Prodan excitation and solvent relaxation workflow

100 (a) (b) (c) (d) 4

S2 S3 2 S1

0 LUMO+1 LUMO (hexane)

−2 LUMO LUMO (water) 84% 83% 42% 66% 25% 24% 27% 70% 80% −4

HOMO −6 S1 S1 S2 S3 S1 S1 HOMO (water) 8 HOMO-1

quasiparticle energy (eV) quasiparticle energy −

10 − absorption HOMO-2 emission emission HOMO (hexane) emission vacuum vacuum hexane water QP Figure 7.7: Quasiparticle energy levels (εi ) and contributions of interlevel transi- tions to the electron-hole wavefunctions for (a) absorption and (b) emission in vaccum on optimized geometries, as well as sample MD/GW -BSE relaxed geometries in (c) hexane, and (d) water solvent. Isosurfaces of selected quasiparticle orbitals (isovalue 10−2, red/blue) and difference densities of the excited state (isovalue 5 · 10−4 e, orange/green) are shown as insets. by a HOMO→LUMO transition (66%) with a smaller contribution (27%) coming from a HOMO→ LUMO+1 transition. After relaxation of the geometry following S1 excitation for emission, as shown in Fig. 7.7(b), the quasiparticle HOMO (LUMO) level results 0.16 eV (0.25 eV) higher (lower) in energy, and the emitting state is determined to 84% by a LUMO→ HOMO transition and is red-shifted by 0.4 eV. Note that even though the quasiparticle excitation energies are changed, the character of the involved orbitals remains practically the same compared to the calculation in the ground state geometry. Turning now to the combined MD/GW -BSE results, Fig. 7.8 shows the evo- lution of the S1 state dipole moment of Prodan (a)/(c) and S1 to S0 emission energy

101 (b)/(d) for the forty nine steps of the iterative relaxation procedure in six differ- ent solvents of differing polarity. The left panel correspond to solvent relaxation of 1 ps in each iteration with 49 iterations and the right panel correspond to solvent relaxation of 20 ps in each iteration with 9 iterations. In every case, both the excited state dipole and the emission energy plateau after (at most) 20 ps, indicating convergence of the iterative MD/GW -BSE relax- ation protocol. During the 1ps MD simulation steps, the solvent nuclear degrees of freedom relax in response to the excited state partial charges. In the least polar solvent (hexane), the dipole moment of the dye in the S1 state changes from 5.5 to 10.98 Debye. In more polar solvents the dipole moment evolves to a larger value (24 Debye in the most polar environment). The excited state dipole polarizes the environment, which stabilizes the charge transfer state, which polarizes the environ- ment more. This process continues until the environment can no longer polarize. The predicted dipole moments for the CT state are within the range reported in the literature for the excited state (see Table 7.3), although the most polarizable environments yield dipole moments at the very upper end of this range. The dipole moments included in the table are obtained via different methods (solvatochromic shift, solvent perturbation, thermochromic shift, and microwave absorption) with the Onsager cavity radius approximated between 4.2 to 6.3 Å. The observable effect of the stabilization of the CT state is the shift tolower emission energies with increasing polarity of the solvent, shown in Fig. 7.8 (b). After the solvent has fully relaxed and the data have plateaued, the S1 to S0 emission energy predicts the steady state emission spectra that is typically reported in the literature as a probe of the solvent environment of the chromophore. A comparison to reported values for the Stokes shift will be revisited in the Discussion.

102 Figure 7.8: (a)/(c) Prodan S1 state dipole moment and (b)/(d) S1 → S0 emission energy in hexane (red), octanol (black), acetone (cyan), ethanol (magenta), methanol (green), and water (blue). Error bars and background shading indicate the standard deviation among 25 different solvent configurations at each time point. The leftand right panels correspond to 1 ps and 20 ps of MD relaxation respectively.

103 Table 7.3: Prodan dipole moments reported in the literature. [2, 3, 4, 5, 6, 7, 8, 9] µG and µE are ground state and excited state dipole moment of Prodan measured in Debye (D).

Reference µG (D) µE(D) Balter et al. [2] 2.9 10.9 Catalan et al. [3] 4.7 11.7 Bunker et al. [4] 2.85 9.8 Kawski [5] 2.1/2.5 6.4/7.4 Kawski [9] 2.14/2.46 6.46/7.37 Kawski et al. [8] 2.45/2.80 6.65/7.6 Samanta, Fessenden [6] 5.2 10.2 Cintia et al. [7] 5.5 20.0

In Fig. 7.7 (c) and (d), we show sample quasiparticle spectra and the com- position of the emitting S1 state for two sample configurations taken from the final step of the MD/GW -BSE relaxation in hexane (least polar) and water (most polar), respectively. The presence of hexane in this particular geometry minimally lowers all quasiparticle energies compared to the vacuum emission case, and the energy and character of the S1 state are hardly affected. In water, one can clearly discern more changes of the quasiparticle spectrum. In general, the level distances are increased and, most importantly, the energies of the empty states are reduced, e.g., the LUMO is lowered by 1.12 eV compared to the vaccum case. Steady state absorption and emission spectra are predicted by broadening the

S0 to S1 and S1 to S0 transitions assuming Gaussian band shape with σ = 0.4 eV , and then averaging the obtained energies for twenty-five different configurations. Fig. 7.9 shows the absorption spectra, the emission spectrum immediately after absorption, and the emission spectrum following the 49 ps of iterative relaxation shown in Fig. 7.8. As expected, the absorption spectra show almost no dependence on the solvent. Similarly, the emission immediately following excitation (and before

104 Figure 7.9: Normalized absorption (dashed lines), emission immediately after ab- sorption (dashed-dotted lines) and after 49 ps MD/GW -BSE relaxation (solid lines) spectra of Prodan in different solvents. the polar solvents can relax and stabilize the CT state) shows little dependence on the solvent environment. However, after the solvent is relaxed via the iterative series of MD/GW -BSE steps the emission in polar solvents displays a strong bathochromic shift as the CT state is progressively stabilized. In the following next sections, a brief description of experimental measurement is provided and the experimental measurement is compared to the simulation result.

105 Figure 7.10: Prodan TA maps in acetone (a), ethanol (b), and methanol (c).

7.4.5 Experimental Measurement: Ultrafast Spectroscopy Prodan was obtained from Invitrogen, and all solutions for transient absorp- tion (TA) measurements were saturated. Spectroscopic grade solvents obtained from Fisher Chemicals and were dried over sodium sulfate before use. 1.25 M solutions of Prodan were prepared from a stock solution in acetone: After allocation, the acetone was allowed to dry off in open air (about 30 s, aliquot was 50 L) leaving Prodan in the vial, to which the desired solvent was added and shaken well. This was done immediately before measurement for each solution. A scanning fluorometer (Fluoromax-4, Horiba) with excitation at 350 and 410 nmanda resolution of 5 nm was used. The spectra were recorded with a resolution of 1 nm. To ensure the system was working properly and consistently a water blank was measured before and after experimental measurements, and the emission peak at 397 nm was observed with consistent wavelength and intensity.

106 Femtosecond transient spectra were recorded using a setup previously dis- cussed [119]. Briefly, the system is based on a 10 kHz regenerative Ti:sapphire am- plifier. White-light was generated in a 1 mm sapphire window and compressed toa group velocity mismatch of less than 2 fs across the spectrum using chirped-mirrors (Laser Quantum) and fused silica wedges. The pump pulse at 340 nm (3.65 eV) was generated by second harmonic generation from the output of a non-collinear para- metric amplifier. The instrument response function of the setup was 25fs. Transient maps (Fig. 7.10) were analyzed by fitting spectra at a range of time steps to a Gaussian function. The peak maximum was plotted as a function of time. The shift of the maximum with time was fitted by one or two exponential functions to extract time-constants.

7.4.6 Solvent relaxation dynamics: Comparison of ultrafast spectroscopy to MM/GW-BSE approach. Absorption and emission spectra of Prodan were measured in two solvents of similar polarity (acetone and methanol), but which differ in their ability to donate a hydrogen bond. (As mentioned in the Introduction, Prodan is a hydrogen bond ac- ceptor (at the carbonyl), and this has an influence on spectral properties which is not captured in the present MD/GW -BSE protocol [10, 120].) Figure 7.11 compares the absorption and emission spectra in acetone (aprotic) and methanol (h-bond donor) to the experimentally measured steady state spectra. No shift or fit parameters are applied to the predicted spectra. Although the absolute positions of the lines are at systematically higher energy in the MD/GW -BSE approach, the Stokes shifts are in nearly quantitative agreement, although the acetone calculation agrees slightly bet- ter (94 nm (simulation) vs. 100.8 nm (experiment)) than the methanol calculation

107 Figure 7.11: Prodan absorption (left) and emission (right) spectra in acetone, ethanol, and methanol from simulation (solid lines) and experimental measurements (dashed lines).

(117.6 nm calculated vs. 113.9 nm measured). Note that hydrogen bonding is not explicitly accounted for in the ab initio calculation. Figure 7.12 compares the time-dependent relaxation of the environment ob- tained by the iterative MD/GW -BSE scheme to ultrafast measurements of the same. In both the experimental and simulated data, the relaxation dynamics in methanol are slower than in acetone. An exponential fit to the acetone data yields a single timescale of 2.63 ps (experiment) and 1.36 ps (simulation). The methanol data are better fit by a fast and a slow decay, with a timescale of 1.03 ps and 15.16 ps(ex- periment) and 1.74 ps and 45.31 ps (simulation). Fast and slow timescales are also observed in ethanol, with timescales of 0.18 ps and 13.44 ps (experiment) and 1.25 ps and 13.61 ps (simulation). The simulations confirm the observation of fast and slow relaxations inthe

108 Figure 7.12: Prodan time resolved emission spectra (black) compared with simulation in acetone (cyan), ethanol (magenta), and methanol (green). alcohols, but not in acetone. Fitting the relaxation of the excited state for the rest of the simulation data reveals an unexpected trend: Three very different solvents (hexane, acetone, and water) all display a single, fast timescale. In contrast all of the hydrocarbon alcohols (octanol, ethanol, methanol) show a fast (ps) and slow (tens of ps) relaxation. This suggests that the slow relaxation is not due not to polarization of the environment, nor to formation of hydrogen bonds. Instead, the only commonality among the solvents with a second, slower timescale is that they are all weakly amphiphilic. We speculate that the slower timescale is due to local rearrangement of the solvent to form a slightly more hydrophilic cavity for the excited

109 Table 7.4: Prodan S1 state relaxation time in simulation and experiment obtained − t − t from single and double exponential fitting function, F (t) = ae τ1 + be τ2 + c

Simulation (Sim) Experiment Sim Sim Asymptotic std error Solvents τ , τ (ps) τ (ps) a,b, (c) 1 2 exp (%) Hexane 0.41 44.91 0.23,(3.28) Octanol 0.89, 29.13 21.62. 53 0.32, 0.16, (3.0) Acetone 1.36 2.63 42.72 0.04 (2.71) Ethanol 1.25, 13.61 0.18, 13.44 22.28, 29.13 0.42, 0.24, (2.77) Methanol 1.74, 45.31 1.03, 15.16 15.75, 119.9 0.56, 0.23, (2.65) Water 1.74 6.56 0.87 (2.57)

Table 7.5: Stokes shift of Prodan spectra from MD/GW -BSE calculated absorption and emission energies compared to experimental measurements from literature [10] and Fig. 7.11.

absorption emission Solvent EMD/GW -BSE (eV) EMD/GW -BSE (eV) ∆EMD/GW -BSE (eV) ∆Eexp(eV) Hexane 3.67 3.26 0.41 0.44 Acetone 3.76 2.88 0.88 0.74 Octanol 3.79 3.03 0.76 0.79 Ethanol 3.75 2.77 0.98 0.91 Methanol 3.71 2.75 0.96 0.93 Water 3.75 2.57 1.18 1.10 chromophore. It will be interesting to see how these observations carry over to ultrafast measurements in lipid environments, which embed the chromophore in a more chemically and dielectrically diverse interface.

110 Figure 7.13: Comparison of Stokes shift obtained in simulation with experimental measurements from the literature [10] and Fig. 7.11.

7.5 Discussion Prediction of the spectral properties of polarity sensitive dyes by a combined MD/GW -BSE approach yields excellent agreement with both steady state and time- resolved, ultrafast measurements. The iterative approach based on alternating MD relaxation of the solvent environment and GW-BSE prediction of the excited state leads to progressive stabilization of a charge-transfer excited state in more polar en- vironments, resulting in lower energy emission as polarity increases. Comparison of the Stokes shift obtained for the six solvents in Fig. 7.8 and Fig. 7.9 to results pre- sented here and in the published literature [10] yields an excellent, linear correlation with a slope of 1 (Fig. 7.13), without any adjustment of the simulated data. Ultrafast spectroscopy directly reveals the evolution of the CT state in polar environments (both protic and aprotic), providing evidence for a fast (ca. 1 ps) and slow (ca. 10-20 ps) process. Note that both of these processes are too fast to be observed by previous applications of other time-resolved spectroscopy to Prodan and

111 Figure 7.14: Panel (a) and (b): Prodan time resolved emission spectra in acetone (cyan) and methanol (green) from simulation for 20 ps and 1 ps MD solvent relaxation between consecutive emission energy calculation compared with experiment(dashed lines). Panel (c): Prodan time resolved emission spectra in acetone (cyan) and methanol (green) from simulation for 20 ps and 1 ps MD solvent relaxation between consecutive emission energy calculation compared with experiment (dashed lines). related dyes [121, 122, 123, 124]. The iterative MD/GW -BSE approach also agrees quantitatively with timescale of relaxation in acetone, ethanol, and methanol, when the duration of the MD steps is appropriately chosen. To test the effect of duration of MD relaxation, theMD/GW -BSE calculation is also repeated for MD relaxation of 0.5 ps and 20 ps. A longer solvent relaxation step arbitrarily increases the simulated relaxation timescale, and a shorter step does not change the observed timescales (see Fig. 7.14) The iterative MD/GW -BSE approach presented here is designed specifically for quantitative prediction of the spectral properties of polarity sensitive dyes that are commonly used to study biomembrane structure and dynamics. The MD portion of the calculation (if the model is well parametrized) accounts for the heterogeneity of

112 the environment, overcoming the limitations of continuum treatments that are more commonly used. Together, these two components yield a very high accuracy, quan- titative method with the potential to reveal the mechanism of other more recently developed, less well understood chromophores. Integrating the calculations with ul- trafast measurements of the excited state dynamics will provide valuable information on the nature of the emitting state for these other dyes.

7.6 Conclusion In this chapter, is shown that the evolution of the Prodan excited state in presence of polar solvent lowers the emission energy. In the next chapter, MD/GW - BSE calculation is repeated for Laurdan in lipid bilayer systems.

113 Chapter 8

MD/QM CALCULATION OF LAURDAN IN SIMPLE LIPID BILAYERS

In last chapter, the result of an iterative molecular dynamics/GW -BSE calcu- lation of Prodan in solvents of different polarity (hexane, acetone, octanol, methanol, ethanol, and water) was presented and compared to the ultrafast spectroscopic mea- surements. It was shown that the interaction with polar solvents can indeed lower the emission energy of polarity sensitive dyes (such as Prodan and Laurdan) as shown in Fig. 2.2 and explained in Chapter 2. In this chapter, the experimental measurement of Laurdan in DOPC vesicle is first described and then the result of theMD/GW -BSE calculation described in chapter 7 is repeated in more complex system (Laurdan in pure DOPC bilayer and DOPC/CHOL mixture)

8.1 Laurdan experimental emission spectrum Figure 3.1 shows the emission spectra of Laurdan in DPPC small unilamelar vesicle (spherical chamber bounded by one layer of lipid bilayer of diameter of 20- 100 nm) in gel phase (35◦C) and liquid crystaline phase (50◦C). The emission peak in liquid phase is 50 nm red shifted which, as mentioned above, is attributed to disordered liquid phase as compared to the ordered gel phase. In order to study the effect the emission energy of Laurdan present inmem- brane environment, two systems, DOPC (disordered) and DOPC/CHOL(ordered),

114 were chosen in which MD/GW -BSE iterative calculation described in chapter 8 was repeated as discussed in next section.

8.2 Laurdan MD/GW -BSE calculation in lipid bilayers 8.2.1 System Setup and sampling heterogeneity using classical MD sim- ulation Simulation systems consisting of 226 DOPC and 9112 water molecules and the mixture of 142 DOPC, 142 CHOL, and 9382 water molecules were built using CHARMM-GUI webserver [125, 126, 127, 128]. Two laurdan molecules were inserted into each leaflet of both DOPC and DOPC/CHOL systems. MD simulations were run using GROMACS 5.1.1 [82] using CHARMM36 force field [77]. The energy minimization was ran using steepest descent algorithm followed by the 30 ns NVT where N is the number of particles, V is the volume of the simulation box, and T is the temperature. Nose-Hoover [83] thermostat was used for temperature coupling at 298 K. NPT (where P is the pressure) equilibration was ran for 30 ns using V-rescale thermostat (298 K) and semi-isotropic Parrinello-Rahman barostat [85](1 atm). The production simulation was performed for 500 ns using Nose-Hoover [83] thermostat and Parrinello-Rahman[85] barostat. Lennard-Jones interactions were cut off using a switching function between 8.0 and 10.0 Å. Particle mesh Ewald [63] method with 10.0 Å cutoff radius was used for long range electrostatics.

8.2.2 Determining size of MM region for MD/GW -BSE calculation Based on the absorption energies of first three excited states for different values of cutoff distance (distance from the Laurdan molecule (in QM region)to outer boundary of MM region (see Fig. 4.8) shown in Table 8.1, the cutoff distance of 2.5 nm is used in MD/GW -BSE calculation.

115 Table 8.1: The first three excited state energies of Laurdan in DOPC membrane for different values of the cutoff distance.

Cutoff dist (nm) S1(eV) S2(eV) S3(eV) 0.5 3.38 3.95 4.00 0.8 3.50 3.83 3.96 0.9 3.52 3.82 3.99 1.0 3.51 3.82 3.95 1.5 3.55 3.77 3.91 2.0 3.55 3.77 3.91 2.5 3.56 3.77 3.91 3.0 3.56 3.76 3.91 3.5 3.56 3.76 3.90

The difference between the structure of Laurdan and Prodan is the extended hydrocarbon tail which is useful to partition the chromophore into membrane envi- ronment but is not crucial to fluorescence (see Fig. 5.1). Inclusion of the hydrocarbon tail in QM region also increases the computational cost. Therefore the hydrocarbon tail is not included in QM calculation and placed in the MM region as shown in Fig. 8.1. The MD/GW -BSE iterative calculation as described in section 7.3.2 and il- lustrated in flow chart 8.2 is performed for Laurdan in DOPC and DOPC/CHOL. In the next section, the result of the simulation is presented.

8.2.3 Simulation Results: Laurdan emission and dipolar evolution in lipid bilayers

Figure 8.3 shows the evolution of the S1 state dipole moment of Prodan (a) and S1 to S0 emission energy (b) for the nineteen steps of the iterative relaxation procedure in DOPC(red) and DOPC/CHOL (blue) lipid bilayers. The excited state

116 Figure 8.1: Panel a) shows molecules included in GW -BSE calculation: Laur- dan(blue; in QM region), DOPC, and water molecules in MM region. Panel b) shows the chromophore included in QM region and the excluded hydrocarbon tail (blue). dipole moment and the emission energy slowly plateau after (at most) 12 ps, indi- cating convergence of the iterative MM/GW -BSE relaxation protocol. The system is allowed to relax during 1 ps MD simulation during which the nuclear degree of freedom of molecules in MM region (Lipids, water, and Laurdan tail) evolved in response to excited partial charges. The S1 state of dipole moment of the Laurdan chromophore in DOPC system increases up to 15 D and DOPC/CHOL system in- creases up to 21 D. The evolved excited state dipole moment stabilizes the charge transfer excitation which evolves the dipole moment again until the calculation con- verges. This process lowers emission energies as expected (Fig. 8.3 (b)). The emission energy is expected to be lower in more disordered membrane as more polar solvent molecules can permeate in disordered phase. This is also seen experimentally in gel and liquid DPPC vesicle as shown in Fig. 3.1 explained in section 8.1. The simulation result (Fig. 8.3), lower in DOPC/CHOL system

117 Classical MD Simulation (500 ns)

Sample 45 configurations

GW -BSE: S0 → S1 Absorption Energy Calculation

GW -BSE: S1 → S0 Emission Energy Calculation

19 Laurdan S1 state par- tial charges extraction iterations

Classical MD Solvent Relax- ation (1 ps) using Laurdan S1 state partial charges

Figure 8.2: Laurdan excitation and solvent relaxation workflow

118 Figure 8.3: (a) Laurdan S1 state dipole moment and (b) S1 → S0 emission energy in DOPC bilayer (blue) and DOPC/CHOL (red). Error bars and background shading indicate the standard deviation among 45 different solvent configurations at each time point.

(2.85 eV) than DOPC (3.15 eV), is backwards compared to emission energy. The emission energy in DOPC/CHOL (ordered membrane) is red-shifted by 47.24 nm. MM/GW -BSE method that agreed with ultrafast measurements for Prodan in bulk solvents is unable to produce similar results for Laurdan in lipid bilayer. Since the presence of water molecules in vicinity of the chromophore is one of the major mechanisms that lowers the emission energy of Laurdan, it is reasonable to question the positioning of the chromophore in the direction perpendicular to the membrane water interface (Z axis), perhaps because the force field requires further

119 Figure 8.4: (a) S1 → S0 emission energy and (b) Laurdan S1 state dipole moment in DOPC bilayer (blue) and DOPC/CHOL (red). Each iteration step in this figure, unlike Fig. 8.3, correspond to 20 ps classical MD. optimization as discussed at the end of Chapter 6. Unlike what was shown in chap- ter 6, if the Laurdan molecule stayed upright closer to the bulk water, the solvent relaxation process due to the evolution of charge transfer state would lower the en- ergy. To check the effect of chromophore position along the direction perpendicular to membrane interface, the biased sampling was used instead of randomly sampling the configurations. Configurations were selected from the Ld trajectory in which the chromopohore samples locations closer to the water phase. The MD/GW -BSE was repeated for 45 system in these configuration which lie closer to the water interface and the result (Fig. 8.4) indeed show the if the chromophore were to sample more solvated configurations in the Ld phase, the predicted emission is 90 nm red-shifted compared to randomly sampled configurations, and 28 nm red-shifted compared to the ordered phase. The MD/GW -BSE calculation is sensitive to the result of MD simulation

120 which depends on the quality of the force field parameters. In our case, only the Laurdan partial charges were systematically optimized as described in Chapter 5 where as all other parameters were used from CGenFF as provided by Paramchem web-server. In the next chapter, a simulation is performed to determine the free energy to transfer the Prodan model from hexadecane to water. This is used to determine whether the partitioning of the model between bulk hydrocarbon and water phases matches experimental measurements, in order to guide further force field development.

121 Chapter 9

PARTITIONING OF PRODAN AT AN OIL-WATER INTERFACE

As shown in the previous chapter, the emission spectrum of Laurdan is sen- sitive to slight differences in in the position of the chromophore along the bilayer normal. We therefore implemented a free energy method to determine the parti- tioning behavior of the Prodan model at an oil-water interface, implemented as an 56 Å thick slab of hexadecane with water on either side (Fig. 9.1). The adaptive biasing force (ABF) [129, 130, 131, 132, 133, 134] method was used to determine the potential of mean force along the direction normal to the slab. In this chapter, the free energy calculation and ABF method are briefly described, and the result of the ABF calculation is presented.

9.1 Free energy calculation The calculation of free energy differences for biochemical processes is a funda- mental problem in biophysics, governing many different aspects of cellular function including molecular interactions (like drug-protein interactions or protein-protein interactions) and partitioning of small molecules into membranes. While there is no “one size fits all” solution to free energy calculation, many numerical techniques have been developed and tailored to specific problems. First the basic thermody- namics are reviewed, then the ABF method is described, then the results for Prodan partitioning in hexadecane/water are presented.

122 In canonical ensemble (ensemble in which a number of particles, N, volume of the system, V , and temperature of the system, T are fixed), the free energy termed as the Helmholtz free energy, A, is the difference of internal energy of the system and the product of entropy and temperature.

A = U − TS (9.1)

In isothermal-isobaric ensemble (ensemble in which number of particles, N, the pressure of the system, P , and temperature of the system, T are fixed), the free energy is termed as Gibbs free energy, G, which is given by,

G = A + PV (9.2) where P and V are pressure and volume, respectively. Helmholtz free energy for the canonical ensemble can also be expressed as

A = −β−1lnQ(N,V,T ) (9.3)

−1 1 In equation 9.3, β = where kB is Boltzmann constant and Q is the canonical kB T partition function given by [135, 136, 137],

∫ 1 Q(N,V,T ) = exp(−βH(x, p))dxdp (9.4) h3N N! In equation 9.4, q and p are 3N coordinates and momenta respectively. According to equation 9.3, calculating the free energy is equivalent to calcu- lating the partition function which is very difficult for many-body molecular systems, as sampling configurations is computationally demanding. Fortunately, most cases of interest in biophysics, we are interested in evaluating the change in free energy

123 rather than the absolute value of the free energy. The change in free energy between states 1 and 2 is given by,

Q Z ∆A = −β−1ln 2 = −β−1ln 2 (9.5) Q Z ∫ 1 1 where ∆A = A2 − A1 and Z(N,V,T ) = exp(−βU(x))dx However, computing the free-energy difference as in equation 9.5 is still chal- lenging numerically, since it requires sampling configurations in both states 1 and2 [138] for the complex system often have high barriers separating different regions of the phase space containing states 1 and 2 [133]. The transition between these regions of the phase space might not occur during a computer simulation or may occur only, sometimes causing the problem of inadequate sampling. A wide range of techniques has been developed to address this issue which tend to be somewhat problem-specific, designed to overcome barriers particular to a class of problems (partitioning, ligand binding, etc) [139]. In the next section, the method of thermodynamic integration to compute free energy differences is discussed.

9.1.1 Thermodynamic Integration to compute free energy differences This method relies on calculating the derivatives of the free energy and in- tegrating it with respect to the order parameter (which may be function of atomic coordinates or just a parameter in the Hamiltonian) along a transformation path that takes the system between the states of interest. Free energy is related to the probability density function, P (ξ) , of the tran- sition coordinate), ξ , can be written as,

A(ξ) = −β−1lnP (ξ) (9.6)

124 In the canonical ensemble, the probability density function is given by,

1 P (x, p) = exp(−βH(x, p)) (9.7) Q Equation 9.6 suggests that probability of sampling conformations with large A(ξ) is very low. Therefore a straightforward method which relies on unbiased sampling of configurations along to obtain the free energy difference between two states willsuffer from poor sampling, as barriers in the free energy barrier reduce transitions between the end states. This demands very long simulation which may not be practical [139]. In the method of thermodynamic integration (TI), we write the free energy difference for a smooth surface as,

∫ ξ1 dA A(ξ1) − A(ξ0) = dξ (9.8) ξ0 dξ Based on equation 9.8, it is possible to first compute the derivative of the free energy and then integrate it to find the free energy. The derivative can be written as,

dA ∂H = ⟨ ⟩ (9.9) dξ ∂ξ ξ In equation 9.9, the partial derivative of the Hamiltonian with respect to transition coordinate for a fixed value of ξ can be thought of as a generalized force. Several methods, constrained and unconstrained, have been developed to com-

∂H pute ∂ξ . In a constrained method such as a blue-moon ensemble method [140], the simulation is performed with a fixed value of ξ by applying an external constraining force. This method allows enough sampling at each location of ξ along the specified interval. However, to apply the constraining force, the system has to be set up and equilibrated for the desired value of ξ. It may also be difficult to sample all the

125 relevant conformations of the system by fixing ξ. These problems can be solved by using the adaptive biasing force (ABF) method [129, 130, 131, 132, 133, 134] which is briefly described in section 9.1.1.1.

9.1.1.1 Adaptive Biasing Force ABF method is based on unconstrained MD simulation in which the mean − dA force acting on ξ, dξ , is computed and removed from the system to obtain uniform sampling along ξ. The removal is implemented by adding an external biasing force

dA equal to dξ ∆ξ. In other words, ABF is the combination of two parts. In the first part, the free energy is calculated by integrating the mean force, collected in bins along transition coordinate, as a function of transition coordinate. In the second part, the sampling is enhanced by applying the negative of the current estimate of the mean force to allow crossing of free energy barriers. Even though the mean force becomes zero, there exist non-zero force fluctu- ating around zero mean, which provides the diffusive dynamics to the system [139]. ABF thus obtains uniform sampling on the fly, and no prior knowledge of the free energy landscape is required. In equation 9.9, calculating the derivative of the free energy requires the computation of partial derivatives of the Hamiltonian with respect to the transition

∂H coordinate, . To obtain this, the generalized coordinates of the form (ξ, q1, ..qN−1) ∂ξ ∫ is used. Together with equation 9.6, 9.7, and Q = exp(−βH(x, p))dxdp, we obtain the definition of free energy in terms of integral in phase space.

∫ e−βH A(ξ) = −k T ln δ(ξ − ξ(x))dxdp (9.10) B Q

126 In equation 9.10, Dirac delta function ensures integration of x such that ξ(x) = ξ. Since the quantity of interest is difference in free energy, the partition function is removed from the discussions below. It is mathematically difficult to deal with delta functions so the set ofgener-

ξ 1 N−1 alized coordinate, (ξ, q1,..qN−1) and (p , p ,..p ) is defined to simplify the integral as,

∫ − −βH ξ q A(ξ) = kBT ln e dq1..dqN−1dp ..dpN−1 (9.11)

The equation of motion of the system is obtained by defining the Hamiltonian in generalized coordinate as the sum of kinetic (T) and potential (U) energy term,

H(q, p) = T (q, p) + U(q, p) (9.12)

The equation of motion are,

dq ∂H i = dt ∂p i (9.13) dp ∂H i = − dt ∂qi Using the above formalism, the derivative of free energy with respect to the transition coordinate can be found, ∫ ∂H −βH q dA e dq1..dqN−1dpξ..dp − ∂H = ∂ξ ∫ N 1 = ⟨ ⟩ (9.14) −βH q ξ dξ e dq1..dqN−1dpξ..dpN−1 ∂ξ

127 In equation 9.14, dA/dξ is a mean force acting on the generalized particle ξ. To simplify the calculation of partial derivative of H with respect to ξ, it can be rewritten as, ⟨ ⟩ ⟨ ⟩ ∂H ∂U ∂ln|J| = − k T = −⟨f ⟩ (9.15) ∂ξ ∂ξ B ∂ξ ξ ξ ξ ξ In equation 9.15, |J| is the determinant of the Jacobian matrix for Cartesian to [ ] ∂xi generalized coordinates given by J(q) = ∂q . In our calculation, the value of ij j Jacobian is 1. In equation 9.15, ∂U/∂ξ represents physical forces acting on the system, ∂ln|J| kBT ∂ξ is the geometric correction term during the Cartesian to generalized co- ordinate transformation, and ⟨fξ⟩ξ is the average force acting along the transition coordinate, ξ. The biasing force is the negative of the average force.

dA(ξ) f = = −⟨f ⟩ (9.16) ABF dξ ξ ξ During numerical calculation, the instantaneous force is collected in small bins of size ∆ξ along the transition coordinate. The biasing force is applied only after the threshold of the instantaneous force measurement is achieved. Harmonic boundary forces are applied at two ends to keep the molecule in the ABF domain , avoiding sampling of uninteresting configurations.

9.2 ABF calculation of Prodan in hexadecane-water slab ABF calculation of Prodan was run in hexadecane-water slab system to deter- mine the partitioning behavior of Prodan in the lipid environment. In this section, details about the ABF calculation and the results are presented.

128 Figure 9.1: The schematic diagram showing ABF calculation setup for Prodan in hexadecane-water slabs. Prodan molecule moves along the direction of transition coordinate (ξ) as shown by a dotted line which is stratified into several bins.

9.2.1 Method A hexadecane-water slab system (Fig. 9.1) with water slabs on either side of hexadecane slab in ξ direction was built using Packmol [116]. The system contained 500 hexadecane molecules and 1101 water molecules, resulting in a hexadecane slab 56 Å thick adjacent to a water slab of thickness 48 Å. The simulations were run using Gromacs 5.1.1 [82] and NAMD [141, 132, 133, 134] with CHARMM36 [77, 78] force field for lipids and TIP3P water model80 [ ]. Energy minimization and the equlibration runs were performed using Gromacs 5.1.1 [82]. Steepest descent energy minimization was performed followed by 30 ns NVT where N is number of particles, V is the volume of the simulation box, and T is

129 the temperature. Nose-Hoover [83] thermostat was used for temperature coupling at 298 K. 30 ns of NPT simulation (P is the pressure) was run using V-rescale thermostat [84] (298 K) and semi-isotropic Parrinello-Rahman barostat [85](1 atm). Lennard- Jones [118] interactions cut off using a 10-12 Å switching function and Particle mesh Ewald [63] method with 10.0 Å cutoff radius was used for long-range electrostatics. A Prodan molecule from the simulated system described in chapter 7 was inserted into hexadecane-water slab approximately 14 Å above hexadecane-water in- terface in the positive ξ direction (Fig. 9.1). The dimensions of the system are about 86 Å in the direction of transition coordinate and 50 Å × 50 Å in the orthogonal di- rections. The ABF calculation was performed in six replicas of the system. Before running the ABF calculation, the system was further minimized and equilibrated for 30 ns. ABF calculation was performed in NAMD (version 2017/07/15) [141, 132, 133, 134] using Colvar module [142]. The ABF algorithm [129, 130, 131, 132, 133, 134] was used to determine the PMF, which was computed by integrating the average force exerted along the transition coordinate. The mean force acting along a transition coordinate was evaluated, as shown in equation 9.15.

The transition coordinate is stratified into several bins in which fξ is estimated. The free energy is flattened by applying the negative of fξ to the system allowing uniform sampling along the transition coordinate (ξ). The free energy along the transition coordinate is obtained by integrating the force which is the derivative of the free energy. The transition coordinate was defined by projecting the distance between the center of mass of central CH2 in all hexadecane molecules and center of mass of Prodan into the direction normal to the hexadecane-water interface. In each hexadecane molecule, only central CH2 was considered for computing the center of mass to reduce the cost of ABF calculation, and the hydrogen atoms of the central

130 carbon were included to avoid issues of net forces from hydrogen constrains (rigid bonds) on ABF calculation. The pathway of Prodan spans 84 Å (-42 Å ≤ ξ ≤ 42 Å) with origin at the center of hexadecane slab (see Fig. 9.1). Transition coordinate was divided into bins of 0.05 Å in which forces are sampled and averaged. The opposite of the averaged force is applied to the system, which flattens the free energy barrier and enhances sampling. In the water slabs at both ends, boundary potential with force constant 2 of 10 kcal/mol/Å was applied to keep the Prodan within the region of interest. To improve performance, 4 fs time step for MD simulation was implemented by using Hydrogen Mass Repartioning (HMR) [143]. Time step of the simulation is constrained by the highest frequency motion in the system which is typically the vibration of hydrogen atoms. Due to these fast vibrations, 1 fs the time step needs to be implemented. However by applying restrains (SHAKE [144] algorithm to non water hydrogen and SETTLE [145] algorithm to water molecules) to the high frequency bonds, 2 fs time step is used. In HMR scheme, atomic masses of hydrogen atoms are increased and masses of heavy atoms to which they are bonded are decreased, keeping the overall mass constant. This permits doubling the time step to 4 fs without using constraint algorithms like SHAKE. The masses of non water hydrogen were tripled and the mass of the nearby atoms were adjusted to nullify the changes made in hydrogen mass. For example, the masses of carbon and non water hydrogen atoms were modified to 7.978 amu and 3.024 amu for CH2 and

5.962 amu and 3.024 amu for CH3. Masses of hydrogen in water were not adjusted as the SETTLE algorithm allows increase in time step beyond 2 fs. Readjusting the masses does not have any effect on thermodynamic averages of observables and should not alter free energy calculation. The ABF MD simulation was run for 2.1 s and the PMF along the transition

131 Figure 9.2: Potential of mean force (PMF) of six replicas along the transition coor- dinate. coordinate was obtained. The water-hexadecane partition coefficient (P ) is related to the free energy difference between water and hexadecane by

P = e−β∆A (9.17)

In equation 9.17, ∆A is the difference in free energy between the bulk water (ξ = 40 Å) and the center of hexadecane slab (ξ = 0 Å). In the next section, the result of the ABF calculation is presented.

9.2.2 Results Figure 9.2 shows the PMF of Prodan underlying its permeation into hexade- cane from bulk water. The PMF profile is similar in all replicas indicating thatthe calculation has started to converge.

132 Figure 9.3: Time series obtained from 475 ns simulation showing the location of center of mass of Prodan moving along the direction of transition coordinate.

Figure 9.3 shows that the center of mass of Prodan moved along the transition coordinate five times from one end to the other during the span of 475 ns simulation of replica 3. The gradient from all replicas for each grid point along the transition coor- dinate was combined by taking the weighting average of mean value of gradients in each bin across all replicas as follows,

⟨grad1⟩n1 + ⟨grad2⟩n2 + ··· + ⟨grad6⟩n6 gradcomb = (9.18) n1 + n2 + ··· + n6 where ⟨gradi⟩ is the mean value of gradients and ni is the total number of samples in ith bins of replica i.

133 (a) Gradient of PMF

(b) PMF

Figure 9.4: Gradient(top) and potential of mean force (bottom) of six replicas along the transition coordinate. The black curve is the error in measurement. The vertical lines represent the hexadecane-water interfaces.

134 Figure 9.5: Symmetrized version of Free-energy profile (PMF) from Fig. 9.4b.

Figure 9.4 shows the combined gradient and the uncertainty associated with the measurements. The standard deviations of the gradients were calculated at each grid point (bins) over all six replicas. We have observed that the differences in the gradient among the replicas is reduced as the simulations are run longer, suggesting that the ABF calculation is converging. The combined PMF (Fig. 9.4) was obtained by integrating the combined gradient. Since the free energy difference is the relevant quantity, the profile is shifted by anchoring the right-most region (ξ = 39-40 Å) to zero. The uncertainty of the gradient was integrated from the anchor point to give the uncertainty in the change of free energy for displacement from the anchor point to a given position ξ. The water-hexadecane interface in the plot is shown by vertical dashed line. The most likely location that Prodan molecule resides about 10.2 Å away from hexadecane- water interfaces in hexadecane slab.

135 Equation 9.17 was used to compute the values of the partition coefficient. In Fig. 9.4b the partitioning of Prodan across the left and right interfaces are not equal, indicating that the PMF has not yet converged. To obtain an estimate for the converged calculation, the combined PMF was also symmetrized as shown in Fig. 9.5. In contrast, the Gundlach Group at the University of Delaware has experimentally measured the value of the partition coefficient of Prodan in the water-hexadecane system as 3.73. The ABF method provided the free energy landscape of Prodan at the water– hexadecane interface, revealing in detail the thermodynamics of its partitioning into a hydrophobic environment. However, it seems that the current model for Prodan is far too hydrophobic, which is likely responsible for the results presented in Chapter 8, in which it was found that Laurdan partitions into the hydrocarbon region in membrane simulations. In the next chapter, the work presented in this dissertation is summarized and the important points about future outlook is provided.

136 Chapter 10

CONCLUSION AND OUTLOOK

The polarity sensitive fluorescent dyes, such as Laurdan and Prodan, pro- vide an effective way to interrogate structural differences between the different lipid phases in both model systems and living cell membranes because the emission spec- tra of these molecules are sensitive to the polarity of the micro-environment of the chromophore. The present thesis provided i) the result of classical MD simulation of Laur- dan in lipid bilayers and ii) the quantitative prediction of the spectral properties of Prodan and Laurdan using MD/GW -BSE calculation in which Prodan and Laurdan are treated quantum mechanically and the environment is relaxed via classical MD simulation, and iii) the Adaptive Biasing Force(ABF) free energy profile of Prodan in the water-hexadecane slab. The ground state partial charges of Laurdan used in MD simulation was obtained by following the optimization protocol provided by the CHARMM family of force fields developers. The scaled HF/6-31G(d) model compound-water interaction energies and the computed dipole moment were used as the target data to fit the partial charges. The first excited state partial charges were determined withthe CHELPG method constraining the partial charges of the terminal hydrogen to 0.09e (elementary charge). The result of MD simulation indicates that Laurdan resides deeper in liquid disordered (Ld) with a greater tilt angle with respect to bilayer normal compared

137 to liquid-ordered (Lo) phase. The tilt brings the oxygen in Laurdan closer to the interface in the Ld phase improving its solvation. The predicted absorption spectra, emission spectra, and the Stokes shift per- formed on Prodan in bulk solvents (water, ethanol, methanol, octanol, acetone, and hexane) are in excellent agreement with ultrafast measurements both in terms of steady-state and time-resolved behavior. The relaxation time of Prodan in all sol- vents obtained in simulations agree with the experiments showing one relaxation time for hexane, acetone, and water and two relaxation times (fast (ps) and slow (tens of ps)) for octanol, ethanol, and methanol. These results suggest that the MD/GW-BSE approach is a reliable and quantiative method for predicting the spectral properties of polarity sensitiuve dyes. The iterative MD/GW -BSE calculation was repeated on ordered (DOPC/CHOL: 50/50) and disordered (DOPC) bilayers. The emission spec- tra and the Stokes shift obtained in simulations are reversed as compared to the experiment. When the MD/GW -BSE was repeated by sampling the biased configurations in which Laurdan molecule interfaces, the Stokes shift in simulation agreed better to the experiment. Thus, the MD/GW -BSE method implemented in this work is sensitive to the position of the Laurdan in trans-membrane direction and can be a useful tool for testing the accuracy of force field parameters. In order to guide future force field optimization, a free energy calculation was performed to determine the partitioning of Prodan at an oil water interface using the Adaptive Biasing Force method. The result indicated that the current Prodan model resides about 10.2 Å away from the water-hexadecane interface in the bulk hexadecane phase but that the model is about 20 times more hydrophobic than is reported by the experimental partition coefficient. The present work can be extended by optimizing the fixed charge Force Field

138 model for Laurdan based on the results of the ABF calculation. Laurdan force field parameters other than partial charges (optimized in the current work) couldbe optimized and compared with target data. Since the cation-π interaction is reported to be stronger than hydrogen bonding [146, 147, 148] and the choline in lipid head group can participate in cation-pi interaction[149, 150], the simple modifications can be made on Lennard-Jones potentials [151] to capture the cation-π interaction of the aromatic ring of Prodan. It might also be that a polarizable force field is required to capture the complex dielectric environment at the water/lipid interface, and that a fixed charge model simply cannot achieve quantitative accuracy for this problem. However, the polarizable force field model for lipids is still in the early phaseof development and is not ready for such calculations. The water model might also be at fault. The current family of CHARMM force fields uses TIP3P water [80] which is considered as the primitive water model developed about 30 years ago. TIP3P water model is non-polarizable and describes the solvent relaxation dynamics poorly. Furthermore, the TIP3P dipole moment (2.3 D) is different from both dipole moment in vacuum (1.85 D) and in liquid state(≈ 3.0 D) [152, 153]. Molecular interactions such as hydrogen bonding may also be important for the Laurdan spectral shift in membranes. Although the present work includes hydrogen-bonded water alongside Prodan in absorption energy calculation, it has not yet been included in MD/GW -BSE emission energy calculation. The effect of solute-solvent interaction, especially hydrogen bonding, has been known to affect the maximum position of emission spectra of Prodan in solvents of similar polarity [81]. In summary, the present work sets a new standard for quantitative accuracy when modeling polarity sensitive dyes like Prodan in bulk solvents. Application to membrane systems both reveals the value of sensitive tests of force field accuracy,

139 and shows that work remains to achieve the same level of quantitative accuracy at interfaces.

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162 Appendix A

SAMPLE MDP FILE USED IN MD SIMULATION

title = Production run of Laurdan in DOPC

; Run parameters integrator = md nsteps = 250000000 dt = 0.002 ; Output control nstlog = 100000 nstxout = 150000 nstvout = 150000 nstfout = 150000 nstcalcenergy = 1000 nstenergy = 150000 nstxout-compressed = 50000 compressed-x-precision = 1000 ; Bond parameters continuation = yes constraint_algorithm = lincs constraints = h-bonds lincs_iter = 1

163 lincs_order = 4 ; Neighborsearching cutoff-scheme = Verlet vdwtype = cutoff vdw-modifier = force-switch ns_type = grid nstlist = 10 rlist = 1.0 rcoulomb = 1.0 rvdw = 1.0 rvdw-switch = 0.8 ; Electrostatics coulombtype = pme pme_order = 4 fourierspacing = 0.16 ; Temperature coupling is on tcoupl = Nose-Hoover tc-grps = DOPC_LAU SOL tau_t = 1.0 1.0 ref_t = 298 298 ; Pressure coupling is on pcoupl = Parrinello-Rahman pcoupltype = semiisotropic tau_p = 5.0 5.0 ref_p = 1.0 1.0 compressibility = 4.5e-5 4.5e-5

164 ; Periodic boundary conditions pbc = xyz ; Dispersion correction DispCorr = no ; Velocity generation gen_vel = no ; COM motion removal nstcomm = 100 comm-mode = linear comm-grps = DOPC_LAU SOL

165 Appendix B

PERMISSIONS

The dissertation reuses figures from my published paper [154] with the per- mission shown in the following image.

Figure B.1: Permission statement to use Fig. 5.3, 5.4, 7.6, 7.7, 7.8, 7.9, 7.10, 7.11, 7.12, 7.13, and 7.14 and Table 7.1, 7.4, and 7.5.

The permissions to reuse the published materials from sources by other au- thors are presented below.

166 Permission to use Fig. 1.4: This Agreement between University of Delaware – Swapnil Baral (”You”) and Elsevier (”Elsevier”) consists of your license details and the terms and conditions provided by Elsevier and Copyright Clearance Center. License Number: 4870550712704 License date: Jul 15, 2020 Licensed Content Publisher: Elsevier Licensed Content Publication: Biophysical Journal Licensed Content Title: Separation of Liquid Phases in Giant Vesicles of Ternary Mixtures of Phospholipids and Cholesterol Licensed Content Author: Sarah L. Veatch, Sarah L. Keller Licensed Content Date: Nov 1, 2003 Licensed Content Volume: 85 Licensed Content Issue: 5 Licensed Content Pages: 10 Start Page: 3074 End Page: 3083 Type of Use reuse: in a thesis/dissertation Portion: figures/tables/illustrations Number of figures/tables/illustrations: 1 Portions: Figure 2b

167 Permission to use Fig. 2.3 and Table. 3.1: Synthesis and spectral properties of a hydrophobic fluorescent probe: 6-propionyl-2-(dimethylamino)naphthalene Author: Gregorio Weber, Fay J. Farris Publication: Biochemistry Publisher: American Chemical Society Date: Jul 1, 1979 Copyright © 1979, American Chemical Society

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• Permission is granted for your request in both print and electronic formats, and translations.

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• Appropriate credit for the requested material should be given as follows: ”Reprinted (adapted) with permission from (COMPLETE REFERENCE CI- TATION). Copyright (YEAR) American Chemical Society.” Insert appropriate information in place of the capitalized words.

• One-time permission is granted only for the use specified in your request. No additional uses are granted (such as derivative works or other editions). For any other uses, please submit a new request. If credit is given to another source for the material you requested, permission must be obtained from that source.

168 Permission to use Fig 3.5, 3.6, and 3.7: This Agreement between University of Delaware – Swapnil Baral (”You”) and Springer Nature (”Springer Nature”) consists of your license details and the terms and conditions provided by Springer Nature and Copyright Clearance Center. License Number 4873250019707 License date Jul 20, 2020 Licensed Content Publisher: Springer Nature Licensed Content Publication: Biophysical Reviews Licensed Content Title: Electric dipole moments of the fluorescent probes Prodan and Laurdan: experimental and theoretical evaluations Licensed Content Author: Cíntia C. Vequi-Suplicy et al Licensed Content Date: Jan 14, 2014 Type of Use: Thesis/Dissertation Requestor type: academic/university or research institute Format: print and electronic Portion: figures/tables/illustrations Number of figures/tables/illustrations: 3

169