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The effect of deuteration on receptor-ligand binding Lee, Anna 2017

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The Effect of Deuteration onReceptor-Ligand BindingbyAnna LeeB.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2017c© Anna Lee 2017AbstractIn this thesis, we use a particle coupled to a phonon bath to accurately model biologicaland chemical reactions. The path decomposition expansion (PDX) formalism is used todetermine the tunneling dynamics of the particle. By decomposing the potential energylandscape into the classically allowed and classically forbidden regions, we can calculate thepath integrals associated with each region and connect them to evaluate the full Green’sfunction.We will also discuss how deuteration of ligand molecules may affect enzyme-substratebinding in GPCR systems. It has been theorized that binding may be dependent on amolecular vibrational component. We investigate this in the β-adrenergic receptor systemusing the deuterated and non-deuterated forms of the ligand epinephrine. The measure-ment for successful binding is determined by the amounts of second messenger cyclic-AMPproduced. However, our results proved inconclusive and a discussion of possible problemsas well as recommendations is included.iiPrefaceThis thesis is original and unpublished work by the author, A. Lee. All of the work presentedin Chapter 3 was conducted in Dr. U. Kumar’s Laboratory at the University of BritishColumbia, Point Grey campus. I was responsible for performing biological assays and theanalysis of all research data.The experimental design was conceived by Dr. B. Finlay, Dr. U. Kumar and Dr. R.Somvanshi (University of British Columbia, Vancouver, Canada). All cells used in thisexperiment were also provided by Dr. U. Kumar.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 G Protein and GPCRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Inelastic Electron Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Tunneling Dynamics of a Particle Coupled to a Phonon Bath . . . . . 152.1 Path Decomposition Expansion . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Quantum Tunneling with Phonon Coupling . . . . . . . . . . . . . . . . . . 202.3 Applying the PDX Formalism to a Multiwell Potential . . . . . . . . . . . 232.3.1 Path Integral in the Classically Allowed Region . . . . . . . . . . . 242.3.2 Path Integral in the Classically Forbidden Region . . . . . . . . . . 262.4 Polaron Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Dual Coupling Model . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.2 The Holstein and Su-Schrieffer Heeger Models . . . . . . . . . . . . 302.4.3 Spectral densities of the Holstein and SSH Polarons . . . . . . . . . 312.4.4 Diagonal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.5 Non-diagonal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.6 Fourier Transform of the Effective Action . . . . . . . . . . . . . . . 333 Experimental Setup and Results . . . . . . . . . . . . . . . . . . . . . . . . 363.1 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51ivTable of Contents4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57vList of Figures1.1 All G protein-coupled receptors are characterized by their seven transmem-brane alpha helices. These helices are connected by extracellular and in-tracellular loops. The N-terminal (NH2) of the receptor is located on theextracellular side of the cell while the C-terminal (COOH) is located on theintracellular side. The above figure depicts the common structural similari-ties in a GPCR although differences may occur between different classes ofreceptors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 (a) The chemical structure of guanosine triphosphate (GTP). GTP consistsof a guanine nucleotide and a triphosphate molecule (both indicated in thefigure). (b) The chemical structure of guanosine diphosphate, which consistsof a guanine nucleotide and a disphosphate molecule. The Gα subunit pos-sesses GTPase activity, allowing it to hydrolyze GTP to GDP by removingan inorganic phosphate group. . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 A schematic of two of the four subfamilies of Gα subunits. Shown aboveare the Gαi/0 and Gαs subunits. Although the Gα12/13 and Gαq/11 subunitsare not shown above, the method of activation for the G protein and thedissociation of the α subunits are similar. The ligand binds to its specificGPCR and causes the dissociation of the Gα subunit from the βγ complex.The Gα subunit and βγ complex go on to activate their specific effectorsand trigger separate signaling pathways. The inhibition or stimulation of aneffector is denoted by a minus or plus sign respectively. . . . . . . . . . . . . 61.4 The mechanism of signal transduction through a GPCR. (a) The ligand bindsto the inactive GPCR, inducing a conformational change in the receptor. TheGPCR interacts with the Gα subunit, causing it to exchange GDP for GTP.(b) The α-GTP complex dissociates from the βγ subunits and each complextargets their downstream effectors. (c) The α subunit eventually hydrolyzesthe GTP to GDP. The α subunit reassociates with the βγ complex and theligand dissociates from the receptor. (d) The GPCR-G protein returns totheir inactive states. A more detailed description is provided in the text. . . 8viList of Figures1.5 The above figure illustrates the biological response of receptors to differenttypes of agonists and antagonists. The basal or constitutive level is thephysiological response that occurs in the absence of a bound ligand. A fullagonist induces a maximal biological response from the receptor. A partialagonist also stimulates a biological response but induces a weaker level ofactivity than a full agonist. A neutral agonist neither stimulates or inhibitsreceptor activity. An inverse agonist can inhibit a biological response andreduce it below the constitutive activity level. . . . . . . . . . . . . . . . . . 91.6 The crystal structure of a human engineered β2 adrenergic receptor bound tothe inverse agonist carazolol (denoted by green and red sticks). This figurewas produced by S. Jahnichen who has released the figure into the publicdomain [24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 A schematic of inelastic electron tunneling. The shaded areas represent thefilled electronic states up to the fermi level. (a) Depicts elastic tunnelingin which an electron that occupies a state below the Fermi level in the leftmetal crosses the barrier without any loss of energy and ends in a unoccupiedstate (white circle) above the Fermi level of the right metal. (b) Illustratesinelastic tunneling where the electron tunnels through to the other side butwith an energy loss of ~ω. Inelastic tunneling is only possible if eV ≥ ~ω. . 111.8 A graph of the current against the potential in a tunnel junction at T = 0and when eV = ~ω. The elastic tunneling process is linearly proportional toV . The presence of inelastic electron tunneling causes the gradient to skewat V = ±~ω/e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.9 A graph of the first derivative of the current against the potential at T = 0and when eV = ~ω. Inelastic electron tunneling causes the first derivative tohave step-like discontinuities at V = ±~ω. . . . . . . . . . . . . . . . . . . . 131.10 A graph of the second derivative of the current against the potential at T = 0and when eV = ~ω. The values of V where the delta functions occur canbe used to calculate the vibrational spectrum of the molecule in the tunneljunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 The configuration space is split into regions, one which is enclosed by thesurface Σ and the other region encompassing all regions which exclude Σ.Paths begin inside Σ at the point xi and can cross the surface any numberof times before finally exiting at the point xσ and then continuing to the endpoint at xf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17viiList of Figures2.2 A sketch of a possible path the particle can take in moving from the pointxi to the point xf . The paths are integrated from −∞ to +∞ up until thepoint xN as at that point the path must be integrated from −∞ to xσ. Pathsafter this are integrated starting from xσ to +∞ as the paths can no longertraverse the surface Σ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 A sketch of a possible path taken by a particle which starts and ends at Σ1.The path runs from Σ1 to Σ2 to Σ3 before finally ending at Σ1. This is oneof many paths that a particle can take starting at Σ1 and then returning tothe same surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Sketch of an asymmetric well with potential minima located at coordinates±q0/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Chemical structures of epinephrine, epinephrine-d3 and epinephrine-d6. Epinephrine-d3 has protons on the methyl group replaced with deuterium. Epinephrine-d6has all protons attached to carbon molecules switched with deuterium exceptthe lone methyl group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Assay 1. Plot of the agonist (epinephrine) concentration versus the percent-age of activity (%). HEK293 cells were transfected with either β1 (left) or β2(right) adrenergic receptors at varying concentrations. Results indicated thatthe percentage of activity increased with increasing epinephrine concentration. 403.3 Assay 1. Plot of the agonist (epinephrine-d3) concentration versus the per-centage of activity (%). HEK293 cells were transfected with either β1 (left) orβ2 (right) adrenergic receptors at varying concentrations. Results indicatedthat the percentage of activity increased with increasing epinephrine-d3 con-centration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 Assay 1. Plot of the agonist (epinephrine-d6) concentration versus the per-centage of activity (%). HEK293 cells were transfected with either β1 (left)or β2 (right) adrenergic receptors at varying concentrations. Results indi-cated that the percentage of activity increased with increasing epinephrineconcentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5 Assay 1. Plot of specific treatments versus the percent increase comparedto the basal level (%). Cells were treated with Forskolin and 1µM concen-trations of deuterated and non-deuterated epinephrine. HEK293 cells weretransfected with β1 adrenergic receptors. In this figure, cells treated withIBMX had an activity level set to 100%. Results indicated that treatmentswith Forskolin produced at least a 90% increase in cAMP production. Italso showed that adding 1µM of agonist to Forskolin did not make affect theactivity level greatly as Forskolin alone saturated cAMP levels. . . . . . . . 43viiiList of Figures3.6 Assay 1. Plot of specific treatments versus the percent increase comparedto the basal level (%). Cells were treated with Forskolin and 1µM concen-trations of deuterated and non-deuterated epinephrine. HEK293 cells weretransfected with β1 adrenergic receptors at varying concentrations. In thisfigure, cells treated with IBMX had an activity level set to 100%. Resultshere were skewed and differed from Fig. 3.5. It is likely that the “FSK” and“FSK + E3 1µM” samples were lower due to human error. The other twotreatments of Forskolin with epinephrine and epinephrine-d6 indicated highlevels of saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.7 Assay 3. Plot of the agonist (epinephrine-d3) concentration versus the per-centage of activity (%). HEK293 cells were transfected with β1 adrenergicreceptors and treated with varying concentrations of agonist. Results shownhere were randomly fluctuating and did not follow a clear trend. . . . . . . 453.8 Assay 3. Plot of the agonist (epinephrine-d6) concentration versus the per-centage of activity (%). HEK293 cells were transfected with β1 adrenergicreceptors and treated with varying concentrations of agonist. Most of theresults shown here were negative, implying that these samples produced lesscAMP than the basal level. This plot may follow a bell curve as there seemedto be an increase in activity up to 2.5µM and then a decrease in activity afterthis point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.9 Assay 3. Plot of the agonist (epinephrine) concentration versus the per-centage of activity (%). HEK293 cells were transfected with β1 adrenergicreceptors and treated with varying concentrations of agonist. Results shownhere gave the typical dose-response curve of increasing activity with increas-ing agonist concentration. This dose-response is what is expected for theepinephrine agonist in the β-AR system. . . . . . . . . . . . . . . . . . . . . 463.10 Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the per-centage of activity (%). HEK293 cells were transfected with β1 adrenergicreceptors and treated with varying concentrations of agonist. The dose-response here increases with increasing agonist concentration. . . . . . . . . 463.11 Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the per-centage of activity (%). HEK293 cells were transfected with β1 adrenergicreceptors, treated with varying concentrations of agonist and Gs inhibitor.The Gs inhibitor did not seem to have worked properly for certain treatments(156.25nM and 625nM treatments) as it should have suppressed cAMP pro-duction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47ixList of Figures3.12 Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the per-centage of activity (%). HEK293 cells were transfected with β1 adrenergicreceptors, treated with varying concentrations of agonist and Gi inhibitor.The results indicated very low levels of percentage of activity which was notexpected with the addition of Gi inhibitor. . . . . . . . . . . . . . . . . . . . 473.13 Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the per-centage of activity (%). HEK293 cells were transfected with β1 adrenergicreceptors, treated with varying concentrations of agonist and Gs and Gi in-hibitors. Here the activity levels are lower than 30%. Although not all ofthe treatments had activity levels lower than the non-inhibited values shownin Fig. 3.10, the Gs inhibitor did seem to suppress cAMP production in the2.5µM treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.14 Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the per-centage increase compared to the basal level (%). HEK293 cells were trans-fected with β1 adrenergic receptors and treated with either Forskolin orForskolin + 625nM of ED3 with or without an inhibitor. The percentageof activity for the sample treated with IBMX was set to 100%. The resultsalso seemed to indicate that the Gi and Gs inhibitors did not work as planned,although Forskolin did stimulate cAMP production above the basal level. . 493.15 Assay 7. Plot of the agonist (epinephrine) concentration versus the per-centage of activity (%). HEK293 cells were transfected with β1 adrenergicreceptors, treated with varying concentrations of agonist and Gs inhibitor.In this assay, the Gs inhibitor seemed to be effective at suppressing cAMPproduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.16 The signal transduction pathways initiated by the Gs and Gi subunits. TheGs subunit activates adenylate cyclase which stimulates the production ofcAMP. Higher concentrations of cAMP in the cell activates protein kinase A(PKA), a family of enzymes capable of phosphorylating other proteins. PKAcan phosphorylate the β-ARs which decreases the receptor’s coupling to Gsand increase coupling to Gi. The multiple arrows in succession representmultiple steps that are not shown. . . . . . . . . . . . . . . . . . . . . . . . 52xAcknowledgementsFirst, I would like to thank my thesis supervisor Dr. Philip Stamp for his guidance andsupport on this project. His knowledge proved invaluable during the course of this thesisand I am grateful for his contributions, patience and enthusiasm.Aside from my supervisor, I would like to thank Dr. Mona Berciu and Dr. StevenPlotkin of the Physics Department for their insightful comments and questions. I wouldlike to thank Dr. Brett Finlay, Dr. Ujendra Kumar and Dr. Rishi Somvanshi for theirexpertise in the biological experiment and for supplying the time and lab space needed forthis experiment.Lastly, I would like to thank my family for their support and encouragement throughoutmy entire academic career. This accomplishment could not have been achieved withoutthem.xiDedicationTo my parents.xiiChapter 1IntroductionEnzymes are essential to a myriad of physiological processes as they catalyze, or accelerate,chemical reactions. Due to their importance in biological systems, enzyme-substrate bindingare widely studied and two prominent theories have been proposed as to how these classof proteins recognize and bind specific substrates. These two theories are the lock-and-keymodel and the induced-fit model. Enzymes contain a region known as the active site wheresubstrates bind and undergo chemical reactions. In the lock-and-key model, the structureof the substrate fits exactly into the enzyme’s active site, allowing the enzyme to recognizeonly certain ligands. In the induced-fit model, the active site does not exactly match theshape of the substrate. The active site has residues or amino acids that may bind to specificsites on the substrate, causing the enzyme to undergo a conformational change. This changeallows the enzyme to bind to the substrate. Once the reaction is completed, the enzymereturns to its initial state. In both models, the success of binding is reliant on the enzyme’sability to recognize the shape and structure of the substrate.Recently, there has been evidence that binding may also be dependent on a molecularvibrational component. Experiments on olfactory mechanisms performed using Drosophilamelanogaster (fruit flies) indicate that these flies can differentiate between odourant (smell)molecules whose hydrogens have been replaced with deuterium, and those that have not.Even though the deuterated and non-deuterated molecules have identical structures, fliescould be trained to discriminate between them. Further experiments also indicated thatmolecules without comparable structures but similar vibrational spectra smelled similar toD. melanogaster [14]. As olfaction involves a G protein-coupled receptor (GPCR) system,it raises questions as to whether other GPCR systems may also bind substrates or ligandsusing a molecular vibration-sensing mechanism.Theories have been developed to explain how changes in molecular vibrational modes canalter receptor-substrate binding. One such theory involves inelastic electron tunneling froma donor site to an acceptor site, a mechanism that is facilitated by the odourant molecule.In this thesis we will study a different model, using a path decomposition approach todetermine the tunneling amplitudes for a multidimensional N -well problem that involveselectron-phonon coupling.Experimental results on another GPCR system, the beta-adrenergic system, will alsobe presented. The effect of deuterated and non-deuterated epinephrine on HEK 293 (hu-man embryonic kidney) cells expressing β1 adrenergic receptors (β1AR) and β2 adrenergic11.1. G Protein and GPCRsreceptors (β2AR) will be discussed. This thesis will present evidence on whether differencesin the vibrational modes of epinephrine will affect the binding of the ligand to the receptor.1.1 G Protein and GPCRsG protein-coupled receptors (GPCRs) form a family of transmembrane proteins that areinstrumental in passing signals and sensory stimuli from the outside of the cell to the inside.The activation of the receptor by a ligand, a biomolecule that binds and forms a complexwith the receptor, results in a cascade of chemical reactions that leads to the activation ofG proteins. Examples of ligands include hormones (signalling molecules), neurotransmit-ters (chemicals that facilitate nerve impulses), ions and odourants which stimulate sensoryreceptors (sight and smell respectively). G proteins are a class of regulatory proteins, whosepurpose is to form an appropriate response to any incoming stimuli and activate the relevantorgans or cells (called effectors) as needed. Essentially, the receptor and G protein form themachinery needed for signal transduction. The GPCR first intercepts the message from theligand and relays it to the G protein. The G protein then determines the course of actionand activates the necessary effectors involved. In this section, more detailed backgroundinformation on GPCRs and G Proteins and their mechanism of signal transduction will beprovided.About 800 human GPCR sequences have been identified with approximately 460 pre-dicted as being olfactory receptors [15]. As they have an important role in mediatingphysiological processes, approximately 30% of pharmaceutical drugs are developed to tar-get GPCRs, as defective receptors can lead to pathologies such as heart disease or cancer[23]. GPCRs can be identified by their structural configuration of seven transmembranealpha-helices connected by alternating intracellular and extracellular loops. The carboxyterminus (C-terminal) is on the inside of the cell while the amino terminus (N-terminal) ison the outside (see Fig. 1.1). GPCRs can be divided into 6 classes based on sequence sim-ilarities and structural motifs unique to that class. A description of each class is providedbelow [10, 19]:• Class A receptors are also known as rhodopsin-like receptors and form the majorityof the GPCR superfamily. This class of receptors includes rhodopsin, a photoreceptorthat enables vision in low-light conditions. Although this class is considered to berhodopsin-like, receptors in this class are diverse and can bind a variety of ligands.These can include peptides and other small biomolecules. Class A receptors sharesequence similarities in their transmembrane regions which may play a critical role intheir binding capabilities.• Class B receptors are known as secretin-like receptors as they contain receptors for thepeptide hormone secretin. Secretin regulates water balance and intestinal and stomach21.1. G Protein and GPCRsI	   II	   III	   IV	   V	   VI	   VII	  NH2	  COOH	  Cell	  Membrane	  Extracellular	  Intracellular	  Figure 1.1: All G protein-coupled receptors are characterized by their seven transmembranealpha helices. These helices are connected by extracellular and intracellular loops. TheN-terminal (NH2) of the receptor is located on the extracellular side of the cell while the C-terminal (COOH) is located on the intracellular side. The above figure depicts the commonstructural similarities in a GPCR although differences may occur between different classesof receptors.secretions. About 20 different receptors in this class interact with various neuropep-tides and hormones with the N-terminus involved in binding. The N-terminus is verylong and contains a network of disulphide bridges. These bridges are strong bondsformed between two cysteine amino acid residues.• Class C receptors are identified as metabotropic glutamate receptors which are pri-marily involved in central and peripheral nervous system functions. This class ofreceptors bind to glutamate, an amino acid that acts as a neurotransmitter. As it ismetabotropic, these receptors activate effectors indirectly through secondary messen-ger molecules. The N-terminus of this class of receptors is also very long (approxi-mately 600 amino acids) and is thought to contain the ligand binding site.• Class D and E are considered minor families found in fungi.• Class F consists of the frizzled/smoothened receptor family and is the most recentaddition to the classification. Both frizzled and smoothened receptors play a role incell and embryonic development though they bind to different ligands.Although the aforementioned receptors and classes all adopt a seven transmembrane31.1. G Protein and GPCRshelical configuration, the sequence similarity between each class is very minimal so they canbe differentiated structurally [10]. An alternative method of classification, called GRAFS,classifies GPCRs into five groups based on their phylogeny or evolutionary development. Inthis classification, the families are Glutamate, Rhodopsin, Adhesion, Frizzled/Taste2 andSecretin, which are similar to the previous classification but with the addition of Adhe-sion and Taste2. Adhesion receptors are involved in proper cell positioning in organs (celladhesion) and migration, while Taste2 receptors facilitate taste sensation [15].In order to transduce signals across the cell membrane, GPCRs must interact with Gproteins. G proteins are considered molecular switches. In their “off” state, G proteinsare inactive. In the “on” state, G proteins can initiate a cascade of chemical reactions inresponse to a stimulus detected by the GPCR. G proteins consist of three subunits labeledα, β and γ. These subunits are single protein molecules that can assemble with other proteinmolecules to form one larger complex. The G protein is considered a heterotrimer as thethree subunits that form it are not identical. The subunits of the G protein are essential tosignal transduction as they relay information from the receptor to various effectors. Thereare 21 Gα subunits, 6 Gβ subunits and 12 Gγ subunits [11]. I will provide a brief overviewof the Gα subunit as this subunit provides most of the basic properties of the G protein[41].All Gα subunits have a region with high binding affinity for guanine nucleotides. Gua-nine is one of the four main nitrogen-containing biological compounds that serves as abuilding block for DNA and RNA. Due to a high binding affinity for guanine nucleotides,the Gα subunit can bind to guanosine triphosphate (GTP) and guanosine diphosphate(GDP) compounds, which are simply guanine bases attached to three or two phosphategroups respectively (see Fig. 1.2(a) and 1.2(b)). When the α subunit is bound to GDP,the subunit exists in a complex with the β and γ subunits so the G protein remains in itsinactive heterotrimeric state. When the α subunit is bound to GTP, it dissociates from theβ and γ subunits and is capable of stimulating effectors. This is considered the “on” stateof the protein. These α subunits are enzymes and have weak GTPase activity which allowsthem to hydrolyze GTP at a slow rate. This allows the α subunits to cleave an inorganicphosphate group from GTP using water and turn it into GDP, thus returning the proteinback to its inactive state.The diverse number of Gα subunits can trigger various signaling pathways. Thesesubunits can be divided into four subfamilies where members of each family will sharestructural similarities and functionalities [41]. Descriptions of these four groups are providedbelow and schematically in Fig. 1.3.• The Gαi/0 family includes a variety of different subunits ranging from subunits thatinhibit adenylyl cyclase (Gαi1, Gαi2 and Gαi3) and α subunits responsible for thesignal transduction of taste and vision. Adenylyl cyclase is an enzyme which stimulatesthe production cyclic AMP (cAMP). Cyclic AMP is an important second messenger41.1. G Protein and GPCRs(a)	  Guanine	  base	  Triphosphate	  (b)	  Guanine	  base	  Diphosphate	  Figure 1.2: (a) The chemical structure of guanosine triphosphate (GTP). GTP consists ofa guanine nucleotide and a triphosphate molecule (both indicated in the figure). (b) Thechemical structure of guanosine diphosphate, which consists of a guanine nucleotide and adisphosphate molecule. The Gα subunit possesses GTPase activity, allowing it to hydrolyzeGTP to GDP by removing an inorganic phosphate group.51.1. G Protein and GPCRsN	  C	  αi βγ Adenylyl Cyclase ATP cAMP PLC-β N	  C	  αs Adenylyl Cyclase ATP cAMP Gi/0 Gs Figure 1.3: A schematic of two of the four subfamilies of Gα subunits. Shown above arethe Gαi/0 and Gαs subunits. Although the Gα12/13 and Gαq/11 subunits are not shownabove, the method of activation for the G protein and the dissociation of the α subunitsare similar. The ligand binds to its specific GPCR and causes the dissociation of the Gαsubunit from the βγ complex. The Gα subunit and βγ complex go on to activate theirspecific effectors and trigger separate signaling pathways. The inhibition or stimulation ofan effector is denoted by a minus or plus sign respectively.that relays signals received from receptors and initiates further intracellular reactions.• The Gαq/11 family includes all subunits that are responsible for the activation ofphospholipase C-β (PLC-β). Similar to adenylyl cyclase, PLC-β also mediates theproduction of a second messenger needed for signal transduction.• The Gα12/13 subunits stimulate regulatory processes such as transcription (DNA se-quences copied into RNA) and cell migration.• The Gαs subunits activate adenylyl cyclase to produce more of the second messengercAMP. The cAMP-dependent pathways regulate signal transduction for olfaction, cellproliferation and differentiation, and maintain physiological conditions at a stable andconstant level (homeostasis).The two subfamilies we will be focusing on in this thesis are the Gαi/0 and Gαs subunits.The Gα subunit acts independently during the active state of the G protein while the Gβ61.1. G Protein and GPCRsand Gγ subunits form a complex and act as a single unit. The βγ complex helps stabilizethe GDP-bound α subunit and also acts as a membrane anchor. In addition to this, thecomplex can also stimulate downstream effectors like adenylyl cyclase and phospholipaseCβ1 to Cβ3 [10, 20].There is a multitude of G proteins that can be made from the various combinations ofα, β and γ subunits available. In addition to that, one G protein can couple with variousGPCRs and thus stimulate a diverse set of pathways. Despite this, the process of activationcan be generalized over all sets of GPCRs, G proteins and ligands. In the inactive state, thereceptor is ligand-free and the G protein exists as a heterotrimer with the α subunit boundto GDP. When a ligand binds to the receptor, a conformational change is induced in thereceptor and it can now act as a guanine exchange factor (GEF). This allows the receptorto exchange GDP for GTP. The ligand-receptor complex interacts with the G Protein andcauses the dissociation of the GDP molecule from the α subunit. With normal cellularconcentrations of guanine nucleotides, GTP fills the site immediately causing the G proteinto “switch on” and reduces the affinity of the α subunit for the βγ subunits (Fig. 1.4a). Thisallows the GTP-α complex to dissociate from the βγ subunits and each complex can targettheir downstream effectors (Fig. 1.4b). Eventually the Gα subunit will hydrolyze the GTPmolecule bound to it to GDP and an inorganic phosphate (Fig. 1.4c). This terminates theactive phase and the GDP-bound α subunit reassociates with the βγ complex (Fig. 1.4d).The ligands which trigger the signal transduction mechanism described above can elicitdifferent responses from the GPCR and G protein systems. When the ligand binds to thereceptor, the system performs a specific action in response to the ligand. Receptor systemshave a constitutive or basal level of activity where the system exhibits a response despitehaving no ligands bound to the receptor. In the presence of a ligand however, the biologicalresponse can be increased or decreased depending on the concentration of ligand present.Bioassays are procedures that are commonly used to measure the biological response as afunction of the concentration of ligands present. The resulting data can be used to producedose-response curves which can determine quantitatively the effect of a ligand on a receptorand whether the ligand is an agonist or antagonist.Agonists are ligands which can activate receptors to stimulate a higher biological re-sponse to the stimuli. They can either be full agonists, which elicit the maximal biologicalresponse a receptor is capable of, or a partial agonist which induces a weaker response.Antagonists are ligands which block agonists from binding to the receptor molecule andcan dampen receptor activity. Inverse agonists are one type of antagonist that can inhibitactivity levels below that of the basal or constitutive levels. Neutral antagonists are inca-pable of eliciting any stimulation or inhibition from a receptor although they bind to thereceptor. Lastly, a biased agonist is a ligand which binds the receptor and signals throughG protein-dependent and independent pathways [38]. A schematic of how these differenttypes of agonists and antagonists affect the biological response are provided in Fig. 1.5.71.1. G Protein and GPCRsα	  β γ α Effector Effector α β γ Ligand GTP GDP (a)	  (b)	  N N C C α	  β γ α	  Effector Effector α β γ (c) (d) GTP Hydrolysis Ligand Dissociation C C N N Figure 1.4: The mechanism of signal transduction through a GPCR. (a) The ligand binds tothe inactive GPCR, inducing a conformational change in the receptor. The GPCR interactswith the Gα subunit, causing it to exchange GDP for GTP. (b) The α-GTP complexdissociates from the βγ subunits and each complex targets their downstream effectors. (c)The α subunit eventually hydrolyzes the GTP to GDP. The α subunit reassociates with theβγ complex and the ligand dissociates from the receptor. (d) The GPCR-G protein returnsto their inactive states. A more detailed description is provided in the text.81.1. G Protein and GPCRs100 50 0Biological Response (%) Increasing Drug Concentration à Full agonist Partial agonist Neutral agonist Inverse Agonist Basal Level Figure 1.5: The above figure illustrates the biological response of receptors to different typesof agonists and antagonists. The basal or constitutive level is the physiological responsethat occurs in the absence of a bound ligand. A full agonist induces a maximal biologicalresponse from the receptor. A partial agonist also stimulates a biological response butinduces a weaker level of activity than a full agonist. A neutral agonist neither stimulatesor inhibits receptor activity. An inverse agonist can inhibit a biological response and reduceit below the constitutive activity level.The signal transduction mechanism is well studied in β-adrenergic receptors (β-AR).Adrenergic receptors are targets of catecholamines (benzene molecules with 2 hydroxygroups and an amine side chain). Some well-known catecholamines are epinephrine, nore-pinephrine and dopamine. There are two main categories of adrenergic receptors, α andβ, with various subtypes. This thesis will focus on both β1 and β2, although a third sub-type exists, called β3. The β1ARs are primarily found in the heart and kidneys wherethey help regulate heart beat and blood pressure. The β2ARs regulate the function of thesmooth muscles of the lung, uterus and blood vessels [5]. Many drugs have been developedthat target both these receptors to treat heart disease, hypertension and asthma [36]. Theβ2AR was the second GPCR sequenced and cloned after rhodopsin, and was one of the firstGPCRs used in radioligand binding assays [27].Both β1 and β2ARs couple to the Gs subunit, although there is some evidence that thecoupling of these receptors can switch to Gi depending on agonist concentrations [16, 21, 33,37]. The binding of epinephrine to the βARs stimulates adenylyl cyclase. Adenylyl cyclaseincreases the intracellular production of second messenger cAMP, which also leads to thedownstream effectors of cAMP (like cAMP-dependent protein kinase, otherwise known asPKA) being activated.In chapter 3, we will look closely at the β-adrenergic system and how it responds todeuterated and non-deuterated epinephrine. We will attempt to determine whether a molec-91.2. Inelastic Electron TunnellingFigure 1.6: The crystal structure of a human engineered β2 adrenergic receptor bound tothe inverse agonist carazolol (denoted by green and red sticks). This figure was producedby S. Jahnichen who has released the figure into the public domain [24].ular vibration sensing mechanism is used in this GPCR system by performing a cAMPbioassay.1.2 Inelastic Electron TunnellingAs mentioned previously, there has been evidence that fruit flies are capable of distinguishingbetween odourants that are non-deuterated (unaltered) and their deuterated counterparts.As the chemical structures of these molecules are identical, it supports the idea that olfactionmay consist of a molecular vibration-sensing component in addition to structural recognition[14]. These experiments have also been performed using humans, although results have beenconflicting. Gane et al., 2013 [18], were able to determine that humans could differentiatecertain musks, although subjects could not distinguish between the odourants acetophenoneand d-8 acetophenone (the deuterated counterpart).Olfaction mechanisms that rely solely on shape and structure recognition are not ca-pable of describing the phenomenon presented above. As deuterated and non-deuteratedmolecules essentially possess the same structural properties, notably the same shape andvery similar mass, theories that account for a vibrational component must also be discussed.Turin has proposed that receptors act as biological spectrometers capable of detecting in-elastic electron tunneling within the system [39].101.2. Inelastic Electron Tunnelling￿ω(a) (b) eVlµLµRLeft Metal Right Metal Barrier Figure 1.7: A schematic of inelastic electron tunneling. The shaded areas represent the filledelectronic states up to the fermi level. (a) Depicts elastic tunneling in which an electronthat occupies a state below the Fermi level in the left metal crosses the barrier without anyloss of energy and ends in a unoccupied state (white circle) above the Fermi level of theright metal. (b) Illustrates inelastic tunneling where the electron tunnels through to theother side but with an energy loss of ~ω. Inelastic tunneling is only possible if eV ≥ ~ω.Inelastic electron tunneling spectroscopy (IETS) is a standard experimental techniquethat is frequently used to study the vibrational modes in a variety of systems. IETS requiresa tunnel junction, where a thin insulating barrier is sandwiched between two metal contacts.A voltage is applied between the contacts, which causes the Fermi energy levels of thecontacts to become separated by an energy of eV . The thin barrier is usually a metal oxidelayer with molecules adsorbed onto it. IETS can be used to study these tunnel junctionsand provide information about the vibrational modes of the molecular adsorbates.Tunneling between the contacts can occur in two ways, through either an elastic orinelastic process (see Fig. 1.7). At T = 0, the states in each metal contact are filled up tothe Fermi energy and are empty above it. The occupied states on the left contact overlapwith the unoccupied states on the right contact. By the exclusion principle, only transitionsthat start from an occupied state and end at an unoccupied state can occur; therefore, thetransitions are determined by the potential difference, V . In an elastic tunneling process, anelectron starts from an occupied state on the left contact, tunnels through the barrier andends in an unoccupied state at the right contact without any loss of energy. The tunnelingcurrent produced from this process can be determined by integrating over the probabilitiesthat a particle starts out in an occupied state, ends in an unoccupied state and that it hasthe ability to penetrate the barrier. We will state the formula here, but more detail can be111.2. Inelastic Electron Tunnellingfound in [4],I = C∫ ∞−∞dE[f(E)− f(E + eV )]∫ E0dE⊥ exp(−2∫ l0Kdx), (1.1)f(E) = [1 + exp(E/kT )]−1, (1.2)K =2m~2√U(x)− (E − E⊥), (1.3)where U(x) is the potential energy, E is the total electron energy and E⊥ is the kineticenergy of the electron that is perpendicular to the barrier and is equivalent to ~2k2⊥/2m.If one approximates that the voltage dependence of the barrier is negligible then thecurrent becomesI = C∫ ∞−∞dE[f(E)− f(E + eV )] = CeV (1.4)such that the elastic tunneling process is ohmic, and that its first derivative is constant.In the inelastic tunneling process, the electron starts off from an occupied state andalso tunnels through the barrier. However, in this process the electron loses a quantumof energy, ~ω, to a local vibrational mode of the barrier as it crosses. The electron endsin an unoccupied state on the right contact with an energy ~ω less than the initial state.The inelastic electron tunneling process is thus facilitated by the emission of a phonon[22]. Inelastic electron tunneling will only occur if there is an unoccupied state on the rightcontact that has an energy exactly ~ω less than the occupied state on the left. This processis only possible if the energy difference between the two Fermi levels fulfills the conditionthat eV ≥ ~ω. This process provides an additional pathway for the particle to tunnel acrossthe barrier and thus alters the I − V relationship slightly by causing breaks or kinks wheneV = ~ω. At T = 0, these kinks introduce discontinuities in the first derivative, ∂I/∂V ,which are then represented as delta functions in the second derivative. Figures 1.8 to 1.10illustrate the changes in the I-V relationship when inelastic electron tunneling is introduced.If the system being studied has many modes, each of these modes will contribute a deltafunction in the second derivative at the associated potential difference. Therefore, one coulddetermine the vibrational spectrum by studying the second derivative of the current withrespect to the potential.Turin theorizes that olfactory receptors function as biological spectrometers. The re-ceptor is analogous to the two metal contacts while the odourant molecule is the substancewithin the gap being studied. As electrons travel from the left contact (donour site) to theright contact (acceptor site), they must pass through the odourant molecule and lose energyto the odourant’s local modes. In this fashion, the receptor is capable of distinguishing theodourant from other molecules based on its vibrational spectrum. Turin notes that olfac-tory receptors fulfill certain requirements needed for electron transfer and inelastic electrontunneling to work in this system. The receptors have conserved binding sites present for121.2. Inelastic Electron Tunnelling−￿ω/e+￿ω/eIVFigure 1.8: A graph of the current against the potential in a tunnel junction at T = 0 andwhen eV = ~ω. The elastic tunneling process is linearly proportional to V . The presenceof inelastic electron tunneling causes the gradient to skew at V = ±~ω/e.−￿ω/e +￿ω/eV∂I∂VFigure 1.9: A graph of the first derivative of the current against the potential at T = 0and when eV = ~ω. Inelastic electron tunneling causes the first derivative to have step-likediscontinuities at V = ±~ω.131.2. Inelastic Electron Tunnelling−￿ω/e+￿ω/eV∂2I∂2VFigure 1.10: A graph of the second derivative of the current against the potential at T = 0and when eV = ~ω. The values of V where the delta functions occur can be used tocalculate the vibrational spectrum of the molecule in the tunnel junction.soluble electron carriers and also binding sites for metal cofactors [39]. Metal cofactors(metal ions) are often required for electron-transfer enzymes to function correctly. Thesecharacteristics help facilitate electron tunneling within the receptors.Calculations have also been done to determine whether IETS is a plausible mechanism inolfaction. While these studies determined that IETS could be possible in this system, manyassumptions were made regarding the properties of the receptor [3, 40]. It is doubtful thatsuch a model could reliably describe the system and mechanism of olfaction. Furthermore,receptors are complex biomolecules and although they may possess certain properties neededfor inelastic electron tunneling, the presence of these is not enough evidence that receptorsact as biological spectrometers. A more realistic model is needed to determine how such asystem works.14Chapter 2Tunneling Dynamics of a ParticleCoupled to a Phonon BathIn the previous chapter, we mentioned that there is currently a lack of a realistic model thatadequately describes biological and chemical reactions. Here, we will attempt to constructsuch a model by using a path integral formulation that includes both diagonal and non-diagonal couplings. We will do this by combining the work done by Auerbach and Kivelson,and Caldeira and Leggett.In section 2.1, we will first discuss the path decomposition expansion (PDX) developedby Auerbach and Kivelson, which uses a connection formula to determine the dynamics ofa particle subject to a complicated potential energy landscape. Section 2.2 will then brieflydiscuss Caldeira and Leggett’s work on a particle coupled to a phonon bath, focusing onthe effective action. In section 2.3, we will see how the PDX formalism can be applied to amultiwell potential and determine the propagators for a particle traveling in the classicallyallowed and forbidden regions. In section 2.4, we outline a dual coupling polaron modelwhich provides the diagonal and non-diagonal couplings which are used to determine thespectral functions. We then end this chapter by determining the full Green’s function forthe particle.2.1 Path Decomposition ExpansionThe path decomposition expansion is a method established by Auerbach and Kivelson toevaluate multidimensional tunneling [2]. There is a variety of instanton techniques thatcan be used to solve these problems but they tend to have limitations. For instance, thesetechniques rely heavily on ground-state tunneling paths and so are not suitable for situationswhere back-scattering is significant or where the initial state is excited [1]. To overcomethese limitations, Auerbach and Kivelson developed a multidimensional connection formulawhich breaks the configuration space into various regions (classically forbidden and allowed)and allows one to solve each region separately. The separation of the regions in configurationspace simplifies the calculations and also allows one to tackle problems where the potentialenergy landscape is very complex. I will first briefly describe the derivation of the pathintegral formalism and how it can be developed into the path decomposition expansion.Path integrals provide the transition amplitudes of a particle that moves from a point152.1. Path Decomposition Expansionxi at time ti to a point xf at time tf . The propagator, Gˆ(tf − ti) = exp(−(i/~)Hˆ(tf − ti),is the time evolution operator that can be applied to the states of the system to determinethe transition amplitude. For example, Gxf ,xi(tf − ti) = 〈xf | Gˆ(tf − ti) |xi〉 is the transitionamplitude (or 1-particle Green’s function) for a particle to move in space-time from (xi, ti)to (xf , tf ).This form can be extended into the Feynman form for path integrals. The propagatorfor a many particle system can be written as,G(Xf , Xi; tf , ti) =∫ x(tf )=Xfx(ti)=XiDx(t) exp((i/~)S[x, x˙] (2.1)where X = (~r1, ~r2, ..., ~rn), and the action is S[x, x˙] =∫ tftidtL(x, x˙; t). The path inte-gration measure,∫Dx(t), is simply the integration of infinitesimal line segments over theentire space. For a 1-dimensional free particle with potential V (x), the Green’s function in1D can be written in terms of infinitesimal line segments as in the following equation:G(xf , xi; tf , ti) = limN→∞t→∞N−1∏j=1∫dxjAN i~dtN−1∑j=1[m2(xj+1 − xjdt)2− V (xj)] (2.2)where dt =tf−tiN , dxj = xj+1 − xj and AN is the normalization factor. This is thestarting point for the PDX formulation.In the PDX formalism the configuration space can be divided into various regions,usually the classically allowed and forbidden zones. I will begin with an example of wherethe configuration contains a surface Σ and consists of two regions, inside the surface andoutside the surface. The paths originate at a point xi located inside the surface and end ata point xf outside the surface. These paths can cross Σ any number of times before exitingthe surface one final time at a point xσ. The point xσ is the last point where the pathtouches the surface Σ, and from there the path continues on to the end point xf (see Fig.2.1).As was explained previously, the path integral for a free particle is usually integratedover the entire surface. In this approach however, the paths that cross the point xσ will notre-enter the interior region of the surface and so the integration limits must reflect this (seeFig. 2.2). The sum over all paths then becomes:GN,n = AN∫ ∞−∞...∫ ∞−∞dx1dx2...dxn−1 exp[−1~S(0, n− 1)](2.3)×∫ xσ−∞dxn∫ ∞xσ...∫ ∞xσdxn+1dxn+2...dxN exp[−1~S(n− 1, N)](2.4)where S(n1, n2) =∑n2j=n1[m(xj+1−xj)22dt + V (xj)dt]. The above equation is for 1D butcan be easily reconfigured for multiple dimensions. The displacements along the surface can162.1. Path Decomposition ExpansionxixfxσΣFigure 2.1: The configuration space is split into regions, one which is enclosed by the surfaceΣ and the other region encompassing all regions which exclude Σ. Paths begin inside Σat the point xi and can cross the surface any number of times before finally exiting at thepoint xσ and then continuing to the end point at xf .2dt0ndtNdtdt 3dtxixσxfxn+∞−∞Figure 2.2: A sketch of a possible path the particle can take in moving from the point xi tothe point xf . The paths are integrated from −∞ to +∞ up until the point xN as at thatpoint the path must be integrated from −∞ to xσ. Paths after this are integrated startingfrom xσ to +∞ as the paths can no longer traverse the surface Σ.172.1. Path Decomposition Expansionbe parametrized by the components xσ,i, where i = 1, 2, ...N with xσ,1 chosen to be alongthe unit vector ~n1 which is normal to the surface Σ. By changing the integration variablesto integrate over the surface, the connection formula becomes:G(xi, xf , T ) =∫ T0dt∫ΣdσG(xi, x, t)[−i~2mnˆ1 · ~∇]G(r)(x, xf , T − t)∣∣x=xσ(2.5)or in terms of the energy Green’s functionG(xi, xf ;E) =∫ΣdσG(xi, x;E)[~2mnˆ1 · ~∇]G(r)(x, xf ;E)∣∣x=xσ(2.6)where G(x2, x1;E) =∫∞0 dt exp(iEt)∫ x2x1DX exp( i~(S[X] + iδt) and δ → 0+. TheGreen’s function G(r) is called the restricted Green’s function and sums over all pathslocated outside the surface Σ. It can be defined as follows:G(r)(x, x′;E) =∫ ∞0dt exp(iEt)∫ x′xDX Θ(X − Σ) exp(i~S[X] + iδt). (2.7)G(r) contains the same action as G and also satisfies the same differential equationeverywhere outside of Σ. It also obeys a Dirichlet boundary condition on the surface whichis given byG(r)(x, x′;E)∣∣x=Σ= 0. (2.8)Eq. 2.6 is the multidimensional connection formula developed by Auerbach and Kivel-son. The term i~2m~∇σ arises from the Jacobian of the change in variables. It can be inter-preted as a flux operator and represents the average incoming or outgoing velocities at thepoint xσ. Its expression is equivalent tof(x)~∇g(x) = [f(x)∂xg(x)− g(x)∂xf(x)] . (2.9)We will express the flux operator as[Σ] = limx,x′→xσ∫Σdσ |xσ〉 〈x|[~2mnˆ1 · ~∇]|x′〉 〈xσ| (2.10)and will use this to simplify the notation for the full Green’s function.The PDX connection formula provided above was only for two regions, however thisformula can easily be extended for multiple decomposition surfaces, Σi, where i = 1, ..., N .We will define the transition matrix tij astij = [Σi]gij [Σj ] (2.11)182.1. Path Decomposition Expansionwhere gij is the restricted Green’s function which starts and ends at points xσ,i andxσ,j , which belongs to surfaces [Σi] and [Σj ] respectively, but does not enter the interior ofany surface. We will first begin with an example of a 2-surface problem which splits theconfiguration space into 3 regions. If the particle were to begin its path starting from x1at Σ1 and end in the same location, the particle can take a multitude of paths, one wherethe particle stays within the well, one where the particle travels to Σ2 and back, and pathswhere it constantly moves back and forth between the two sites. This is summarized in thefollowing equation:G11 = 〈x1| g1 + g1t12g2t21g1 + g1t12g2t21g1t12g2t21g1 + ... |x1〉 (2.12)where here g1 is the local Green’s function for the site centred at Σ1. We can see thatthis is a geometric series and that the full Green’s function can also be rewritten asG11 = 〈x1| [g−11 − t12g2t21]−1 |x1〉 . (2.13)We will now try to generalize the full Green’s function for cases where there are morethan 2 potentials and where the initial and final points are not identical. We note that fora full Green’s function the sum can be expressed asGαβ = gαδγβ + gαtαβgβ + gαtαγ1gγ1tγ1βgβ + gαtαγ1gγ1tγ1γ2gγ2tγ2βgβ + ... (2.14)where α represents the initial site, β the final site, and γn (where n = 1, 2, 3...) representsthe intermediate sites the particle travels to. One can see here that only certain paths areallowed within the full Green’s function, as indicated by the indices, and so the summationmust be done carefully. We note that the above expression contains the pattern t + tgt +tgtgt+ ... which allows us to write two recursion formulas for the full Green’s function:Tαβ = tαβ + tαγGγδtδβ, (2.15)Gαβ = gαδαβ + gαTαβgβ. (2.16)Eq. 2.15 and 2.16 ensure that only allowed paths are included in the full Green’sfunction.The path decomposition expansion is very useful for multidimensional tunneling prob-lems with complicated potential landscapes. While there is a multitude of instanton tech-niques available to solve these problems, they tend to focus on the forbidden region andignore the classical regions. The PDX formalism allows one to decompose a complicatedproblem into manageable pieces that can be solved separately. This allows one to evaluateboth the forbidden and classically allowed regions which can provide more physical insightto the problem.Using the PDX formalism, we will attempt to realistically model and solve a multiwell192.2. Quantum Tunneling with Phonon CouplingΣ1Σ2Σ3Figure 2.3: A sketch of a possible path taken by a particle which starts and ends at Σ1.The path runs from Σ1 to Σ2 to Σ3 before finally ending at Σ1. This is one of many pathsthat a particle can take starting at Σ1 and then returning to the same surface.quantum tunneling problem. We will look at the multiwell problem as done by Auerbachand Kivelson but also include the system’s interaction with an environment consisting ofphonons. This will ensure that we realistically model what occurs in chemical and biologicalreactions.2.2 Quantum Tunneling with Phonon CouplingIn chemical and biological reactions, the interaction between the system and environmentmay play a significant role in the reaction dynamics. To realistically simulate these inter-actions, we will include an environment consisting of harmonic oscillators coupled to theparticle system. We will introduce one restriction to simplify the calculations: the system isonly weakly coupled to each environmental mode, ensuring that each environmental modeis only weakly perturbed. This restriction will lead to a Lagrangian that can be used torepresent various dissipative systems [6].The phonon coupling can be included in the PDX method mentioned in the previoussection to determine how phonon modes can mediate the tunneling of a particle throughthe potential energy landscape. We will approach this problem using the Lagrangian for-mulation of quantum mechanics and make use of influence functionals, which were definedby Feynman and Vernon (1963) [13]. We will then look at the effective action that resultsand see how this will fit into the PDX formalism.202.2. Quantum Tunneling with Phonon CouplingA Lagrangian which describes a system interacting with an environment of phonons is:L =12Mq˙2 − V (q) + 12∑j(mj x˙2j −mjω2jx2j )−∑jFj(q)xj −∑jF 2j (q)/2mjω2j (2.17)where the first two terms describe the particle (system) whose displacement is denoted by q,the third term describes the harmonic oscillators (environment) denoted by xj , the fourthterm describes the interaction between the system and environment and the last term isa counter-term which renormalizes the potential. We choose the term∑j Fj(q)xj to bebilinear such thatFj(q) = Cjq. (2.18)All information about the effects of the environment on the system is then incorporatedin the spectral densityJ(ω) ≡ pi2∑jC2jmjωjδ(ω − ωj). (2.19)With the above general Lagrangian, we can determine the density matrix in equilibriumfor the system and environment asρ(qi, {xji}; qf , {xjf};β) ≡∑nψ∗n(qi, {xji})ψn(qf , {xjf}) exp(−βEn), (2.20)where qi and qf represent the initial and final values for the particle coordinates respec-tively, xji and xjf represent the endpoints for the harmonic oscillators and ψn represents thecombined state of the system and environment. The general Lagrangian for this problemcan be split into two parts, one which is dependent on the system only and another whichdepends on the oscillator coordinates and the bilinear interaction. As we are only interestedin the dynamics of the system, the environmental coordinates can be traced over to obtainthe following propagatorG(qi, qf , T ) =∫ ∏jdxji∫ q(T )=qfq(0)=qiDq(τ)∏j∫ xj(T )=xjixj(0)=xjiDxj(τ) (2.21)× exp(−∫ T0L(q, q˙; {xj , x˙j})dτ/~)(2.22)where L is the Lagrangian given in Eq. 2.17 with the bilinear coupling.We can perform this trace by noting that the environment coordinates xj and x˙j are in212.2. Quantum Tunneling with Phonon Couplingeither quadratic or linear powers. The integral to be evaluated isQ(τ) =∫dxi∫ x(T )=xfx(0)=xiDx(τ) exp{−1~∫ T0(12mx˙2 +12mω2x2 − Cqx)dτ}(2.23)which contains the action for a driven harmonic oscillator. This integral can be solvedin various ways. One method that provides the simplest result is detailed in [12]. We referthe reader to [12] for further details and will state the result here:Q(τ) = exp{− 14mω~∫ T0∫ T0coshω(|τ − τ ′| − 12T )sinh 12ωTq(τ)q(τ ′)dτdτ ′}. (2.24)If we specify that q(τ) outside of 0 ≤ τ ≤ T be periodic with period T , then theexpression above can be simplified toQ(τ) =12cosechωT2exp{C24mω~∫ ∞−∞dτ ′∫ T0dτe−ω|τ−τ′|q(τ)q(τ ′)}. (2.25)Now inserting this term into the propagator we findG(qi, qf ;T ) = G0(T )∫ q(T )=qfq(0)=qiDq(τ) exp[−Ssys/~] exp[Λ/~] (2.26)whereG0(T ) =∏j12cosechωjT2, (2.27)Ssys[q(τ)] =∫ T0(12Mq˙2 + V (q))dτ, (2.28)Λ[q(τ)] =∫ T012M∆ω2q2(τ)dτ +∑j{C2j4mjωj∫ ∞−∞dτ ′∫ T0dτe−ωj |τ−τ′|q(τ)q(τ ′)}.(2.29)The form of Λ[q(τ)] can be simplified by noting thatq(τ)q(τ ′) =12{q2(τ) + q2(τ ′)− (q(τ)− q(τ ′))2} (2.30)and by inserting this into Λ[q(τ)] we can integrate q2(τ)and q2(τ ′) over dτ and dτ ′respectively. Performing this integration results in canceling out the first term in Λ[q(τ)]so that the resulting form isΛ[q(τ)] = −12∫ ∞−∞dτ ′∫ T0dτ α(τ − τ ′){q(τ)− q(τ ′)}2, (2.31)222.3. Applying the PDX Formalism to a Multiwell Potentialwhereα(τ − τ ′) ≡∑jCj4mjωjexp(−ωj |τ − τ ′|) (2.32)≡ 12pi∫ ∞0J(ω) exp(−ω|τ − τ ′|)dω ≥ 0 (2.33)where J(ω) is the spectral density defined in Eq. 2.19. The propagator for a particle linearlycoupled to a bath of harmonic oscillators is expressed asG(qi, qf , T ) = G0(T )∫ q(T )=qfq(0)=qiDq(τ) exp[−Seff[q(τ)]/~] (2.34)where the effective action is given bySeff[q(τ)] =∫ T0{12Mq˙2 + V (q)}dτ +12∫ ∞−∞dτ ′∫ T0dτα(τ − τ ′){q(τ)− q(τ ′)}2. (2.35)The above result has been used in various papers to determine the tunneling dynamicsof a particle coupled to a phonon bath. In [6] and [29], the authors determine the dynamicsby first truncating the high frequency modes of the harmonic oscillators in order to generatea two-state system called the spin-boson system. While we will not describe the details ofthe truncation in this thesis, we will mention that in this method they introduce an artificialcutoff for the spectral density in order to split the low and high frequency modes of thebath.Instead of introducing an artificial cutoff, we will determine the effective action for aparticle traveling in a phonon bath by adopting a dual coupling polaron model. This methodwill better determine the dynamics as it includes both diagonal and non-diagonal couplingsand does not include an unphysical cutoff.2.3 Applying the PDX Formalism to a Multiwell PotentialIn this section, we will see how the effective action can be used to determine the propagatorsfor regions in the classically allowed and forbidden regions. Only a brief overview will beprovided so the reader is encouraged to see [6] and [29] for more detail.As was mentioned previously, the spectral density incorporates all effects the environ-ment, in this case the phonon bath, has on the system. This includes the effects of boththe diagonal and non-diagonal couplings. In the classically allowed regions, the diagonalcouplings can be used to determine the spectral density while the forbidden regions areassociated with the non-diagonal couplings. By inserting these spectral densities into thepropagator, we can determine the propagators for a particle coupled to a phonon bath.To apply the PDX formalism to a multidimensional well, one must split the configuration232.3. Applying the PDX Formalism to a Multiwell Potentialspace into various regions. For example, Fig. 2.4 depicts an asymmetrical well in which theground state energies are separated by an energy ˜ and the minima are located at ±q0/2. Forsuch a potential, we can first split the potential into three separate regions. Regions I and IIIcomprise of surfaces which enclose the potential wells while region II encompasses the areabetween the two wells (e.g. the forbidden region). One can solve the path integral withineach region and then simply connect the propagators via the PDX formalism described inthe previous section. We will see how one calculates the path integrals within the classicallyallowed and forbidden regions in the next two subsections.￿˜V (q)−q02+q02qFigure 2.4: Sketch of an asymmetric well with potential minima located at coordinates±q0/2.2.3.1 Path Integral in the Classically Allowed RegionWe will begin by determining the path integrals for a particle coupled to bath oscillatorswithin the potential wells. The propagator, as we have seen previously, can be representedas followsG(qi, qf ;β) = const exp[−Scl(qi, qf ;β)] (2.36)where Scl is the action evaluated along the classical path starting at qi at time zero andending at qf at time β. To simplify the expression of the effective action, we will take its242.3. Applying the PDX Formalism to a Multiwell PotentialFourier transform. The Fourier transforms areQ(ω) =∫ ∞−∞q(τ)e−iωτ , (2.37)q(τ) =12pi∫ ∞−∞dωQ(ω)eiωτ . (2.38)Inserting these fourier transforms into the effective action, we obtainSeff[Q(ω)] =12pi∫ ∞−∞dω∫ ∞−∞dω′12M(−ωω′)Q(ω)Q(ω′)δ(ω + ω′) + SV+12∫ ∫ ∫ ∫dτdτ ′dω′dω′′∫ ∞0dω(2pi)3J(ω)e−ω|τ−τ′|Q(ω′)(Q(ω′′)eiτ(ω+ω′)×(1− eiω′(τ ′−τ) − eiω′′(τ ′−τ) + ei(ω′+ω′′)(τ ′−τ)).(2.39)In the last term, we let s = τ ′ − τ and then integrate over dτ . The last term then takesthe form12∫ ∫ ∫dsdω′dω′′∫ ∞0dω(2pi)2J(ω)e−ω|−s|Q(ω′)Q(ω′′)δ(ω′ + ω′′)× {1− eiω′s − eiω′′s + ei(ω′+ω′′)s}=12(2pi)2∫ ∫ ∫dωdω′Q(ω′)Q(−ω′){2∫ ∞−∞dsJ(ω)e−ω|−s|−2∫ ∞−∞dsJ(ω)eω|−s|e−iω′s}=1(2pi)2∫ ∫dωdω′Q(ω′)Q(−ω′)J(ω){∫ 0−∞eωsds+∫ ∞0e−ωsds∫ ∞0es(ω−iω′)ds+∫ 0−∞eis(ω+iω′)ds}=12pi2∫ ∫dωdω′Q(ω′)Q(−ω′)J(ω){1ω− ωω + ω′2}=12pi2∫ ∫dωdω′Q(ω′)Q(−ω′)J(ω)(ω′2ω(ω2 + ω′2).(2.40)The full effective action written in terms of double integrals and frequency ω is thenSeff[Q(ω)] =12pi∫ ∞−∞dω′12Mω′2Q(ω′)Q(−ω′) + SV+12pi2∫ ∞−∞dω′∫ ∞0dωQ(ω′)Q(−ω′)J(ω){ω′2ω(ω2 + ω′2)} (2.41)where SV is the Fourier transform of V (q). After determining the spectral density, wecan use Eq. 2.41 to fully determine the propagator for the paths the particle takes inside a252.3. Applying the PDX Formalism to a Multiwell Potentialwell, or as was discussed previously, the Green’s function for the particle that is localizedto a surface before it finally leaves at the point xσ.2.3.2 Path Integral in the Classically Forbidden RegionIn this section we will use instanton techniques to determine the restricted Green’s functionwhich includes all paths that do not cross the enclosed surfaces Σi and Σi′ where i and i′are labels for different sites. We will provide the main results of the instanton techniquehere as a detailed explanation can be found in [7] and [8]. In the instanton method, thetunneling amplitude Γ is defined asΓ = A exp(−B/~), (2.42)where the factors A and B areA =(B2pi~)1/2 ∣∣∣∣ det Dˆ0det′ Dˆ1∣∣∣∣1/2, (2.43)B =∫ T0{12Mq˙2 + V (q)}dτ +12∫ ∞−∞dτ ′∫ T−∞dτα(τ − τ ′){q(τ)− q(τ ′)}2. (2.44)The quantities Dˆ0 and Dˆ1 are differential operators and are defined asDˆ0 ≡(− d2dτ2+ ω2)+1M∫ ∞−∞α(τ − τ ′)[q(τ)− q(τ ′)]dτ ′, (2.45)Dˆ1 ≡(− d2dτ2+1MV ′′|q¯(τ)|)+1M∫ ∞−∞α(τ − τ ′)[q(τ)− q(τ ′)]dτ ′ (2.46)and the prime on the determinant indicates that the zero eigenvalue is removed.The above results are obtained by noting that the path integral is dominated by thestationary or critical points of the action. The stationary point or classical path, denotedq¯(τ), satisfies the equations of motion so that the first variational derivative of the actionδSδq¯ = 0. The classical path q¯(τ) satisfies the boundary conditions such that q(−T/2) = q(i)σand q(T/2) = q(i′)σ during time T . While the classical path dominates, small fluctuationscan contribute to the path integral which are given by the second variational derivative ofthe action. These can be represented by the differential operators mentioned above. InCallan and Coleman’s description of the instanton method, multiple “bounces” may takeplace in which the particle travels back and forth across the barrier. Here, we are onlyinterested in one bounce or one instanton.The restricted Green’s function in the forbidden region is expressed asgii′(q(i)σ , q(i′)σ , τ) =∫Dq(τ)e−SE [q(τ)]/~ (2.47)262.4. Polaron Modelwhere q(i)σ is the point at which the particle exits the surface Σi and never returns, q(i′)σ isthe point at which the particle enters the surface Σi′ , τ represents imaginary time and SEis the Euclidean action. For one bounce, the restricted Green’s function takes the formg(q(i)σ , q(i′)σ , τ) = NT(B2pi~)1/2e−B/~(′det[Dˆ1])−1/2. (2.48)We now have the amplitudes for each region and may determine the total Green’sfunction using the propagators found in subsections 2.3.1 and 2.3.2. In the next section, wewill determine the spectral densities by employing a dual coupling polaron model.2.4 Polaron ModelSo far we have discussed the PDX formalism, the effective action for a particle coupled toa phonon bath and have determined the propagators that may be used for the classicallyallowed and forbidden regions. In this section, we will now determine the diagonal andnon-diagonal couplings that will be used to determine the spectral density for the particleas it travels in a multiwell potential landscape. These couplings can be determined by usingthe Holstein and Peierls coupling. A dual coupling model will be discussed in this sectionwhich combines both the Holstein and Su-Schrieffer-Heeger model.2.4.1 Dual Coupling ModelThe coupling between electrons and phonons has been well-studied for decades as the in-teraction leads to various interesting phenomena. An electron traveling through a solid canlead to the formation of a polaron, which is an electron surrounded by a cloud of optical oracoustic phonons. Polarons are interesting subjects as they can be used to determine thephysics of various insulating and semiconducting materials. Most models used to describethe coupling between electrons and phonons generally look at diagonal couplings, such asthe Holstein model. However, recent studies indicate that off-diagonal couplings may alsocontribute significantly to polaron properties and their dynamics ([31], [32]). In this thesiswe will look at a dual coupling model which includes both Holstein and Peierls coupling,following the prescriptions set forth in [32].The interaction of electrons and phonons can be represented by the Hamiltonian below:H0 = −∑ijtij({bλ})(c†icj + c†jci) +∑ii({bλ})c†ici +∑λωλb†λbλ. (2.49)The above Hamiltonian represents a particle hopping between different lattice sitesi, j with a hopping amplitude of tij . The particle interacts with a bath of phonons withfrequencies ωλ where the index λ describes the quantum numbers of the phonons (e.g.λ = {~q, µ}). Both the hopping amplitude and the onsite energies are dependent on the272.4. Polaron Modelbosonic variables through the displacement operator, xˆλ =√~2Mω0(bλ + b†λ). One canexpand the onsite and hopping energies in terms of the bosonic variables. The onsiteenergy can then be given as:i = 0 +∑λU(1)i (λ)(bλ + b†λ) +∑λ,λ′U(2)i (λ, λ′)(bλ + b†λ)(bλ′ + b†λ′) + ... (2.50)where the quantities U(1)i and U(2)i represent the 1-phonon and 2-phonon diagonal cou-plings. The hopping terms may also be expanded, although its expansion is quite differentfrom the onsite energies. As the hopping amplitudes tij represent the electron tunnelingamplitudes, it will vary as an exponential function of the bosonic variables. The hoppingenergy can then be written astij({bλ) = t0 exp[−∑λVij(λ)ωλ(bλ + b†λ)](2.51)= t0[1−∑λVij(λ)ωλ(bλ + b†λ)](2.52)where we have expanded the exponential and kept the linear coupling. We can now seehow the displacement operators affect the diagonal and off-diagonal couplings. In the onsiteenergy, the diagonal terms come from the polarization of the lattice by the electron. Forthe hopping energy, the phonons modify the distance between lattice sites as they vibratewhich in turn modulates the tunneling amplitude required by the electron to travel betweensites.To study the Hamiltonian we will first take its Fourier transform and make a few simpli-fying approximations. We will only include linear couplings of the electron to the phononsand only consider the optical branch. The Fourier transform for the creation operator isc†k =1√N∑rie−ik·rci(ri) (2.53)and the Fourier transform for the Hamiltonian is thenH =∑kkc†kck +∑qωqb†qbq +1√N∑k,qV (k,q)c†k-qck(b†q + b−q). (2.54)The quantity V (k,q) is the sum of the Fourier transforms for the diagonal and non-diagonal coupling terms, c†k and b†q are the creation operators for the electrons and phononsrespectively, k and q are the momenta for the electron and phonons respectively which aresummed over the first Brillouin zone and N is the number of lattice sites where N →∞.282.4. Polaron ModelThe interaction term V (k,q) can be split into the diagonal and non-diagonal terms asV (k,q) = g1(q) + g2(k,q) (2.55)where the diagonal coupling, g1(q), is dependent only on the phonon momentum whilethe non-diagonal coupling, g2(k,q), depends on both phonon and electron momenta. Tostudy these results, we will use the following properties in our model. We take the latticeconstant to be a = 1, assume only nearest neighbour hopping (tij = t0δi±1,j), and as men-tioned previously, we will consider coupling to Einstein phonons only where the frequencyωq = Ω0. The diagonal or non-interacting part of the Hamiltonian is simplyH0 = −2t0∑kcos(k)c†kck +∑qΩ0b†qbq (2.56)where the cosine term comes from the Fourier transformation of the hopping termsc†icj+c†jci in the original site Hamiltonian. To determine the couplings between the electronand phonons, we will use the Holstein and Su-Shrieffer-Heeger (SSH) models for the diagonaland non-diagonal couplings respectively. For the diagonal coupling, the Holstein model givesg1(q) = g0, a constant with no momentum dependence. In the SSH model we modify thehopping element slightly so that the hopping amplitude becomesti,i+1 = t0 − α0(xˆi+1 − xˆi), (2.57)where xˆi is the displacement operator. By Fourier transforming the Hamiltonian withthis form of the hopping, the non-diagonal coupling g2(k,q) isg2(k,q) = 2iα0[sin(k− q)− sin(k)]. (2.58)We now have the complete form of the Hamiltonian for the dual coupling model. TheHamiltonian for a particle interacting with a bath of phonons with diagonal and non-diagonal couplings included isH = −2t0∑kcos(k)c†kck +∑qΩ0b†qbq +1√N∑k, q(g0 (2.59)+2i√Nα0[sin(k− q)− sin(k)])c†k−qck(b†q + b−q). (2.60)We can define the adiabaticity parameter as Λ0 = Ω0/t0. When Λ0  1, then theelectron dynamics are fast while the phonon dynamics are slow.The model described above is one of the simplest models one can study to determinethe dynamics of a polaron with diagonal and off-diagonal couplings. Together with theAuerbach and Kivelson PDX method, we will use this model to determine the full Green’s292.4. Polaron Modelfunction for a particle coupled to a bosonic bath tunneling between multiple wells. We willfirst study the Holstein and SSH models separately to determine how the combined modelwith both couplings will affect the particle dynamics.2.4.2 The Holstein and Su-Schrieffer Heeger ModelsIn the Holstein model, the Hamiltonian contains only the constant coupling between theelectrons and phonons giving it the following form:H = −t∑<i,j>(c†icj + c†jci) + Ω0∑ib†ibi + g∑ic†ici(b†i + bi), (2.61)which in momentum space isH =∑kkc†kck + Ω0∑qb†qbq +g√N∑k,qc†k−qck(b†q + b−q). (2.62)In a system with only diagonal couplings, phonons can only be created and absorbedwhen an electron stays on the same site. In the weak coupling limit, the coupling termcan be treated as a perturbation. With g = 0, the eigenstates and eigenenergies of theHamiltonian are simply:c†k |0〉 c†k−qb†q |0〉 c†k−q−q′b†qb†q′ |0〉 c†k−q−q′−q′′b†qb†q′b†q′′ |0〉 ... (2.63)k k−q + Ω0 k−q−q′ + 2Ω0 k−q−q′−q′′ + 3Ω0 ... (2.64)where k = −2t∑di=1 coski and d = 1, 2, 3. The higher energy states will contain oneelectron and n phonons. The effect of the phonons will be to narrow the band of the energyspectrum and shift it vertically, however, the energy spectrum will still maintain a cosineform. If the coupling g is turned on, the free electron state can mix with the higher energystates above it.In the strong coupling limit, it is the hopping term that becomes the perturbation. Onecan use the Lang-Firsov canonical transformation ([25]) in order to diagonalize the secondand third term in the Hamiltonian. Then the small polaron energy up to first order isEk = − g2Ω0− exp(− g2Ω20)2td∑i=1cos(ki) (2.65)where we have taken into account the energy mediated by the quantities c†icj . We seethat up to first order the energy spectrum still takes the form of a tight binding model butthe bandwidth is exponentially suppressed and shifted by −g2/Ω0.In the SSH model, the coupling between the electron and phonons are off-diagonal.302.4. Polaron ModelRecall that the Hamiltonian for this model isH =∑kkc†kck + Ω0∑qb†qbq +1√N∑k,qg(k,q)c†k−qck(b†q + bq) (2.66)where g(k,q) = 2iα0[sin(k− q)− sin(k)] and represents the off-diagonal coupling.In the Holstein model, the effective hopping terms are exponentially suppressed whenthere are diagonal couplings between the electron and phonons. Here, the non-diagonalcouplings allow the particle to hop to a neighbouring site, create a phonon, and then hopto the next site while absorbing the phonon that was created in the previous site. Thenon-diagonal couplings then mediate the hopping of the particle from site to site. Thesecouplings can be used to calculate the spectral function, J(ω).2.4.3 Spectral densities of the Holstein and SSH PolaronsIn this subsection, we will calculate the spectral densities of the Holstein and SSH polaronswhich can be inserted into the effective action as determined by Caldeira-Leggett. Thepurpose of this is to include the diagonal and non-diagonal couplings into the path integralformalism devised by Auerbach and Kivelson. By including these couplings, we can thencreate a more realistic model for biological and chemical reactions.The spectral density is defined asJ(ω) =pi2∑q|cq|2mqωqδ(ω − ωq) (2.67)where |cq|2 is the coupling, mq is the phonon mass and ωq is the phonon frequency. Todetermine the spectral density for the Holstein and SSH models, we first have to examinethe interactions between the particle and phonons. We will begin with the Holstein modelbefore moving onto the SSH model.2.4.4 Diagonal CouplingThe spectral density for the diagonal coupling begins by starting withcq =g0√Nδ(ω − ωq), (2.68)which can be inserted into the spectral density. As we are only coupling the particle tooptical phonons, the Einstein model is used to approximate the phonon frequency, ωq, as312.4. Polaron ModelΩ0. The spectral density then takes the following formJ(ω) =pi2∑qg20NmΩ0δ(ω − Ω0) (2.69)=pi2g20mΩδ(ω − Ω0) (2.70)where the sum over the delta function gives N times the delta function. Generalizingthis to d dimensions provides the following resultJ(ω) =pi2g20dmΩ0δ(ω − Ω0). (2.71)2.4.5 Non-diagonal CouplingThe spectral density using the non-diagonal coupling is determined similarly to the diagonalcoupling although it is slightly more involved. The coupling here isg(k,q) =2i√Nα0[sin(k− q)− sin(k)]2δ(ω − Ω0), (2.72)which is dependent on k, the momentum of the particle. The spectral density in 1D is thenJ(1D)k (ω) =pi2∑q|g(k, q)|2mΩδ(ω − Ω) (2.73)=pi2∑q4N[sin(k − q)− sin(k)]2mΩδ(ω − Ω) (2.74)=pi2α20∫ pi−pidq2pi4NmΩ{sin2(k − q)− 2 sin(k − q) sin(k) + sin2(k)}δ(ω − Ω) (2.75)=piα20mΩ(1 + 2 sin2(k))δ(ω − Ω). (2.76)Similar calculations can be done in 2D and 3D by taking into account the vector com-ponents. In the 2D and 3D case, the coupling takes the following form.c(2D)q =2i√Nα0[sin(kx − qx) + sin(ky − qy)− sin(kx)− sin(ky)]2δ(ω − Ω) (2.77)c(3D)q =2i√Nα0 [sin(kx − qx) + sin(ky − qy) + sin(kz − qz) (2.78)− sin(kx)− sin(ky)− sin(kz)]2 δ(ω − Ω) (2.79)322.4. Polaron ModelThe integration for the 2D spectral density can then be carried out as followsJ(2D)k (ω) =pi24α20NmΩ∑q{sin(kx − qx) + sin(ky − qy)− sin(kx)− sin(ky)}2δ(ω − Ω)=pi24α20mΩ1(2pi)2∫ pi−pidqx∫ pi−pidqy{sin2(kx − qx) + 2 sin(kx − qx) sin(ky − qy)−2 sin(kx − qx) sin(kx)− 2 sin(kx − qx) sin(ky) + sin2(ky − qy)−2 sin(ky − qy) sin(kx)− 2 sin(ky − qy) sin(ky)+ sin2(kx) + 2 sin(kx) sin(ky) + sin2(ky)}=pi24α20mΩ(1 + sin2(k)δ(ω − Ω)=piα20mΩ(2 + 2 sin2(k)δ(ω − Ω).(2.80)The same calculation in 3D can be carried out and it yieldsJ(3D)k (ω) =piα20mΩ(3 + 2 sin2(k)δ(ω − Ω). (2.81)The spectral density then gives this general form for d = 1, 2 and 3 dimensionsJ(d)k (ω) =piα20mΩ(d+ 2 sin2(k))δ(ω − Ω). (2.82)In this last spectral function, we note that it not only depends on frequency, ω, butalso on the electron momentum k. As a result, the effective action in which this spectralfunction is inserted into, will also be dependent on the electron momentum.2.4.6 Fourier Transform of the Effective ActionAs seen previously, the full effective action written in terms of double integrals using ω isSeff[Q(ω)] =12pi∫ ∞−∞dω12Mω2Q(ω)Q(−ω) + SV+12pi2∫ ∞−∞dω∫ ∞0dω′Q(ω)Q(−ω)J(ω′) ω2ω′(ω′2 + ω2)(2.83)where SV is the Fourier transform of V (q) defined asSV (Q(ω)) ≡∫ ∞−∞dτV (q(τ)). (2.84)The effective actions written as integrals over time and frequency for the diagonal cou-332.4. Polaron Modelpling areSeff[q(τ)] =∫ ∞−∞{12Mq˙2 + V (q)}dτ+12∫ ∞−∞dτ ′∫ ∞−∞dτg20d4mΩe−Ω|τ−τ′|{q(τ)− q(τ ′)}2Θ(Ω),(2.85)Seff[Q(ω)] =12pi∫ ∞−∞dω12Mω2Q(ω)Q(−ω) + SV+12pi2∫ ∞−∞dωQ(ω)Q(−ω) pidg202mΩ2ω2(Ω2 + ω2)Θ(Ω)(2.86)and for the non-diagonal coupling areSeff[q(τ)] =∫ ∞−∞{12Mq˙2 + V (q)}dτ +12∫ ∞−∞dτ ′∫ ∞−∞dτα202mΩ× (d+ 2 sin2(k))e−Ω|τ−τ ′|{q(τ)− q(τ ′)}2Θ(Ω),(2.87)Seff[Q(ω)] =12pi∫ ∞−∞dω12Mω2Q(ω)Q(−ω) + SV+12pi2∫ ∞−∞dωQ(ω)Q(−ω) piα20mΩ2ω2(Ω2 + ω2)(d+ 2 sin2(k))Θ(Ω).(2.88)Here we note that the effective action with the non-diagonal coupling is dependent onthe particle momentum k.Given that we have all of the pieces for determining the path integrals, we can now de-termine the expressions for the propagators in the classically allowed and forbidden regions.The propagator for the particle in the classically allowed region (or inside a potential well)isgi(E) =∏j12cosech(ωjT2)∫ qσ,iqiDq(τ) exp(i~∫ ∞−∞{12Mq˙2 + V (q)}dτ+12∫ ∞−∞dτ ′∫ ∞−∞dτg20d4mΩe−Ω|τ−τ′|{q(τ)− q(τ ′)}2Θ(Ω)),(2.89)while the propagator for a particle in the forbidden region isgij(E) =(B2pi~)1/2 ∣∣∣∣ det Dˆ0det′ Dˆ1∣∣∣∣1/2 ∫ qσ,jqσ,iDq(τ) exp(∫ ∞−∞{12Mq˙2 + V (q)}dτ+12∫ ∞−∞dτ ′∫ ∞−∞dτα202mΩ(d+ 2 sin2(k))e−Ω|τ−τ′|{q(τ)− q(τ ′)}2Θ(Ω)) (2.90)342.4. Polaron ModelwhereB =∫ T0{12Mq˙2 + V (q)}dτ +12∫ ∞−∞dτ ′∫ T−∞dτα(τ − τ ′){q(τ)− q(τ ′)}2, (2.91)Dˆ0 ≡(− d2dτ2+ ω2)+1M∫ ∞−∞α(τ − τ ′)[q(τ)− q(τ ′)]dτ ′, (2.92)Dˆ1 ≡(− d2dτ2+1MV ′′|q¯(τ)|)+1M∫ ∞−∞α(τ − τ ′)[q(τ)− q(τ ′)]dτ ′. (2.93)These expressions for the propagators can then be inserted into the Auerbach and Kivel-son PDX formalism in order to determine the total Green’s function for a particle that startsat some site i and then ends at site i′. The following equations which were provided in earliersections can then be fully determined:tij = [Σi]gij [Σj ] (2.94)Tij = tij + timGmntnj (2.95)Gij = giδij + giTijgj (2.96)where the flux operator [Σi] was defined as[Σi] = limx,x′→xσ,i∫Σidσ |xσ,i〉 〈x|[~2m~∇σ]|x′〉 〈xσ,i| . (2.97)In summary, we have determined the tunneling amplitude for a particle coupled to aphonon bath in a multiwell potential. Despite being coupled to a phonon bath, one cantrace over the environment and determine the environment’s effects on the system. Previouswork done to determine the tunneling amplitude of a particle coupled to a phonon bathrequired the use of truncation methods which introduced artificial cutoffs. By using the PDXformalism, we were able to avoid this by deconstructing the potential energy landscape intosimpler pieces where each region could be solved using different methods. The full Green’sfunction could then be determined using the connection formula constructed by Auerbachand Kivelson. By constructing and solving the problem in this way, we have prepared amodel to describe a particle coupled to a phonon bath traveling in a multiwell potential.35Chapter 3Experimental Setup and ResultsThe objective of this experiment was to determine whether the deuteration of moleculescould affect the binding of a ligand to its receptor. Previous studies had shown that apossible molecular vibration-sensing mechanism was involved in olfaction for fruit flies.Our experiment was designed to determine quantitatively whether such a mechanism waspresent in other GPCR systems. The β1 and β2 adrenergic receptor system, the agonistepinephrine and its deuterated counterparts were chosen to study these effects as the β-ARsystem is well-studied.Our experiment studied the effects that non-deuterated and deuterated epinephrinecompounds had on the production of cAMP. A quantitative analysis was performed byusing an enzyme-linked immunosorbent assay (ELISA) to determine the levels of cAMP.Our experimental results, however, were very inconsistent and troubleshooting indicatedthe possibility that the preparation of the agonists was not suitable for our experiment. Forthis reason, we were not able to establish any definite conclusions from the experiments. Adetailed explanation of the experimental setup and a discussion of the results are providedin this chapter.3.1 Materials and MethodsThe agonist epinephrine was chosen for this experiment, as it is a known agonist for theβ-AR system. To avoid differences in compound and sample preparation, all agonists werepurchased from the same company in the same form. Epinephrine, epinephrine-d3 andepinephrine-d6 (see Fig. 3.1) were all purchased from Medical Isotopes Inc. in powderform and were dissolved in 100% Dimethyl sulfoxide (DMSO). Epinephrine is soluble inhydrochloric acid (HCl) but DMSO was used instead as it was unknown whether the additionof acid would negatively affect cell growth. DMSO is a universal solvent as it dissolves bothpolar and nonpolar compounds. All agonists were dissolved in the lowest volume of DMSOpossible to maintain low concentrations of DMSO in the final volume of growth medium.HEK293 cells (derived from human embryonic kidney cells) were introduced to DNAthat coded for either β1ARs, β2ARs or both. This process of deliberately introducingDNA to the cell, called transfection, allows the cells to take up the DNA and express theseβ-adrenergic receptors. The cells were cultured in Dulbecco’s Modified Eagle Medium(DMEM), a liquid or gel that contains substances required to support cell growth such as363.1. Materials and MethodsEpinephrine (E)   Epinephrine-d6 (ED6)   Epinephrine-d3 (ED3) Figure 3.1: Chemical structures of epinephrine, epinephrine-d3 and epinephrine-d6.Epinephrine-d3 has protons on the methyl group replaced with deuterium. Epinephrine-d6 has all protons attached to carbon molecules switched with deuterium except the lonemethyl group.glucose and amino acids. Cells were grown until 90-100% of the plate surface was coveredin adhered cells. Two controls were used in this experiment: a sample treated with only3-isobutyl-1-methylxanthine (IBMX) and a sample treated with only Forskolin (FSK). Thefirst control served as a negative control as IBMX prevents the degradation of cAMP. Thesecond control acted as a positive control as it maximally stimulates cAMP production.Forskolin also activates adenylyl cyclase directly and not through a G protein-dependentpathway [26, 35].Cells were treated with Gi or Gs inhibitors before treatment with agonists to block theαi or αs subunit function. The Gi inhibitor used was pertussis toxin and the Gs inhibitorswere either mellitin or G protein antagonist peptide (GPAP). Pertussis toxin was added tothe cells 16 hours before treatment with agonist and Gs inhibitors were added to cells 2hours before treatment. The purpose of these inhibitors was to determine if the couplingof the receptor to specific α subunits affected cAMP production.Cells were treated with varying concentrations of epinephrine or deuterated epinephrinefor 30 minutes and then destroyed and lysed with lysis buffer. They were further sonicated(agitated at high frequencies using sound energy) for 5 seconds to ensure cells had beenruptured. Samples were centrifuged at 2500rpm for 5 minutes at 4◦C to separate the373.1. Materials and Methodscellular debris from the cellular solution which contained the cAMP and other proteins.Cell debris was discarded and a Bradford protein assay was performed on the remainingsamples. The Bradford assay was done to determine the total concentration of protein(µg/µL) in the samples. Samples were then diluted and normalized down to the samplewith the lowest protein concentration. This ensured that differences in cAMP productionwere due to agonist efficacy and not due to a difference in the amount of protein or cellularcontent in each sample.The cyclic AMP assay kit, purchased from Cell Signaling, is a competition enzyme-linkedimmunosorbent assay (ELISA) that utilizes antibodies and colour changes on a 96-well plateto determine the concentration and identity of a substance. The 96 well-plate was coatedwith immobilized rabbit antibodies which identify and bind to cAMP. 50µL of each preparedsample was added to the plate in triplicate. 50µL of HRP-linked cAMP (cAMP bound tohorse radish peroxidase enzyme) was also added to each well containing a sample. Standardsamples with known concentrations of cAMP could also added to the plate. Depending onthe availability of space in the plate, the standard samples were not always included. ThecAMP in our sample and the HRP-linked cAMP compete with each other in binding to theantibodies in the plate. The plate was left to incubate at room temperature for 3 hours ona plate shaker to ensure proper mixing of all solutions.After incubation, the wells were washed three times with a wash buffer to remove anyexcess cAMP (both the sample cAMP and the HRP-linked cAMP) not bound to the anti-bodies. A substance called TMB substrate was then added. TMB substrate is a compoundacted on by the HRP enzyme in the HRP-linked cAMP. The interaction between the TMBsubstrate and the HRP enzyme induces a colour change that can be detected via spec-troscopy. Wells that showed an intense colour change indicated that there was much moreHRP-linked cAMP bound to the antibodies than the cAMP obtained from the cell. Italso implied that there was a smaller concentration of cAMP in the sample. A less intensecolour change indicated that the samples had a much higher concentration of cAMP. Thecolour change was analyzed using a spectrophotometer and the absorbance of each sampleat 450nm was read.The biological response, or the percentage of activity as we call it here, is the productionof cAMP. The percentage of activity, as defined by the manufacturer, was calculated asfollows:100× A−AbasalAmax −Abasal (3.1)where A is the absorbance of the sample, Abasal is the absorbance of the sample with thebasal level of cAMP, and Amax is the absorbance of the sample with the highest amount ofcAMP. The values for Abasal and Amax were provided by the control samples treated withIBMX and FSK respectively. The absorbance spectrum of each sample gives an indicationof how much cAMP is present. The quantity Amax−Abasal thus should give the total rangeof cAMP produced in the experiment. The percentage of activity as defined above in Eq.383.2. Results3.1 thus gives the relative increase or decrease of cAMP compared to the basal level.3.2 ResultsThe first assay was used to determine the concentration range that should be used forsubsequent experiments. Epinephrine, epinephrine-d3 (ED3) and epinephrine-d6 (ED6)were used to treat cells at concentrations of 10nM, 100nM and 1µM. These concentrationswere chosen as previous experiments showed that concentrations of epinephrine in thisrange produced significant levels of cAMP [16, 30]. The treatments were performed oncells with either β1 or β2 adrenergic receptors. The results of the first assay indicated thatconcentrations between 100nM and 1µM would provide optimal results so concentrationswithin this range were used for the following assays (Fig. 3.2 to 3.4). The treatment withFSK also indicated that concentrations in the micromolar range would most likely saturatecAMP production (Fig. 3.5 and 3.6). This first assay also indicated that there were possibledifferences in efficacy between the three agonists. At a concentration of 1µM, cells withβ1AR and treated with ED3 increased by 9.1% compared to the basal level while agonistsepinephrine and ED6 increased production of cAMP by 102.7% and 103.3% respectively.In β2AR, the percentage increases for epinephrine, ED3 and ED6 were 40.6%, 11.5% and98.5% respectively.Assays two and three, which tested all 3 compounds at varying concentrations, indicatedthat not all of the results were following the dose-response curve that was expected ofepinephrine when acting on β-adrenergic receptors. Usually, one should see increasingactivity with increasing concentration of agonist until it reaches a point where the activityplateaus. The results showed that the deuterated compounds tended to give results thatfluctuated even with increasing concentration (see Fig. 3.7 and 3.8). Many of these resultsalso indicated that the percentage of activity was below 0%, which implied that thesesamples had levels of cAMP lower than the basal level. Fig. 3.8 could be interpreted asshowing the typical dose-response as the percentage of activity does increase with increasingED6 concentration that peaks at 2.5µM. On the other hand, the results for cells treated withregular epinephrine and the cAMP standard samples were typical of what was seen in theliterature (see Fig. 3.9). It is possible that human error contributed to these inconsistenciesseen here. However, it is strange that half of the samples worked while the others did not,especially since all samples were prepared at the same time using the same methods. Thisleads us to believe that there is another issue affecting the results besides human error.The ensuing assays used pertussis toxin (Gi inhibitor) and melittin or GPAP (Gs in-hibitors) to determine if the coupling of the receptor to the Gα subunit was causing thefluctuating results. Previous research showed that high concentrations of cAMP can causethe β-ARs to switch the coupling from Gs to Gi [16, 21, 33, 37]. A switch in coupling to393.2. Results10nM 100nM 1µM−30−20−100102030405060708090100110120Agonist ConcentrationPercentage of Activity (%)Epinephrine Stimulation in ß1AR10nM 100nM 1µM−30−20−100102030405060708090100110120Agonist ConcentrationPercentage of Activity (%)Epinephrine Stimulation in ß2ARFigure 3.2: Assay 1. Plot of the agonist (epinephrine) concentration versus the percentageof activity (%). HEK293 cells were transfected with either β1 (left) or β2 (right) adrener-gic receptors at varying concentrations. Results indicated that the percentage of activityincreased with increasing epinephrine concentration.the Gi subunit could inhibit adenylyl cyclase and decrease cAMP production. Pertussistoxin inhibits the Gi subunit by adding a molecule to the G protein. This addition preventsthe receptor from interacting with the G protein [20]. Melittin inhibits Gs by reducing itsaffinity for both GTP and GDP [17]. Lastly, GPAP binds to the region of the G proteinthat interacts with the receptor which prevents activation [34].As epinephrine binding to β-adrenergic receptors should stimulate the Gs subunit,adding Gi inhibitor should theoretically not affect the activity of the agonist. Thus, re-sults of cells treated with Gi should give very similar results to those cells with no addedinhibitors. In contrast, adding Gs inhibitor should show decreased levels of activity as lesscAMP is produced. Assays were performed using these inhibitors and the agonists at vary-ing concentrations to determine if these results would be consistent with what was knownin the literature. The results of the assay using epinephrine-d3 is provided in figures 3.10to 3.13.The results for figures 3.10 to 3.14 conflict with what is expected. Firstly, figure 3.10indicates that the treatment using ED3 without inhibitors gave the trend we were expect-ing to see in the previous assays (assays 2 and 3), an increasing response with increasingconcentration. The results from assay 3 had shown randomly fluctuating levels of activity.403.2. Results100nM 1µM−20−15−10−505101520Agonist ConcentrationPercentage of Activity (%)Epinephrine−d3 Stimulation in ß1AR100nM 1µM−20−15−10−505101520Agonist ConcentrationPercentage of Activity (%)Epinephrine−d3 Stimulation in ß2ARFigure 3.3: Assay 1. Plot of the agonist (epinephrine-d3) concentration versus the per-centage of activity (%). HEK293 cells were transfected with either β1 (left) or β2 (right)adrenergic receptors at varying concentrations. Results indicated that the percentage ofactivity increased with increasing epinephrine-d3 concentration.This indicates that there are consistency problems between the experiments that could beunrelated to Gs or Gi coupling. The treatments with the inhibitors also revealed that theinhibitors did not work as we had hoped. While the Gs inhibitor did appear to decreasethe activity level of the cells, the cells treated with Gi inhibitor did not show the samedose-response as the cells treated with no inhibitors. These results were in contrast withassay 7 where the Gs inhibitor did hinder cAMP production (Fig. 3.15). The Gs inhibitedcells show that cAMP production is decreased by 3-14% compared to the non-inhibitedcells.In general, the results from all assays are very inconsistent. It is difficult to draw anysignificance from the experiments as they conflict with information gathered from previousstudies that also used the β-adrenergic system, epinephrine and HEK293 cells. Also noclear trend can be seen from these results. The initial experiment and some treatmentsdid show the dose-response that agreed with the existing literature. In certain instances,it also appears as if the deuterated and non-deuterated epinephrine are following a bell-shaped curve, where they initially increase with increasing concentration but then slowlydecrease in activity after a certain point. However, whether this is a significant outcomeor not is doubtful as the results themselves are not stable and do not agree with what413.2. Results100nM 1µM−20−100102030405060708090100110120Agonist ConcentrationPercentage of Activity (%)Epinephrine−d6 Stimulation in ß1AR100nM 1µM−20−100102030405060708090100110120Agonist ConcentrationPercentage of Activity (%)Epinephrine−d6 Stimulation in ß2ARFigure 3.4: Assay 1. Plot of the agonist (epinephrine-d6) concentration versus the per-centage of activity (%). HEK293 cells were transfected with either β1 (left) or β2 (right)adrenergic receptors at varying concentrations. Results indicated that the percentage ofactivity increased with increasing epinephrine concentration.is expected. These results indicate that there may be a problem with the setup that iscausing inconsistencies throughout the experiment. The following section will elaborate onthe possible problems within the experiment that caused these inconsistencies as well aswhat can be done to improve the experimental design.423.2. ResultsIBMX FSK FSK + E 1µM FSK + E3 1µM FSK + E6 1µM020406080100120140160180200TreatmentPercent Increase Compared to Basal Level (%)Forskolin Treatment Compared to Basal Level in ß1ARFigure 3.5: Assay 1. Plot of specific treatments versus the percent increase compared to thebasal level (%). Cells were treated with Forskolin and 1µM concentrations of deuterated andnon-deuterated epinephrine. HEK293 cells were transfected with β1 adrenergic receptors.In this figure, cells treated with IBMX had an activity level set to 100%. Results indicatedthat treatments with Forskolin produced at least a 90% increase in cAMP production. Italso showed that adding 1µM of agonist to Forskolin did not make affect the activity levelgreatly as Forskolin alone saturated cAMP levels.433.2. ResultsIBMX FSK FSK + E 1µM FSK + E3 1µM FSK + E6 1µM020406080100120140160180200TreatmentPercent Increase Compared to Basal Level (%)Forskolin Treatment Compared to Basal Level in ß2ARFigure 3.6: Assay 1. Plot of specific treatments versus the percent increase compared to thebasal level (%). Cells were treated with Forskolin and 1µM concentrations of deuterated andnon-deuterated epinephrine. HEK293 cells were transfected with β1 adrenergic receptorsat varying concentrations. In this figure, cells treated with IBMX had an activity level setto 100%. Results here were skewed and differed from Fig. 3.5. It is likely that the “FSK”and “FSK + E3 1µM” samples were lower due to human error. The other two treatmentsof Forskolin with epinephrine and epinephrine-d6 indicated high levels of saturation.443.2. Results156.25nM 312.5nM 625nM 1.25µM 2.5µM 5µM 10µM−40−20020406080100Agonist ConcentrationPercentage of Activity (%)Epinephrine−d3 Stimulation in ß1ARFigure 3.7: Assay 3. Plot of the agonist (epinephrine-d3) concentration versus the percent-age of activity (%). HEK293 cells were transfected with β1 adrenergic receptors and treatedwith varying concentrations of agonist. Results shown here were randomly fluctuating anddid not follow a clear trend.156.25nM 312.5nM 625nM 1.25µM 2.5µM 5µM 10µM−60−40−20020406080Agonist ConcentrationPercentage of Activity (%)Epinephrine−d6 Stimulation in ß1ARFigure 3.8: Assay 3. Plot of the agonist (epinephrine-d6) concentration versus the percent-age of activity (%). HEK293 cells were transfected with β1 adrenergic receptors and treatedwith varying concentrations of agonist. Most of the results shown here were negative, im-plying that these samples produced less cAMP than the basal level. This plot may followa bell curve as there seemed to be an increase in activity up to 2.5µM and then a decreasein activity after this point.453.2. Results156.25nM 312.5nM 625nM 1.25µM 2.5µM 5µM 10µM−40−20020406080100120Agonist ConcentrationPercentage of Activity (%)Epinephrine Stimulation in ß1ARFigure 3.9: Assay 3. Plot of the agonist (epinephrine) concentration versus the percentageof activity (%). HEK293 cells were transfected with β1 adrenergic receptors and treatedwith varying concentrations of agonist. Results shown here gave the typical dose-responsecurve of increasing activity with increasing agonist concentration. This dose-response iswhat is expected for the epinephrine agonist in the β-AR system.156.25nM 312.5nM 625nM 1.25µM 2.5µM−30−20−100102030405060Agonist ConcentrationPercentage of Activity (%)Epinephrine−d3 Stimulation in ß1AR with No InhibitorsFigure 3.10: Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the percent-age of activity (%). HEK293 cells were transfected with β1 adrenergic receptors and treatedwith varying concentrations of agonist. The dose-response here increases with increasingagonist concentration.463.2. Results156.25nM 312.5nM 625nM 1.25µM 2.5µM−30−20−100102030405060Agonist ConcentrationPercentage of Activity (%)Epinephrine−d3 Stimulation in ß1AR with Gs InhibitorFigure 3.11: Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the percent-age of activity (%). HEK293 cells were transfected with β1 adrenergic receptors, treatedwith varying concentrations of agonist and Gs inhibitor. The Gs inhibitor did not seem tohave worked properly for certain treatments (156.25nM and 625nM treatments) as it shouldhave suppressed cAMP production.156.25nM 312.5nM 625nM 1.25µM 2.5µM−30−20−100102030405060Agonist ConcentrationPercentage of Activity (%)Epinephrine−d3 Stimulation in ß1AR with Gi InhibitorFigure 3.12: Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the percent-age of activity (%). HEK293 cells were transfected with β1 adrenergic receptors, treatedwith varying concentrations of agonist and Gi inhibitor. The results indicated very lowlevels of percentage of activity which was not expected with the addition of Gi inhibitor.473.2. Results156.25nM 312.5nM 625nM 1.25µM 2.5µM−30−20−100102030405060Agonist ConcentrationPercentage of Activity (%)Epinephrine−d3 Stimulation in ß1AR with Gi and Gs InhibitorsFigure 3.13: Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the percent-age of activity (%). HEK293 cells were transfected with β1 adrenergic receptors, treatedwith varying concentrations of agonist and Gs and Gi inhibitors. Here the activity levelsare lower than 30%. Although not all of the treatments had activity levels lower thanthe non-inhibited values shown in Fig. 3.10, the Gs inhibitor did seem to suppress cAMPproduction in the 2.5µM treatment.483.2. ResultsIBMX FSK FSK + 625nM050100150200TreatmentPercent Increase Compared to Basal Level (%) Forskolin Treatment with No InhibitorsCompared to Basal Level in ß1ARIBMX FSK FSK + 625nM + Gi Inhibitor050100150200TreatmentPercent Increase Compared to Basal Level (%) Forskolin Treatment with Gi InhibitorCompared to Basal Level in ß1ARIBMX FSK FSK + 625nM + Gs Inhibitor050100150200TreatmentPercent Increase Compared to Basal Level (%) Forskolin Treatment with Gs InhibitorCompared to Basal Level in ß1ARIBMX FSK FSK + 625nM + Gi/Gs Inhibitor050100150200TreatmentPercent Increase Compared to Basal Level (%) Forskolin Treatment with Gi and Gs InhibitorCompared to Basal Level in ß1ARFigure 3.14: Assay 4. Plot of the agonist (epinephrine-d3) concentration versus the per-centage increase compared to the basal level (%). HEK293 cells were transfected with β1adrenergic receptors and treated with either Forskolin or Forskolin + 625nM of ED3 withor without an inhibitor. The percentage of activity for the sample treated with IBMX wasset to 100%. The results also seemed to indicate that the Gi and Gs inhibitors did not workas planned, although Forskolin did stimulate cAMP production above the basal level.493.2. Results625nM 1.25µM 2.5µM−10−50510152025303540Agonist ConcentrationPercent of Activity (%)Epinephrine Stimulation in ß1AR with No Inhibitors625nM 1.25µM 2.5µM−10−50510152025303540Agonist ConcentrationPercent of Activity (%)Epinephrine Stimulation in ß1AR with Gs InhibitorFigure 3.15: Assay 7. Plot of the agonist (epinephrine) concentration versus the percentageof activity (%). HEK293 cells were transfected with β1 adrenergic receptors, treated withvarying concentrations of agonist and Gs inhibitor. In this assay, the Gs inhibitor seemedto be effective at suppressing cAMP production.503.3. Discussion3.3 Discussionβ1- and β2-adrenergic receptors are highly expressed in the heart and at varying levelsof expression in the rest of the body. They are targeted by catecholamines, hormonesproduced by the adrenal glands, that mediate many important physiological processes.Although studies on β2ARs primarily focus on its association with the Gs protein, variousexperiments have shown that this coupling is not specific. For example, β2-ARs appearto have a biphasic effect on cardiac myocyte apoptosis (cell death), as it can cause eitherpro- or anti-apoptotic effects. This dual phase effect has been attributed to the receptorsability to switch coupling from Gs to Gi [33] through phosphorylation of the receptor byprotein kinase A (PKA) [9]. This phenomenon has been established in reconstituted systemswith recombinant proteins, cultured cells and in transgenic mice that overexpress β2-ARs.While β1-ARs have not been known to couple to Gi under normal circumstances, therehas been evidence that it is capable of switching from Gs to Gi through a phosphorylationmechanism as well [33]. It is because of this evidence that we incorporated Gs and Giinhibitors into our experimental procedures. Our initial results were indeterminate and noclear conclusion could be drawn as to whether or not the agonists were effective. It washypothesized that Gs/Gi switching was occurring and was possibly affecting the amount ofcAMP being produced.The Gs/Gi switch occurs through feedback regulation. When an agonist such as epinephrinebinds to a β-adrenergic receptor, Gs-mediated stimulation of adenylyl cyclase occurs andleads to an increase in the intracellular concentration of cAMP. The accumulation of cAMPactivates PKA, a family of enzymes that have several important functions in the cell such asthe regulation of glycogen and lipid metabolism. Both β-ARs have PKA phosphorylationsites (specific peptide sequences) located in the third intracellular loop and the carboxy-terminal tail of the receptor. Phosphorylation of these sites by PKA have been shown todecrease coupling between Gs and increase coupling to Gi for β-ARs [9, 42]. Fig. 3.16depicts parts of the signal transduction pathways regulated by the Gs and Gi subunitsthat culminate in the production of extracellular signal-regulated kinase (ERK), a proteininvolved in regulating cell growth. The Gi subunit can activate ERK but can also inhibitadenylyl cyclase [28].If a switch in coupling was occurring, addition of Gi inhibitors should prevent an increasein Gi-receptor coupling and prevent levels of cAMP from decreasing unexpectedly. Cellstreated with Gs inhibitors should show very low levels of cAMP, while cells treated with Gishould produce levels of cAMP that increase with increasing agonist concentration until itplateaus at saturation levels. Despite the inclusion of Gs and Gi inhibitors, we did not seeany clear evidence of a Gs/Gi switch occurring, as we did not consistently see the expectedtrends.There have been disagreements that such a switch in G protein coupling occurs. Astudy performed by Friedman et al. 2002, suggests that there is a lack of evidence that513.3. DiscussionGs AC cAMP PKA Gi Gβγ ERK PKA phosphorylates sites on receptor, leading to: -Decreased coupling to Gs -Increased coupling to Gi Figure 3.16: The signal transduction pathways initiated by the Gs and Gi subunits. TheGs subunit activates adenylate cyclase which stimulates the production of cAMP. Higherconcentrations of cAMP in the cell activates protein kinase A (PKA), a family of enzymescapable of phosphorylating other proteins. PKA can phosphorylate the β-ARs which de-creases the receptor’s coupling to Gs and increase coupling to Gi. The multiple arrows insuccession represent multiple steps that are not shown.523.3. Discussionthe β2AR associates with the Gi protein. Their study, which also used the HEK293 cellline, indicated that a downstream effector, extracellular signal-regulated kinase (ERK), isactivated solely through the Gs protein which disagrees with studies put forth by othersthat indicated the possible Gs/Gi switch [9, 42]. However, it has been argued that HEK293cell lines do not share uniform properties as they can vary from cell line to cell line. Thisvariability may be caused by prolonged culture where the lines may alter in chromosomenumber, gene expression signaling pathways and structure [28]. As to whether or not thevariability in the HEK293 cells used in our experiment have affected the results is unknown.This cell line has been used before in other experiments and no major effects caused by thecell line have been noted.Aside from a possible Gs/Gi switch, there are two other speculations as to what hascaused the inconsistencies in the results. Firstly, the epinephrine compounds that were usedin this experiment were dissolved in 100% DMSO. Epinephrine is soluble in hydrochloricacid (HCl) and is usually prepared in a salt form, either as epinephrine hydrochloride or asan epinephrine bitartrate salt. These compounds are easily dissolved in water and will notprecipitate out of solution. However, deuterated epinephrine compounds are usually notprepared in salt form. We avoided acquiring compounds in different phases or solutions asdifferences in the preparation could have contributed to differences in the experiment. Forthis reason, all epinephrine compounds were purchased in powder form and prepared in thelaboratory using 100% DMSO. DMSO was added in small increments until the epinephrinecompound just dissolved. The purpose of doing this was to minimize the amount of DMSOthat would eventually be added into the cell culture medium as high concentrations ofDMSO can cause cell death. It is not advisable to have more than 0.5% of DMSO in theresulting medium as levels above this are cytotoxic to cells.Despite DMSO being a universal solvent, we found that unless adequate mixing andagitation was performed, precipitates would form in the solution. It is possible that whiletreating the cells, the addition of the epinephrine solutions into the medium caused theepinephrine to precipitate out and reduce its effectiveness. As to how much epinephrinecould have precipitated out of solution, this is unclear and could explain why the resultsoften followed no trend. Hydrochloric acid could have been used to prepare the compoundsinstead of DMSO to fully dissolve the epinephrine into solution. However, doing so couldhave affected the pH of the cells and also promote cell death.The second issue that could have affected the consistency of the results is the sensitivityof the kit used to quantify the amount of cAMP produced. Based on the manufacturer’sdata, the kit quantifies cAMP in the nanomolar range. If our samples have concentrationsof cAMP lower than the nanomolar range, it would explain why the higher concentrationsof epinephrine and Forskolin samples tend to generally show the high saturation levels thatwe expect while the lower concentrations tend to have very minimal activity percentages.The assay kit also does not list a total protein concentration that should be loaded into533.3. Discussioneach well of the plate and only suggests that 6-100 x 103 cells/well be used. Overloadingthe plate with high concentrations of protein affects the accuracy of the results as it causeseach of the wells to reach the maximum saturation levels. For our experiments, the proteinamount loaded into each well was normalized to the sample with the lowest protein amount.This protein concentration varied between assays but would generally range from 40-50µg ofprotein per 50µL sample. Normalizing the samples beforehand ensures that all samples havethe same protein concentration and that differences in the results would not be attributedto certain samples having higher cellular content. We calculated the protein concentrationthat would result from the manufacturer’s suggested number of cells/well and this rangewas found to be around 10µg of protein per 50µL of sample.As can be seen, there is still a considerable amount of trouble-shooting and problem-solving that must be done to perfect the experimental setup and to obtain consistent andaccurate results. A simple experiment can be done to determine whether the compounds orthe kit are affecting the current experiments. One would need an assay kit that can detectcAMP in the picomolar range and an epinephrine salt compound. The epinephrine salt andthe epinephrine/DMSO solution can be used to treat the cells used previously and the sameprocedures and methods can be used to prepare the samples for an ELISA. Running thetwo compounds in parallel in an assay kit with better sensitivity should allow one to seethe differences between the two epinephrine solutions and assay kits. If it is the epinephrinesolubility that is affecting the outcome, then more studies will have to be done to determinehow to properly prepare the epinephrine solutions. For instance, if the compounds mustbe dissolved in HCl, then a major issue may be determining how to alter the pH so that itdoes not affect cell growth and development.54Chapter 4Conclusion4.1 SummaryIn this thesis we have discussed both theoretical and experimental work. In the theoreticalsection, we recognized that there is a lack of theoretical models that can accurately describebiological and chemical reactions. Our goal was to develop such a model by merging thework done by Caldeira and Leggett regarding particles coupled to phonon baths and themethod developed by Auerbach and Kivelson which decomposes a complicated potentialenergy landscape into manageable pieces. We were also able to include both diagonal andnon-diagonal couplings in our model by considering Holstein and Peierls coupling in oursystem. This allowed us to calculate the path integrals inside and outside a potential wellfor a particle coupled to a phonon bath. The path decomposition expansion formalismdeveloped by Auerbach and Kivelson then allowed us to connect these pieces together inorder to determine the full dynamics of the system.In the experimental portion of this thesis, our goal was to determine whether enzyme-substrate binding would be affected by deuteration, or changes in molecular vibrationalmodes. We studied the effects of deuterated and non-deuterated epinephrine on the β-AR system. We performed a biological assay by exposing deuterated and non-deuteratedepinephrine to β1 and β2 adrenergic receptors and then determined the amount of cAMPproduced. Our aim was to determine whether deuterated compounds could significantlyaffect second messenger production. If a significant difference was seen, this could indicateevidence that enzyme-substrate binding may rely on more than just shape for effectivebinding but also on the vibrational spectra of the molecules involved. Unfortunately, ourresults were inconsistent and no clear conclusions could be drawn as to whether a molecularvibration-sensing mechanism occurred here.4.2 Future WorkThe theoretical model discussed in the previous chapters can be used to model variousbiological and chemical reactions with complicated potential energy landscapes. While wehave provided the theoretical framework here, experimental or numerical work should bedone to determine the validity of this model.As for the experimental portion of this work, there are a few possible issues that need554.2. Future Workto be considered in order for this work to continue forward. These are the possibilityof Gi/Gs switching, poor sensitivity of the assay kit and epinephrine’s weak solubility inDMSO. The solubility issues are most likely what caused the nonsensical nature of ourresults as the random precipitation of epinephrine from the solution would have affectedcAMP levels. Troubleshooting and revision of the experiment must be done before we candraw conclusions about the validity of a molecular vibration-sensing mechanism in GPCRsystems. If these issues can be resolved, then the results should provide a better indicationof whether deuteration can affect enzyme-substrate binding in this system.56Bibliography[1] A. Auerbach and S. Kivelson. Path decomposition for multdimensional tunneling.Physical Review Letters, 64(5):411, 1984.[2] A. Auerbach and S. Kivelson. The path decomposition expansion and multidimensionaltunneling. Nuclear Physics B, 257:799–858, 1985.[3] J. C. Brookes, F. Hartoutsiou, A. P. Horsfield, and A. M. Stoneham. Could humansrecognize odor by phonon assisted tunneling? Physical Review Letters, 98(3):1–10,2007.[4] E. Burnstein and S. Lundqvist, editors. Tunneling Phenomena in Solids. Plenum Press,New York, 1969.[5] D. B. Bylund, D. C. Eikenberg, J. P. Hieble, S. Z. Langer, R. J. Lefkowitz, K. P.Minneman, P. B. Molinoff, R. R. Ruffolo, and U. Trendelenburg. of Pharmacology ofAdrenoceptors. Pharmacological reviews, 46(2):121–136, 1994.[6] A. O. Caldeira and A. J. Leggett. Quantum tunnelling in dissipative systems. Annalsof Physics, 149(2):374–456, 1983.[7] C. G. Callan and S. Coleman. Fate of the false vacuum. II. First quantum corrections.Physical Review D, 16(6):1762–1768, 1977.[8] S. R. Coleman. The Fate of the False Vacuum. 1. Semiclassical Theory. Physical ReviewD, 15(4):2929–2936, 1977.[9] Y. Daaka, L. M. Luttrell, and R. J. Lefkowitz. Switching of the coupling of the beta(2)-adrenergic receptor to different G proteins by protein kinase A. Nature, 390(6655):88–91, 1997.[10] Lakshmi A. Devi, editor. The G Protein-Coupled Receptors Handbook. Humana PressInc., Totowa, New Jersey, 2005.[11] G. B. Downes and N. Gautam. The G protein subunit gene families. Genomics,62(3):544–552, 1999.[12] R. P. Feynman. Statistical Mechanics. W. A. Benjamin, Inc., New York, 1972.57Bibliography[13] R. P . Feynman and F. L . Vernon. The theory of a general quantum system interactingwith a linear dissipative system. Annals of Physics, 24:118–173, 1963.[14] M. I. Franco, L. Turin, A. Mershin, and E. M. C. Skoulakis. Molecular vibration-sensing component in Drosophila melanogaster olfaction. Proceedings of the NationalAcademy of Sciences of the United States of America, 108(9):3797–3802, 2011.[15] R. Fredriksson, M. C. Lagerstro¨m, L. Lundin, and H. B. Schio¨th. The G-protein-coupled receptors in the human genome form five main families. Phylogenetic analysis,paralogon groups, and fingerprints. Molecular pharmacology, 63(6):1256–1272, 2003.[16] J. Friedman, B. Babu, and R. B. Clark. Beta(2)-adrenergic receptor lacking the cyclicAMP-dependent protein kinase consensus sites fully activates extracellular signal-regulated kinase 1/2 in human embryonic kidney 293 cells: lack of evidence forG(s)/G(i) switching. Molecular pharmacology, 62(5):1094–102, 2002.[17] N. Fukushima, M. Kohno, T. Kato, S. Kawamoto, K. Okuda, Y. Misu, and H. Ueda.Melittin, a metabostatic peptide inhibiting G(s) activity. Peptides, 19(5):811–819,1998.[18] S. Gane, D. Georganakis, K. Maniati, M. Vamvakias, N. Ragoussis, E. M. C. Skoulakis,and L. Turin. 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