1
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Wu D, Salsbury FR. Allosteric Modulation of Thrombin by Thrombomodulin: Insights from Logistic Regression and Statistical Analysis of Molecular Dynamics Simulations. ACS OMEGA 2024; 9:23086-23100. [PMID: 38826540 PMCID: PMC11137727 DOI: 10.1021/acsomega.4c03375] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/10/2024] [Revised: 05/01/2024] [Accepted: 05/08/2024] [Indexed: 06/04/2024]
Abstract
Thrombomodulin (TM), a transmembrane receptor integral to the anticoagulant pathway, governs thrombin's substrate specificity via interaction with thrombin's anion-binding exosite I. Despite its established role, the precise mechanisms underlying this regulatory function are yet to be fully unraveled. In this study, we deepen the understanding of these mechanisms through eight independent 1 μs all-atom simulations, analyzing thrombin both in its free form and when bound to TM fragments TM456 and TM56. Our investigations revealed distinct and significant conformational changes in thrombin mediated by the binding of TM56 and TM456. While TM56 predominantly influences motions within exosite I, TM456 orchestrates coordinated alterations across various loop regions, thereby unveiling a multifaceted modulatory role that extends beyond that of TM56. A highlight of our study is the identification of critical hydrogen bonds that undergo transformations during TM56 and TM456 binding, shedding light on the pivotal allosteric influence exerted by TM4 on thrombin's structural dynamics. This work offers a nuanced appreciation of TM's regulatory role in blood coagulation, paving the way for innovative approaches in the development of anticoagulant therapies and expanding the horizons in oncology therapeutics through a deeper understanding of molecular interactions in the coagulation pathway.
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Affiliation(s)
- Dizhou Wu
- Department of Physics, Wake
Forest University, Winston-Salem, North Carolina 27106, United
States
| | - Freddie R. Salsbury
- Department of Physics, Wake
Forest University, Winston-Salem, North Carolina 27106, United
States
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2
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Singh D. Macromolecular Polymer Based Complexes: A Diverse Strategy for the Delivery of Nucleotides. Protein Pept Lett 2024; 31:586-601. [PMID: 39177133 DOI: 10.2174/0109298665310091240809103048] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2024] [Revised: 06/06/2024] [Accepted: 07/24/2024] [Indexed: 08/24/2024]
Abstract
This review explores the burgeoning field of macromolecular polymer-based complexes, highlighting their revolutionary potential for the delivery of nucleotides for therapeutic applications. These complexes, ingeniously crafted from a variety of polymers, offer a unique solution to the challenges of nucleotide delivery, including protection from degradation, targeted delivery, and controlled release. The focus of this report is primarily on the design principles, encapsulation strategies, and biological interactions of these complexes, with an emphasis on their biocompatibility, biodegradability, and ability to form diverse structures, such as nanoparticles and micelles. Significant attention is paid to the latest advancements in polymer science that enable the precise tailoring of these complexes for specific nucleotides, such as DNA, RNA, and siRNA. The review discusses the critical role of surface modifications and the incorporation of targeting ligands in enhancing cellular uptake and ensuring delivery to specific tissues or cells, thereby reducing off-target effects and improving therapeutic efficacy. Clinical applications of these polymer-based delivery systems are thoroughly examined with a focus on their use in treating genetic disorders, cancer, and infectious diseases. The review also addresses the challenges and limitations currently faced in this field, such as scalability, manufacturing complexities, and regulatory hurdles. Overall, this review provides a comprehensive overview of the current state and future prospects of macromolecular polymer-based complexes in nucleotide delivery. It underscores the significance of these systems in advancing the field of targeted therapeutics and their potential to reshape the landscape of medical treatment for a wide range of diseases.
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Affiliation(s)
- Dilpreet Singh
- Department of Pharmaceutics, University Institute of Pharma Sciences, Chandigarh University, Gharuan, Mohali, 140413, India
- University Centre for Research and Development, Chandigarh University, Gharuan, Mohali, 140413, India
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3
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Wu D, Salsbury FR. Unraveling the Role of Hydrogen Bonds in Thrombin via Two Machine Learning Methods. J Chem Inf Model 2023; 63:3705-3718. [PMID: 37285464 PMCID: PMC11164249 DOI: 10.1021/acs.jcim.3c00153] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
Hydrogen bonds play a critical role in the folding and stability of proteins, such as proteins and nucleic acids, by providing strong and directional interactions. They help to maintain the secondary and 3D structure of proteins, and structural changes in these molecules often result from the formation or breaking of hydrogen bonds. To gain insights into these hydrogen bonding networks, we applied two machine learning models - a logistic regression model and a decision tree model - to study four variants of thrombin: wild-type, ΔK9, E8K, and R4A. Our results showed that both models have their unique advantages. The logistic regression model highlighted potential key residues (GLU295) in thrombin's allosteric pathways, while the decision tree model identified important hydrogen bonding motifs. This information can aid in understanding the mechanisms of folding in proteins and has potential applications in drug design and other therapies. The use of these two models highlights their usefulness in studying hydrogen bonding networks in proteins.
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Affiliation(s)
- Dizhou Wu
- Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27106, United States
| | - Freddie R Salsbury
- Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27106, United States
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4
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Lima MA, Rudd TR, Fernig DG, Yates EA. Phosphorylation and sulfation share a common biosynthetic pathway, but extend biochemical and evolutionary diversity of biological macromolecules in distinct ways. JOURNAL OF THE ROYAL SOCIETY, INTERFACE 2022; 19:20220391. [PMID: 35919982 PMCID: PMC9346353 DOI: 10.1098/rsif.2022.0391] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Phosphate and sulfate groups are integral to energy metabolism and introduce negative charges into biological macromolecules. One purpose of such modifications is to elicit precise binding/activation of protein partners. The physico-chemical properties of the two groups, while superficially similar, differ in one important respect—the valency of the central (phosphorus or sulfur) atom. This dictates the distinct properties of their respective esters, di-esters and hence their charges, interactions with metal ions and their solubility. These, in turn, determine the contrasting roles for which each group has evolved in biological systems. Biosynthetic links exist between the two modifications; the sulfate donor 3′-phosphoadenosine-5′-phosphosulfate being formed from adenosine triphosphate (ATP) and adenosine phosphosulfate, while the latter is generated from sulfate anions and ATP. Furthermore, phosphorylation, by a xylosyl kinase (Fam20B, glycosaminoglycan xylosylkinase) of the xylose residue of the tetrasaccharide linker region that connects nascent glycosaminoglycan (GAG) chains to their parent proteoglycans, substantially accelerates their biosynthesis. Following observations that GAG chains can enter the cell nucleus, it is hypothesized that sulfated GAGs could influence events in the nucleus, which would complete a feedback loop uniting the complementary anionic modifications of phosphorylation and sulfation through complex, inter-connected signalling networks and warrants further exploration.
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Affiliation(s)
- M A Lima
- Centre for Glycosciences, Keele University, Keele ST5 5BG, UK.,School of Life Sciences, Keele University, Keele ST5 5BG, UK
| | - T R Rudd
- Analytical and Biological Science Department, National Institute of Biological Standards and Control (NIBSC), Blanche Lane, South Mimms, Potters Bar EN6 3QG, UK.,Department of Biochemistry and Systems Biology, ISMIB, University of Liverpool, Liverpool L69 7ZB, UK
| | - D G Fernig
- Department of Biochemistry and Systems Biology, ISMIB, University of Liverpool, Liverpool L69 7ZB, UK
| | - E A Yates
- School of Life Sciences, Keele University, Keele ST5 5BG, UK.,Department of Biochemistry and Systems Biology, ISMIB, University of Liverpool, Liverpool L69 7ZB, UK
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5
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Bengtsen T, Holm VL, Kjølbye LR, Midtgaard SR, Johansen NT, Tesei G, Bottaro S, Schiøtt B, Arleth L, Lindorff-Larsen K. Structure and dynamics of a nanodisc by integrating NMR, SAXS and SANS experiments with molecular dynamics simulations. eLife 2020; 9:e56518. [PMID: 32729831 PMCID: PMC7426092 DOI: 10.7554/elife.56518] [Citation(s) in RCA: 45] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2020] [Accepted: 07/28/2020] [Indexed: 12/16/2022] Open
Abstract
Nanodiscs are membrane mimetics that consist of a protein belt surrounding a lipid bilayer, and are broadly used for characterization of membrane proteins. Here, we investigate the structure, dynamics and biophysical properties of two small nanodiscs, MSP1D1ΔH5 and ΔH4H5. We combine our SAXS and SANS experiments with molecular dynamics simulations and previously obtained NMR and EPR data to derive and validate a conformational ensemble that represents the structure and dynamics of the nanodisc. We find that it displays conformational heterogeneity with various elliptical shapes, and with substantial differences in lipid ordering in the centre and rim of the discs. Together, our results reconcile previous apparently conflicting observations about the shape of nanodiscs, and pave the way for future integrative studies of larger complex systems such as membrane proteins embedded in nanodiscs.
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Affiliation(s)
- Tone Bengtsen
- Structural Biology and NMR Laboratory and Linderstrøm-Lang Centre for Protein Science, Department of Biology, University of CopenhagenCopenhagenDenmark
| | - Viktor L Holm
- Structural Biophysics, X-ray and Neutron Science, Niels Bohr Institute, University of CopenhagenCopenhagenDenmark
| | | | - Søren R Midtgaard
- Structural Biophysics, X-ray and Neutron Science, Niels Bohr Institute, University of CopenhagenCopenhagenDenmark
| | - Nicolai Tidemand Johansen
- Structural Biophysics, X-ray and Neutron Science, Niels Bohr Institute, University of CopenhagenCopenhagenDenmark
| | - Giulio Tesei
- Structural Biology and NMR Laboratory and Linderstrøm-Lang Centre for Protein Science, Department of Biology, University of CopenhagenCopenhagenDenmark
| | - Sandro Bottaro
- Structural Biology and NMR Laboratory and Linderstrøm-Lang Centre for Protein Science, Department of Biology, University of CopenhagenCopenhagenDenmark
| | | | - Lise Arleth
- Structural Biophysics, X-ray and Neutron Science, Niels Bohr Institute, University of CopenhagenCopenhagenDenmark
| | - Kresten Lindorff-Larsen
- Structural Biology and NMR Laboratory and Linderstrøm-Lang Centre for Protein Science, Department of Biology, University of CopenhagenCopenhagenDenmark
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6
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González-Alemán R, Hernández-Castillo D, Caballero J, Montero-Cabrera LA. Quality Threshold Clustering of Molecular Dynamics: A Word of Caution. J Chem Inf Model 2019; 60:467-472. [PMID: 31532987 DOI: 10.1021/acs.jcim.9b00558] [Citation(s) in RCA: 40] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
Abstract
Clustering Molecular Dynamics trajectories is a common analysis that allows grouping together similar conformations. Several algorithms have been designed and optimized to perform this routine task, and among them, Quality Threshold stands as a very attractive option. This algorithm guarantees that in retrieved clusters no pair of frames will have a similarity value greater than a specified threshold, and hence, a set of strongly correlated frames are obtained for each cluster. In this work, it is shown that various commonly used software implementations are flawed by confusing Quality Threshold with another simplistic well-known clustering algorithm published by Daura et al. (Daura, X.; van Gunsteren, W. F.; Jaun, B.; Mark, A. E.; Gademann, K.; Seebach, D. Peptide Folding: When Simulation Meets Experiment. Angew. Chemie Int. Ed. 1999, 38 (1/2), 236-240). Daura's algorithm does not impose any quality threshold for the frames contained in retrieved clusters, bringing unrelated structural configurations altogether. The advantages of using Quality Threshold whenever possible to explore Molecular Dynamic trajectories is exemplified. An in-house implementation of the original Quality Threshold algorithm has been developed in order to illustrate our comments, and its code is freely available for further use by the scientific community.
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Affiliation(s)
- Roy González-Alemán
- Laboratorio de Química Computacional y Teórica, Facultad de Química , Universidad de La Habana , 10400 La Habana , Cuba
| | - David Hernández-Castillo
- Laboratorio de Química Computacional y Teórica, Facultad de Química , Universidad de La Habana , 10400 La Habana , Cuba
| | - Julio Caballero
- Centro de Bioinformática y Simulación Molecular, Facultad de Ingeniería en Bioinformática , Universidad de Talca , 2 Norte 685, Casilla 721 , Talca , Chile
| | - Luis A Montero-Cabrera
- Laboratorio de Química Computacional y Teórica, Facultad de Química , Universidad de La Habana , 10400 La Habana , Cuba
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7
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Xiao J, Melvin RL, Salsbury FR. Mechanistic insights into thrombin's switch between "slow" and "fast" forms. Phys Chem Chem Phys 2017; 19:24522-24533. [PMID: 28849814 PMCID: PMC5719506 DOI: 10.1039/c7cp03671j] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023]
Abstract
Thrombin is a multifunctional enzyme that plays an important role in blood coagulation, cell growth, and metastasis. Depending upon the binding of sodium ions, thrombin presents significantly different enzymatic activities. In the environment with sodium ions, thrombin is highly active in cleaving the coagulated substrates and this is referred to as the "fast" form; in the environment without sodium ions, thrombin turns catalytically less active and is in the "slow" form. Although many experimental studies over the last two decades have attempted to reveal the structural and kinetic differences between these two forms, it remains vague and disputed how the functional switch between the "fast" and "slow" forms is mediated by Na+ cations. In this work, we employ microsecond-scale all-atom molecular dynamics simulations to investigate the differences in the structural ensembles in sodium-bound/unbound and potassium-bound/unbound thrombin. Our calculations indicate that the regulatory regions, including the 60s, γ loops, and exosite I and II, are primarily affected by both the bound and unbound cations. Conformational free energy surfaces, estimated from principal component analysis, further reveal the existence of multiple conformational states. The binding of a cation introduces changes in the distribution of these states. Through comparisons with potassium-binding, the binding of sodium ions appears to shift the population toward conformational states that might be catalytically favorable. Our study of thrombin in the presence of sodium/potassium ions suggests Na+-mediated generalized allostery is the mechanism of thrombin's functional switch between the "fast" and "slow" forms.
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Affiliation(s)
- Jiajie Xiao
- Department of Physics, Wake Forest University, Winston Salem, NC, USA.
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8
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Melvin RL, Gmeiner WH, Salsbury FR. All-atom MD indicates ion-dependent behavior of therapeutic DNA polymer. Phys Chem Chem Phys 2017; 19:22363-22374. [PMID: 28805211 PMCID: PMC5600158 DOI: 10.1039/c7cp03479b] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2022]
Abstract
Understanding the efficacy of and creating delivery mechanisms for therapeutic nucleic acids requires understanding structural and kinetic properties which allow these polymers to promote the death of cancerous cells. One molecule of interest is a 10 mer of FdUMP (5-fluoro-2'-deoxyuridine-5'-O-monophosphate) - also called F10. Here we investigate the structural and kinetic behavior of F10 in intracellular and extracellular solvent conditions along with non-biological conditions that may be efficacious in in vitro preparations of F10 delivery systems. From our all-atom molecular dynamics simulations totaling 80 microseconds, we predict that F10's phosphate groups form close-range interactions with calcium and zinc ions, with calcium having the highest affinity of the five ions investigated. We also predict that F10's interactions with magnesium, potassium and sodium are almost exclusively long-range interactions. In terms of intramolecular interactions, we find that F10 is least structured (in terms of hydrogen bonds among bases) in the 150 mM NaCl (extracellular-like solvent conditions) and most structured in 150 mM ZnCl2. Kinetically, we see that F10 is unstable in the presence of magnesium, sodium or potassium, finding stable kinetic traps in the presence of calcium or zinc.
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Affiliation(s)
- Ryan L Melvin
- Department of Physics, Wake Forest University, Winston Salem, NC, USA.
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9
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Melvin RL, Gmeiner WH, Salsbury FR. All-Atom MD Predicts Magnesium-Induced Hairpin in Chemically Perturbed RNA Analog of F10 Therapeutic. J Phys Chem B 2017; 121:7803-7812. [PMID: 28745046 DOI: 10.1021/acs.jpcb.7b04724] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Given their increasingly frequent usage, understanding the chemical and structural properties which allow therapeutic nucleic acids to promote the death of cancer cells is critical for medical advancement. One molecule of interest is a 10-mer of FdUMP (5-fluoro-2'-deoxyuridine-5'-O-monophosphate) also called F10. To investigate causes of structural stability, we have computationally restored the 2' oxygen on each ribose sugar of the phosphodiester backbone, creating FUMP[10]. Microsecond time-scale, all-atom, simulations of FUMP[10] in the presence of 150 mM MgCl2 predict that the strand has a 45% probability of folding into a stable hairpin-like secondary structure. Analysis of 16 μs of data reveals phosphate interactions as likely contributors to the stability of this folded state. Comparison with polydT and polyU simulations predicts that FUMP[10]'s lowest order structures last for one to 2 orders of magnitude longer than similar nucleic acid strands. Here we provide a brief structural and conformational analysis of the predicted structures of FUMP[10], and suggest insights into its stability via comparison to F10, polydT, and polyU.
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Affiliation(s)
- Ryan L Melvin
- Department of Physics, Wake Forest University , Winston-Salem, North Carolina 27109, United States.,Department of Mathematics and Statistics, Wake Forest University , Winston-Salem, North Carolina 27109, United States
| | - William H Gmeiner
- Department of Cancer Biology, Wake Forest University School of Medicine , Winston-Salem North Carolina 27101, United States
| | - Freddie R Salsbury
- Department of Physics, Wake Forest University , Winston-Salem, North Carolina 27109, United States
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10
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Melvin RL, Thompson WG, Godwin RC, Gmeiner WH, Salsbury FR. MutS α's Multi-Domain Allosteric Response to Three DNA Damage Types Revealed by Machine Learning. FRONTIERS IN PHYSICS 2017; 5:10. [PMID: 31938712 PMCID: PMC6959842 DOI: 10.3389/fphy.2017.00010] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
MutSα is a key component in the mismatch repair (MMR) pathway. This protein is responsible for initiating the signaling pathways for DNA repair or cell death. Herein we investigate this heterodimer's post-recognition, post-binding response to three types of DNA damage involving cytotoxic, anti-cancer agents-carboplatin, cisplatin, and FdU. Through a combination of supervised and unsupervised machine learning techniques along with more traditional structural and kinetic analysis applied to all-atom molecular dynamics (MD) calculations, we predict that MutSα has a distinct response to each of the three damage types. Via a binary classification tree (a supervised machine learning technique), we identify key hydrogen bond motifs unique to each type of damage and suggest residues for experimental mutation studies. Through a combination of a recently developed clustering (unsupervised learning) algorithm, RMSF calculations, PCA, and correlated motions we predict that each type of damage causes MutSα to explore a specific region of conformation space. Detailed analysis suggests a short range effect for carboplatin-primarily altering the structures and kinetics of residues within 10 angstroms of the damaged DNA-and distinct longer-range effects for cisplatin and FdU. In our simulations, we also observe that a key phenylalanine residue-known to stack with a mismatched or unmatched bases in MMR-stacks with the base complementary to the damaged base in 88.61% of MD frames containing carboplatinated DNA. Similarly, this Phe71 stacks with the base complementary to damage in 91.73% of frames with cisplatinated DNA. This residue, however, stacks with the damaged base itself in 62.18% of trajectory frames with FdU-substituted DNA and has no stacking interaction at all in 30.72% of these frames. Each drug investigated here induces a unique perturbation in the MutSα complex, indicating the possibility of a distinct signaling event and specific repair or death pathway (or set of pathways) for a given type of damage.
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Affiliation(s)
- Ryan L. Melvin
- Salsbury Group, Department of Physics, Wake Forest University, Winston-Salem, NC, USA
| | - William G. Thompson
- Salsbury Group, Department of Physics, Wake Forest University, Winston-Salem, NC, USA
| | - Ryan C. Godwin
- Salsbury Group, Department of Physics, Wake Forest University, Winston-Salem, NC, USA
| | - William H. Gmeiner
- Gmeiner Laboratory, Department of Cancer Biology, Wake Forest University School of Medicine, Winston-Salem, NC, USA
| | - Freddie R. Salsbury
- Salsbury Group, Department of Physics, Wake Forest University, Winston-Salem, NC, USA
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11
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Melvin RL, Godwin RC, Xiao J, Thompson WG, Berenhaut KS, Salsbury FR. Uncovering Large-Scale Conformational Change in Molecular Dynamics without Prior Knowledge. J Chem Theory Comput 2016; 12:6130-6146. [PMID: 27802394 PMCID: PMC5719493 DOI: 10.1021/acs.jctc.6b00757] [Citation(s) in RCA: 39] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
Abstract
As the length of molecular dynamics (MD) trajectories grows with increasing computational power, so does the importance of clustering methods for partitioning trajectories into conformational bins. Of the methods available, the vast majority require users to either have some a priori knowledge about the system to be clustered or to tune clustering parameters through trial and error. Here we present non-parametric uses of two modern clustering techniques suitable for first-pass investigation of an MD trajectory. Being non-parametric, these methods require neither prior knowledge nor parameter tuning. The first method, HDBSCAN, is fast-relative to other popular clustering methods-and is able to group unstructured or intrinsically disordered systems (such as intrinsically disordered proteins, or IDPs) into bins that represent global conformational shifts. HDBSCAN is also useful for determining the overall stability of a system-as it tends to group stable systems into one or two bins-and identifying transition events between metastable states. The second method, iMWK-Means, with explicit rescaling followed by K-Means, while slower than HDBSCAN, performs well with stable, structured systems such as folded proteins and is able to identify higher resolution details such as changes in relative position of secondary structural elements. Used in conjunction, these clustering methods allow a user to discern quickly and without prior knowledge the stability of a simulated system and identify both local and global conformational changes.
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Affiliation(s)
- Ryan L. Melvin
- Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, United States
| | - Ryan C. Godwin
- Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, United States
| | - Jiajie Xiao
- Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, United States
| | - William G. Thompson
- Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, United States
| | - Kenneth S. Berenhaut
- Department of Mathematics & Statistics, Wake Forest University, Winston-Salem, North Carolina 27109, United States
| | - Freddie R. Salsbury
- Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, United States
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