The Gigii-Bapiimin Study: resilience and the impacts of COVID-19 on health and wellbeing of Indigenous people living with HIV in Manitoba and Saskatchewan.

The Gigii-Bapiimin study explored the impacts of the COVID-19 pandemic on the health and wellbeing of First Nations, Inuit, and Métis people living with HIV in Manitoba and Saskatchewan, two provinces in Canada with alarmingly high rates of HIV infections. Participants (n = 28 in Manitoba and n = 23 in Saskatchewan) were recruited using various methods, including flyers, community organizations, peers, and social media. The qualitative interviews focused on the pandemic's impact on health, access to services, and ceremonies. The data were analyzed using inductive thematic analysis. The study identified three key themes: (a) resilience and coping; (b) negative impacts on health and substance use; (c) decreased access to health services, HIV care and harm reduction. The participants shared their experiences of social isolation and the loss of community support, which had deleterious effects on their mental health and substance use. The impacts on access to HIV care were exacerbated by poverty, homelessness, and distress over inadvertent disclosure of HIV status. Participants mitigated these impacts by relying on Indigenous knowledges, ceremonies, and resilience within their communities. Service providers must address the impacts of the COVID-19 pandemic on Indigenous people living with HIV and their access to HIV services and ceremonies.

Resisting and disrupting HIV-related stigma: a photovoice study.

BACKGROUND: The stigma associated with human immunodeficiency virus (HIV) is a significant global public health concern. Health care providers and policy makers continue to struggle with understanding and implementing strategies to reduce HIV-related stigma in particular contexts and at the intersections of additional oppressions. Perspectives and direction from people living with HIV are imperative. METHODS: In this project we amplified the voices of people living with HIV about their experiences of HIV-related stigma in Manitoba, Canada. We used an arts-based qualitative case study research design using photovoice and narrative interviews. Adults living with HIV participated by taking pictures that represented their stigma experiences. The photos were a catalyst for conversations about HIV and stigma during follow-up individual narrative interviews. Journaling provided opportunities for participants to reflect on their experiences of, and resistance to, stigma. Interviews were audio recorded and transcribed. Photos, journals, and transcribed interviews were analyzed using inductive qualitative methods RESULTS: Through pictures and dialogue, participants (N = 11; 64% women) expressed the emotional and social impacts of stigmas that were created and supported by oppressive structures and interpersonal attitudes and behaviours. These experiences were compounded by intersecting forms of oppression including racism, sexism, and homophobia. Participants also relayed stories of their personal strategies and transitions toward confronting stigma. Strategies were themed as caring for oneself, caring for children and pets, reconstituting social support networks, and resisting and disrupting stigma. Participants made important recommendations for system and policy change. CONCLUSIONS: These stories of oppression and resistance can inspire action to reduce HIV-related stigma. People living with HIV can consider the strategies to confront stigma that were shared in these stories. Health care providers and policy makers can take concerted actions to support peoples' transitions to resisting stigmas. They can facilitate supportive and anti-oppressive health and social service systems that address medical care as well as basic needs for food, shelter, income, and positive social and community connections.

The CECAM electronic structure library and the modular software development paradigm.

First-principles electronic structure calculations are now accessible to a very large community of users across many disciplines, thanks to many successful software packages, some of which are described in this special issue. The traditional coding paradigm for such packages is monolithic, i.e., regardless of how modular its internal structure may be, the code is built independently from others, essentially from the compiler up, possibly with the exception of linear-algebra and message-passing libraries. This model has endured and been quite successful for decades. The successful evolution of the electronic structure methodology itself, however, has resulted in an increasing complexity and an ever longer list of features expected within all software packages, which implies a growing amount of replication between different packages, not only in the initial coding but, more importantly, every time a code needs to be re-engineered to adapt to the evolution of computer hardware architecture. The Electronic Structure Library (ESL) was initiated by CECAM (the European Centre for Atomic and Molecular Calculations) to catalyze a paradigm shift away from the monolithic model and promote modularization, with the ambition to extract common tasks from electronic structure codes and redesign them as open-source libraries available to everybody. Such libraries include "heavy-duty" ones that have the potential for a high degree of parallelization and adaptation to novel hardware within them, thereby separating the sophisticated computer science aspects of performance optimization and re-engineering from the computational science done by, e.g., physicists and chemists when implementing new ideas. We envisage that this modular paradigm will improve overall coding efficiency and enable specialists (whether they be computer scientists or computational scientists) to use their skills more effectively and will lead to a more dynamic evolution of software in the community as well as lower barriers to entry for new developers. The model comes with new challenges, though. The building and compilation of a code based on many interdependent libraries (and their versions) is a much more complex task than that of a code delivered in a single self-contained package. Here, we describe the state of the ESL, the different libraries it now contains, the short- and mid-term plans for further libraries, and the way the new challenges are faced. The ESL is a community initiative into which several pre-existing codes and their developers have contributed with their software and efforts, from which several codes are already benefiting, and which remains open to the community.

The ONETEP linear-scaling density functional theory program.

We present an overview of the onetep program for linear-scaling density functional theory (DFT) calculations with large basis set (plane-wave) accuracy on parallel computers. The DFT energy is computed from the density matrix, which is constructed from spatially localized orbitals we call Non-orthogonal Generalized Wannier Functions (NGWFs), expressed in terms of periodic sinc (psinc) functions. During the calculation, both the density matrix and the NGWFs are optimized with localization constraints. By taking advantage of localization, onetep is able to perform calculations including thousands of atoms with computational effort, which scales linearly with the number or atoms. The code has a large and diverse range of capabilities, explored in this paper, including different boundary conditions, various exchange-correlation functionals (with and without exact exchange), finite electronic temperature methods for metallic systems, methods for strongly correlated systems, molecular dynamics, vibrational calculations, time-dependent DFT, electronic transport, core loss spectroscopy, implicit solvation, quantum mechanical (QM)/molecular mechanical and QM-in-QM embedding, density of states calculations, distributed multipole analysis, and methods for partitioning charges and interactions between fragments. Calculations with onetep provide unique insights into large and complex systems that require an accurate atomic-level description, ranging from biomolecular to chemical, to materials, and to physical problems, as we show with a small selection of illustrative examples. onetep has always aimed to be at the cutting edge of method and software developments, and it serves as a platform for developing new methods of electronic structure simulation. We therefore conclude by describing some of the challenges and directions for its future developments and applications.

A qualitative study on the intersectional social determinants for indigenous people who become infected with HIV in their youth.

BACKGROUND: Indigenous young people are currently highly overrepresented in the HIV epidemic in Canada, especially in the Prairie Provinces, such as Manitoba. Understanding HIV-vulnerability in Indigenous peoples must begin with understanding that social determinants are intersectional and linked to the historical legacy of European colonization. In this paper findings that detail the influence of the intersectional social determinants on Indigenous people who become infected with HIV in their youth are presented. METHODS: The qualitative research design of phenomenology was used as it afforded the opportunity to understand Indigenous young people from their frames of reference and experiences of reality, resulting in a phenomenological understanding of their perspectives and experiences of the early years of living with HIV. A total of 21 Indigenous young people took part open-ended interviews. RESULTS: The stories that the Indigenous young people shared revealed their deeply interconnected social worlds, and how social determinants including abuse, trauma, being part of the child welfare system, and housing and food security were connected throughout various stages of their lives. Such stages included childhood, adolescence and young adulthood (the time of HIV infection), and later adulthood for older participants with the social determinants having multiple influences on their health trajectories. CONCLUSIONS: The findings highlight the need for policies and programs that are broadly focused, addressing multiple social determinants together. Overall, there needs to be more emphasis on the multiple social determinants in the life situations of all Indigenous youth. Reducing the health and social disparities in Indigenous youth is key to reducing the number of young Indigenous people diagnosed with HIV. The findings also shed light on the importance of listening to young Indigenous people who have experienced HIV diagnosis and life following diagnosis.

Renormalization of myoglobin-ligand binding energetics by quantum many-body effects.

We carry out a first-principles atomistic study of the electronic mechanisms of ligand binding and discrimination in the myoglobin protein. Electronic correlation effects are taken into account using one of the most advanced methods currently available, namely a linear-scaling density functional theory (DFT) approach wherein the treatment of localized iron 3d electrons is further refined using dynamical mean-field theory. This combination of methods explicitly accounts for dynamical and multireference quantum physics, such as valence and spin fluctuations, of the 3d electrons, while treating a significant proportion of the protein (more than 1,000 atoms) with DFT. The computed electronic structure of the myoglobin complexes and the nature of the Fe-O2 bonding are validated against experimental spectroscopic observables. We elucidate and solve a long-standing problem related to the quantum-mechanical description of the respiration process, namely that DFT calculations predict a strong imbalance between O2 and CO binding, favoring the latter to an unphysically large extent. We show that the explicit inclusion of the many-body effects induced by the Hund's coupling mechanism results in the correct prediction of similar binding energies for oxy- and carbonmonoxymyoglobin.

Sexually transmitted and blood-borne infections by sex, methamphetamine use, and houselessness before, at, and after HIV diagnosis in Manitoba, Canada.

Objectives: Describe the proportion of people newly living with HIV with sexually transmitted and blood-borne infections (STBBIs) before, at, and after HIV diagnosis in Manitoba, Canada. Methods: A retrospective cohort study reviewed clinical charts of all 404 people ≥18 years old newly diagnosed with HIV in Manitoba, Canada between 2018 and 2021. Syphilis, hepatitis C and B, gonorrhea, and chlamydia infections before, at, and after HIV diagnosis were recorded and analyzed by sex at birth, injection drug use status, use of methamphetamines, and housing status. Results: A total of 53% of people were diagnosed with syphilis, 44.1% with gonorrhea, 42.8% with chlamydia, and 40.6% with hepatitis C at least once. Among females, 64.1% had at least one or more STBBIs diagnoses before HIV diagnosis compared with 44.8% of males. Over 70% of people experiencing houselessness had at least one STBBI diagnosis before their HIV diagnosis compared with 43.9% of people not houseless. Among people who used methamphetamines, 68.3% had one or more STBBIs before HIV diagnosis compared with 28.9% of people who do not use methamphetamines. In a multivariable analysis houselessness, methamphetamine use, and younger age were associated with increased risk of any STBBIs. Conclusions: In our Manitoba cohort of people living with HIV, disproportionately more females, people experiencing houselessness, and those who use methamphetamine were diagnosed with STBBIs. The proportion of new infections before HIV diagnoses highlights a missed opportunity to provide prevention modalities, including pre-exposure prophylaxis, and the proportion after HIV diagnosis emphasizes the importance of enhancing engagement, repeated testing, and educational strategies to ameliorate ongoing exposures.

Sex differences in houselessness, injection drug use, and mental health conditions among people newly diagnosed with HIV in Manitoba, Canada from 2018 to 2021: a retrospective cohort study.

Background: Manitoba saw the highest number of new HIV diagnoses in the province's history in 2021 and is the only Canadian province not meeting any of the previous UNAIDS 90-90-90 targets. Our goal was to describe sex differences and syndemic conditions within an incident HIV cohort in Manitoba, and the HIV treatment initiation and undetectable viral load outcomes. Methods: This was a retrospective cohort study of all people 18 years and older newly diagnosed with HIV in Manitoba, Canada between January 1st, 2018 and December 31st, 2021. Data was collected as follows: before HIV diagnosis: chlamydia, gonorrhoea, syphilis, and/or hepatitis C antibodies. At the time of HIV diagnosis: age, sex, gender, race/ethnicity, sexual orientation. During follow-up: CD4 counts, viral load, HIV treatment, hospitalizations, and number of visits to HIV care. Main exposures evaluated: methamphetamine use, injection drug use, houselessness, and mental health conditions. Outcomes: started antiretroviral treatment and achieved an undetectable viral load. A descriptive statistical analysis was used. Findings: There were 404 new HIV diagnoses in Manitoba from 2018 to 2021; 44.8% were female, 55.2% male; 76.% self-identified as Indigenous, 13.4% white/European, 4.7% African/black; 86.6% cis-gender; 60.9% heterosexual, 13.4% gay, bisexual and men who have sex with men, and 1.7% lesbian. Injection drug use was reported by 71.8% and 43.5% of females and males respectively. Methamphetamine was the most frequently injected drug (62.4%). Amongst females, 81.8% experienced at least one of the following: houselessness (43.1%), mental health comorbidities (46.4%), and injection drug use (71.8%). Only 64.9% of all individuals had an undetectable viral load (61.1% females and 67.9% males), 56.5% among people experiencing houselessness, 59% among young people (≤29 years), and 60.1% among people who inject drugs. Interpretation: People newly diagnosed with HIV in Manitoba are disproportionately experiencing houselessness, mental illness, and injection drug use (mostly methamphetamine). This pattern is more pronounced for female individuals. These findings highlight the need for syndemic and gender-specific approaches, simultaneously addressing social and health conditions, to treat HIV. Funding: This work was supported by the Canadian Institutes of Health Research, The Manitoba Medical Service Foundation, The James Farley Memorial Fund and the Canada Research Chairs Program.

Natural bond orbital analysis in the ONETEP code: applications to large protein systems.

First principles electronic structure calculations are typically performed in terms of molecular orbitals (or bands), providing a straightforward theoretical avenue for approximations of increasing sophistication, but do not usually provide any qualitative chemical information about the system. We can derive such information via post-processing using natural bond orbital (NBO) analysis, which produces a chemical picture of bonding in terms of localized Lewis-type bond and lone pair orbitals that we can use to understand molecular structure and interactions. We present NBO analysis of large-scale calculations with the ONETEP linear-scaling density functional theory package, which we have interfaced with the NBO 5 analysis program. In ONETEP calculations involving thousands of atoms, one is typically interested in particular regions of a nanosystem whilst accounting for long-range electronic effects from the entire system. We show that by transforming the Non-orthogonal Generalized Wannier Functions of ONETEP to natural atomic orbitals, NBO analysis can be performed within a localized region in such a way that ensures the results are identical to an analysis on the full system. We demonstrate the capabilities of this approach by performing illustrative studies of large proteins--namely, investigating changes in charge transfer between the heme group of myoglobin and its ligands with increasing system size and between a protein and its explicit solvent, estimating the contribution of electronic delocalization to the stabilization of hydrogen bonds in the binding pocket of a drug-receptor complex, and observing, in situ, the n → π* hyperconjugative interactions between carbonyl groups that stabilize protein backbones.

Importance of many-body effects in the Kernel of hemoglobin for ligand binding.

We propose a mechanism for binding of diatomic ligands to heme based on a dynamical orbital selection process. This scenario may be described as bonding determined by local valence fluctuations. We support this model using linear-scaling first-principles calculations, in combination with dynamical mean-field theory, applied to heme, the kernel of the hemoglobin metalloprotein central to human respiration. We find that variations in Hund's exchange coupling induce a reduction of the iron 3d density, with a concomitant increase of valence fluctuations. We discuss the comparison between our computed optical absorption spectra and experimental data, our picture accounting for the observation of optical transitions in the infrared regime, and how the Hund's coupling reduces, by a factor of 5, the strong imbalance in the binding energies of heme with CO and O(2) ligands.

Vanadium dioxide: a Peierls-Mott insulator stable against disorder.

Vanadium dioxide undergoes a first order metal-insulator transition at 340 K. In this Letter, we develop and carry out state-of-the-art linear scaling density-functional theory calculations refined with nonlocal dynamical mean-field theory. We identify a complex mechanism, a Peierls-assisted orbital selection Mott instability, which is responsible for the insulating M(1) phase, and which furthermore survives a moderate degree of disorder.

Interrogation of the protein-protein interactions between human BRCA2 BRC repeats and RAD51 reveals atomistic determinants of affinity.

The breast cancer suppressor BRCA2 controls the recombinase RAD51 in the reactions that mediate homologous DNA recombination, an essential cellular process required for the error-free repair of DNA double-stranded breaks. The primary mode of interaction between BRCA2 and RAD51 is through the BRC repeats, which are ∼35 residue peptide motifs that interact directly with RAD51 in vitro. Human BRCA2, like its mammalian orthologues, contains 8 BRC repeats whose sequence and spacing are evolutionarily conserved. Despite their sequence conservation, there is evidence that the different human BRC repeats have distinct capacities to bind RAD51. A previously published crystal structure reports the structural basis of the interaction between human BRC4 and the catalytic core domain of RAD51. However, no structural information is available regarding the binding of the remaining seven BRC repeats to RAD51, nor is it known why the BRC repeats show marked variation in binding affinity to RAD51 despite only subtle sequence variation. To address these issues, we have performed fluorescence polarisation assays to indirectly measure relative binding affinity, and applied computational simulations to interrogate the behaviour of the eight human BRC-RAD51 complexes, as well as a suite of BRC cancer-associated mutations. Our computational approaches encompass a range of techniques designed to link sequence variation with binding free energy. They include MM-PBSA and thermodynamic integration, which are based on classical force fields, and a recently developed approach to computing binding free energies from large-scale quantum mechanical first principles calculations with the linear-scaling density functional code onetep. Our findings not only reveal how sequence variation in the BRC repeats directly affects affinity with RAD51 and provide significant new insights into the control of RAD51 by human BRCA2, but also exemplify a palette of computational and experimental tools for the analysis of protein-protein interactions for chemical biology and molecular therapeutics.

QM/MM simulation of liquid water with an adaptive quantum region.

The simulation of complex chemical systems often requires a multi-level description, in which a region of special interest is treated using a computationally expensive quantum mechanical (QM) model while its environment is described by a faster, simpler molecular mechanical (MM) model. Furthermore, studying dynamic effects in solvated systems or bio-molecules requires a variable definition of the two regions, so that atoms or molecules can be dynamically re-assigned between the QM and MM descriptions during the course of the simulation. Such reassignments pose a problem for traditional QM/MM schemes by exacerbating the errors that stem from switching the model at the boundary. Here we show that stable, long adaptive simulations can be carried out using density functional theory with the BLYP exchange-correlation functional for the QM model and a flexible TIP3P force field for the MM model without requiring adjustments of either. Using a primary benchmark system of pure water, we investigate the convergence of the liquid structure with the size of the QM region, and demonstrate that by using a sufficiently large QM region (with radius 6 Å) it is possible to obtain radial and angular distributions that, in the QM region, match the results of fully quantum mechanical calculations with periodic boundary conditions, and, after a smooth transition, also agree with fully MM calculations in the MM region. The key ingredient is the accurate evaluation of forces in the QM subsystem which we achieve by including an extended buffer region in the QM calculations. We also show that our buffered-force QM/MM scheme is transferable by simulating the solvated Cl(-) ion.

Doping Engineering of Single-Walled Carbon Nanotubes by Nitrogen Compounds Using Basicity and Alignment.

Charge transport properties in single-walled carbon nanotubes (SWCNTs) can be significantly modified through doping, tuning their electrical and thermoelectric properties. In our study, we used more than 40 nitrogen-bearing compounds as dopants and determined their impact on the material's electrical conductivity. The application of nitrogen compounds of diverse structures and electronic configurations enabled us to determine how the dopant nature affects the SWCNTs. The results reveal that the impact of these dopants can often be anticipated by considering their Hammett's constants and pKa values. Furthermore, the empirical observations supported by first-principles calculations indicate that the doping level can be tuned not only by changing the type and the concentration of dopants but also by varying the orientation of nitrogen compounds around SWCNTs.

Computational analysis of phosphopeptide binding to the polo-box domain of the mitotic kinase PLK1 using molecular dynamics simulation.

The Polo-Like Kinase 1 (PLK1) acts as a central regulator of mitosis and is over-expressed in a wide range of human tumours where high levels of expression correlate with a poor prognosis. PLK1 comprises two structural elements, a kinase domain and a polo-box domain (PBD). The PBD binds phosphorylated substrates to control substrate phosphorylation by the kinase domain. Although the PBD preferentially binds to phosphopeptides, it has a relatively broad sequence specificity in comparison with other phosphopeptide binding domains. We analysed the molecular determinants of recognition by performing molecular dynamics simulations of the PBD with one of its natural substrates, CDC25c. Predicted binding free energies were calculated using a molecular mechanics, Poisson-Boltzmann surface area approach. We calculated the per-residue contributions to the binding free energy change, showing that the phosphothreonine residue and the mainchain account for the vast majority of the interaction energy. This explains the very broad sequence specificity with respect to other sidechain residues. Finally, we considered the key role of bridging water molecules at the binding interface. We employed inhomogeneous fluid solvation theory to consider the free energy of water molecules on the protein surface with respect to bulk water molecules. Such an analysis highlights binding hotspots created by elimination of water molecules from hydrophobic surfaces. It also predicts that a number of water molecules are stabilized by the presence of the charged phosphate group, and that this will have a significant effect on the binding affinity. Our findings suggest a molecular rationale for the promiscuous binding of the PBD and highlight a role for bridging water molecules at the interface. We expect that this method of analysis will be very useful for probing other protein surfaces to identify binding hotspots for natural binding partners and small molecule inhibitors.

Interaction between surface waves on wire lines.

This paper investigates coupling between electromagnetic surface waves on parallel wires. Finite-element method (FEM)-based and analytic models are developed for single- and double-wire Sommerfeld and Goubau lines. Models are validated via measurements for Goubau lines and a comparison between the analytic and the FEM-based computations for coupled Sommerfeld- and Goubau-type lines is carried out. The measurements and calculations show remarkable agreement. The FEM-based and analytic models match remarkably well too. The results exhibit new favourable effects for surface waves propagation over multiple conductors. The short-range behaviour of the coupled wires and, consequently, the existence of an optimum separation of coupled wires is one of the most significant findings of this paper. We comment on the relevance of our results, particularly in relation to applications of high bandwidth demands and cross-coupling effects.

Enhancing thermoelectric properties of single-walled carbon nanotubes using halide compounds at room temperature and above.

Carbon nanotubes (CNTs) are materials with exceptional electrical, thermal, mechanical, and optical properties. Ever since it was demonstrated that they also possess interesting thermoelectric properties, they have been considered a promising solution for thermal energy harvesting. In this study, we present a simple method to enhance their performance. For this purpose, thin films obtained from high-quality single-walled CNTs (SWCNTs) were doped with a spectrum of inorganic and organic halide compounds. We studied how incorporating various halide species affects the electrical conductivity, the Seebeck coefficient, and the Power Factor. Since thermoelectric devices operate under non-ambient conditions, we also evaluated these materials' performance at elevated temperatures. Our research shows that appropriate dopant selection can result in almost fivefold improvement to the Power Factor compared to the pristine material. We also demonstrate that the chemical potential of the starting CNT network determines its properties, which is important for deciphering the true impact of chemical and physical functionalization of such ensembles.

Time averaging of NMR chemical shifts in the MLF peptide in the solid state.

Since experimental measurements of NMR chemical shifts provide time and ensemble averaged values, we investigated how these effects should be included when chemical shifts are computed using density functional theory (DFT). We measured the chemical shifts of the N-formyl-L-methionyl-L-leucyl-L-phenylalanine-OMe (MLF) peptide in the solid state, and then used the X-ray structure to calculate the (13)C chemical shifts using the gauge including projector augmented wave (GIPAW) method, which accounts for the periodic nature of the crystal structure, obtaining an overall accuracy of 4.2 ppm. In order to understand the origin of the difference between experimental and calculated chemical shifts, we carried out first-principles molecular dynamics simulations to characterize the molecular motion of the MLF peptide on the picosecond time scale. We found that (13)C chemical shifts experience very rapid fluctuations of more than 20 ppm that are averaged out over less than 200 fs. Taking account of these fluctuations in the calculation of the chemical shifts resulted in an accuracy of 3.3 ppm. To investigate the effects of averaging over longer time scales we sampled the rotameric states populated by the MLF peptides in the solid state by performing a total of 5 micros classical molecular dynamics simulations. By averaging the chemical shifts over these rotameric states, we increased the accuracy of the chemical shift calculations to 3.0 ppm, with less than 1 ppm error in 10 out of 22 cases. These results suggests that better DFT-based predictions of chemical shifts of peptides and proteins will be achieved by developing improved computational strategies capable of taking into account the averaging process up to the millisecond time scale on which the chemical shift measurements report.

Gaussian approximation potentials: the accuracy of quantum mechanics, without the electrons.

We introduce a class of interatomic potential models that can be automatically generated from data consisting of the energies and forces experienced by atoms, as derived from quantum mechanical calculations. The models do not have a fixed functional form and hence are capable of modeling complex potential energy landscapes. They are systematically improvable with more data. We apply the method to bulk crystals, and test it by calculating properties at high temperatures. Using the interatomic potential to generate the long molecular dynamics trajectories required for such calculations saves orders of magnitude in computational cost.

Analytic modelling of a planar Goubau line with circular conductor.

Planar Goubau lines show promise as high frequency, low-loss waveguides on a substrate. However, to date only numerical simulations and experimental measurements have been performed. This paper analytically investigates the surface wave mode propagating along a planar Goubau line consisting of a perfectly conducting circular wire on top of a dielectric substrate of finite thickness but infinite width. An approximate equation for the propagation constant is derived and solved through numerical integration. The dependence of the propagation constant on various system parameters is calculated and the results agree well with full numerical simulations. In addition, the spatial distribution of the longitudinal electric field is reported and excellent agreement with a numerical simulation and previous studies is found. Moreover, validation against experimental phase velocity measurements is also reported. Finally, insights gained from the model are considered for a Goubau line with a rectangular conductor. The analytic model reveals that the propagating mode of a planar Goubau line is hybrid in contrast to the transverse magnetic mode of a classic Goubau line.