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This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semi-groups can be extended to all semigroups. In the second part we show that if a sub-semigroup S of a group G generates G in the sense that G = S · S−1, then a generalized polynomial on S with values in an Abelian group H can be extended to a generalized polynomial on G into H. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials.

We consider two variants of the sine subtraction law on a semi-group S. The main objective is to solve f(xy∗ ) = f(x)g(y) − g(x)f(y) for unknown functions f, g : S → ℂ, where x ↦ x* is an anti-homomorphic involution. Until now this equation was not solved even when S is a non-Abelian group and x* = x−1. We find the solutions assuming that f is central. A secondary objective is to solve f(xσ(y)) = f(x)g(y) − g(x)f(y), where σ : S → S is a homomorphic involution. Until now this variant was solved assuming that S has an identity element. We also find the continuous solutions of these equations on topological semigroups.

We consider the Levi–Civita equation f(xy)=g1(x)h1(y)+g2(x)h2(y)f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right) for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.

The primary object of study is the “cosine-sine” functional equation f(xy) = f(x)g(y)+g(x)f(y)+h(x)h(y) for unknown functions f, g, h : S → ℂ, where S is a semigroup. The name refers to the fact that it contains both the sine and cosine addition laws. This equation has been solved on groups and on semigroups generated by their squares. Here we find the solutions on a larger class of semigroups and discuss the obstacles to finding a general solution for all semigroups. Examples are given to illustrate both the results and the obstacles.

Objectives: The objective of this study is to investigate early-to-late postdoctoral clinical academic progression and the experiences of NIHR Clinical Lectureship (CL) fellows, considering enablers and barriers to success, and identifying the factors associated with immediate progression to a clinical academic role following completion of the award., Setting: Datasets of CL awardees across the UK., Participants: For semistructured interviews, n=40 CL awardees that had finished their award within the previous 5 years. For quantitative analysis, n=1226 completed or currently active CL awardees., Outcome Measures: The responses from the semistructured interviews to the defined questions on experiences during the award, postaward progression, and enablers and barriers to academic progression. Other primary outcome measures were quantitative data on first destinations postaward, demographic data, and whether an awardee had previously held an NIHR Academic Clinical Fellowship (ACF) or was a recipient of the Academy of Medical Sciences (AMS) Starter Grant., Results: CL awardees identified numerous benefits to the award, with the majority achieving their aims. Most awardees progressed to a clinical academic role; however, some returned to a clinical only position, citing concerns around the time pressure associated with balancing clinical and academic responsibilities, and the competition to attain further postdoctoral awards. The region of the award partnership, year of award end and success in applying for an AMS Starter Grant were associated with progression to a clinical academic role. Gender, holding an ACF and having a craft or non-craft specialty had no independent statistical association with clinical academic progression., Conclusions: The CL is a valued element of the Integrated Academic Pathway. By addressing issues around later postdoctoral progression opportunities, responding to challenges experienced by CLs, and by understanding the factors identified in this study associated with clinical academic progression, it should be possible to increase the proportion of CLs that become fully independent clinical academic research leaders., Participants: 1226 NIHR CLs active or completed on the award between 2006 and 2020., Competing Interests: Competing interests: JH, MRM and HH were employed by the University of Leeds while working on the manuscript, JH was a member then Chair of a selection panel for NIHR Academy Fellowships between 2006 and 2014. MMR and CM are employed by the Academy of Medical Sciences. The remaining authors are all employed by NIHR CC., (© Author(s) (or their employer(s)) 2023. Re-use permitted under CC BY-NC. No commercial re-use. See rights and permissions. Published by BMJ.)

High-resolution respirometry methods allow for the assessment of oxygen consumption by the electron transfer systems within cells, tissue samples, and isolated mitochondrial preparations. As mitochondrial integrity is compromised by the process of cryopreservation, these methods have been limited to fresh samples. Here we present a simple method to assess the activity of mitochondria respiratory complexes I and II in previously cryopreserved murine skeletal muscle tissue homogenates, as well as previously frozen D. melanogaster, as a function of oxygen consumption., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2023 Ebanks et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

The globin protein superfamily has diverse functions. Haemoglobin has been found in non-erythroid locations, including within the mitochondria. Using co-immunoprecipitation and in silico methods, we investigated the interaction of mitochondrial haemoglobin with ATP synthase and its associated proteins, including inhibitory factor 1 (IF1). We measured the expression of mitochondrial haemoglobin in response to hypoxia. In vitro and in silico evidence of interactions between mitochondrial haemoglobin and ATP synthase were found, and we report upregulated mitochondrial haemoglobin expression in response to hypoxia within skeletal muscle tissue. Our observations indicate that mitochondrial pH and ATP synthase activity are implicated in the mitochondrial haemoglobin response to hypoxia.

Antarctic notothenioids have unique adaptations that allow them to thrive in subzero Antarctic waters. Within the suborder Notothenioidei, species of the family Channichthyidae (icefish) lack hemoglobin and in some instances myoglobin too. In studies of mitochondrial function of notothenioids, few have focused specifically on ATP synthase. In this study, we find that the icefish Champsocephalus gunnari has a significantly higher level of ATP synthase subunit α expression than the red-blooded Notothenia rossii , but a much smaller interactome than the other species. We characterize the interactome of ATP synthase subunit α in two red-blooded species Trematomus bernacchii, N. rossii , and in the icefish Chionodraco rastrospinosus and C. gunnari and find that, in comparison with the other species, reactome enrichment for C. gunnari lacks chaperonin-mediated protein folding, and fewer oxidative-stress-associated proteins are present in the identified interactome of C. gunnari . Reactome enrichment analysis also identifies a transcript-specific translational silencing pathway for the iron oxidase protein ceruloplasmin, which has previously been reported in studies of icefish as distinct from other red-blooded fish and vertebrates in its activity and RNA transcript expression. Ceruloplasmin protein expression is detected by Western blot in the liver of T. bernacchii , but not in N. rossii , C. rastrospinosus , and C. gunnari. We suggest that the translation of ceruloplasmin transcripts is silenced by the identified pathway in icefish notothenioids, which is indicative of altered iron metabolism and Fe(II) detoxification.

Antarctic icefish are extraordinary in their ability to thrive without haemoglobin. We wanted to understand how the mitochondrial proteome has adapted to the loss of this protein. Metabolic pathways that utilise oxygen are most likely to be rearranged in these species. Here, we have defined the mitochondrial proteomes of both the red and white muscle of two different icefish species ( Champsocephalus gunnari and Chionodraco rastrospinosus) and compared these with two related red-blooded Notothenioids ( Notothenia rossii , Trematomus bernacchii ). Liquid Chromatography-Mass spectrometry (LC-MS/MS) was used to generate and examine the proteomic profiles of the two groups. We recorded a total of 91 differentially expressed proteins in the icefish red muscle mitochondria and 89 in the white muscle mitochondria when compared with the red-blooded related species. The icefish have a relatively higher abundance of proteins involved with Complex V of oxidative phosphorylation, RNA metabolism, and homeostasis, and fewer proteins for striated muscle contraction, haem, iron, creatine, and carbohydrate metabolism. Enrichment analyses showed that many important pathways were different in both red muscle and white muscle, including the citric acid cycle, ribosome machinery and fatty acid degradation. Life in the Antarctic waters poses extra challenges to the organisms that reside within them. Icefish have successfully inhabited this environment and we surmise that species without haemoglobin uniquely maintain their physiology. Our study highlights the mitochondrial protein pathway differences between similar fish species according to their specific tissue oxygenation idiosyncrasies.

The mitochondrial ATP synthase is responsible for the production of cellular ATP, and it does so by harnessing the membrane potential of the mitochondria that is produced by the sequential oxidation of select cellular metabolites. Since the structural features of ATP synthase were first resolved nearly three decades ago, significant progress has been made in understanding its role in health and disease. Mitochondrial dysfunction is common to neurodegeneration, with elevated oxidative stress a hallmark of this dysfunction. The patterns of this oxidative stress, including molecular targets and the form of oxidative modification, can vary widely. In this mini review we discuss the oxidative modifications of ATP synthase that have been observed in Alzheimer's disease, Parkinson's disease, and Huntington's disease. Oxidative modifications of ATP synthase in Alzheimer's disease are well-documented, and there is a growing body of knowledge on the subject in Parkinson's disease. The consideration of ATP synthase as a pharmacological target in a variety of diseases underlines the importance of understanding these modifications, both as a potential target, and also as inhibitors of any pharmacological intervention., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2022 Ebanks and Chakrabarti.)

Mitochondria are intracellular organelles responsible for cellular respiration with one of their major roles in the production of energy in the form of ATP. Activities with increased energetic demand are especially dependent on efficient ATP production, hence sufficient mitochondrial function is fundamental. In bees, flight muscle and the brain have particularly high densities of mitochondria to facilitate the substantial ATP production required for flight activity and neuronal signalling. Neonicotinoids are systemic synthetic insecticides that are widely utilised against crop herbivores but have been reported to cause, by unknown mechanisms, mitochondrial dysfunction, decreasing cognitive function and flight activity among pollinating bees. Here we explore, using high-resolution respirometry, how the neonicotinoid imidacloprid may affect oxidative phosphorylation in the brain and flight muscle of the buff-tailed bumblebee, Bombus terrestris . We find that acute exposure increases routine oxygen consumption in the flight muscle of worker bees. This provides a candidate explanation for prior reports of early declines in flight activity following acute exposure. We further find that imidacloprid increases the maximum electron transport capacity in the brain, with a trend towards increased overall oxygen consumption. However, intra-individual variability is high, limiting the extent to which apparent effects of imidacloprid on brain mitochondria are shown conclusively. Overall, our results highlight the necessity to examine tissue-specific effects of imidacloprid on respiration and energy production., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2021 Sargent, Ebanks, Hardy, Davies, Chakrabarti and Stöger.)

Ageing is a major risk factor for many of the most prevalent diseases, including neurodegenerative diseases, cancer, and heart disease. As the global population continues to age, behavioural interventions that can promote healthy ageing will improve quality of life and relieve the socioeconomic burden that comes with an aged society. Exercise is recognised as an effective intervention against many diseases of ageing, but we do not know the stage in an individual's lifetime at which exercise is most effective at promoting healthy ageing, and whether or not it has a direct effect on lifespan. We exercised w 1118 Drosophila melanogaster , investigating the effects of sex and group size at different stages of their lifetime, and recorded their lifespan. Climbing scores at 30 days were measured to record differences in fitness in response to exercise. We also assessed the mitochondrial proteome of w 1118 Drosophila that had been exercised for one week, alongside mitochondrial respiration measured using high-resolution respirometry, to determine changes in mitochondrial physiology in response to exercise. We found that age-targeted exercise interventions improved the lifespan of both male and female Drosophila , and grouped males exercised in late life had improved climbing scores when compared with those exercised throughout their entire lifespan. The proteins of the electron transport chain were significantly upregulated in expression after one week of exercise, and complex-II-linked respiration was significantly increased in exercised Drosophila . Taken together, our findings provide a basis to test specific proteins, and complex II of the respiratory chain, as important effectors of exercise-induced healthy ageing.

Mitochondrial changes such as tight coupling of the mitochondria have facilitated sustained oxygen and respiratory activity in haemoglobin-less icefish of the Channichthyidae family. We aimed to characterise features in the sequence and structure of the proteins directly involved in proton transport, which have potential physiological implications. ATP synthase subunit a (ATP6) and subunit 8 (ATP8) are proteins that function as part of the F0 component (proton pump) of the F0F1complex. Both proteins are encoded by the mitochondrial genome and involved in oxidative phosphorylation. To explore mitochondrial sequence variation for ATP6 and ATP8 we analysed sequences from C. gunnari and C. rastrospinosus and compared them with their closely related red-blooded species and eight other vertebrate species. Our comparison of the amino acid sequence of these proteins reveals important differences that could underlie aspects of the unique physiology of the icefish. In this study we find that changes in the sequence of subunit a of the icefish C. gunnari at position 35 where there is a hydrophobic alanine which is not seen in the other notothenioids we analysed. An amino acid change of this type is significant since it may have a structural impact. The biology of the haemoglobin-less icefish is necessarily unique and any insights about these animals will help to generate a better overall understanding of important physiological pathways., Competing Interests: The authors have declared that no competing interests exist.

One of the genes which has been linked to the onset of juvenile/early onset Parkinson's disease (PD) is PINK1. There is evidence that supports the therapeutic potential of exercise in the alleviation of PD symptoms. It is possible that exercise may enhance synaptic plasticity, protect against neuro-inflammation and modulate L-Dopa regulated signalling pathways. We explored the effects of exercise on Pink1 deficient Drosophila melanogaster which undergo neurodegeneration and muscle degeneration. We used a 'power-tower' type exercise platform to deliver exercise activity to Pink1 - and age matched wild-type Drosophila . Mitochondrial proteomic profiles responding to exercise were obtained. Of the 516 proteins identified, 105 proteins had different levels between Pink1 - and wild-type non-exercised Drosophila . Gene ontology enrichment analysis and STRING network analysis highlighted proteins and pathways with altered expression within the mitochondrial proteome. Comparison of the Pink1 - exercised proteome to wild-type proteomes showed that exercising the Pink1 - Drosophila caused their proteomic profile to return towards wild-type levels.