Formulas vs. Circuits for Small Distance Connectivity

We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance $k(n)$ Connectivity, which asks whether two specified nodes in a graph of size $n$ are connected by a path of length at most $k(n)$. This problem is solvable (by the recursive doubling technique) on {\bf circuits} of depth $O(\log k)$ and size $O(kn^3)$. In contrast, we show that solving this problem on {\bf formulas} of depth $\log n/(\log\log n)^{O(1)}$ requires size $n^{\Omega(\log k)}$ for all $k(n) \leq \log\log n$. As corollaries: (i) It follows that polynomial-size circuits for Distance $k(n)$ Connectivity require depth $\Omega(\log k)$ for all $k(n) \leq \log\log n$. This matches the upper bound from recursive doubling and improves a previous $\Omega(\log\log k)$ lower bound of Beame, Pitassi and Impagliazzo [BIP98]. (ii) We get a tight lower bound of $s^{\Omega(d)}$ on the size required to simulate size-$s$ depth-$d$ circuits by depth-$d$ formulas for all $s(n) = n^{O(1)}$ and $d(n) \leq \log\log\log n$. No lower bound better than $s^{\Omega(1)}$ was previously known for any $d(n) \nleq O(1)$. Our proof technique is centered on a new notion of pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size $n$ via the operations of union and relational join, subject to certain density constraints. Half of our proof shows that bounded-depth formulas solving Distance $k(n)$ Connectivity imply upper bounds on pathset complexity. The other half is a combinatorial lower bound on pathset complexity

Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms

We present a theorem on the unitarizability of loop group valued monodromy representations and apply this to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply-connected 3-dimensional space forms $\R^3$, \bbS^3 and \bbH^3. Additionally, we compute the extended frame for any associated family of Delaunay surfaces.Comment: 18 pages, revised versio

Constant mean curvature surfaces of any positive genus

We show the existence of several new families of non-compact constant mean curvature surfaces: (i) singly-punctured surfaces of arbitrary genus $g \geq 1$, (ii) doubly-punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.Comment: 14 pages, 10 figure

An O(n^3)-Time Algorithm for Tree Edit Distance

The {\em edit distance} between two ordered trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. In this paper, we present a worst-case $O(n^3)$-time algorithm for this problem, improving the previous best $O(n^3\log n)$-time algorithm~\cite{Klein}. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems (which is interesting in its own right), together with a deeper understanding of the previous algorithms for the problem. We also prove the optimality of our algorithm among the family of \emph{decomposition strategy} algorithms--which also includes the previous fastest algorithms--by tightening the known lower bound of $\Omega(n^2\log^2 n)$~\cite{Touzet} to $\Omega(n^3)$, matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds of $\Theta(n m^2 (1 + \log \frac{n}{m}))$ when the two trees have different sizes $m$ and~$n$, where $m < n$.Comment: 10 pages, 5 figures, 5 .tex files where TED.tex is the main on

Vesuvianite From Pajsberg, Sweden, and the Role of Be In the Vesuvianite Structure

Vesuvianite from Pajsberg, Sweden contains about one atom of Mn, based on 50 cations per formula unit, and small amounts of Be, B, and As. Optical absorption analysis suggests that the Mn is predominantly or entirely trivalent. Crystal-structure analysis indicates that Mn is housed at the general octahedral site Y3, which exhibits only minor distortion from ideal octahedral symmetry. Arsenic is housed at Y2 and Z2, and the formula derived from electron microprobe and LA-ICP-MS analyses suggests minor substitution of Al for Si, also at Z2. Beryllium and B are at T1, between the edge-sharing trimers Y3Y2Y3, as is the case for B in the boron-dominant vesuvianite species wiluite. The total content at T1 is interpreted as 0.82Be, 0.34B, and 0.037Fe^(3+)

Senior Capstone Team Formation Based on Project Interest: Team Selection by Students Compared to Team Selection by Instructors

Assigning teams in large courses is logistically challenging and students are sometimes unhappy with their assigned team. This is exacerbated when the project work extends over multiple terms and teams have unique projects. Giving students some agency in team and project selection is one way to improve their project experience. This paper examines two key questions: (a) What is the best way to incorporate student interests into the team-forming process? (b) What impact does the team-forming process have on the student experience throughout the project? We consider two different approaches to giving students agency in the team formation / project selection process that have been implemented in our capstone course. One approach has faculty forming teams outside of class based on student surveys of project interests, skills, time availability, and team preferences. The alternative method enables students to form their own teams in a dynamic faculty-guided setting: Students place nametags on their top project posters, speak with other interested students, and move their nametags as needed until each project had teams with the appropriate size and skillset. Teams formed using these two approaches have completed a full year-long senior design project experience. Throughout these experiences, we collected data to help answer our two key questions. We used student surveys about the experience and the class, peer feedback on team dynamics, focus group discussions, and faculty observations. The results are inconclusive: The differences between the two approaches are small, indicating that either approach could be used to enable student agency in the team-forming process

Parameterized complexity of DPLL search procedures

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas

Camaronesite, [Fe^(3+)(H_2O)_2(PO_3OH)]_2(SO_4)•1-2H_2O, a new phosphate-sulfate from the Camarones Valley, Chile, structurally related to taranakite

Camaronesite (IMA 2012-094), [Fe^(3+)(H_2O)_2(PO_3OH)]_2(SO_4)•1-2H_2O, is a new mineral from near the village of Cuya in the Camarones Valley, Arica Province, Chile. The mineral is a low-temperature, secondary mineral occurring in a sulfate assemblage with anhydrite, botryogen, chalcanthite, copiapite, halotrichite, hexahydrite, hydroniumjarosite, pyrite, römerite, rozenite and szomolnokite. Lavender-coloured crystals up to several mm across form dense intergrowths. More rarely crystals occur as drusy aggregates of tablets up to 0.5 mm in diameter and 0.02 mm thick. Tablets are flattened on {001} and exhibit the forms {001}, {104}, {015} and {018}. The mineral is transparent with white streak and vitreous lustre. The Mohs hardness is 2½, the tenacity is brittle and the fracture is irregular, conchoidal and stepped. Camaronesite has one perfect cleavage on {001}. The measured and calculated densities are 2.43(1) and 2.383 g/cm^3, respectively. The mineral is optically uniaxial (+) with ω = 1.612(1) and ε = 1.621(1) (white light). The pleochroism is O (pale lavender) > E (colourless). Electron-microprobe analyses provided Fe_2O_331.84, P_2O_529.22, SO_315.74, H_2O 23.94 (based on O analyses), total 100.74 wt.%. The empirical formula (based on 2 P a.p.f.u.) is: Fe_(1.94)(PO_3OH)_2(S_(0.96)O_4)(H_2O)_4•1.46H_2O. The mineral is slowly soluble in concentrated HCl and extremely slowly soluble in concentrated H_2SO_4. Camaronesite is trigonal, R32, with cell parameters:a = 9.0833(5), c = 42.944(3) Å, V = 3068.5(3) Å3 and Z = 9. The eight strongest lines in the X-ray powder diffraction pattern are [d_(obs) Å (I)(hkl)]: 7.74(45)(101), 7.415(100)(012), 4.545(72)(110), 4.426(26)(018), 3.862(32)(021,202,116), 3.298(93)(027,119), 3.179(25)(208) and 2.818(25)(1•1•12,125). In the structure of camaronesite (R_1 = 2.28% for 1138 F_o > 4σF), three types of Fe octahedra are linked by corner sharing with (PO_3OH) tetrahedra to form polyhedral layers perpendicular to c with composition [Fe^(3+)(H_2O)_2(PO_3OH)]. Two such layers are joined through SO_4 tetrahedra (in two half-occupied orientations) to form thick slabs of composition [Fe^(3+)(H_2O)_2(PO_3OH)]_2(SO_4). Between the slabs are partially occupied H_2O groups. The only linkages between the slabs are hydrogen bonds. The most distinctive component in the structure consists of two Fe octahedra linked to one another by three PO_4 tetrahedra yielding an [Fe_2(PO_4)_3] unit. This unit is also the key component in the sodium super-ionic conductor (NASICON) structure and has been referred to as the lantern unit. The polyhedral layers in the structure of camaronesite are similar to those in the structure of taranakite. The Raman spectrum exhibits peaks consistent with sulfate, phosphate, water and OH groups

Delimitation of Neonectria and Cylindrocarpon (Nectriaceae, Hypocreales, Ascomycota) and related genera with Cylindrocarpon-like anamorphs

Neonectria is a cosmopolitan genus and it is, in part, defined by its link to the anamorph genus Cylindrocarpon. Neonectria has been divided into informal groups on the basis of combined morphology of anamorph and teleomorph. Previously, Cylindrocarpon was divided into four groups defined by presence or absence of microconidia and chlamydospores. Molecular phylogenetic analyses have indicated that Neonectria sensu stricto and Cylindrocarpon sensu stricto are phylogenetically congeneric. In addition, morphological and molecular data accumulated over several years have indicated that Neonectria sensu lato and Cylindrocarpon sensu lato do not form a monophyletic group and that the respective informal groups may represent distinct genera. In the present work, a multilocus analysis (act, ITS, LSU, rpb1, tef1, tub) was applied to representatives of the informal groups to determine their level of phylogenetic support as a first step towards taxonomic revision of Neonectria sensu lato. Results show five distinct highly supported clades that correspond to some extent with the informal Neonectria and Cylindrocarpon groups that are here recognised as genera: (1) N. coccinea-group and Cylindrocarpon groups 1 & 4 (Neonectria/Cylindrocarpon sensu stricto); (2) N. rugulosa-group (Rugonectria gen. nov.); (3) N. mammoidea/N. veuillotiana-groups and Cylindrocarpon group 2 (Thelonectria gen. nov.); (4) N. radicicola-group and Cylindrocarpon group 3 (Ilyonectria gen. nov.); and (5) anamorph genus Campylocarpon. Characteristics of the anamorphs and teleomorphs correlate with the five genera, three of which are newly described. New combinations are made for species where their classification is confirmed by phylogenetic data

Targeting mitochondrial fitness as a strategy for healthy vascular aging.

Cardiovascular diseases (CVD) are the leading cause of death worldwide and aging is the primary risk factor for CVD. The development of vascular dysfunction, including endothelial dysfunction and stiffening of the large elastic arteries (i.e., the aorta and carotid arteries), contribute importantly to the age-related increase in CVD risk. Vascular aging is driven in large part by oxidative stress, which reduces bioavailability of nitric oxide and promotes alterations in the extracellular matrix. A key upstream driver of vascular oxidative stress is age-associated mitochondrial dysfunction. This review will focus on vascular mitochondria, mitochondrial dysregulation and mitochondrial reactive oxygen species (ROS) production and discuss current evidence for prevention and treatment of vascular aging via lifestyle and pharmacological strategies that improve mitochondrial health. We will also identify promising areas and important considerations ('research gaps') for future investigation.REVIEW ARTICLE| JUNE 25 2020 Targeting mitochondrial fitness as a strategy for healthy vascular aging Matthew J. Rossman ; Rachel A. Gioscia-Ryan ; Zachary S. Clayton ; Michael P. Murphy ; Douglas R. Seals Crossmark: Check for Updates Clin Sci (Lond) (2020) 134 (12): 1491–1519. https://doi.org/10.1042/CS20190559 Article history Split-Screen Views Icon Views PDF Link PDF Share Icon Share Cite Icon Cite Get Permissions Abstract Cardiovascular diseases (CVD) are the leading cause of death worldwide and aging is the primary risk factor for CVD. The development of vascular dysfunction, including endothelial dysfunction and stiffening of the large elastic arteries (i.e., the aorta and carotid arteries), contribute importantly to the age-related increase in CVD risk. Vascular aging is driven in large part by oxidative stress, which reduces bioavailability of nitric oxide and promotes alterations in the extracellular matrix. A key upstream driver of vascular oxidative stress is age-associated mitochondrial dysfunction. This review will focus on vascular mitochondria, mitochondrial dysregulation and mitochondrial reactive oxygen species (ROS) production and discuss current evidence for prevention and treatment of vascular aging via lifestyle and pharmacological strategies that improve mitochondrial health. We will also identify promising areas and important considerations (‘research gaps’) for future investigation. Keywords:arterial stiffness, endothelial function, mitophagy, oxidative stress, reactive oxygen species Subjects:Aging, Cardiovascular System & Vascular Biology, Translational Science Cardiovascular diseases (CVD) remain the largest contributor to morbidity and mortality in both developed and many developing nations [1,2]. Aging is by far the strongest risk factor for CVD, with >90% of all deaths occurring in adults 50 years of age and older [1,2]. Importantly, the changing demographics of aging characterized by a shift toward older populations [3] predicts a progressive, marked increase in prevalence of CVD in the absence of effective intervention [4]. A key mechanism by which aging increases CVD risk is the development of vascular dysfunction [5,6]. A number of adverse changes to the vasculature occur with aging, but two major clinically relevant expressions are endothelial dysfunction, as assessed by reduced arterial dilation in response to endothelium-derived nitric oxide (NO), and stiffening of the large elastic arteries (i.e., the aorta and carotid arteries) [5,6]. In combination, endothelial dysfunction and arterial stiffening contribute to a ‘vascular aging’ phenotype that drives much of the adverse effects of age on CVD. Vascular endothelial dysfunction The vascular endothelium is a single-cell layer lining the lumen of blood vessels. Endothelial cells play a critical role regulating vasomotor tone, metabolism, immune function, thrombosis and many other processes via synthesis and release of a variety of vasoactive molecules [7]. A major vasodilatory and largely vasoprotective molecule released by endothelial cells is NO, which is produced in response to mechanical (i.e., blood flow) and chemical (e.g., acetylcholine [ACh]) stimuli by the enzyme nitric oxide synthase (eNOS); eNOS catalyzes the generation of NO from L-arginine and oxygen, with NO subsequently diffusing to vascular smooth muscle cells where it induces vascular smooth muscle relaxation and vasodilation [7]. Endothelial dysfunction occurs with aging and is characterized by a decline in endothelium-dependent dilation (EDD), largely as a consequence of reductions in NO, although changes in concentrations of vasoactive factors such as prostaglandins, endothelin-1, norepinephrine and angiotensin II also contribute [7]. NO-mediated EDD can be determined in pre-clinical models by assessing changes in artery diameter in response to flow in vivo [8,9] or changes in diameter of isolated artery segments ex vivo in response to mechanical or pharmacological stimuli, such as ACh [10]. In humans, the gold-standard non-invasive assessment of NO-mediated EDD is brachial artery flow-mediated dilation (FMD), in which the change in brachial artery diameter in response to increases in blood flow is determined [10,11]. Brachial artery FMD primarily assesses macrovascular (conduit artery) function. Microvascular (resistance vessel) function can be determined by measuring changes in blood flow in response to intra-arterial infusions of ACh and is primarily assessed in the forearm [10,11]. These experimental approaches all demonstrate reduced endothelial function with aging in pre-clinical models and humans [12–17]. Endothelial dysfunction is the major antecedent of atherosclerosis [5,18] and both reduced brachial artery FMD and lower forearm blood flow responses to ACh are independent predictors of CV events and CVD in middle-aged and older adults free from clinical disease in large, community-based cohort studies [19–21]. Large elastic artery stiffening The aorta and carotid arteries expand and recoil as blood is ejected into the arterial system by the left ventricle during systole [22]. This action limits arterial pulsatile pressures by providing a dampening function and protects the downstream microvasculature from potentially damaging fluctuations in blood pressure and flow [23]. Moreover, the elastic recoil of the aorta aids in the propulsion of blood to the periphery and maintains perfusion of the heart during diastole [22]. With aging, aortic stiffening results in blood being ejected into a stiffer aorta, which augments central systolic blood pressure because the ejected pressure wave travels at a higher velocity in stiffer arteries and is reflected by points of impedance such that the returning pressure wave reaches the heart at mid-to-late systole [22,24]. In addition, the greater forward moving pressure wave amplitude (from systolic ejection, prior to the return of wave reflections) is a major contributor to the age-related increase in central systolic blood pressure after age 60, particularly in women, as a consequence of a plateau or decrease in reflected wave amplitude [25,26]. The augmented systolic blood pressure, in turn, contributes to isolated systolic hypertension and results in a loss of diastolic pressure augmentation, such that aortic pulse pressure is widened [22,24]. Aortic stiffening therefore increases left ventricular afterload during systole, promoting left ventricular hypertrophy and dysfunction, and compromises coronary perfusion during diastole because of the reduced augmentation of diastolic pressure [24,27]. The loss of pulsatility-dampening effects of the aorta and the carotid artery also allows for transmission of high pulsatile pressures to the delicate small vessels in the microcirculation, which is particularly harmful for high-flow, low-resistance organs such as the brain and kidney, and a potential causative factor in target organ damage [23]. Structural changes to arteries, functional influences (i.e., factors influencing vascular smooth muscle tone) and the stiffness of vascular smooth muscle cells contribute to large elastic artery stiffening with aging [28,29]. The primary structural changes mediating arterial stiffening occur in the extracellular matrix and include degradation/fragmentation of elastin (e.g., by matrix metalloproteinases), an increase in the deposition of collagen and formation of advanced glycation end products (AGEs), which cross-link collagen fibers, increasing their stiffness [5,30,31]. Increased vascular smooth muscle tone is a consequence of changes such as reductions in NO and increased sympathetic nervous system, endothelin-1 and renin–angiotensin aldosterone system activity [32–34]. These factors also influence the intrinsic stiffness of the vascular smooth muscle cells, which adds to the stiffness of the arterial wall [29]. The mechanical stiffness of the large elastic arteries can be determined ex vivo in pre-clinical models by directly measuring properties such as compliance by creating stress-strain curves [35,36]. In vivo, arterial stiffness can be assessed in pre-clinical settings and humans with pulse wave velocity (PWV), which is a measure of the (regional) speed of the pulse wave generated by the heart when blood is ejected into the arterial system [22]. Aortic PWV is the predominant measure in rodents and carotid-femoral PWV is the reference standard measure of aortic stiffness in humans [10,22]. Carotid-femoral PWV increases with aging and is a strong, independent predictor of CVD risk in older adults [37,38]. Moreover, consistent with aortic stiffness-associated end organ damage, growing evidence supports an association between elevated carotid-femoral PWV and other age-related clinical disorders such as cognitive decline, dementia, including Alzheimer’s disease, and decreases in renal function/chronic kidney disease [39–43]. The local distensibility of the carotid artery can also be determined in humans by measuring carotid artery compliance (the change in artery diameter for a given change in arterial pressure) and determining the carotid distensibility coefficient (i.e., changes in artery diameter normalized to diastolic lumen diameter) and/or carotid beta-stiffness index, which is largely independent of blood pressure [10,22]. Carotid artery compliance is associated with incident stroke, independent of aortic stiffness [44]. Mechanisms of vascular dysfunction with aging The primary molecular mechanisms of vascular aging are oxidative stress and chronic, low grade inflammation [45,46] (Figure 1). Excessive production of reactive oxygen species (ROS) in combination with unchanged or decreased abundance/activity of antioxidant enzymes (e.g., superoxide dismutase, SOD) results in the development of oxidative stress in arteries with aging [24,45]. Excess superoxide rapidly reacts with NO to form the secondary reactive species peroxynitrite (ONOO−), decreasing the bioavailability of NO [24,45], causing endothelial dysfunction. Peroxynitrite is also the primary molecule that reacts with and oxidizes tetrahydrobiopterin (BH4), an essential co-factor for NO production by eNOS [47]. Loss of BH4 leads to eNOS uncoupling, whereby eNOS produces more superoxide and less NO, exacerbating oxidative stress and decreasing bioavailable NO and endothelial cell function [47]. Excess ROS also can activate pro-inflammatory networks such as those regulated by the transcription factor nuclear factor kappa B (NF-kB), which up-regulates the production of pro-inflammatory cytokines that can impair vascular function and activate other ROS producing systems and enzymes, creating an adverse feed-forward (vicious) cycle of inflammation and oxidative stress [24,45]. Figure 1 Aging is associated with mitochondrial dysfunction-induced increases in reactive oxygen species (ROS) and oxidative stress and increases in pro-inflammatory cytokine signaling and chronic low-grade inflammation. Together, these processes induce vascular dysfunction, featuring: (lower left) large elastic artery stiffening mediated by degradation of elastin fibers (blue), increased deposition of collagen (brown), and greater cross-linking of structural proteins by advanced glycation end-products (dashed connecting lines); and (right) vascular endothelial dysfunction characterized by reduced nitric oxide (NO) bioavailability and endothelium-dependent dilation. These and other changes to arteries, in turn, increase the risk of developing cardiovascular diseases, chronic kidney disease, and Alzheimer’s disease and related dementias. VIEW LARGEDOWNLOAD SLIDE Mechanisms of age-associated vascular dysfunction and related clinical disorders Aging is associated with mitochondrial dysfunction-induced increases in reactive oxygen species (ROS) and oxidative stress and increases in pro-inflammatory cytokine signaling and chronic low-grade inflammation. Together, these processes induce vascular dysfunction, featuring: (lower left) large elastic artery stiffening mediated by degradation of elastin fibers (blue), increased deposition of collagen (brown), and greater cross-linking of structural proteins by advanced glycation end-products (dashed connecting lines); and (right) vascular endothelial dysfunction characterized by reduced nitric oxide (NO) bioavailability and endothelium-dependent dilation. These and other changes to arteries, in turn, increase the risk of developing cardiovascular diseases, chronic kidney disease, and Alzheimer’s disease and related dementias. This overall state of oxidative stress and inflammation also contributes to arterial stiffening with aging by altering the structural properties of the arterial wall. Production of collagen by fibroblasts is stimulated by superoxide-related oxidative stress [30,48,49]. Matrix metalloproteinases are up-regulated and elastin content is lower in aorta of SOD-deficient mice, consistent with the concept that elastin degradation is induced by oxidative stress [50]. Vascular oxidative stress also promotes transforming growth factor β signaling and this, in turn, stimulates inflammation, which further reinforces arterial stiffness via activation of the pro-oxidant enzyme, NADPH oxidase [48]. AGEs interact with the receptor for AGEs to activate NFkB-regulated pro-inflammatory pathways and oxidative stress, which ultimately perpetuates arterial stiffening and further increases production of AGEs [51]. Mitochondrial dysfunction is emerging as a key source of vascular oxidative stress and contributor to age-related vascular dysfunction. The remaining sections of this article will focus on mitochondrial dysfunction as a driver of vascular aging and review current evidence for prevention/treatment of age-associated vascular dysfunction via lifestyle and pharmacological strategies that improve mitochondrial health. We will also discuss current ‘research gaps’ and future directions for the field. Vascular mitochondria, mitochondrial dysregulation and ROS Mitochondria are cytoplasmic organelles that are present in the majority of cell types in the human body, including vascular endothelial and smooth muscle cells. Mitochondria are often referred to as the ‘powerhouse’ of the cell for their role in ATP production by oxidative phosphorylation, which occurs via a series of electron transfers through the respiratory chain in the mitochondrial inner membrane that is coupled to ATP synthesis by the FoF1-ATP synthase by the protonmotive force across the inner membrane. However, mitochondria are also vital for a number of additional cellular processes, including regulation of metabolism, calcium homeostasis, immune function, cell growth and stem cell function, and cell death pathways. Although mitochondrial density in vascular tissues is considerably lower than other tissues such as skeletal muscle, liver and heart [52,53], increasing evidence indicates that these organelles are critical for maintenance of cellular and tissue homeostasis in the vasculature. This topic has been reviewed in detail elsewhere [54–61], but below we briefly summarize some of the key roles of mitochondria in the vasculature. A first important distinction is to consider the vascular cell type in question, as the density and subcellular distribution of mitochondria vary between endothelial and vascular smooth muscle cells, and indeed even among the same cell types in different vascular beds [54,60]. In general, unlike in highly metabolically active tissues with greater ATP demand, the principal role of mitochondria in the vasculature appears to be cellular signaling rather than energy provision [54]. Cellular energy demand is quite low in endothelial cells, and ATP demand is met primarily via glycolysis. However, endothelial mitochondria are critical in the regulation of calcium homeostasis, apoptosis/necrosis, cellular response to stress, and immune and inflammatory pathways. An essential feature of these roles is the regulated production of signaling molecules including redox-active molecules (reactive oxygen, nitrogen and other species; mtROS), mitochondrial DNA, mitochondria-derived peptides and damage-associated molecular pattern molecules (DAMPs), which exert effects intra- and extra-cellularly [62]. Importantly, there is cross-talk between mitochondrial and nuclear signaling pathways, whereby mitochondria-derived signaling is both influenced by and can influence nuclear events including gene expression [63]. Similarly, in vascular smooth muscle cells, mitochondria have an important role in cellular signaling. Mitochondria are involved in signaling pathways for regulation of vascular smooth muscle cell growth and proliferation (e.g., TGF-β activity) [64], as well as maintenance of the dynamic balance among synthesis and breakdown of extracellular structural proteins, including collagen and elastin (e.g., matrix metalloproteinase enzyme activities) [65]. There is also emerging evidence demonstrating interplay between mtROS signaling and inflammatory pathways known to be important for regulating vascular smooth muscle cell function, including those involving NFkB and the NLRP3-inflammasome [66–69], further highlighting the crucial role of mtROS in vascular homeostasis. Mitochondrial ROS The signaling functions of vascular mitochondria are thought to be mediated in large part by the production of ROS at low, physiological levels. However, the dysregulation of this mtROS production also has the potential to lead to pathophysiological sequelae that disrupt other mitochondrial functions, cellular homeostasis, and ultimately vascular function. The production of ROS by mitochondria can occur at several sites (Figure 2), including but not limited to the electron transport proteins, and this topic has been reviewed in detail elsewhere [54,60,70]. The most important sites for ROS production within mitochondria appear to be complexes I and III. These ROS are thought to be critical transducers of signaling mediated by mitochondria, leading to post-transcriptional modification of proteins and interactions with immune and inflammatory cellular pathways, although the mechanistic details are still uncertain. In the vasculature, the proximal mtROS species is superoxide, which is generated primarily at the electron transport chain in the mitochondrial inner membrane via interaction between oxygen and unpaired electrons, influenced by the proton motive force and the redox state of the coenzyme Q pool and integrity of intrinsic electron transport chain proteins [54,58,60,70]. Superoxide is released into the matrix (complex I) or into both the matrix and intermembrane space (complex III); it can also undergo dismutation to hydrogen peroxide by the antioxidant enzyme manganese superoxide dismutase (MnSOD) [59,60,62,70]. Hydrogen peroxide is also generated de novo on the surface of the mitochondrial outer membrane or in the intermembrane space mitochondria by p66SHC, a growth factor adapter protein referred to as a sensor/marker and ‘master regulator’ of mitochondrial redox signaling whose activity is indicative of the rate of mtROS production [71]. In addition, NADPH oxidase 4 (NOX4) is viewed as a primarily mitochondrial isoform of the NOX monoamine oxidase family of enzymes that contributes to mitochondrial hydrogen peroxide generation [72], although more research is needed to confirm the mitochondrial specificity of NOX4. Figure 2 Aging is associated with dysregulated mitochondrial quality control featuring reduced mitochondrial biogenesis (upper left) and reduced mitophagy (upper right), increased mitochondrial fission (upper middle right), reduced mitochondrial fusion (lower middle right), reduced mitochondrial stress resistance (lower right), increased mitochondrial DNA damage (middle left of mitochondria image) and increased bioactivity of mitochondrial reactive oxygen species (e.g., superoxide and other reactive oxygen species [ROS], middle of mitochondria image) relative to antioxidant defenses (e.g., manganese superoxide dismutase [SOD], lower right of mitochondria). VIEW LARGEDOWNLOAD SLIDE Mechanisms of age-associated mitochondrial dysfunction Aging is associated with dysregulated mitochondrial quality control featuring reduced mitochondrial biogenesis (upper left) and reduced mitophagy (upper right), increased mitochondrial fission (upper middle right), reduced mitochondrial fusion (lower middle right), reduced mitochondrial stress resistance (lower right), increased mitochondrial DNA damage (middle left of mitochondria image) and increased bioactivity of mitochondrial reactive oxygen species (e.g., superoxide and other reactive oxygen species [ROS], middle of mitochondria image) relative to antioxidant defenses (e.g., manganese superoxide dismuta