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Amyotrophic lateral sclerosis (ALS) is a complex neurodegenerative disorder, in which the clinical manifestations may be influenced by genetic and unknown environmental factors. Here we show that ALS-prone Sod1 transgenic ( Sod1 -Tg) mice have a pre-symptomatic, vivarium-dependent dysbiosis and altered metabolite configuration, coupled with an exacerbated disease under germ-free conditions or after treatment with broad-spectrum antibiotics. We correlate eleven distinct commensal bacteria at our vivarium with the severity of ALS in mice, and by their individual supplementation into antibiotic-treated Sod1 -Tg mice we demonstrate that Akkermansia muciniphila (AM) ameliorates whereas Ruminococcus torques and Parabacteroides distasonis exacerbate the symptoms of ALS. Furthermore, Sod1 -Tg mice that are administered AM are found to accumulate AM-associated nicotinamide in the central nervous system, and systemic supplementation of nicotinamide improves motor symptoms and gene expression patterns in the spinal cord of Sod1 -Tg mice. In humans, we identify distinct microbiome and metabolite configurations—including reduced levels of nicotinamide systemically and in the cerebrospinal fluid—in a small preliminary study that compares patients with ALS with household controls. We suggest that environmentally driven microbiome–brain interactions may modulate ALS in mice, and we call for similar investigations in the human form of the disease. A study of the functional microbiome in a mouse model of ALS shows that several gut bacteria may modulate the severity of the disease.

Longitudinal ’omics analytical methods are extensively used in the evolving field of precision medicine, by enabling ‘big data’ recording and high-resolution interpretation of complex datasets, driven by individual variations in response to perturbations such as disease pathogenesis, medical treatment or changes in lifestyle. However, inherent technical limitations in biomedical studies often result in the generation of feature-rich and sample-limited datasets. Analyzing such data using conventional modalities often proves to be challenging since the repeated, high-dimensional measurements overload the outlook with inconsequential variations that must be filtered from the data in order to find the true, biologically relevant signal. Tensor methods for the analysis and meaningful representation of multiway data may prove useful to the biological research community by their advertised ability to tackle this challenge. In this study, we present tcam —a new unsupervised tensor factorization method for the analysis of multiway data. Building on top of cutting-edge developments in the field of tensor-tensor algebra, we characterize the unique mathematical properties of our method, namely, 1) preservation of geometric and statistical traits of the data, which enable uncovering information beyond the inter-individual variation that often takes over the focus, especially in human studies. 2) Natural and straightforward out-of-sample extension, making tcam amenable for integration in machine learning workflows. A series of re-analyses of real-world, human experimental datasets showcase these theoretical properties, while providing empirical confirmation of tcam ’s utility in the analysis of longitudinal ’omics data.

Cigarette smoking constitutes a leading global cause of morbidity and preventable death 1 , and most active smokers report a desire or recent attempt to quit 2 . Smoking-cessation-induced weight gain (SCWG; 4.5 kg reported to be gained on average per 6–12 months, >10 kg year –1 in 13% of those who stopped smoking 3 ) constitutes a major obstacle to smoking abstinence 4 , even under stable 5 , 6 or restricted 7 caloric intake. Here we use a mouse model to demonstrate that smoking and cessation induce a dysbiotic state that is driven by an intestinal influx of cigarette-smoke-related metabolites. Microbiome depletion induced by treatment with antibiotics prevents SCWG. Conversely, fecal microbiome transplantation from mice previously exposed to cigarette smoke into germ-free mice naive to smoke exposure induces excessive weight gain across diets and mouse strains. Metabolically, microbiome-induced SCWG involves a concerted host and microbiome shunting of dietary choline to dimethylglycine driving increased gut energy harvest, coupled with the depletion of a cross-regulated weight-lowering metabolite, N -acetylglycine, and possibly by the effects of other differentially abundant cigarette-smoke-related metabolites. Dimethylglycine and N -acetylglycine may also modulate weight and associated adipose-tissue immunity under non-smoking conditions. Preliminary observations in a small cross-sectional human cohort support these findings, which calls for larger human trials to establish the relevance of this mechanism in active smokers. Collectively, we uncover a microbiome-dependent orchestration of SCWG that may be exploitable to improve smoking-cessation success and to correct metabolic perturbations even in non-smoking settings. A study of mice exposed to cigarette smoke suggests that smoking-cessation-induced weight gain is associated with a dysbiotic state that is driven by smoking-related metabolites.

We report the results of a first exploratory study testing the use of vaginal microbiome transplantation (VMT) from healthy donors as a therapeutic alternative for patients suffering from symptomatic, intractable and recurrent bacterial vaginosis (ClinicalTrials.gov NCT02236429). In our case series, five patients were treated, and in four of them VMT was associated with full long-term remission until the end of follow-up at 5–21 months after VMT, defined as marked improvement of symptoms, Amsel criteria, microscopic vaginal fluid appearance and reconstitution of a Lactobacillus-dominated vaginal microbiome. One patient presented with incomplete remission in clinical and laboratory features. No adverse effects were observed in any of the five women. Notably, remission in three patients necessitated repeated VMT, including a donor change in one patient, to elicit a long-standing clinical response. The therapeutic efficacy of VMT in women with intractable and recurrent bacterial vaginosis should be further determined in randomized, placebo-controlled clinical trials.

Increasing evidence suggests that the gut microbiome may modulate the efficacy of cancer immunotherapy. In a B cell lymphoma patient cohort from five centers in Germany and the United States (Germany, n  = 66; United States, n  = 106; total, n  = 172), we demonstrate that wide-spectrum antibiotics treatment (‘high-risk antibiotics’) prior to CD19-targeted chimeric antigen receptor (CAR)-T cell therapy is associated with adverse outcomes, but this effect is likely to be confounded by an increased pretreatment tumor burden and systemic inflammation in patients pretreated with high-risk antibiotics. To resolve this confounding effect and gain insights into antibiotics-masked microbiome signals impacting CAR-T efficacy, we focused on the high-risk antibiotics non-exposed patient population. Indeed, in these patients, significant correlations were noted between pre-CAR-T infusion Bifidobacterium longum and microbiome-encoded peptidoglycan biosynthesis, and CAR-T treatment-associated 6-month survival or lymphoma progression. Furthermore, predictive pre-CAR-T treatment microbiome-based machine learning algorithms trained on the high-risk antibiotics non-exposed German cohort and validated by the respective US cohort robustly segregated long-term responders from non-responders. Bacteroides , Ruminococcus , Eubacterium and Akkermansia were most important in determining CAR-T responsiveness, with Akkermansia also being associated with pre-infusion peripheral T cell levels in these patients. Collectively, we identify conserved microbiome features across clinical and geographical variations, which may enable cross-cohort microbiome-based predictions of outcomes in CAR-T cell immunotherapy. Conserved microbiome features across clinical and geographical variations may enable microbiome-based predictions of outcomes in CD19-targeted CAR-T cell immunotherapy

A valuable feature of the tubal tensor framework is that many familiar constructions from matrix algebra carry over to tensors, including SVD and notions of rank. Most importantly, it has been shown that for a specific family of tubal products, an Eckart-Young type theorem holds, i.e., the best low-rank approximation of a tensor under the Frobenius norm is obtained by truncating its tubal SVD. In this paper, we provide a complete characterization of the family of tubal products that yield an Eckart-Young type result. We demonstrate the practical implications of our theoretical findings by conducting experiments with video data and data-driven dynamical systems.

Developed in a series of seminal papers in the early 2010s, the tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features such as a tensor Singular Value Decomposition and Eckart-Young-like optimality results. It has proven to be a powerful tool for analyzing inherently multilinear data arising in hyperspectral imaging, medical imaging, neural dynamics, scientific simulations, and more. At the heart of tubal tensor algebra lies a special tensor-tensor product: originally the t-product, later generalized into a full family of products via the \\(\\star_M\\)-product. Though initially defined through the multiplication of a block-circulant unfolding of one tensor by a matricization of another, it was soon observed that the t-product can be interpreted as standard matrix multiplication where the scalars are tubes-i.e., real vectors twisted ``inward.'' Yet, a fundamental question remains: why is this the ``right'' way to define a tensor-tensor product in the tubal setting? In this paper, we show that the t-product and its \\(\\star_M\\) generalization arise naturally when viewing third-order tensors as matrices of tubes, together with a small set of desired algebraic properties. Furthermore, we prove that the \\(\\star_M\\)-product is, in fact, the only way to define a tubal product satisfying these properties. Thus, while partly expository in nature - aimed at presenting the foundations of tubal tensor algebra in a cohesive and accessible way - this paper also addresses theoretical gaps in the tubal tensor framework, proves new results, and provides justification for the tubal tensor framework central constructions, thereby shedding new light on it.

The tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features like Singular Value Decomposition and Eckart-Young-like optimality results. Underlying the tubal tensor framework is a view of a tensor as a matrix of finite sized tubes. In this work, we lay the mathematical and computational foundations for working with tensors with infinite size tubes: matrices whose elements are elements from a separable Hilbert space. A key challenge is that existence of important desired matrix-mimetic features of tubal tensors rely on the existence of a unit element in the ring of tubes. Such unit element cannot exist for tubes which are elements of an infinite-dimensional Hilbert space. We sidestep this issue by embedding the tubal space in a commutative unital C*-algebra of bounded operators. The resulting quasitubal algebra recovers the structural properties needed for decomposition and low-rank approximation. In addition to laying the theoretical groundwork for working with tubal tensors with infinite dimensional tubes, we discuss computational aspects of our construction, and provide a numerical illustration where we compute a finite dimensional approximation to a infinitely-sized synthetic tensor using our theory. We believe our theory opens new exciting avenues for applying matrix mimetic tensor framework in the context of inherently infinite dimensional problems.

The Trust Region Subproblem is a fundamental optimization problem that takes a pivotal role in Trust Region Methods. However, the problem, and variants of it, also arise in quite a few other applications. In this article, we present a family of iterative Riemannian optimization algorithms for a variant of the Trust Region Subproblem that replaces the inequality constraint with an equality constraint, and converge to a global optimum. Our approach uses either a trivial or a non-trivial Riemannian geometry of the search-space, and requires only minimal spectral information about the quadratic component of the objective function. We further show how the theory of Riemannian optimization promotes a deeper understanding of the Trust Region Subproblem and its difficulties, e.g., a deep connection between the Trust Region Subproblem and the problem of finding affine eigenvectors, and a new examination of the so-called hard case in light of the condition number of the Riemannian Hessian operator at a global optimum. Finally, we propose to incorporate preconditioning via a careful selection of a variable Riemannian metric, and establish bounds on the asymptotic convergence rate in terms of how well the preconditioner approximates the input matrix.