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PNS and CNS myelin contain large amounts of galactocerebroside and sulfatide with 2-hydroxylated fatty acids. The underlying hydroxylation reaction is catalyzed by fatty acid 2-hydroxylase (FA2H). Deficiency in this enzyme causes a complicated hereditary spastic paraplegia, SPG35, which is associated with leukodystrophy. Mass spectrometry-based proteomics of purified myelin isolated from sciatic nerves of Fa2h-deficient (Fa2h−/−) mice revealed an increase in the concentration of the three proteins Cadm4, Mpp6 (Pals2), and protein band 4.1G (Epb41l2) in 17-month-old, but not in young (4 to 6-month-old), Fa2h−/− mice. These proteins are known to form a complex, together with the protein Lin7, in Schmidt-Lanterman incisures (SLIs). Accordingly, the number of SLIs was significantly increased in 17-month-old but not 4-month-old Fa2h−/− mice compared to age-matched wild-type mice. On the other hand, the relative increase in the SLI frequency was less pronounced than expected from Cadm4, Lin7, Mpp6 (Pals2), and band 4.1G (Epb41l2) protein levels. This suggests that the latter not only reflect the higher SLI frequency but that the concentration of the Cadm4 containing complex itself is increased in the SLIs or compact myelin of Fa2h−/− mice and may potentially play a role in the pathogenesis of the disease. The proteome data are available via ProteomeXchange with identifier PXD030244.

SummaryOligodendrocytes are generated via a two-step mechanism from pluripotent neural stem cells (NSCs): after differentiation of NSCs to oligodendrocyte precursor/NG2 cells (OPCs), they further develop into mature oligodendrocytes. The first step of this differentiation process is only incompletely understood. In this study, we utilized the neurosphere assay to investigate NSC to OPC differentiation in a time course-dependent manner by mass spectrometry-based (phospho-) proteomics. We identify double cortin like kinase 1 (Dclk1) as one of the most prominently regulated proteins in both datasets, and show that it undergoes a gradual transition between its short/long isoform during NSC to OPC differentiation. This is regulated by phosphorylation of its SP-rich region, resulting in inhibition of proteolytic Dclk1 long cleavage, and therefore Dclk1 short generation. Through interactome analyses of different Dclk1 isoforms by proximity biotinylation, we characterize their individual putative interaction partners and substrates.

Impaired proinsulin-to-insulin processing in pancreatic β-cells is a key defective step in both type 1 diabetes and type 2 diabetes (T2D) (refs. 1,2), but the mechanisms involved remain to be defined. Altered metabolism of sphingolipids (SLs) has been linked to development of obesity, type 1 diabetes and T2D (refs. 3–8); nonetheless, the role of specific SL species in β-cell function and demise is unclear. Here we define the lipid signature of T2D-associated β-cell failure, including an imbalance of specific very-long-chain SLs and long-chain SLs. β-cell-specific ablation of CerS2, the enzyme necessary for generation of very-long-chain SLs, selectively reduces insulin content, impairs insulin secretion and disturbs systemic glucose tolerance in multiple complementary models. In contrast, ablation of long-chain-SL-synthesizing enzymes has no effect on insulin content. By quantitatively defining the SL–protein interactome, we reveal that CerS2 ablation affects SL binding to several endoplasmic reticulum–Golgi transport proteins, including Tmed2, which we define as an endogenous regulator of the essential proinsulin processing enzyme Pcsk1. Our study uncovers roles for specific SL subtypes and SL-binding proteins in β-cell function and T2D-associated β-cell failure.

Spastic paraplegia 35 (SPG35) (OMIM: 612319) or fatty acid hydroxylase-associated neurodegeneration (FAHN) is caused by deficiency of fatty acid 2-hydroxylase (FA2H). This enzyme synthesizes sphingolipids containing 2-hydroxylated fatty acids, which are particularly abundant in myelin. Fa2h-deficient (Fa2h−/−) mice develop symptoms reminiscent of the human disease and therefore serve as animal model of SPG35. In order to understand further the pathogenesis of SPG35, we compared the proteome of purified CNS myelin isolated from wild type and Fa2h−/− mice at different time points of disease progression using tandem mass tag labeling. Data analysis with a focus on myelin membrane proteins revealed a significant increase of the oligodendrocytic myelin paranodal and inner loop protein (Opalin) in Fa2h−/− mice, whereas the concentration of other major myelin proteins was not significantly changed. Western blot analysis revealed an almost 6-fold increase of Opalin in myelin of Fa2h−/− mice aged 21–23 months. A concurrent unaltered Opalin gene expression suggested a decreased turnover of the Opalin protein in Fa2h−/− mice. Supporting this hypothesis, Opalin protein half-life was reduced significantly when expressed in CHO cells synthesizing 2-hydroxylated sulfatide, compared to cells synthesizing only non-hydroxylated sulfatide. Degradation of Opalin was inhibited by inhibitors of lysosomal degradation but unaffected by proteasome inhibitors. Taken together, these results reveal a new function of 2-hydroxylated sphingolipids namely affecting the turnover of a myelin membrane protein. This may play a role in the pathogenesis of SPG35.

The fatty acid 2-hydroxylase (FA2H) is essential for synthesis of 2-hydroxylated fatty acids in myelinating and other cells, and deficiency of this enzyme causes a complicated form of hereditary spastic paraplegia also known as fatty acid hydroxylase-associated neurodegeneration. Despite its important role in sphingolipid metabolism, regulation of FA2H and its interaction with other proteins involved in the same or other metabolic pathways is poorly understood. To identify potential interaction partners of the enzyme, quantitative mass spectrometry using stable isotope labeling of cells was combined with formaldehyde cross-linking and proximity biotinylation, respectively. Besides other enzymes involved in sphingolipid synthesis and intermembrane transfer of ceramide, and putative redox partners of FA2H, progesterone receptor membrane component-1 (PGRMC1) and PGRMC2 were identified as putative interaction partners. These two related heme-binding proteins are known to regulate several cytochrome P450 enzymes. Bimolecular fluorescence complementation experiments confirmed the interaction of FA2H with PGRMC1. Moreover, the PGRMC1 inhibitor AG-205 significantly reduced synthesis of hydroxylated ceramide and glucosylceramide in FA2H-expressing cells. This suggests that PGRMC1 may regulate FA2H activity, possibly through its heme chaperone activity.

The mass spectrometry-compatible surfactant RapiGest promotes the enzymatic digestion of proteins by facilitating their unfolding while retaining enzymatic activity. RapiGest consists of a hydrophilic head and a hydrophobic tail, which can be separated by acid hydrolysis. This allows for removal of RapiGest prior to mass spectrometric analysis via precipitation and solid phase extraction. During in-solution digestion experiments with RapiGest, we noticed a high variability in the formation of precipitates after acid hydrolysis, implying that RapiGest precipitation is sample-dependent. We show that RapiGest hydrolyses efficiently under acidic conditions and that differences in precipitation are solely due to protein/peptide concentration. Furthermore, we demonstrate that RapiGest precipitation can be triggered by the addition of intact proteins, providing a strategy for its efficient removal from highly diluted samples. Data are available via ProteomeXchange with identifier PXD025982.

CDC14A codes for a conserved proline-directed phosphatase, and mutations in the gene are associated with autosomal-recessive severe to profound deafness, due to defective kinocilia. A role of CDC14A in cilia formation has also been described in other organisms. However, how human CDC14A impacts on cilia formation remains unclear. Here, we show that human RPE1 hCDC14A(PD) cells, encoding a phosphatase dead version of hCDC14A, have longer cilia than wild-type cells, while hCDC14A overexpression reduces cilia formation. Phospho-proteome analysis of ciliated RPE1 cells identified actin-associated and microtubule binding proteins regulating cilia length as hCDC14A substrates, including the actin-binding protein drebrin. Indeed, we find that hCDC14A counteracts the CDK5-dependent phosphorylation of drebrin at S142 during ciliogenesis. Further, we show that drebrin and hCDC14A regulate the recruitment of the actin organizer Arp2 to centrosomes. In addition, during ciliogenesis hCDC14A also regulates endocytosis and targeting of myosin Va vesicles to the basal body in a drebrin-independent manner, indicating that it impacts primary cilia formation in a multilayered manner.

CDC14 is an essential dual-specificity phosphatase that counteracts CDK1 activity during anaphase to promote mitotic exit in Saccharomyces cerevisiae. Surprisingly, human CDC14A is not essential for cell cycle progression. Instead, it regulates cell migration and cell adhesion. Little is known about the substrates of hCDC14A and the counteracting kinases. Here, we combine phospho-proteome profiling and proximity-dependent biotin identification to identify hCDC14A substrates. Among these targets were actin regulators, including the tumor suppressor eplin. hCDC14A counteracts EGF-induced rearrangements of actin cytoskeleton by dephosphorylating eplin at two known extracellular signal-regulated kinase sites, serine 362 and 604. hCDC14APD and eplin knockout cell lines exhibited down-regulation of E-cadherin and a reduction in α/β-catenin at cell–cell adhesions. Reduction in the levels of hCDC14A and eplin mRNA is frequently associated with colorectal carcinoma and is correlated with poor prognosis. We therefore propose that eplin dephosphorylation by hCDC14A reduces actin dynamics to restrict tumor malignancy.

For a complete normed Abelian group G , we show that a flat G chain (respectively, normal flat G chain) of codimension 0 in a finite dimensional Banach space corresponds to a G valued functions that is H m summable (respectively, of bounded variation). We also prove a constancy theorem for an m dimensional flat G chain T in a metric space assuming supp T is contained in the closure of a connected m dimensional Lipschitz submanifold M with M ∩ supp ∂ T = ∅ and H m ( M ) = H m ( Clos M ) ∞ .

We prove that the zero set of a minimizer of the modified Ericksen energy for nematic liquid crystals with variable degree of orientation consists locally of isolated points and a finite union of H\"older continuous curves with only finitely many crossings., Comment: 32 pages; minor revisions; typographical errors corrected; new references added

Chains, in a complete metric space, which have coefficients in a normed abelian group $G$ are studied. An $m$ dimensional rectifiable chain is the Lipschitz push-forward of a region in ${\Bbb R}^m$ equipped with a measurable $G$-valued density. Flat chains are obtained by completion using a certain flat norm on polyhedral or Lipschitz chain approximations. Numerous basic results of geometric measure theory for these chains are derived including the rectifiability of finite mass flat chains provided that $G$ contains no nonconstant Lipschitz curves. The work here generalizes, and uses many ideas from, the 1999 paper of B. White on at $G$ chains in ${\Bbb R}^n$ and the 2000 paper of L. Ambrosio and B. Kirchheim on currents in a metric space. The use of flat and rectifiable chains in geometric variational problems or in defining geometric homology theories may reveal geometric properties of spaces.

The existence of Dirichlet minimizing multiple-valued functions for given boundary data has been known since pioneering work of F. Almgren. Here we prove a multiple-valued analogue of the classical Plateau problem of the existence of area-minimizing mappings of the disk. Specifically, we find, for $K \in \mathbb N,$ $k_1,...,k_K\in \mathbb N$ with sum $Q$ and any collection of $K$ disjoint Lipschitz Neighborhood Retract Jordan curves, optimal multiple-valued boundary data with these multiplicities which extends to a Dirichlet minimizing $Q$-valued function with minimal Dirichlet energy among all possible monotone parameterizations of the boundary curves. Under a condition analogous to the Douglas condition for minimizers from planar domains, conformality of the minimizer follows from topological methods and some complex analysis. Finally, we analyze two particular cases: in contrast to single-valued Douglas solutions, we first give a class of examples for which our multiple-valued Plateau solution has branch points. Second, we give examples of a degenerate behavior, illustrating the weakness of the multiple-valued maximum principle and provide motivation for our analogous Douglas condition., Comment: 18 pages, 2 figures, changed the proof of Theorem 1.2

This article surveys research on the existence, structure, behavior, and asymptotics of singularities of harmonic maps.

Consider the space \(W^{2,2}(\Omega ;N)\) of second order Sobolev mappings \(\ v\ \) from a smooth domain \(\Omega \subset \mathbb {R}^m\) to a compact Riemannian manifold N whose Hessian energy \(\int _\Omega |\nabla ^2 v|^2\, dx\) is finite. Here we are interested in relations between the topology of N and the \(W^{2,2}\) strong or weak approximability of a \(W^{2,2}\) map by a sequence of smooth maps from \(\Omega \) to N. We treat in detail \(W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)\) where we establish the sequential weak \(W^{2,2}\) density of \(W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)\,\cap \,{\mathcal C}^\infty \). The strong \(W^{2,2}\) approximability of higher order Sobolev maps has been studied in the recent paper of Bousquet et al. (J. Eur. Math. Soc. (JEMS) 17(4), 763–817, 2015). For an individual map \(v\in W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)\), we define a number L(v) which is approximately the total length required to connect the isolated singularities of a strong approximation u of v either to each other or to \(\partial Bx^5\). Then \(L(v)=0\) if and only if v admits \(W^{2,2}\) strongly approximable by smooth maps. Our critical result, obtained by constructing specific curves connecting the singularities of u, is the bound \(\ L(u)\le c+c\int _{\mathbb {B}^5}|\nabla ^2 u|^2\, dx\ \). This allows us to construct, for the given Sobolev map \(v\in W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)\), the desired \(W^{2,2}\) weakly approximating sequence of smooth maps. To find suitable connecting curves for u, one uses the twisting of a u pull-back normal framing on a suitable level surface of u.

(1994). An evolution problem for linear growth functionals. Communications in Partial Differential Equations: Vol. 19, No. 11-12, pp. 1879-1907.

Given a surface of revolution with boundary, we study the extrinsic energy of smooth tangent unit-length vector fields. Fixing continuous tangent unit-length vector fields on the boundary of the surface of revolution, we ask if there is a unique smooth tangent unit-length vector field continuously achieving the boundary data and minimizing energy amongst all smooth tangent unit-length vector fields also continuously achieving the boundary data.

Suppose Ω is a smooth domain in Rm,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu eH tou ¦ϖΩ is Fredholm of index 0.

Here we obtain various results on the class of axially symmetric harmonic maps from B3 to S2. We find some new classes of non-minimizing harmonic maps exhibiting unusual singular behavior. Optimal partial regularity estimates are obtained for mappings which minimize, among axially symmetric maps, various relaxed energies which have been studied in [4]and [11].

Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W 1,p (R p+1, N) be the Sobolev space of measurable maps from R p+1 into N whose gradients are in L p . The restriction of u to almost every p-dimensional sphere S in R p+1 is in W 1,p (S, N) and defines an homotopy class in π p (N) (White 1988). Evaluating a fixed element z of Hom(π p (N), R) on this homotopy class thus gives a real number Φ z,u (S). The main result of the paper is that any W 1,p -weakly convergent limit u of a sequence of smooth maps in C ∞(R p+1, N), Φ z,u has a rectifiable Poincare dual $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ . Here Γ is a a countable union of C 1 curves in R p+1 with Hausdorff $ {\user1{\mathcal{H}}}^{1} $ -measurable orientation $ {\overrightarrow{\Gamma }} :\Gamma \to S^{p} $ and density function θ: Γ→R. The intersection number between $ {\left( {\Gamma ,{\overrightarrow{\Gamma }} ,\theta } \right)} $ and S evaluates Φ z,u (S), for almost every p-sphere S. Moreover, we exhibit a non-negative integer n z , depending only on homotopy operation z, such that $ {\int_\Gamma {{\left| \theta \right|}^{{p \mathord{\left/ {\vphantom {p {{\left( {p + n_{z} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {p + n_{z} } \right)}}}} d{\user1{\mathcal{H}}}^{1} < \infty } } $ even though the mass $ {\int_\Gamma {{\left| \theta \right|}d{\user1{\mathcal{H}}}^{1} } } $ may be infinite. We also provide cases of N, p and z for which this rational power p/(p + n z ) is optimal. The construction of this Poincare dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Riviere (2003). We also describe how to generalize these results to R m for any m ⩾ p + 1, in which case the bubbling is described by an (m–p)-rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p-sphere.

We complete the details of a theory outlined by Kontsevich and Soibelman that associates to a semi-algebraic set a certain graded commutative differential algebra of "semi-algebraic differential forms" in a functorial way. This algebra encodes the real homotopy type of the semi-algebraic set in the spirit of the DeRham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich's proof of the formality of the little cubes operad., Comment: 58 pages. Cosmetic changes with respect to previous version. Submitted

The singular set of an energy minimizing map from a four dimensional domain toS2 consists locally of a finite set and a finite union of Holder continuous curves.

Calculus of variations: What does "variations" mean? by F. Jones How many equilibria are there? An introduction to Morse theory by R. Forman Aye, there's the rub. An inquiry into why a plucked string comes to rest by S. J. Cox Proof of the double bubble conjecture by F. Morgan Minimal surfaces, flat cone spheres and moduli spaces of staircases by M. Wolf Hold that light! Modeling of traffic flow by differential equations by B. L. Keyfitz.

In this note we describe the notion of a rectifiable scan and consider some applications [DH1], [DH2] to Plateau-type minimization problems. “Scans” were first introduced in the work [HR1] of Tristan Riviere and the second author to adequately describe certain bubbling phenomena. There, the behaviour of certain W 1,3 weakly convergent sequences of smooth maps from four-dimensional domains into S 2 led to the consideration of a necessarily infinite mass generalization of a rectifiable current. The definition of a scan is motivated by the fact that a rectifiable current can be expressed in terms of its lower-dimensional slices by oriented affine subspaces. By integral geometry, the slicing function for the rectifiable current is a mass integrable function of the subspaces. With a scan one considers more general such functions that are not necessarily mass integrable.

This is a survey of recent developments in a number of areas of nonlinear partial differential equations in differential geometry. The authors of the articles are not only excellent expositors, but are alos leaders in this field of research. All of the articles provide in-depth treatment of the topics and require few prerequisites and less background than current research articles.

Here we discuss recent results of [HLP] on the class of axially symmetric harmonic maps from B3 to S2. We find some new classes of non-minimizing harmonic maps exhibiting unusual singular behavior. Optimal partial regularity estimates are obtained for mappings which minimize, among axially symmetric maps, various relaxed energies [BBC].

A liquid crystal is generally understood to be a mesomorphic state of matter which flows like a liquid and which exhibits some anisotropic behavior. See [E], [EK], [C], [DG]. The liquid crystal phase usually lies between a solid phase and an isotropic liquid phase with phase transition being induced by temperature change. A static model typically involves a kinematic variable n(x), called the director, which is a unit vector defined for x in a spatial region Ω.

Here we make several observations on a variety of special classes of harmonic maps from domains in IR3 to S 2. Such maps are relevant for the study of liquid crystals; see e.g. [HK1], [HKL1], [HKLu]. Part of our work is motivated by questions about the nonuniqueness and the number of harmonic maps having fixed boundary data. Here we show that the number may actually be infinite. For example, by Corollary 3.2 below there exists a one parameter family of distinct energy-minimizers each having the same Dirichlet boundary data.