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SUMMARY Mitochondrial Ca2+ overload is a critical, preceding event in neuronal damage encountered during neurodegenerative and ischemic insults. We found that loss of PTEN-induced putative kinase 1 (PINK1) function, implicated in Parkinson disease, inhibits the mitochondrial Na+/Ca2+ exchanger (NCLX), leading to impaired mitochondrial Ca2+ extrusion. NCLX activity was, however, fully rescued by activation of the protein kinase A (PKA) pathway. We further show that PKA rescues NCLX activity by phosphorylating serine 258, a putative regulatory NCLX site. Remarkably, a constitutively active phosphomimetic mutant of NCLX (NCLXS258D) prevents mitochondrial Ca2+ overload and mitochondrial depolarization in PINK1 knockout neurons, thereby enhancing neuronal survival. Our results identify an mitochondrial Ca2+ transport regulatory pathway that protects against mitochondrial Ca2+ overload. Because mitochondrial Ca2+ dyshomeostasis is a prominent feature of multiple disorders, the link between NCLX and PKA may offer a therapeutic target.

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In this paper, we introduce several various classes of c-almost periodic type functions and their Stepanov generalizations, where c ∈ ℂ and |c| = 1. We also consider the corresponding classes of c-almost periodic type functions depending on two variables and prove several related composition principles. Plenty of illustrative examples and applications are presented.

Convoluted C-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated Ccosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted C-cosine functions and analytic convoluted C-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups. Several examples of operators generating convoluted cosine functions, (analytic) convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors. As an application, it is proved that the polyharmonic operator Δ 2 n , n ∈ N, acting on L 2 [0, π] with appropriate boundary conditions, generates an exponentially bounded K n -convoluted cosine function, and consequently, an exponentially bounded analytic K n+1 -convoluted semigroup of angle π 2 , for suitable exponentially bounded kernels K n and K n+1 .

Highlights d Mitochondrial membrane potential (DJm) allosterically inhibits NCLX d Phosphorylation of NCLX at Ser258 can override DJmdriven NCLX regulation d DJm regulation of NCLX requires Ser258 interaction with positively charged residues d NCLX regulation by DJm links mitochondrial metabolism and Ca 2+ signaling

In this paper, we analyze multi-dimensional (R X , B)-almost periodic type functions and multi-dimensional Bohr B-almost periodic type functions. The main structural characterizations and composition principles for the introduced classes of almost periodic functions are established. Several applications of our abstract theoretical results to the abstract Volterra integrodifferential equations in Banach spaces are provided, as well. Examples and applications to the abstractVolterra integro-differential equations 3.1. Application to nonautonomous retarded functional evolution equations 4. Appendix 4.1. n-Parameter strongly continuous semigroups 4.2. Multivariate trigonometric polynomials and approximations of periodic functions of several real variables References 2010 Mathematics Subject Classification. 42A75, 43A60, 47D99. Key words and phrases. (R, B)-Multi-almost periodic type functions, (R X , B)-multi-almost periodic type functions, Bohr B-almost periodic type functions, composition principles, abstract Volterra integro-differential equations. Marko Kostić is partially supported by grant 451-03-68/2020/14/200156 of Ministry of Science and Technological Development, Republic of Serbia. Manuel Pinto is partially supported by Fondecyt 1170466.The Euler Gamma function is denoted by Γ(•). If t 0 ∈ R n and ǫ > 0, then we set B(t 0 , ǫ)Now we are ready to briefly explain the organization and main ideas of this paper. In Subsection 1.1, we recall the basic facts and definitions about vectorvalued almost periodic functions of several real variables; in Subsection 1.2, we recall some applications of vector-valued almost periodic functions of several real variables made so far. Definition 2.1 and Definition 2.2 introduce the notion of (R, B)-multi-almost periodicity and the notion of (R X , B)-multi-almost periodicity for a continuous function F : I × X → Y, respectively. The convolution invariance of space consisting of all (R X , B)-multi-almost periodic functions is stated in Proposition 2.5, while the supremum formula for the class of (R, B)-multi-almost periodic functions is stated in Proposition 2.6.The notion of Bohr B-almost periodicity and the notion of B-uniform recurrence for a continuous function F : I × X → Y are introduced in Definition 2.9, provided that the region I satisfies the semigroup property I + I ⊆ I. Numerous illustrative examples of Bohr B-almost periodic functions and B-uniformly recurrent functions are presented in Example 2.12 and Example 2.13. In Definition 2.14, we introduce the notion of Bohr (B, I ′ )-almost periodicity and (B, I ′ )-uniform recurrence, provided that ∅ = I ′ ⊆ I ⊆ R n , F : I × X → Y is a continuous function and I + I ′ ⊆ I. After that, we provide several examples of Bohr (B, I ′ )-almost periodic functions and (B, I ′ )-uniformly recurrent functions in Example 2.15. The relative compactness of range F (I × B) for a Bohr B-almost periodic function F : I × X → Y,where B ∈ B, is analyzed in Proposition 2.16. The Bochner criterion for Bohr B-almost periodic functions is sta...

The main aim of this paper is to consider the classes of quasiasymptotically almost periodic functions and Stepanov quasi-asymptotically almost periodic functions in Banach spaces. These classes extend the well known classes of asymptotically almost periodic functions, Stepanov asymptotically almost periodic functions and S-asymptotically ω-periodic functions with values in Banach spaces. We investigate the invariance of introduced properties under the action of finite and inifinite convolution products, providing also an illustrative application to abstract nonautonomous differential equations of first order.2010 Mathematics Subject Classification. 47A16, 47B37, 47D06. Key words and phrases. Quasi-asymptotically almost periodic functions, Stepanov quasiasymptotically almost periodic functions, convolution products, evolution systems, abstract nonautonomous differential equations of first order.