Computing maximum matchings in temporal graphs

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ ∈ ℕ is given. The requirement that a vertex cannot be matched twice in any Δ-window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms

Computing maximum matchings in temporal graphs.

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ ∈ ℕ is given. The requirement that a vertex cannot be matched twice in any Δ-window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms

Neurological deficits and glycosphingolipid accumulation in saposin B deficient mice

Saposin B derives from the multi-functional precursor, prosaposin, and functions as an activity enhancer for several glycosphingolipid (GSL) hydrolases. Mutations in saposin B present in humans with phenotypes resembling metachromatic leukodystrophy. To gain insight into saposin B's physiological functions, a specific deficiency was created in mice by a knock-in mutation of an essential cysteine in exon 7 of the prosaposin locus. No saposin B protein was detected in the homozygotes (B−/−) mice, whereas prosaposin, and saposins A, C and D were at normal levels. B−/− mice exhibited slowly progressive neuromotor deterioration and minor head tremor by 15 months. Excess hydroxy and non-hydroxy fatty acid sulfatide levels were present in brain and kidney. Alcian blue positive (sulfatide) storage cells were found in the brain, spinal cord and kidney. Ultrastructural analyses showed lamellar inclusion material in the kidney, sciatic nerve, brain and spinal cord tissues. Lactosylceramide (LacCer) and globotriaosylceramide (TriCer) were increased in various tissues of B−/− mice supporting the in vivo role of saposin B in the degradation of these lipids. CD68 positive microglial cells and activated GFAP positive astrocytes showed a proinflammatory response in the brains of B−/− mice. These findings delineate the roles of saposin B for the in vivo degradation of several GSLs and its primary function in maintenance of CNS function. B−/− provide a useful model for understanding the contributions of this saposin to GSL metabolism and homeostasis

Specific saposin C deficiency: CNS impairment and acid β-glucosidase effects in the mouse

Saposins A, B, C and D are derived from a common precursor, prosaposin (psap). The few patients with saposin C deficiency develop a Gaucher disease-like central nervous system (CNS) phenotype attributed to diminished glucosylceramide (GC) cleavage activity by acid β-glucosidase (GCase). The in vivo effects of saposin C were examined by creating mice with selective absence of saposin C (C−/−) using a knock-in point mutation (cysteine-to-proline) in exon 11 of the psap gene. In C−/− mice, prosaposin and saposins A, B and D proteins were present at near wild-type levels, but the saposin C protein was absent. By 1 year, the C−/− mice exhibited weakness of the hind limbs and progressive ataxia. Decreased neuromotor activity and impaired hippocampal long-term potentiation were evident. Foamy storage cells were observed in dorsal root ganglion and there was progressive loss of cerebellar Purkinje cells and atrophy of cerebellar granule cells. Ultrastructural analyses revealed inclusions in axonal processes in the spinal cord, sciatic nerve and brain, but no excess of multivesicular bodies. Activated microglial cells and astrocytes were present in thalamus, brain stem, cerebellum and spinal cord, indicating regional pro-inflammatory responses. No storage cells were found in visceral organs of these mice. The absence of saposin C led to moderate increases in GC and lactosylceramide (LacCer) and their deacylated analogues. These results support the view that saposin C has multiple roles in glycosphingolipid (GSL) catabolism as well as a prominent function in CNS and axonal integrity independent of its role as an optimizer/stabilizer of GCase

Interference-free walks in time: Temporally disjoint paths

We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never visit the same vertex at the same time; otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. On the one extreme, we show that on general graphs the problem is computationally hard. The “walk version” is W[1]-hard when parameterized by the number of walks. However, it is polynomialtime solvable for any constant number of walks. The “path version” remains NP-hard even if we want to find only two temporally disjoint paths. On the other extreme, restricting the input temporal graph to have a path as underlying graph, quite counter-intuitively, we find NP-hardness in general but also identify natural tractable cases

Structure and assembly of scalable porous protein cages

Proteins that self-assemble into regular shell-like polyhedra are useful, both in nature and in the laboratory, as molecular containers. Here we describe cryo-electron microscopy (EM) structures of two versatile encapsulation systems that exploit engineered electrostatic interactions for cargo loading. We show that increasing the number of negative charges on the lumenal surface of lumazine synthase, a protein that naturally assembles into a ∼1-MDa dodecahedron composed of 12 pentamers, induces stepwise expansion of the native protein shell, giving rise to thermostable ∼3-MDa and ∼6-MDa assemblies containing 180 and 360 subunits, respectively. Remarkably, these expanded particles assume unprecedented tetrahedrally and icosahedrally symmetric structures constructed entirely from pentameric units. Large keyhole-shaped pores in the shell, not present in the wild-type capsid, enable diffusion-limited encapsulation of complementarily charged guests. The structures of these supercharged assemblies demonstrate how programmed electrostatic effects can be effectively harnessed to tailor the architecture and properties of protein cages

Structure and assembly of scalable porous protein cages

Proteins that self-assemble into regular shell-like polyhedra are useful, both in nature and in the laboratory, as molecular containers. Here we describe cryo-electron microscopy (EM) structures of two versatile encapsulation systems that exploit engineered electrostatic interactions for cargo loading. We show that increasing the number of negative charges on the lumenal surface of lumazine synthase, a protein that naturally assembles into a ∼1-MDa dodecahedron composed of 12 pentamers, induces stepwise expansion of the native protein shell, giving rise to thermostable ∼3-MDa and ∼6-MDa assemblies containing 180 and 360 subunits, respectively. Remarkably, these expanded particles assume unprecedented tetrahedrally and icosahedrally symmetric structures constructed entirely from pentameric units. Large keyhole-shaped pores in the shell, not present in the wild-type capsid, enable diffusion-limited encapsulation of complementarily charged guests. The structures of these supercharged assemblies demonstrate how programmed electrostatic effects can be effectively harnessed to tailor the architecture and properties of protein cages