On Integral Means for Fractional Calculus Operators of Multivalent Functions

2000 Mathematics Subject Classification: Primary 30C45, Secondary 26A33, 30C80Integral means inequalities are obtained for the fractional derivatives and the fractional integrals of multivalent functions. Relevant connections with various known integral means inequalities are also pointed out

Site-specific inductive and inhibitory activities of MMP-2 and MMP-3 orchestrate mammary gland branching morphogenesis

During puberty, mouse mammary epithelial ducts invade the stromal mammary fat pad in a wave of branching morphogenesis to form a complex ductal tree. Using pharmacologic and genetic approaches, we find that mammary gland branching morphogenesis requires transient matrix metalloproteinase (MMP) activity for invasion and branch point selection. MMP-2, but not MMP-9, facilitates terminal end bud invasion by inhibiting epithelial cell apoptosis at the start of puberty. Unexpectedly, MMP-2 also represses precocious lateral branching during mid-puberty. In contrast, MMP-3 induces secondary and tertiary lateral branching of ducts during mid-puberty and early pregnancy. Nevertheless, the mammary gland is able to develop lactational competence in MMP mutant mice. Thus, specific MMPs refine the mammary branching pattern by distinct mechanisms during mammary gland branching morphogenesis

Global parameter search reveals design principles of the mammalian circadian clock

Background: Virtually all living organisms have evolved a circadian (~24 hour) clock that controls physiological and behavioural processes with exquisite precision throughout the day/night cycle. The suprachiasmatic nucleus (SCN), which generates these ~24 h rhythms in mammals, consists of several thousand neurons. Each neuron contains a gene-regulatory network generating molecular oscillations, and the individual neuron oscillations are synchronised by intercellular coupling, presumably via neurotransmitters. Although this basic mechanism is currently accepted and has been recapitulated in mathematical models, several fundamental questions about the design principles of the SCN remain little understood. For example, a remarkable property of the SCN is that the phase of the SCN rhythm resets rapidly after a 'jet lag' type experiment, i.e. when the light/ dark (LD) cycle is abruptly advanced or delayed by several hours. Results: Here, we describe an extensive parameter optimization of a previously constructed simplified model of the SCN in order to further understand its design principles. By examining the top 50 solutions from the parameter optimization, we show that the neurotransmitters' role in generating the molecular circadian rhythms is extremely important. In addition, we show that when a neurotransmitter drives the rhythm of a system of coupled damped oscillators, it exhibits very robust synchronization and is much more easily entrained to light/dark cycles. We were also able to recreate in our simulations the fast rhythm resetting seen after a 'jet lag' type experiment. Conclusion: Our work shows that a careful exploration of parameter space for even an extremely simplified model of the mammalian clock can reveal unexpected behaviours and non-trivial predictions. Our results suggest that the neurotransmitter feedback loop plays a crucial role in the robustness and phase resetting properties of the mammalian clock, even at the single neuron level

How to Achieve Fast Entrainment? The Timescale to Synchronization

Entrainment, where oscillators synchronize to an external signal, is ubiquitous in nature. The transient time leading to entrainment plays a major role in many biological processes. Our goal is to unveil the specific dynamics that leads to fast entrainment. By studying a generic model, we characterize the transient time to entrainment and show how it is governed by two basic properties of an oscillator: the radial relaxation time and the phase velocity distribution around the limit cycle. Those two basic properties are inherent in every oscillator. This concept can be applied to many biological systems to predict the average transient time to entrainment or to infer properties of the underlying oscillator from the observed transients. We found that both a sinusoidal oscillator with fast radial relaxation and a spike-like oscillator with slow radial relaxation give rise to fast entrainment. As an example, we discuss the jet-lag experiments in the mammalian circadian pacemaker

Activation of AMPA Receptors in the Suprachiasmatic Nucleus Phase-Shifts the Mouse Circadian Clock In Vivo and In Vitro

The glutamatergic neurotransmission in the suprachiasmatic nucleus (SCN) plays a central role in the entrainment of the circadian rhythms to environmental light-dark cycles. Although the glutamatergic effect operating via NMDAR (N-methyl D-aspartate receptor) is well elucidated, much less is known about a role of AMPAR (α-amino-3-hydroxy-5-methylisoxazole-4-propionic acid receptor) in circadian entrainment. Here we show that, in the mouse SCN, GluR2 and GluR4 AMPAR subtypes are abundantly expressed in the retinorecipient area. In vivo microinjection of AMPA in the SCN during the early subjective night phase-delays the behavioral rhythm. In the organotypic SCN slice culture, AMPA application induces phase-dependent phase-shifts of core-clock gene transcription rhythms. These data demonstrate that activation of AMPAR is capable of phase-shifting the circadian clock both in vivo and in vitro, and are consistent with the hypothesis that activation of AMPA receptors is a critical step in the transmission of photic information to the SCN