responses mediate cell cycle arrest at G1. Department of Neurology and Neuroscience Mechanisms for Smad-mediated repression of c-myc Weill Medical College of Cornell University and Id have been elucidated (Chen et al., 2002; Kang et 1300 York Avenue al., 2003; Siegel et al., 2003), but less is known about New York, New York 10021 Smad complexes mediating p21Cip1 and p15Ink4b activation. c-Myc levels must be below a certain threshold in order for p21Cip1 and p15Ink4b to be activated by Summary TGF-␤ and other cytostatic signals. When present at high levels, as in mitogen-stimulated cells, c-Myc binds FoxO Forkhead transcription factors are shown here to the p21Cip1 and p15Ink4b promoters via the zinc to act as signal transducers at the confluence of Smad, finger protein Miz-1, interfering with activation of these PI3K, and FoxG1 pathways. Smad proteins activated genes by TGF-␤ (Seoane et al., 2002, 2001; Staller et by TGF-␤ form a complex with FoxO proteins to turn al., 2001), the tumor suppressor p53 (Seoane et al., on the growth inhibitory gene p21Cip1. This process 2002), and other signals (Herold et al., 2002; van de is negatively controlled by the PI3K pathway, a known Wetering et al., 2002; Wu et al., 2003). c-Myc downreguinhibitor of FoxO localization in the nucleus, and by lation by TGF-␤ renders p21Cip1 and p15Ink4b compethe telencephalic development factor FoxG1, which tent for activation. However, relief from c-Myc represwe show binds to FoxO-Smad complexes and blocks sion is not sufficient for activation (Seoane et al., 2002, p21Cip1 expression. We suggest that the activity of this 2001). It has been proposed that activation of p21Cip1 network confers resistance to TGF-␤-mediated cytoand p15Ink4b by TGF-␤ additionally requires the input of stasis during the development of the telencephalic a transactivation complex (Feng et al., 2000; Moustakas neuroepithelium and in glioblastoma brain tumor cells. and Kardassis, 1998; Seoane et al., 2001). Here, we identify FoxO proteins as key partners of Introduction Smad3 and Smad4 in the TGF-␤-dependent generation of a p21Cip1 activation complex. FoxO factors are mem-TGF-␤ is a pleiotropic cytokine that delivers cytostatic bers of the Forkhead box (Fox) family, which figures signals to epithelial, neuronal, and immune cells (Siegel prominently in the control of cell and organismal growth, and Massague ´, 2003; Wakefield and Roberts, 2002). development, metabolism, and longevity (Lee et al., TGF-␤ activates a membrane receptor serine/threonine 2003; Libina et al., 2003; Murphy et al., 2003; Tran et al., kinase complex that phosphorylates Smad2 and Smad3 2003; Czech, 2003). FoxO factors are under negative (Shi and Massague ´, 2003). Once activated, Smads accucontrol by the PI3K (phosphatidyl insositol 3-kinase) mulate in the nucleus and form transcriptional comgrowth-promoting pathway (Tran et al., 2003; Czech, plexes with Smad4 and different DNA binding partners, 2003; Vivanco and Sawyers, 2002). In response to mitocoactivators, and corepressors. These complexes ta...

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Abstract. Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. This leads to the question how the geometric structures behave under such sufficient statistics. While this is well studied in the finite sample size case, in the infinite case, we encounter technical problems concerning the appropriate topologies. Here, we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations. Within this framework, we can then handle the topological issues and show that the Fisher metric and the AmariChentsov tensor on statistical models in the class of symmetric 2-tensor fields and 3-tensor fields can be uniquely (up to a constant) characterized by their invariance under sufficient statistics, thereby achieving a full generalization of the original result of Chentsov to infinite sample sizes. More generally, we decompose Markov morphisms between statistical models in terms of statistics. In particular, a monotonicity result for the Fisher information naturally follows. MSC2010: 53C99, 62B05

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