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Additional file 5: Figure. S1 Carbon Nanoparticles (indicated as CNP, or CNDs) induce a novel cell death of carbopoptosis. a. Investigation of CNPs derived from soybean and beet by AFM and distribution of size counting. b. Soybean CNP induces osmosis-related cell death in PC3 cells treated with 0.25 mg/mL for times indicated upon pretreatment with 2.5M and 10M glucose, respectively. c. CNPs induced cell death at 0.1 mg/mL which can be prevented by Matrigel and fibronectin. d. Single floating cells were collected and formed spheres upon continue culture by removing CNPs. PC3 cells were collected after treatment of overnight by vehicle or 0.3 and 0.9 mg/mL and continue culture for days indicated. e, f. Cell cycle, cell death analysis by flow cytometer, or cell analyzer. PC3 or NRK cells were treated by CNPs for 2 h and subjected to analysis. Figure. S2 Supporting Fig 5d, e. Figure. S3 Nanozyme activity of Beet SS carbon nanoparticles which exhibit higher phosphatase activity and enhanced anchorage-independent sphere growth with less cell death. a. Nanozyme assay with the condition at pH 7 using NBT/BCIP as substrate. NBT/BCIP, and beet/beet SS, at 0.25 mg/mL in the reaction. b. A soft agar assay.

Kullback-Leibler (KL) divergence is one of the most important divergence measures between probability distributions. In this paper, we prove several properties of KL divergence between multivariate Gaussian distributions. First, for any two $n$-dimensional Gaussian distributions $\mathcal{N}_1$ and $\mathcal{N}_2$, we give the supremum of $KL(\mathcal{N}_1||\mathcal{N}_2)$ when $KL(\mathcal{N}_2||\mathcal{N}_1)\leq \varepsilon\ (\varepsilon>0)$. For small $\varepsilon$, we show that the supremum is $\varepsilon + 2\varepsilon^{1.5} + O(\varepsilon^2)$. This quantifies the approximate symmetry of small KL divergence between Gaussians. We also find the infimum of $KL(\mathcal{N}_1||\mathcal{N}_2)$ when $KL(\mathcal{N}_2||\mathcal{N}_1)\geq M\ (M>0)$. We give the conditions when the supremum and infimum can be attained. Second, for any three $n$-dimensional Gaussians $\mathcal{N}_1$, $\mathcal{N}_2$, and $\mathcal{N}_3$, we find an upper bound of $KL(\mathcal{N}_1||\mathcal{N}_3)$ if $KL(\mathcal{N}_1||\mathcal{N}_2)\leq \varepsilon_1$ and $KL(\mathcal{N}_2||\mathcal{N}_3)\leq \varepsilon_2$ for $\varepsilon_1,\varepsilon_2\ge 0$. For small $\varepsilon_1$ and $\varepsilon_2$, we show the upper bound is $3\varepsilon_1+3\varepsilon_2+2\sqrt{\varepsilon_1\varepsilon_2}+o(\varepsilon_1)+o(\varepsilon_2)$. This reveals that KL divergence between Gaussians follows a relaxed triangle inequality. Importantly, all the bounds in the theorems presented in this paper are independent of the dimension $n$. Finally, We discuss the applications of our theorems in explaining counterintuitive phenomenon of flow-based model, deriving deep anomaly detection algorithm, and extending one-step robustness guarantee to multiple steps in safe reinforcement learning., arXiv admin note: text overlap with arXiv:2002.03328

SummaryTuberous sclerosis (TSC) is a tumor syndrome caused by mutation in TSC1 or TSC2 genes. TSC tumorigenesis is not always accompanied by loss of heterozygosity (LOH). Recently, extracellular signal-regulated kinase (Erk) has been found activated in TSC lesions lacking TSC1 or TSC2 LOH. Here, we show that Erk may play a critical role in TSC progression through posttranslational inactivation of TSC2. Erk-dependent phosphorylation leads to TSC1-TSC2 dissociation and markedly impairs TSC2 ability to inhibit mTOR signaling, cell proliferation, and oncogenic transformation. Importantly, expression of an Erk nonphosphorylatable TSC2 mutant in TSC2+/− tumor cells where Erk is constitutively activated blocks tumorigenecity in vivo, while wild-type TSC2 is ineffective. Our findings position the Ras/MAPK pathway upstream of the TSC complex and suggest that Erk may modulate mTOR signaling and contribute to disease progression through phosphorylation and inactivation of TSC2.