We suggest a method to construct a homomorphic encryption scheme for approximate arithmetic. It supports an approximate addition and multiplication of encrypted messages, together with a new rescaling procedure for managing the magnitude of plaintext. This procedure truncates a ciphertext into a smaller modulus, which leads to rounding of plaintext. The main idea is to add a noise following significant figures which contain a main message. This noise is originally added to the plaintext for security, but considered to be a part of error occurring during approximate computations that is reduced along with plaintext by rescaling. As a result, our decryption structure outputs an approximate value of plaintext with a predetermined precision.We also propose a new batching technique for a RLWE-based construction. A plaintext polynomial is an element of a cyclotomic ring of characteristic zero and it is mapped to a message vector of complex numbers via complex canonical embedding map, which is an isometric ring homomorphism. This transformation does not blow up the size of errors, therefore enables us to preserve the precision of plaintext after encoding. In our construction, the bit size of ciphertext modulus grows linearly with the depth of the circuit being evaluated due to rescaling procedure, while all the previous works either require an exponentially large size of modulus or expensive computations such as bootstrapping or bit extraction. One important feature of our method is that the precision loss during evaluation is bounded by the depth of a circuit and it exceeds at most one more bit compared to unencrypted approximate arithmetic such as floating-point operations. In addition to the basic approximate circuits, we show that our scheme can be applied to the efficient evaluation of transcendental functions such as multiplicative inverse, exponential function, logistic function and discrete Fourier transform.

Of 17 genes annotated in the Arabidopsis genome database as cinnamyl alcohol dehydrogenase (CAD) homologues, an in silico analysis revealed that 8 genes were misannotated. Of the remaining nine, six were catalytically competent for NADPH-dependent reduction of p-coumaryl, caffeyl, coniferyl, 5-hydroxyconiferyl, and sinapyl aldehydes, whereas three displayed very low activity and only at very high substrate concentrations. Of the nine putative CADs, two (AtCAD5 and AtCAD4) had the highest activity and homology (Ϸ83% similarity) relative to bona fide CADs from other species. AtCAD5 used all five substrates effectively, whereas AtCAD4 (of lower overall catalytic capacity) poorly used sinapyl aldehyde; the corresponding 270-fold decrease in kenz resulted from higher Km and lower kcat values, respectively. No CAD homologue displayed a specific requirement for sinapyl aldehyde, which was in direct contrast with unfounded claims for a so-called sinapyl alcohol dehydrogenase in angiosperms. AtCAD2, 3, as well as AtCAD7 and 8 (highest homology to sinapyl alcohol dehydrogenase) were catalytically less active overall by at least an order of magnitude, due to increased Km and lower kcat values. Accordingly, alternative and͞or bifunctional metabolic roles of these proteins in plant defense cannot be ruled out. Comprehensive analyses of lignified tissues of various Arabidopsis knockout mutants (for AtCAD5, 6, and 9) at different stages of growth͞ development indicated the presence of functionally redundant CAD metabolic networks. Moreover, disruption of AtCAD5 expression had only a small effect on either overall lignin amounts deposited, or on syringyl-guaiacyl compositions, despite being the most catalytically active form in vitro.

Hepatocellular carcinoma (HCC) is one of the most common causes of cancer death worldwide. HCC can be cured by radical therapies if early diagnosis is done while the tumor has remained of small size. Unfortunately, diagnosis is commonly late when the tumor has grown and spread. Thus, palliative approaches are usually applied such as transarterial intrahepatic chemoembolization and sorafenib, an anti-angiogenic agent and MAP kinase inhibitor. This latter is the only targeted therapy that has shown significant, although moderate, efficiency in some individuals with advanced HCC. This highlights the need to develop other targeted therapies, and to this goal, to identify more and more pathways as potential targets. The Wnt pathway is a key component of a physiological process involved in embryonic development and tissue homeostasis. Activation of this pathway occurs when a Wnt ligand binds to a Frizzled (FZD) receptor at the cell membrane. Two different Wnt signaling cascades have been identified, called non-canonical and canonical pathways, the latter involving the β-catenin protein. Deregulation of the Wnt pathway is an early event in hepatocarcinogenesis and has been associated with an aggressive HCC phenotype, since it is implicated both in cell survival, proliferation, migration and invasion. Thus, component proteins identified in this pathway are potential candidates of pharmacological intervention. This review focuses on the characteristics and functions of the molecular targets of the Wnt signaling cascade and how they may be manipulated to achieve anti-tumor effects.

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