Differentiating densities on smooth manifolds

Lebesgue integration of derivatives of strongly-oscillatory functions is a recurring challenge in computational science and engineering. Integration by parts is an effective remedy for huge computational costs associated with Monte Carlo integration schemes. In case of Lebesgue integrals over a smooth manifold, however, integration by parts gives rise to a derivative of the density implied by charts describing the domain manifold. This paper focuses on the computation of that derivative, which we call the density gradient function, on general smooth manifolds. We analytically derive formulas for the density gradient and present examples of manifolds determined by popular differential equation-driven systems. We highlight the significance of the density gradient by demonstrating a numerical example of Monte Carlo integration involving oscillatory integrands.Comment: 24 pages, 11 figures, submitted to journa

A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds

SRB measures are limiting stationary distributions describing the statistical behavior of chaotic dynamical systems. Directional derivatives of SRB measure densities conditioned on unstable manifolds are critical in the sensitivity analysis of hyperbolic chaos. These derivatives, known as the SRB density gradients, are by-products of the regularization of Lebesgue integrals appearing in the original linear response expression. In this paper, we propose a novel trajectory-driven algorithm for computing the SRB density gradient defined for systems with high-dimensional unstable manifolds. We apply the concept of measure preservation together with the chain rule on smooth manifolds. Due to the recursive one-step nature of our derivations, the proposed procedure is memory-efficient and can be naturally integrated with existing Monte Carlo schemes widely used in computational chaotic dynamics. We numerically show the exponential convergence of our scheme, analyze the computational cost, and present its use in the context of Monte Carlo integration.Comment: 25 pages, 9 figures, submitted to journal, under revie

Ergodic Sensitivity Analysis of One-Dimensional Chaotic Maps

Sensitivity analysis in chaotic dynamical systems is a challenging task from a computational point of view. In this work, we present a numerical investigation of a novel approach, known as the space-split sensitivity or S3 algorithm. The S3 algorithm is an ergodic-averaging method to differentiate statistics in ergodic, chaotic systems, rigorously based on the theory of hyperbolic dynamics. We illustrate S3 on one-dimensional chaotic maps, revealing its computational advantage over naive finite difference computations of the same statistical response. In addition, we provide an intuitive explanation of the key components of the S3 algorithm, including the density gradient function.Comment: 20 pages, 17 figure

Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems

An assumption of smooth response to small parameter changes, of statistics or long-time averages of a chaotic system, is generally made in the field of sensitivity analysis, and the parametric derivatives of statistical quantities are critically used in science and engineering. In this paper, we propose a numerical procedure to assess the differentiability of statistics with respect to parameters in chaotic systems. We numerically show that the existence of the derivative depends on the Lebesgue-integrability of a certain density gradient function, which we define as the derivative of logarithmic SRB density along the unstable manifold. We develop a recursive formula for the density gradient that can be efficiently computed along trajectories, and demonstrate its use in determining the differentiability of statistics. Our numerical procedure is illustrated on low-dimensional chaotic systems whose statistics exhibit both smooth and rough regions in parameter space.Comment: 32 pages, 13 figures, submitted to journal, under revie

Likelihood-based molecular-replacement solution for a highly pathological crystal with tetartohedral twinning and sevenfold translational noncrystallographic symmetry.

Translational noncrystallographic symmetry (tNCS) is a pathology of protein crystals in which multiple copies of a molecule or assembly are found in similar orientations. Structure solution is problematic because this breaks the assumptions used in current likelihood-based methods. To cope with such cases, new likelihood approaches have been developed and implemented in Phaser to account for the statistical effects of tNCS in molecular replacement. Using these new approaches, it was possible to solve the crystal structure of a protein exhibiting an extreme form of this pathology with seven tetrameric assemblies arrayed along the c axis. To resolve space-group ambiguities caused by tetartohedral twinning, the structure was initially solved by placing 56 copies of the monomer in space group P1 and using the symmetry of the solution to define the true space group, C2. The resulting structure of Hyp-1, a pathogenesis-related class 10 (PR-10) protein from the medicinal herb St John's wort, reveals the binding modes of the fluorescent probe 8-anilino-1-naphthalene sulfonate (ANS), providing insight into the function of the protein in binding or storing hydrophobic ligands

ANS complex of St John's wort PR-10 protein with 28 copies in the asymmetric unit: a fiendish combination of pseudosymmetry with tetartohedral twinning.

Hyp-1, a pathogenesis-related class 10 (PR-10) protein from St John's wort (Hypericum perforatum), was crystallized in complex with the fluorescent probe 8-anilino-1-naphthalene sulfonate (ANS). The highly pseudosymmetric crystal has 28 unique protein molecules arranged in columns with sevenfold translational noncrystallographic symmetry (tNCS) along c and modulated X-ray diffraction with intensity crests at l = 7n and l = 7n ± 3. The translational NCS is combined with pseudotetragonal rotational NCS. The crystal was a perfect tetartohedral twin, although detection of twinning was severely hindered by the pseudosymmetry. The structure determined at 2.4 Å resolution reveals that the Hyp-1 molecules (packed as β-sheet dimers) have three novel ligand-binding sites (two internal and one in a surface pocket), which was confirmed by solution studies. In addition to 60 Hyp-1-docked ligands, there are 29 interstitial ANS molecules distributed in a pattern that violates the arrangement of the protein molecules and is likely to be the generator of the structural modulation. In particular, whenever the stacked Hyp-1 molecules are found closer together there is an ANS molecule bridging them.Financial support for this project was provided by the European Union within the European Regional Developmental Fund and by the Polish Ministry of Science and Higher Education (grant No. NN 301 003739) and National Science Center (2013/10/M/NZ1/00251). RJR was supported by a Principal Research Fellowship from theWellcome Trust (grant No. 082961/Z/07/Z). ZD was supported in part by the Intramural Research Program of the National Cancer Institute, Center for Cancer Research.This is the final published version. It first appeared at http://scripts.iucr.org/cgi-bin/paper?S1399004715001388

A perspective on Hypericum perforatum Genetic transformation

Hypericum perforatum (St John's wort) is a reservoir of diverse classes of biologically active and high value secondary metabolites, which captured the interest of both researchers and the pharmaceutical industry alike. Several studies and clinical trials have shown that H. perforatum extracts possess an astounding array of pharmacological properties. These properties include antidepressant, anti-inflammatory, antiviral, anti-cancer, and antibacterial activities; and are largely attributed to the naphtodianthrones and xanthones found in the genus. Hence, improving their production via genetic manipulation is an important strategy. In spite of the presence of contemporary genome editing tools, genetic improvement of this genus remains challenging without robust transformation methods in place. In the recent past, we found that H. perforatum remains recalcitrant to Agrobacterium tumefaciens mediated transformation partly due to the induction of plant defense responses coming into play. However, H. perforatum transformation is possible via a non-biological method, biolistic bombardment. Some research groups have observed the induction of hairy roots in H. perforatum after Agrobacterium rhizogenes co-cultivation. In this review, we aim at updating the available methods for regeneration and transformation of H. perforatum. In addition, we also propose a brief perspective on certain novel strategies to improve transformation efficiency in order to meet the demands of the pharmaceutical industry via metabolic engineering.GF and PS are financed from the BIOTALENT project (GA621321) funded by the European Union Seventh Framework Programme (FP7) ERA Chairs Pilot Call and co-financed by funds allocated for education through project no W26/7.PR/2015 [GA 3413/7.PR/2015/2] for the years 2015-2019. This work was partially supported by Fundacao para a Ciencia e a Tecnologia (FCT) project (PTDC/AGR-GPL/119211/2010). WEE acknowledges the financial support provided by the FCT (SFRH/BD/52561/2014), under the Doctoral Programme "Agricultural Production Chains-from fork to farm" (PD/00122/2012).info:eu-repo/semantics/publishedVersio

Likelihood-based molecular-replacement solution for a highly pathological crystal with tetartohedral twinning and sevenfold translational noncrystallographic symmetry

Translational noncrystallographic symmetry (tNCS) is a pathology of protein crystals in which multiple copies of a molecule or assembly are found in similar orientations. Structure solution is problematic because this breaks the assumptions used in current likelihood-based methods. To cope with such cases, new likelihood approaches have been developed and implemented in Phaser to account for the statistical effects of tNCS in molecular replacement. Using these new approaches, it was possible to solve the crystal structure of a protein exhibiting an extreme form of this pathology with seven tetrameric assemblies arrayed along the c axis. To resolve space-group ambiguities caused by tetartohedral twinning, the structure was initially solved by placing 56 copies of the monomer in space group P1 and using the symmetry of the solution to define the true space group, C2. The resulting structure of Hyp-1, a pathogenesis-related class 10 (PR-10) protein from the medicinal herb St John’s wort, reveals the binding modes of the fluorescent probe 8-­anilino-1-naphthalene sulfonate (ANS), providing insight into the function of the protein in binding or storing hydrophobic ligands