Second order adjoints for solving PDE-constrained optimization problems

Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as PDE-constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost functional subject to the constraints posed by the model equations. The numerical solution of such optimization problems requires the computation of derivatives of the model output with respect to model parameters. The first order derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through first order adjoint sensitivity analysis. Second order adjoint models give second derivative information in the form of matrix-vector products between the Hessian of the cost functional and user defined vectors. Traditionally, the construction of second order derivatives for large scale models has been considered too costly. Consequently, data assimilation applications employ optimization algorithms that use only first order derivative information, like nonlinear conjugate gradients and quasi-Newton methods. In this paper we discuss the mathematical foundations of second order adjoint sensitivity analysis and show that it provides an efficient approach to obtain Hessian-vector products. We study the benefits of using of second order information in the numerical optimization process for data assimilation applications. The numerical studies are performed in a twin experiment setting with a two-dimensional shallow water model. Different scenarios are considered with different discretization approaches, observation sets, and noise levels. Optimization algorithms that employ second order derivatives are tested against widely used methods that require only first order derivatives. Conclusions are drawn regarding the potential benefits and the limitations of using high-order information in large scale data assimilation problems

Discrete Second Order Adjoints in Atmospheric Chemical Transport Modeling

Atmospheric chemical transport models (CTMs) are essential tools for the study of air pollution, for environmental policy decisions, for the interpretation of observational data, and for producing air quality forecasts. Many air quality studies require sensitivity analyses, i.e., the computation of derivatives of the model output with respect to model parameters. The derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through adjoint sensitivity analysis. While the traditional (first order) adjoint models give the gradient of the cost functional with respect to parameters, second order adjoint models give second derivative information in the form of products between the Hessian of the cost functional and a user defined vector. In this paper we discuss the mathematical foundations of the discrete second order adjoint sensitivity method and present a complete set of computational tools for performing second order sensitivity studies in three-dimensional atmospheric CTMs. The tools include discrete second order adjoints of Runge Kutta and of Rosenbrock time stepping methods for stiff equations together with efficient implementation strategies. Numerical examples illustrate the use of these computational tools in important applications like sensitivity analysis, optimization, uncertainty quantification, and the calculation of directions of maximal error growth in three-dimensional atmospheric CTMs

Forward and Inverse Analysis of Chemical Transport Models

Assessing the discrepancy between modeled and observed distributions of aerosols is a persistent problem on many scales. Tools for analyzing the evolution of aerosol size distributions using the adjoint method are presented in idealized box model calculations. The ability to recover information about aerosol growth rates and initial size distributions is assessed given a range of simulated observations of evolving systems. While such tools alone could facilitate analysis of chamber measurements, improving estimates of aerosol sources on regional and global scales requires explicit consideration of many additional chemical and physical processes that govern secondary formation of atmospheric aerosols from emissions of gas-phase precursors. The adjoint of the global chemical transport model GEOS-Chem is derived, affording detailed analysis of the relationship between gas-phase aerosol precursor emissions (SOx, NOx, and NH3) and the subsequent distributions of sulfate - ammonium - nitrate aerosol. Assimilation of surface measurements of sulfate and nitrate aerosol is shown to provide valuable constraints on emissions of ammonia. Adjoint sensitivities are used to propose strategies for air quality control, suggesting, for example, that reduction of SOx emissions in the summer and NH3 emissions in the winter would most effectively reduce non-attainment of aerosol air quality standards. The ability of this model to estimate global distributions of carbonaceous aerosol is also addressed. Based on new yield data from environmental chamber studies, mechanisms for incorporating the dependence of secondary organic aerosol (SOA) formation on NOx concentrations are developed for use in global models. When NOx levels are appropriately accounted for, it is demonstrated that sources such as isoprene and aromatics, previously neglected as sources of aerosol in global models, significantly contribute to predicted SOA burdens downwind of polluted areas (owing to benzene and toluene) and in the free troposphere (owing to isoprene)

Use of Epivolve phage display to generate a monoclonal antibody with opsonic activity directed against a subdominant epitope on extracellular loop 4 of Treponema pallidum BamA (TP0326)

IntroductionSyphilis, a sexually transmitted infection caused by the spirochete Treponema pallidum (Tp), is resurging globally. Tp’s repertoire of outer membrane proteins (OMPs) includes BamA (β-barrel assembly machinery subunit A/TP0326), a bipartite protein consisting of a 16-stranded β-barrel with nine extracellular loops (ECLs) and five periplasmic POTRA (polypeptide transport-associated) domains. BamA ECL4 antisera promotes internalization of Tp by rabbit peritoneal macrophages.MethodsThree overlapping BamA ECL4 peptides and a two-stage, phage display strategy, termed “Epivolve” (for epitope evolution) were employed to generate single-chain variable fragments (scFvs). Additionally, antisera generated by immunizing mice and rabbits with BamA ECL4 displayed by a Pyrococcus furiosus thioredoxin scaffold (PfTrxBamA/ECL4). MAbs and antisera reactivities were evaluated by immunoblotting and ELISA. A comparison of murine and rabbit opsonophagocytosis assays was conducted to evaluate the functional ability of the Abs (e.g., opsonization) and validate the mouse assay. Sera from Tp-infected mice (MSS) and rabbits (IRS) were evaluated for ECL4-specific Abs using PfTrxBamA/ECL4 and overlapping ECL4 peptides in immunoblotting and ELISA assays.ResultsEach of the five mAbs demonstrated reactivity by immunoblotting and ELISA to nanogram amounts of PfTrxBamA/ECL4. One mAb, containing a unique amino acid sequence in both the light and heavy chains, showed activity in the murine opsonophagocytosis assay. Mice and rabbits hyperimmunized with PfTrxBamA/ECL4 produced opsonic antisera that strongly recognized the ECL presented in a heterologous scaffold and overlapping ECL4 peptides, including S2. In contrast, Abs generated during Tp infection of mice and rabbits poorly recognized the peptides, indicating that S2 contains a subdominant epitope.DiscussionEpivolve produced mAbs target subdominant opsonic epitopes in BamA ECL4, a top syphilis vaccine candidate. The murine opsonophagocytosis assay can serve as an alternative model to investigate the opsonic potential of vaccinogens. Detailed characterization of BamA ECL4-specific Abs provided a means to dissect Ab responses elicited by Tp infection

The roles of periplasmic chaperones and the β-barrel assembly machinery complex in outer membrane protein biogenesis

The outer membrane of bacteria is a complex and important structure representing the first (and most ’fortified’) line of defense against insults from the extracellular medium as well as a platform for adhesion, recognition, and nutrient acquisition. As such, the outer membrane is a prime target for rational design of new drugs - both in terms of designing new classes of bacteriocidal agents, and also to sensitize bacteria to improve the efficacy of existing antibiotics. Our current knowledge suggests that all essential routes to the assembly of the outer membrane itself (both the lipids and proteins that form its structure) pass through the β-barrel assembly machinery complex (BAM complex), either directly or indirectly. This nanoscale machine is conserved across all bacteria containing an outer membrane and although the exact constituent parts vary, there is a common architecture scaffolded around one completely conserved protein, BamA. The BAM complex is responsible for the ATP- and protonmotive-independent assembly of integral transmembrane β-barrel proteins commonly refered to as outer membrane proteins (OMPs). Despite its essential role, the availability of high resolution structures, and over 15 years of biochemical studies, many questions about its mechanism of action remain unanswered. In this thesis new methods for studying OMP biogenesis through the use of crosslinking mass spectrometry and cryogenic super-resolution microscopy have been developed and applied to study the interaction with a model OMP, OmpA, and its chaperones Skp and SurA, as well as mapping its interaction with the BAM complex during folding, giving new insights into the mechanism of SurA chaperoning and suggesting a possible mechanism and route for the transit of an OMP from SurA and through BAM. Cryogenic super-resolution microscopy is used to provide preliminary insights into the nanoscale organisation of the BAM complex and OmpA, as well as two-colour co-localisation of these proteins. Kinetic assays are used alongside fluorescent probes of lipid order and single-molecule FRET to study the role of the lipid environment on BAM-catalysed, BamA-catalysed, and uncatalysed folding of tOmpA, both with and without SurA. In Chapter 3, a new crosslinking method was developed and validated on the Skp-OmpA chaperone-substrate pair. This was then applied to the other major OMP chaperone, SurA, where it could be shown that the binding activity of SurA resides almost exclusively in the core N- and C-terminal domains. Finally, this approach was used to try and capture a folding intermediate of OmpA as it was passed from SurA through the BAM complex and then analyse the interactions from OmpA to these partners during folding. A position at the bottom of the first (N-terminal) β-strand of OmpA makes crosslinks with the POTRA1, 4, and 5 domains of BamA as well as with the N-terminus and P2 of SurA, suggesting a greater recruitment of SurA P2 during OMP folding. This pattern of crosslinks from OmpA also implies a possible route from POTRA1, via BamD near the interface with BamA-POTRA5, onto β1 of BamA, with the crosslinks at POTRA4 formed last as the final β-strand is appended to the nascent barrel. In Chapter 4, the hypothesis that BAM functions by disordering lipids in the membrane was tested by using a number of techniques. DMPC was used as a model bilayer to be able to control the phase of lipids by conducting experiments at, below, or above, the transition temperature of 24 °C. The kinetics of tOmpA folding into DMPC liposomes showed that the full BAM complex is a much better catalyst for OMP folding (as measured by t50) than BamA alone when below or at the transition temperature of DMPC, and slightly better when above. While the BAM complex could accelerate the formation of folded tOmpA almost 16X over uncatalyzed folding at the transition temperature (24 °C), folding via BamA was only marginally faster (at 1.5 fold the uncatalyzed t50) which prompted further studies into the ability of these proteins to affect lipid order. The packing of lipids was assessed directly using the lipid order probe, laurdan, and the dynamics and conformational ensemble of the BAM complex was measured at a single-molecule level using FRET (smFRET). Laurdan experiments found that the presence of the BAM complex causes a broadening of the phase transition region as well as a 2 °C fall in the transition temperature implying a stabilisation of the liquid phase by BAM. smFRET studies showed that two populations of the BAM complex exist in solution, corresponding to the predicted FRET efficiencies of the open and closed states and these do not appear to interconvert on a 100s of μs to 100s of ms timescale. In Chapter 5, the organisation of the BAM complex is probed by using a novel method in super-resolution microscopy, cryoSTORM. By plunge-freezing samples of E. coli expressing BamA and/or OmpA fused with fluorescent proteins or the self-labelling HaloTag protein, these proteins could be visualised in their assembled state on the surface of a bacteria trapped in a frozen-hydrated state with sub 5 nm localisation precision. This showed the arrangement of molecules of BAM into discrete ‘islands’ spotted throughout the cell surface as well as smaller islands formed by OmpA showing that OMPs are prone to cluster together in small islands. Initial two-colour studies of OmpA and BamA suggest a relatively low degree of co-localisation for these proteins

Optimal Multi-UAV Trajectory Planning for Filming Applications

Teams of multiple Unmanned Aerial Vehicles (UAVs) can be used to record large-scale outdoor scenarios and complementary views of several action points as a promising system for cinematic video recording. Generating the trajectories of the UAVs plays a key role, as it should be ensured that they comply with requirements for system dynamics, smoothness, and safety. The rise of numerical methods for nonlinear optimization is finding a ourishing field in optimization-based approaches to multi- UAV trajectory planning. In particular, these methods are rather promising for video recording applications, as they enable multiple constraints and objectives to be formulated, such as trajectory smoothness, compliance with UAV and camera dynamics, avoidance of obstacles and inter-UAV con icts, and mutual UAV visibility. The main objective of this thesis is to plan online trajectories for multi-UAV teams in video applications, formulating novel optimization problems and solving them in real time. The thesis begins by presenting a framework for carrying out autonomous cinematography missions with a team of UAVs. This framework enables media directors to design missions involving different types of shots with one or multiple cameras, running sequentially or concurrently. Second, the thesis proposes a novel non-linear formulation for the challenging problem of computing optimal multi-UAV trajectories for cinematography, integrating UAV dynamics and collision avoidance constraints, together with cinematographic aspects such as smoothness, gimbal mechanical limits, and mutual camera visibility. Lastly, the thesis describes a method for autonomous aerial recording with distributed lighting by a team of UAVs. The multi-UAV trajectory optimization problem is decoupled into two steps in order to tackle non-linear cinematographic aspects and obstacle avoidance at separate stages. This allows the trajectory planner to perform in real time and to react online to changes in dynamic environments. It is important to note that all the methods in the thesis have been validated by means of extensive simulations and field experiments. Moreover, all the software components have been developed as open source.Los equipos de vehículos aéreos no tripulados (UAV) son sistemas prometedores para grabar eventos cinematográficos, en escenarios exteriores de grandes dimensiones difíciles de cubrir o para tomar vistas complementarias de diferentes puntos de acción. La generación de trayectorias para este tipo de vehículos desempeña un papel fundamental, ya que debe garantizarse que se cumplan requisitos dinámicos, de suavidad y de seguridad. Los enfoques basados en la optimización para la planificación de trayectorias de múltiples UAVs se pueden ver beneficiados por el auge de los métodos numéricos para la resolución de problemas de optimización no lineales. En particular, estos métodos son bastante prometedores para las aplicaciones de grabación de vídeo, ya que permiten formular múltiples restricciones y objetivos, como la suavidad de la trayectoria, el cumplimiento de la dinámica del UAV y de la cámara, la evitación de obstáculos y de conflictos entre UAVs, y la visibilidad mutua. El objetivo principal de esta tesis es planificar trayectorias para equipos multi-UAV en aplicaciones de vídeo, formulando novedosos problemas de optimización y resolviéndolos en tiempo real. La tesis comienza presentando un marco de trabajo para la realización de misiones cinematográficas autónomas con un equipo de UAVs. Este marco permite a los directores de medios de comunicación diseñar misiones que incluyan diferentes tipos de tomas con una o varias cámaras, ejecutadas de forma secuencial o concurrente. En segundo lugar, la tesis propone una novedosa formulación no lineal para el difícil problema de calcular las trayectorias óptimas de los vehículos aéreos no tripulados en cinematografía, integrando en el problema la dinámica de los UAVs y las restricciones para evitar colisiones, junto con aspectos cinematográficos como la suavidad, los límites mecánicos del cardán y la visibilidad mutua de las cámaras. Por último, la tesis describe un método de grabación aérea autónoma con iluminación distribuida por un equipo de UAVs. El problema de optimización de trayectorias se desacopla en dos pasos para abordar los aspectos cinematográficos no lineales y la evitación de obstáculos en etapas separadas. Esto permite al planificador de trayectorias actuar en tiempo real y reaccionar en línea a los cambios en los entornos dinámicos. Es importante señalar que todos los métodos de la tesis han sido validados mediante extensas simulaciones y experimentos de campo. Además, todos los componentes del software se han desarrollado como código abierto

Structure Preserving and Scalable Simulation of Colliding Systems

Predictive computational tools to study granular materials are important in fields ranging from the geosciences and civil engineering to computer graphics. The simulation of granular materials, however, presents many challenges. The behavior of a granular medium is fundamentally multi-scale, with pair-wise interactions between discrete granules able to influence the continuum-scale evolution of a bulk material. Computational techniques for studying granular materials must therefore contend with this multi-scale nature. This research first addresses both the question of how to accurately model interactions between grains and the question of how to achieve multi-scale simulations of granular materials. We propose a novel rigid body contact model and a time integration technique that, for the first time, are able to simultaneously capture five key features of rigid body impact. We further validate this new model and time integration method by reproducing computationally challenging phenomena from granular physics. We next propose a technique to couple discrete and continuum models of granular materials to one another. This hybrid model reveals a family of possible discretizations suitable for simulation. We derive an explicit integration technique from this framework that is able to capture phenomena previously reserved for discrete treatments, including frictional jamming, while treating bulk regions of the material with a continuum model. To effectively handle the large plastic deformations inherent in the evolution of a granular medium, we further propose a method to dynamically update which regions are treated with a discrete model and which regions are treated with a continuum model. We demonstrate that hybrid simulations of a dynamically evolving granular material are possible and practical, and lay the foundation for further algorithmic development in this space. Finally, as the the tools used in computational science and engineering become progressively more complex, the ability to effectively train students in the field becomes increasingly important. We address the question of how to train students from a computer science background in numerical computation techniques by proposing a new system to automatically vet and identify problems in numerical simulations. This system has been deployed at the undergraduate and graduate level in a course on physical simulation at Columbia University, and has increased both student retention and student satisfaction with the course

NMR Studies of Membrane Associating Peptides and Implications in Autotransporter Function

Membrane associating peptides such as antimicrobial peptides and viral fusion peptides are involved in a diverse set of physiological processes. Their functions often require a change in the structure of the peptide, caused by interactions between the peptide and the biological membrane. This change in structure can be investigated in vitro, by performing circular dichroism (CD) and nuclear magnetic resonance (NMR) spectroscopy experiments with peptide solutions in membrane mimetic such as detergent micelles and in organic solvents. NalP is an outer membrane autotransporter protein from N. meningitides that transports its serine protease passenger domain across the outer membrane of the cell. The secondary structure of a linker peptide from this protein was determined in an aqueous medium, in sodium dodecyl sulfate (SDS) detergent micelles and in trifluoroethanol (TFE). The peptide exhibits a random coiled secondary structure in the aqueous medium and has a partial helical character in SDS detergent micelles. In TFE, the peptide has an α-helical secondary structure, and this structure was determined by NMR spectroscopy. The difference in structure of the peptide in the detergent micelle and in the hydrophobic organic solvent, when compared to the aqueous medium, suggests that the linker might interact with the biological membrane during the protein transport event. The stability of the α-helix formed by this peptide in TFE was determined by investigating the overall and residue specific effects of temperature on the secondary structure of the peptide. Partial loss of secondary structure is observed when the peptide is heated to a temperature of 348 K. Nuclear Overhauser Effect (NOE) crosspeaks that had high relative intensities at elevated temperatures were observed on a stretch of residues located in the middle of the α-helix, suggesting that this region of the α-helix is comparatively more stable and that the unfolding is initiated at both termini of the helix. The effect of electrostatic interactions on thermally induced unfolding of charged helical peptides was determined in detergent micelles of different charge. In the presence of a similar charge on the peptide and the micelle, the antimicrobial peptide Magainin2 and a mutant of the viral fusion peptide HA G1V showed a greater curvature in the temperature dependence of CD signal at 222 nm, suggesting an increased co-operativity in the helix coil transition of the peptide. The residue specific effects of electrostatic interactions were also determined by measuring the temperature dependence of chemical shifts and NOE intensities of Magainin2 peptide in SDS and dodecylphosphocholine (DPC) detergent micelles

Structural investigation of two supramolecular complexes of the eukaryotic cell

Preface This study is focussed on the structural investigation of large molecular assemblies such as the 26S proteasome and the translocation machinery of the outer mitochondrial membrane. It is divided in two chapters and in both parts the structural and further functional analysis is based on X-ray crystallography. Chapter 1: Structural investigation of Rpn13, the multifunctional adaptor protein of the 26S proteasome The results in chapter 1 reveal that the multifunctional adaptor protein Rpn13 acts as a novel ubiquitin receptor of the 26S proteasome and deliver structural and biophysical details of its interaction with ubiquitin and with other proteasomal subunits. The crystal structure of the ubiquitin binding domain of Rpn13 reveals the molecular architecture of a Pleckstrin Homology (PH) domain and the NMR structure of the complex with ubiquitin shows a novel ubiquitin-binding mode. Additional NMR studies and domain mapping by truncation analysis provide further insights in the domain architecture of Rpn13 and the interaction with its partners Rpn2 and Uch37. Chapter 2: Crystallographic studies of the TOM core complex Chapter 2 presents the purification and crystallization of the mitochondrial protein translocase, the TOM core complex, from Neurospora crassa. Preliminary crystallographic data lead to the determination of space group and cell dimensions. This chapter also describes various experiments to improve the diffraction quality of the crystals and the co-crystallization of TOM core complex with specific monoclonal antibody fragments. Furthermore, expression and refolding of the main component Tom40 is raised as an alternative approach in structural investigation of the TOM complex