Protein Design is NP-hard

Biologists working in the area of computational protein design have never doubted the seriousness of the algorithmic challenges that face them in attempting in silico sequence selection. It turns out that in the language of the computer science community, this discrete optimization problem is NP-hard. The purpose of this paper is to explain the context of this observation, to provide a simple illustrative proof and to discuss the implications for future progress on algorithms for computational protein design

Analysis of Adjoint Error Correction for Superconvergent Functional Estimates

Earlier work introduced the notion of adjoint error correction for obtaining superconvergent estimates of functional outputs from approximate PDE solutions. This idea is based on a posteriori error analysis suggesting that the leading order error term in the functional estimate can be removed by using an adjoint PDE solution to reveal the sensitivity of the functional to the residual error in the original PDE solution. The present work provides a priori error analysis that correctly predicts the behaviour of the remaining leading order error term. Furthermore, the discussion is extended from the case of homogeneous boundary conditions and bulk functionals, to encompass the possibilities of inhomogeneous boundary conditions and boundary functionals. Numerical illustrations are provided for both linear and nonlinear problems.\ud \ud This research was supported by EPSRC under grant GR/K91149, and by NASA/Ames Cooperative Agreement No. NCC 2-5431

An introduction to the adjoint approach to design

Optimal design methods involving the solution of an adjoint system of equations are an active area of research in computational fluid dynamics, particularly for aeronautical applications. This paper presents an introduction to the subject, emphasising the simplicity of the ideas when viewed in the context of linear algebra. Detailed discussions also include the extension to p.d.e.'s, the construction of the adjoint p.d.e. and its boundary conditions, and the physical significance of the adjoint solution. The paper concludes with examples of the use of adjoint methods for optimising the design of business jets.\ud \ud This research was supported by funding from Rolls-Royce plc, BAe Systems plc and EPSRC grants GR/K91149 and GR/L95700

Analytic Adjoint Solutions for the Quasi-1D Euler Equations

The analytic properties of adjoint solutions are examined for the quasi-1D Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green's function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging-diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock.\ud \ud This research was supported by EPSRC under grant GR/K9114

Adjoint recovery of superconvergent functionals from PDE approximations

Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

Multiplexed miRNA northern blots via hybridization chain reaction

Northern blots enable detection of a target RNA of interest in a biological sample using standard benchtop equipment. miRNAs are the most challenging targets as they must be detected with a single short nucleic acid probe. With existing approaches, it is cumbersome to perform multiplexed blots in which several RNAs are detected simultaneously, impeding the study of interacting regulatory elements. Here, we address this shortcoming by demonstrating multiplexed northern blotting based on the mechanism of hybridization chain reaction (HCR). With this approach, nucleic acid probes complementary to RNA targets trigger chain reactions in which fluorophore-labeled DNA hairpins self-assemble into tethered fluorescent amplification polymers. The programmability of HCR allows multiple amplifiers to operate simultaneously and independently within a blot, enabling straightforward multiplexing. We demonstrate simultaneous detection of three endogenous miRNAs in total RNA extracted from 293T and HeLa cells. For a given target, HCR signal scales linearly with target abundance, enabling relative and absolute quantitation. Using non-radioactive HCR, sensitive and selective miRNA detection is achieved using 2′OMe-RNA probes. The HCR northern blot protocol takes ∼1.5 days independent of the number of target RNAs

Paradigms for computational nucleic acid design

The design of DNA and RNA sequences is critical for many endeavors, from DNA nanotechnology, to PCR‐based applications, to DNA hybridization arrays. Results in the literature rely on a wide variety of design criteria adapted to the particular requirements of each application. Using an extensively studied thermodynamic model, we perform a detailed study of several criteria for designing sequences intended to adopt a target secondary structure. We conclude that superior design methods should explicitly implement both a positive design paradigm (optimize affinity for the target structure) and a negative design paradigm (optimize specificity for the target structure). The commonly used approaches of sequence symmetry minimization and minimum free‐energy satisfaction primarily implement negative design and can be strengthened by introducing a positive design component. Surprisingly, our findings hold for a wide range of secondary structures and are robust to modest perturbation of the thermodynamic parameters used for evaluating sequence quality, suggesting the feasibility and ongoing utility of a unified approach to nucleic acid design as parameter sets are refined further. Finally, we observe that designing for thermodynamic stability does not determine folding kinetics, emphasizing the opportunity for extending design criteria to target kinetic features of the energy landscape

A Synthetic DNA Walker for Molecular Transport

Inspired by kinesin movement along a microtubule, we demonstrate a processive bipedal DNA walker. Powered by externally controlled DNA fuel strands, the walker locomotes with a 5 nm stride by advancing the trailing foot to the lead at each step. Real-time monitoring of specific bidirectional walker movement is achieved via multiplexed fluorescence quenching

Preconditioning on stretched meshes

High aspect ratio cells in a computational mesh compound the inherent stiffness in the Euler and Navier-Stokes equations which arises from a disparity in the propagative speeds of convective and acoustic modes. A mesh-aligned preconditioning strategy is examined which is intended to improve multigrid performance in two ways: a) enhancing propagation of disturbances by shaping wave front envelopes to match cell aspect ratios, b) clustering high frequency components of the spatial Fourier footprint away from the origin for effective damping by an optimized Runge-Kutta time stepping scheme. In contrast to previous approaches, the method is robust when used in conjunction with high resolution schemes on fine meshes and with multigrid. Results are provided for a number of standard airfoil test cases