Niles A. Pierce

Programming biomolecular self-assembly pathways

In nature, self-assembling and disassembling complexes of proteins and nucleic acids bound to a variety of ligands perform intricate and diverse dynamic functions. In contrast, attempts to rationally encode structure and function into synthetic amino acid and nucleic acid sequences have largely focused on engineering molecules that self-assemble into prescribed target structures, rather than on engineering transient system dynamics. To design systems that perform dynamic functions without human intervention, it is necessary to encode within the biopolymer sequences the reaction pathways by which self-assembly occurs. Nucleic acids show promise as a design medium for engineering dynamic functions, including catalytic hybridization, triggered self-assembly and molecular computation. Here, we program diverse molecular self-assembly and disassembly pathways using a ‘reaction graph’ abstraction to specify complementarity relationships between modular domains in a versatile DNA hairpin motif. Molecular programs are executed for a variety of dynamic functions: catalytic formation of branched junctions, autocatalytic duplex formation by a cross-catalytic circuit, nucleated dendritic growth of a binary molecular ‘tree’, and autonomous locomotion of a bipedal walker.

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.

NUPACK: Analysis and design of nucleic acid systems

The Nucleic Acid Package (NUPACK) is a growing software suite for the analysis and design of nucleic acid systems. The NUPACK web server (http://www.nupack.org) currently enables: Analysis: thermodynamic analysis of dilute solutions of interacting nucleic acid strands. Design: sequence design for complexes of nucleic acid strands intended to adopt a target secondary structure at equilibrium. Utilities: evaluation, display, and annotation of equilibrium properties of a complex of nucleic acid strands. NUPACK algorithms are formulated in terms of nucleic acid secondary structure. In most cases, pseudoknots are excluded from the structural ensemble.

Programmable in situ amplification for multiplexed imaging of mRNA expression

In situ hybridization methods enable the mapping of mRNA expression within intact biological samples. With current approaches, it is challenging to simultaneously map multiple target mRNAs within whole-mount vertebrate embryos, representing a significant limitation in attempting to study interacting regulatory elements in systems most relevant to human development and disease. Here, we report a multiplexed fluorescent in situ hybridization method based on orthogonal amplification with hybridization chain reactions (HCR). With this approach, RNA probes complementary to mRNA targets trigger chain reactions in which fluorophore-labeled RNA hairpins self-assemble into tethered fluorescent amplification polymers. The programmability and sequence specificity of these amplification cascades enable multiple HCR amplifiers to operate orthogonally at the same time in the same sample. Robust performance is achieved when imaging five target mRNAs simultaneously in fixed whole-mount and sectioned zebrafish embryos. HCR amplifiers exhibit deep sample penetration, high signal-to-background ratios and sharp signal localization.

A Partition Function Algorithm for Nucleic Acid Secondary Structure Including Pseudoknots

Nucleic acid secondary structure models usually exclude pseudoknots due to the difficulty of treating these nonnested structures efficiently in structure prediction and partition function algorithms. Here, the standard secondary structure energy model is extended to include the most physically relevant pseudoknots. We describe an O(N5) dynamic programming algorithm, where N is the length of the strand, for computing the partition function and minimum energy structure over this class of secondary structures. Hence, it is possible to determine the probability of sampling the lowest energy structure, or any other structure of particular interest. This capability motivates the use of the partition function for the design of DNA or RNA molecules for bioengineering applications.

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.

An autonomous polymerization motor powered by DNA hybridization

We present a synthetic molecular motor capable of autonomous nanoscale transport in solution. Inspired by bacterial pathogens such as Rickettsia rickettsii, which locomote by inducing the polymerization of the protein actin at their surfaces to form ‘comet tails’1, the motor operates by polymerizing a double-helical DNA tail2. DNA strands are propelled processively at the living end of the growing polymers, demonstrating autonomous locomotion powered by the free energy of DNA hybridization.

Thermodynamic Analysis of Interacting Nucleic Acid Strands

Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for single-stranded structures. We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes. This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex. Alternatively, for large systems (e.g., a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem. Partition function and concentration information can then be used to calculate equilibrium base-pairing observables. The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality.

Conformational splitting: A more powerful criterion for dead‐end elimination

Dead‐end elimination (DEE) is a powerful theorem for selecting optimal protein side‐chain orientations from a large set of discrete conformations. The present work describes a new approach to dead‐end elimination that effectively splits conformational space into partitions to more efficiently eliminate dead‐ending rotamers. Split DEE makes it possible to complete protein design calculations that were previously intractable due to the combinatorial explosion of intermediate conformations generated during the convergence process.

Adjoint equations in CFD: duality, boundary conditions and solution behaviour

The first half of this paper derives the adjoint equations for inviscid and viscous compressible flow, with the emphasis being on the correct formulation of the adjoint boundary conditions and restrictions on the permissible choice of operators in the linearised functional. It is also shown that the boundary conditions for the adjoint problem can be simplified through the use of a linearised perturbation to generalised coordinates. The second half of the paper constructs the Greens functions for the quasi-lD and 2D Euler equations. These are used to show that the adjoint variables have a logarithmic singularity at the sonic line in the quasi-lD case, and a weak inverse square-root singularity at the upstream stagnation streamline in the 2D case, but are continuous at shocks in both cases.