A Mosquito Pick-and-Place System for PfSPZ-based Malaria Vaccine Production.

The treatment of malaria is a global health challenge that stands to benefit from the widespread introduction of a vaccine for the disease. A method has been developed to create a live organism vaccine using the sporozoites (SPZ) of the parasite Plasmodium falciparum (Pf), which are concentrated in the salivary glands of infected mosquitoes. Current manual dissection methods to obtain these PfSPZ are not optimally efficient for large-scale vaccine production. We propose an improved dissection procedure and a mechanical fixture that increases the rate of mosquito dissection and helps to deskill this stage of the production process. We further demonstrate the automation of a key step in this production process, the picking and placing of mosquitoes from a staging apparatus into a dissection assembly. This unit test of a robotic mosquito pick-and-place system is performed using a custom-designed micro-gripper attached to a four degree of freedom (4-DOF) robot under the guidance of a computer vision system. Mosquitoes are autonomously grasped and pulled to a pair of notched dissection blades to remove the head of the mosquito, allowing access to the salivary glands. Placement into these blades is adapted based on output from computer vision to accommodate for the unique anatomy and orientation of each grasped mosquito. In this pilot test of the system on 50 mosquitoes, we demonstrate a 100% grasping accuracy and a 90% accuracy in placing the mosquito with its neck within the blade notches such that the head can be removed. This is a promising result for this difficult and non-standard pick-and-place task. NOTE TO PRACTITIONERS­: Automated processes could help increase malaria vaccine production to global scale. Currently, production requires technicians to manually dissect mosquitoes, a process that is slow, tedious, and requires a lengthy training regimen. This paper presents an an improved manual fixture and procedure that reduces technician training time. Further, an approach to automate this dissection process is proposed and the critical step of robotic manipulation of the mosquito with the aid of computer vision is demonstrated. Our approach may serve as a useful example of system design and integration for practitioners that seek to perform new and challenging pick-and-place tasks with small, non-uniform, and highly deformable objects.

Continuous body 3-D reconstruction of limbless animals.

Limbless animals such as snakes, limbless lizards, worms, eels and lampreys move their slender, long bodies in three dimensions to traverse diverse environments. Accurately quantifying their continuous body's 3-D shape and motion is important for understanding body-environment interactions in complex terrain, but this is difficult to achieve (especially for local orientation and rotation). Here, we describe an interpolation method to quantify continuous body 3-D position and orientation. We simplify the body as an elastic rod and apply a backbone optimization method to interpolate continuous body shape between end constraints imposed by tracked markers. Despite over-simplifying the biomechanics, our method achieves a higher interpolation accuracy (∼50% error) in both 3-D position and orientation compared with the widely used cubic B-spline interpolation method. Beyond snakes traversing large obstacles as demonstrated, our method applies to other long, slender, limbless animals and continuum robots. We provide codes and demo files for easy application of our method.

Rate of Entropy Production in Stochastic Mechanical Systems.

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker-Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group.

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker-Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

Mathematical aspects of molecular replacement. V. Isolating feasible regions in motion spaces.

This paper mathematically characterizes the tiny feasible regions within the vast 6D rotation-translation space in a full molecular replacement (MR) search. The capability to a priori isolate such regions is potentially important for enhancing robustness and efficiency in computational phasing in macromolecular crystallography (MX). The previous four papers in this series have concentrated on the properties of the full configuration space of rigid bodies that move relative to each other with crystallographic symmetry constraints. In particular, it was shown that the configuration space of interest in this problem is the right-coset space Γ\G, where Γ is the space group of the chiral macromolecular crystal and G is the group of rigid-body motions, and that fundamental domains FΓ\G can be realized in many ways that have interesting algebraic and geometric properties. The cost function in MR methods can be viewed as a function on these fundamental domains. This, the fifth and final paper in this series, articulates the constraints that bodies packed with crystallographic symmetry must obey. It is shown that these constraints define a thin feasible set inside a motion space and that they fall into two categories: (i) the bodies must not interpenetrate, thereby excluding so-called `collision zones' from consideration in MR searches; (ii) the bodies must be in contact with a sufficient number of neighbors so as to form a rigid network leading to a physically realizable crystal. In this paper, these constraints are applied using ellipsoidal proxies for proteins to bound the feasible regions. It is shown that the volume of these feasible regions is small relative to the total volume of the motion space, which justifies the use of ellipsoids as proxies for complex proteins in MR searches, and this is demonstrated with P1 (the simplest space group) and with P212121 (the most common space group in MX).

Lateral Oscillation and Body Compliance Help Snakes and Snake Robots Stably Traverse Large, Smooth Obstacles.

Snakes can move through almost any terrain. Similarly, snake robots hold the promise as a versatile platform to traverse complex environments such as earthquake rubble. Unlike snake locomotion on flat surfaces which is inherently stable, when snakes traverse complex terrain by deforming their body out of plane, it becomes challenging to maintain stability. Here, we review our recent progress in understanding how snakes and snake robots traverse large, smooth obstacles such as boulders and felled trees that lack "anchor points" for gripping or bracing. First, we discovered that the generalist variable kingsnake combines lateral oscillation and cantilevering. Regardless of step height and surface friction, the overall gait is preserved. Next, to quantify static stability of the snake, we developed a method to interpolate continuous body in three dimensions (3D) (both position and orientation) between discrete tracked markers. By analyzing the base of support using the interpolated continuous body 3-D kinematics, we discovered that the snake maintained perfect stability during traversal, even on the most challenging low friction, high step. Finally, we applied this gait to a snake robot and systematically tested its performance traversing large steps with variable heights to further understand stability principles. The robot rapidly and stably traversed steps nearly as high as a third of its body length. As step height increased, the robot rolled more frequently to the extent of flipping over, reducing traversal probability. The absence of such failure in the snake with a compliant body inspired us to add body compliance to the robot. With better surface contact, the compliant body robot suffered less roll instability and traversed high steps at higher probability, without sacrificing traversal speed. Our robot traversed large step-like obstacles more rapidly than most previous snake robots, approaching that of the animal. The combination of lateral oscillation and body compliance to form a large, reliable base of support may be useful for snakes and snake robots to traverse diverse 3-D environments with large, smooth obstacles.

Quantizing Euclidean Motions via Double-Coset Decomposition.

Concepts from mathematical crystallography and group theory are used here to quantize the group of rigid-body motions, resulting in a "motion alphabet" with which robot motion primitives are expressed. From these primitives it is possible to develop a dictionary of physical actions. Equipped with an alphabet of the sort developed here, intelligent actions of robots in the world can be approximated with finite sequences of characters, thereby forming the foundation of a language in which robot motion is articulated. In particular, we use the discrete handedness-preserving symmetries of macromolecular crystals (known in mathematical crystallography as Sohncke space groups) to form a coarse discretization of the space SE(3) of rigid-body motions. This discretization is made finer by subdividing using the concept of double-coset decomposition. More specifically, a very efficient, equivolumetric quantization of spatial motion can be defined using the group-theoretic concept of a double-coset decomposition of the form Γ\SE(3)/Δ, where Γ is a Sohncke space group and Δ is a finite group of rotational symmetries such as those of the icosahedron. The resulting discrete alphabet is based on a very uniform sampling of SE(3) and is a tool for describing the continuous trajectories of robots and humans. An efficient coarse-to-fine search algorithm is presented to round off any motion sampled from the continuous group of motions to the nearest element of our alphabet. It is shown that our alphabet and this efficient rounding algorithm can be used as a geometric data structure to accelerate the performance of other sampling schemes designed for desirable dispersion or discrepancy properties. Moreover, the general "signals to symbols" problem in artificial intelligence is cast in this framework for robots moving continuously in the world.

Phantom with multiple active points for ultrasound calibration.

Accurate tracking and localization of ultrasound (US) images are used in various computer-assisted interventions. US calibration is a preoperative procedure to recover the transformation bridging the tracking sensor and the US image coordinate systems. Although many calibration phantom designs have been proposed, a limitation that hinders the resulted calibration accuracy is US elevational beam thickness. Previous studies have proposed an active-echo (AE)-based calibration concept to overcome this limitation by utilizing dynamic active US feedback from a single PZT element-based phantom, which assists in placing the phantom within the US elevational plane. However, the process of searching elevational midplane is time-consuming and requires dedicated hardware to enable "AE" functionality. Extending this active phantom, we present a US calibration concept and associated mathematical framework enabling fast and accurate US calibration using multiple "active" points. The proposed US calibration can simplify the calibration procedure by minimizing the number of times midplane search is performed and shortening calibration time. This concept is demonstrated with a configuration mechanically tracking a US probe using a robot arm. We validated the concept through simulation and experiment, and achieved submillimeter calibration accuracy. This result indicates that the multiple active-point phantom has potential to provide superior calibration performance for applications requiring high tracking accuracy.

Symmetrical Parameterization of Rigid Body Transformations for Biomolecular Structures.

Assessing preferred relative rigid body position and orientation is important in the description of biomolecular structures (such as proteins) and their interactions. In this article, we extend and apply the "symmetrical parameterization," which we recently introduced in the kinematics community, to address problems in structural biology. We also review parameterization methods that are widely used in structural biology to describe relative rigid body motions (in particular, orientations) as a basis for comparison. The new symmetrical parameterization is useful in describing the relative biomolecular rigid body motions, where the parameters are symmetrical in the sense that the subunits of a complex biomolecular structure are described in the same way for the corresponding motion and its inverse. The properties of this new parameterization, singularity analysis, and inverse kinematics are also investigated in more detail. Finally, parameterization is applied to real biomolecular structures and a potential application to structure modeling of symmetric macromolecules to show the efficacy of the symmetrical parameterization in the field of computational structural biology.

Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces.

In molecular-replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid-body models of macromolecules in crystallographic asymmetric units. The properties of the space of non-redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three-dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension.

Cross-Validation of Data Compatibility Between Small Angle X-ray Scattering and Cryo-Electron Microscopy.

Cryo-electron microscopy (EM) and small angle X-ray scattering (SAXS) are two different data acquisition modalities often used to glean information about the structure of large biomolecular complexes in their native states. A SAXS experiment is generally considered fast and easy but unveils the structure at very low resolution, whereas a cryo-EM experiment needs more extensive preparation and postacquisition computation to yield a three-dimensional (3D) density map at higher resolution. In certain applications, we may need to verify whether the data acquired in the SAXS and cryo-EM experiments correspond to the same structure (e.g., before reconstructing the 3D density map in EM). In this article, a simple and fast method is proposed to verify the compatibility of the SAXS and EM experimental data. The method is based on averaging the two-dimensional correlation of EM images and the Abel transform of the SAXS data. Orientational preferences are known to exist in cryo-EM experiments, and we also consider these effects on our method. The results are verified on simulations of conformational states of large biomolecular complexes.

Computational Analysis of SAXS Data Acquisition.

Small-angle x-ray scattering (SAXS) is an experimental biophysical method used for gaining insight into the structure of large biomolecular complexes. Under appropriate chemical conditions, the information obtained from a SAXS experiment can be equated to the pair distribution function, which is the distribution of distances between every pair of points in the complex. Here we develop a mathematical model to calculate the pair distribution function for a structure of known density, and analyze the computational complexity of these calculations. Efficient recursive computation of this forward model is an important step in solving the inverse problem of recovering the three-dimensional density of biomolecular structures from their pair distribution functions. In particular, we show that integrals of products of three spherical-Bessel functions arise naturally in this context. We then develop an algorithm for the efficient recursive computation of these integrals.

Framed curves and knotted DNA.

The present mini-review covers the local and global geometry of framed curves and the computation of twist and writhe in knotted DNA circles. Classical inequalities relating the total amount of bending of a closed space curve and associated knot parameters are also explained.

Multiscale modeling of double-helical DNA and RNA: a unification through Lie groups.

Several different mechanical models of double-helical nucleic-acid structures that have been presented in the literature are reviewed here together with a new analysis method that provides a reconciliation between these disparate models. In all cases, terminology and basic results from the theory of Lie groups are used to describe rigid-body motions in a coordinate-free way, and when necessary, coordinates are introduced in a way in which simple equations result. We consider double-helical DNAs and RNAs which, in their unstressed referential state, have backbones that are either straight, slightly precurved, or bent by the action of a protein or other bound molecule. At the coarsest level, we consider worm-like chains with anisotropic bending stiffness. Then, we show how bi-rod models converge to this for sufficiently long filament lengths. At a finer level, we examine elastic networks of rigid bases and show how these relate to the coarser models. Finally, we show how results from molecular dynamics simulation at full atomic resolution (which is the finest scale considered here) and AFM experimental measurements (which is at the coarsest scale) relate to these models.

Mathematical aspects of molecular replacement. II. Geometry of motion spaces.

Molecular replacement (MR) is a well established computational method for phasing in macromolecular crystallography. In MR searches, spaces of motions are explored for determining the appropriate placement of rigid models of macromolecules in crystallographic asymmetric units. In the first paper of this series, it was shown that this space of motions, when endowed with an appropriate composition operator, forms an algebraic structure called a quasigroup. In this second paper, the geometric properties of these MR search spaces are explored and analyzed. This analysis includes the local differential geometry, global geometry and symmetry properties of these spaces.

Mathematical aspects of molecular replacement. I. Algebraic properties of motion spaces.

Molecular replacement (MR) is a well established method for phasing of X-ray diffraction patterns for crystals composed of biological macromolecules of known chemical structure but unknown conformation. In MR, the starting point is known structural domains that are presumed to be similar in shape to those in the macromolecular structure which is to be determined. A search is then performed over positions and orientations of the known domains within a model of the crystallographic asymmetric unit so as to best match a computed diffraction pattern with experimental data. Unlike continuous rigid-body motions in Euclidean space and the discrete crystallographic space groups, the set of motions over which molecular replacement searches are performed does not form a group under the operation of composition, which is shown here to lack the associative property. However, the set of rigid-body motions in the asymmetric unit forms another mathematical structure called a quasigroup, which can be identified with right-coset spaces of the full group of rigid-body motions with respect to the chiral space group of the macromolecular crystal. The algebraic properties of this space of motions are articulated here.

A Stochastic Kinematic Model of Class Averaging in Single-Particle Electron Microscopy.

Single-particle electron microscopy is an experimental technique that is used to determine the 3D structure of biological macromolecules and the complexes that they form. In general, image processing techniques and reconstruction algorithms are applied to micrographs, which are two-dimensional (2D) images taken by electron microscopes. Each of these planar images can be thought of as a projection of the macromolecular structure of interest from an a priori unknown direction. A class is defined as a collection of projection images with a high degree of similarity, presumably resulting from taking projections along similar directions. In practice, micrographs are very noisy and those in each class are aligned and averaged in order to reduce the background noise. Errors in the alignment process are inevitable due to noise in the electron micrographs. This error results in blurry averaged images. In this paper, we investigate how blurring parameters are related to the properties of the background noise in the case when the alignment is achieved by matching the mass centers and the principal axes of the experimental images. We observe that the background noise in micrographs can be treated as Gaussian. Using the mean and variance of the background Gaussian noise, we derive equations for the mean and variance of translational and rotational misalignments in the class averaging process. This defines a Gaussian probability density on the Euclidean motion group of the plane. Our formulation is validated by convolving the derived blurring function representing the stochasticity of the image alignments with the underlying noiseless projection and comparing with the original blurry image.

Modeling loop entropy.

Proteins fold from a highly disordered state into a highly ordered one. Traditionally, the folding problem has been stated as one of predicting "the" tertiary structure from sequential information. However, new evidence suggests that the ensemble of unfolded forms may not be as disordered as once believed, and that the native form of many proteins may not be described by a single conformation, but rather an ensemble of its own. Quantifying the relative disorder in the folded and unfolded ensembles as an entropy difference may therefore shed light on the folding process. One issue that clouds discussions of "entropy" is that many different kinds of entropy can be defined: entropy associated with overall translational and rotational Brownian motion, configurational entropy, vibrational entropy, conformational entropy computed in internal or Cartesian coordinates (which can even be different from each other), conformational entropy computed on a lattice, each of the above with different solvation and solvent models, thermodynamic entropy measured experimentally, etc. The focus of this work is the conformational entropy of coil/loop regions in proteins. New mathematical modeling tools for the approximation of changes in conformational entropy during transition from unfolded to folded ensembles are introduced. In particular, models for computing lower and upper bounds on entropy for polymer models of polypeptide coils both with and without end constraints are presented. The methods reviewed here include kinematics (the mathematics of rigid-body motions), classical statistical mechanics, and information theory.

The Path-of-Probability Algorithm for Steering and Feedback Control of Flexible Needles.

In this paper we develop a new framework for path planning of flexible needles with bevel tips. Based on a stochastic model of needle steering, the probability density function for the needle tip pose is approximated as a Gaussian. The means and covariances are estimated using an error propagation algorithm which has second order accuracy. Then we adapt the path-of-probability (POP) algorithm to path planning of flexible needles with bevel tips. We demonstrate how our planning algorithm can be used for feedback control of flexible needles. We also derive a closed-form solution for the port placement problem for finding good insertion locations for flexible needles in the case when there are no obstacles. Furthermore, we propose a new method using reference splines with the POP algorithm to solve the path planning problem for flexible needles in more general cases that include obstacles.

Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms.

This paper presents an efficient group-theoretic approach for computing the statistics of non-reversal random walks (NRRW) on lattices. These framed walks evolve on proper crystallographic space groups. In a previous paper we introduced a convolution method for computing the statistics of NRRWs in which the convolution product is defined relative to the space-group operation. Here we use the corresponding concept of the fast Fourier transform for functions on crystallographic space groups together with a non-Abelian version of the convolution theorem. We develop the theory behind this technique and present numerical results for two-dimensional and three-dimensional lattices (square, cubic and diamond). In order to verify our results, the statistics of the end-to-end distance and the probability of ring closure are calculated and compared with results obtained in the literature for the random walks for which closed-form expressions exist.