Paul J. Atzberger

Micromagnetic selection of aptamers in microfluidic channels

Aptamers are nucleic acid molecules that have been selected in vitro to bind to their molecular targets with high affinity and specificity. Typically, the systematic evolution of ligands by exponential enrichment (SELEX) process is used for the isolation of specific, high-affinity aptamers. SELEX, however, is an iterative process requiring multiple rounds of selection and amplification that demand significant time and labor. Here, we describe an aptamer discovery system that is rapid, highly efficient, automatable, and applicable to a wide range of targets, based on the integration of magnetic bead-based SELEX process with microfluidics technology. Our microfluidic SELEX (M-SELEX) method exploits a number of unique phenomena that occur at the microscale and implements a design that enables it to manipulate small numbers of beads precisely and isolate high-affinity aptamers rapidly. As a model to demonstrate the efficiency of the M-SELEX process, we describe here the isolation of DNA aptamers that tightly bind to the light chain of recombinant Botulinum neurotoxin type A (with low-nanomolar dissociation constant) after a single round of selection.

A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales

In modeling many biological systems, it is important to take into account flexible structures which interact with a fluid. At the length scale of cells and cell organelles, thermal fluctuations of the aqueous environment become significant. In this work, it is shown how the immersed boundary method of C.S. Peskin, The immersed boundary method, Acta Num. 11 (2002) 1-39.] for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic (?-3/2) decay for long times B.J. Alder, T.E. Wainwright, Decay of the Velocity Autocorrelation Function, Phys. Rev. A 1(1) (1970) 18-21]. A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method. Specifically, we present simulations of osmotic effects of molecular dimers, worm-like chain polymer knots, and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force. The theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales.

Stochastic Eulerian Lagrangian methods for fluid-structure interactions with thermal fluctuations

A computational approach is introduced for the study of the rheological properties of complex fluids and soft materials. The approach allows for a consistent treatment of microstructure elastic mechanics, hydrodynamic coupling, thermal fluctuations, and externally driven shear flows. A mixed description in terms of Eulerian and Lagrangian reference frames is used for the physical system. Microstructure configurations are represented in a Lagrangian reference frame. Conserved quantities, such as momentum of the fluid and microstructures, are represented in an Eulerian reference frame. The mathematical formalism couples these different descriptions using general operators subject to consistency conditions. Thermal fluctuations are taken into account in the formalism by stochastic driving fields introduced in accordance with the principles of statistical mechanics. To study the rheological responses of materials subject to shear, generalized periodic boundary conditions are developed where periodic images are shifted relative to the unit cell to induce shear. Stochastic numerical methods are developed for the formalism. As a demonstration of the methods, results are presented for the shear responses of a polymeric fluid, lipid vesicle fluid, and a gel-like material.

Shape matters in protein mobility within membranes

Significance Lateral Brownian diffusion of proteins in lipid membranes has been predicted by Saffman and Delbrück to depend only on protein size and on the viscosity of the membrane and of the surrounding medium. Using a single-molecule tracking technique on two transmembrane proteins that bend the membrane differently and are reconstituted in giant unilamellar vesicles, we show that the mobility of a membrane protein is crucially dependent on the local membrane deformation self-generated around the protein, which can be tuned by adjusting membrane tension. The feedback between membrane shaping and mobility is well explained by analytical and numerical models that include the friction of the deformed membrane patch with the surrounding medium and the friction internal to the bilayer. The lateral mobility of proteins within cell membranes is usually thought to be dependent on their size and modulated by local heterogeneities of the membrane. Experiments using single-particle tracking on reconstituted membranes demonstrate that protein diffusion is significantly influenced by the interplay of membrane curvature, membrane tension, and protein shape. We find that the curvature-coupled voltage-gated potassium channel (KvAP) undergoes a significant increase in protein mobility under tension, whereas the mobility of the curvature-neutral water channel aquaporin 0 (AQP0) is insensitive to it. Such observations are well explained in terms of an effective friction coefficient of the protein induced by the local membrane deformation.

Influence of Target Concentration and Background Binding on In Vitro Selection of Affinity Reagents

Nucleic acid-based aptamers possess many useful features that make them a promising alternative to antibodies and other affinity reagents, including well-established chemical synthesis, reversible folding, thermal stability and low cost. However, the selection process typically used to generate aptamers (SELEX) often requires significant resources and can fail to yield aptamers with sufficient affinity and specificity. A number of seminal theoretical models and numerical simulations have been reported in the literature offering insights into experimental factors that govern the effectiveness of the selection process. Though useful, these previous models have not considered the full spectrum of experimental factors or the potential impact of tuning these parameters at each round over the course of a multi-round selection process. We have developed an improved mathematical model to address this important question, and report that both target concentration and the degree of non-specific background binding are critical determinants of SELEX efficiency. Although smaller target concentrations should theoretically offer superior selection outcome, we show that the level of background binding dramatically affect the target concentration that will yield maximum enrichment at each round of selection. Thus, our model enables experimentalists to determine appropriate target concentrations as a means for protocol optimization. Finally, we perform a comparative analysis of two different selection methods over multiple rounds of selection, and show that methods with inherently lower background binding offer dramatic advantages in selection efficiency.

Spectral Analysis Methods for the Robust Measurement of the Flexural Rigidity of Biopolymers

The mechanical properties of biopolymers can be determined from a statistical analysis of the ensemble of shapes they exhibit when subjected to thermal forces. In practice, extracting information from fluorescence microscopy images can be challenging due to low signal/noise ratios and other artifacts. To address these issues, we develop a suite of tools for image processing and spectral data analysis that is based on a biopolymer contour representation expressed in a spectral basis of orthogonal polynomials. We determine biopolymer shape and stiffness using global fitting routines that optimize a utility function measuring the amount of fluorescence intensity overlapped by such contours. This approach allows for filtering of high-frequency noise and interpolation over sporadic gaps in fluorescence. We use benchmarking to demonstrate the validity of our methods, by analyzing an ensemble of simulated images generated using a simulated biopolymer with known stiffness and subjected to various types of image noise. We then use these methods to determine the persistence lengths of taxol-stabilized microtubules. We find that single microtubules are well described by the wormlike chain polymer model, and that ensembles of chemically identical microtubules show significant heterogeneity in bending stiffness, which cannot be attributed to sampling or fitting errors. We expect these approaches to be useful in the study of biopolymer mechanics and the effects of associated regulatory molecules.

Spatially adaptive stochastic numerical methods for intrinsic fluctuations in reaction-diffusion systems

Stochastic partial differential equations are introduced for the continuum concentration fields of reaction-diffusion systems. The stochastic partial differential equations account for fluctuations arising from the finite number of molecules which diffusively migrate and react. Spatially adaptive stochastic numerical methods are developed for approximation of the stochastic partial differential equations. The methods allow for adaptive meshes with multiple levels of resolution, Neumann and Dirichlet boundary conditions, and domains having geometries with curved boundaries. A key issue addressed by the methods is the formulation of consistent discretizations for the stochastic driving fields at coarse-refined interfaces of the mesh and at boundaries. Methods are also introduced for the efficient generation of the required stochastic driving fields on such meshes. As a demonstration of the methods, investigations are made of the role of fluctuations in a biological model for microorganism direction sensing based on concentration gradients. Also investigated, a mechanism for spatial pattern formation induced by fluctuations. The discretization approaches introduced for SPDEs have the potential to be widely applicable in the development of numerical methods for the study of spatially extended stochastic systems.

A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether

The kinesin family of motor proteins are involved in a variety of cellular processes that transport materials and generate force. With recent advances in experimental techniques, such as optical tweezers can probe individual molecules, there has been an increasing interest in understanding the mechanisms by which motor proteins convert chemical energy into mechanical work. Here we present a mathematical model for the chemistry and three dimensional mechanics of the kinesin motor protein which captures many of the force dependent features of the motor. For the elasticity of the tether that attaches cargo to the motor we develop a method for deriving the non-linear force-extension relationship from optical trap data. For the kinesin heads, cargo, and microscope stage we formulate a three dimensional Brownian Dynamics model that takes into account excluded volume interactions. To efficiently compute statistics from the model, an algorithm is proposed which uses a two step protocol that separates the simulation of the mechanical features of the model from the chemical kinetics of the model. Using this approach for a bead transported by the motor, the force dependent average velocity and randomness parameter are computed and compared with the experimental data.

Stochastic reduction method for biological chemical kinetics using time-scale separation.

Many processes in cell biology encode and process information and enact responses by modulating the concentrations of biological molecules. Such modulations serve functions ranging from encoding and transmitting information about external stimuli to regulating internal metabolic states. To understand how such processes operate requires gaining insights into the basic mechanisms by which biochemical species interact and respond to internal and external perturbations. One approach is to model the biochemical species concentrations through the van Kampen Linear Noise Equations, which account for the change in biochemical concentrations from reactions and account for fluctuations in concentrations. For many systems, the Linear Noise Equations exhibit stiffness as a consequence of the chemical reactions occurring at significantly different rates. This presents challenges in the analysis of the kinetics and in performing efficient numerical simulations. To deal with this source of stiffness and to obtain reduced models more amenable to analysis, we present a systematic procedure for obtaining effective stochastic dynamics for the chemical species having relatively slow characteristic time scales while eliminating representations of the chemical species having relatively fast characteristic time scales. To demonstrate the applicability of this multiscale technique in the context of Linear Noise Equations, the reduction is applied to models of gene regulatory networks. Results are presented which compare numerical results for the full system to the reduced descriptions. The presented stochastic reduction procedure provides a potentially versatile tool for systematically obtaining reduced approximations of Linear Noise Equations.

Stochastic Reductions for Inertial Fluid-Structure Interactions Subject to Thermal Fluctuations

We present analysis for the reduction of an inertial description of uid-structure in- teractions subject to thermal uctuations. We show how the viscous coupling between the immersed structures and the uid can be simplied in the regime where this coupling becomes increasingly strong. Many descriptions in uid mechanics and in the formulation of computational methods ac- count for uid-structure interactions through viscous drag terms to tranfer momentum from the uid to immersed structures. In the inertial regime, this coupling often introduces a prohibitively small time-scale into the temporal dynamics of the uid-structure system. This is further exacerbated in the presence of thermal uctuations. We discuss here a systematic reduction technique for the full inertial equations to obtain a simplied description where this coupling term is eliminated. This approach also accounts for the eective stochastic equations for the uid-structure dynamics. The analysis is based on use of the innitesmal generator of the SPDEs and a singular perturbation anal- ysis of the backward kolomogorov PDEs. We also discuss the physical motivations and interpretation of the obtained reduced description of the uid-structure system.