Protein and Drug Design Algorithms Using Improved Biophysical Modeling

This thesis focuses on the development of algorithms that will allow protein design calculations to incorporate more realistic modeling assumptions. Protein design algorithms search large sequence spaces for protein sequences that are biologically and medically useful. Better modeling could improve the chance of success in designs and expand the range of problems to which these algorithms are applied. I have developed algorithms to improve modeling of backbone flexibility (DEEPer) and of more extensive continuous flexibility in general (EPIC and LUTE). I’ve also developed algorithms to perform multistate designs, which account for effects like specificity, with provable guarantees of accuracy (COMETS), and to accommodate a wider range of energy functions in design (EPIC and LUTE).

Using Helix-coil Models to Study Protein Unfolded States

An abstract of a thesis devoted to using helix-coil models to study unfolded states.\\

Research on polypeptide unfolded states has received much more attention in the last decade or so than it has in the past. Unfolded states are thought to be implicated in various

misfolding diseases and likely play crucial roles in protein folding equilibria and folding rates. Structural characterization of unfolded states has proven to be

much more difficult than the now well established practice of determining the structures of folded proteins. This is largely because many core assumptions underlying

folded structure determination methods are invalid for unfolded states. This has led to a dearth of knowledge concerning the nature of unfolded state conformational

distributions. While many aspects of unfolded state structure are not well known, there does exist a significant body of work stretching back half a century that

has been focused on structural characterization of marginally stable polypeptide systems. This body of work represents an extensive collection of experimental

data and biophysical models associated with describing helix-coil equilibria in polypeptide systems. Much of the work on unfolded states in the last decade has not been devoted

specifically to the improvement of our understanding of helix-coil equilibria, which arguably is the most well characterized of the various conformational equilibria

that likely contribute to unfolded state conformational distributions. This thesis seeks to provide a deeper investigation of helix-coil equilibria using modern

statistical data analysis and biophysical modeling techniques. The studies contained within seek to provide deeper insights and new perspectives on what we presumably

know very well about protein unfolded states. \\

Chapter 1 gives an overview of recent and historical work on studying protein unfolded states. The study of helix-coil equilibria is placed in the context

of the general field of unfolded state research and the basics of helix-coil models are introduced.\\

Chapter 2 introduces the newest incarnation of a sophisticated helix-coil model. State of the art modern statistical techniques are employed to estimate the energies

of various physical interactions that serve to influence helix-coil equilibria. A new Bayesian model selection approach is utilized to test many long-standing

hypotheses concerning the physical nature of the helix-coil transition. Some assumptions made in previous models are shown to be invalid and the new model

exhibits greatly improved predictive performance relative to its predecessor. \\

Chapter 3 introduces a new statistical model that can be used to interpret amide exchange measurements. As amide exchange can serve as a probe for residue-specific

properties of helix-coil ensembles, the new model provides a novel and robust method to use these types of measurements to characterize helix-coil ensembles experimentally

and test the position-specific predictions of helix-coil models. The statistical model is shown to perform exceedingly better than the most commonly used

method for interpreting amide exchange data. The estimates of the model obtained from amide exchange measurements on an example helical peptide

also show a remarkable consistency with the predictions of the helix-coil model. \\

Chapter 4 involves a study of helix-coil ensembles through the enumeration of helix-coil configurations. Aside from providing new insights into helix-coil ensembles,

this chapter also introduces a new method by which helix-coil models can be extended to calculate new types of observables. Future work on this approach could potentially

allow helix-coil models to move into use domains that were previously inaccessible and reserved for other types of unfolded state models that were introduced in chapter 1.

On Provable Algorithms for Determination of Continuous Protein Interdomain Motions from Residual Dipolar Couplings

Dynamics of biomolecules over various spatial and time scales are essential for biological functions such as molecular recognition, catalysis and signaling. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. Unfortunately, these distributions cannot be fully constrained by the limited information from experiments, making the problem an ill-posed one in the terminology of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem needs to be regularized by making assumptions, which inevitably introduce biases into the result.

Here, I present two continuous probability density function approaches to solve an important inverse problem called the RDC trigonometric moment problem. By focusing on interdomain orientations we reduced the problem to determination of a distribution on the 3D rotational space from residual dipolar couplings (RDCs). We derived an analytical equation that relates alignment tensors of adjacent domains, which serves as the foundation of the two methods. In the first approach, the ill-posed nature of the problem was avoided by introducing a continuous distribution model, which enjoys a smoothness assumption. To find the optimal solution for the distribution, we also designed an efficient branch-and-bound algorithm that exploits the mathematical structure of the analytical solutions. The algorithm is guaranteed to find the distribution that best satisfies the analytical relationship. We observed good performance of the method when tested under various levels of experimental noise and when applied to two protein systems. The second approach avoids the use of any model by employing maximum entropy principles. This 'model-free' approach delivers the least biased result which presents our state of knowledge. In this approach, the solution is an exponential function of Lagrange multipliers. To determine the multipliers, a convex objective function is constructed. Consequently, the maximum entropy solution can be found easily by gradient descent methods. Both algorithms can be applied to biomolecular RDC data in general, including data from RNA and DNA molecules.

A New Method for Modeling Free Surface Flows and Fluid-structure Interaction with Ocean Applications

The computational modeling of ocean waves and ocean-faring devices poses numerous challenges. Among these are the need to stably and accurately represent both the fluid-fluid interface between water and air as well as the fluid-structure interfaces arising between solid devices and one or more fluids. As techniques are developed to stably and accurately balance the interactions between fluid and structural solvers at these boundaries, a similarly pressing challenge is the development of algorithms that are massively scalable and capable of performing large-scale three-dimensional simulations on reasonable time scales. This dissertation introduces two separate methods for approaching this problem, with the first focusing on the development of sophisticated fluid-fluid interface representations and the second focusing primarily on scalability and extensibility to higher-order methods.

We begin by introducing the narrow-band gradient-augmented level set method (GALSM) for incompressible multiphase Navier-Stokes flow. This is the first use of the high-order GALSM for a fluid flow application, and its reliability and accuracy in modeling ocean environments is tested extensively. The method demonstrates numerous advantages over the traditional level set method, among these a heightened conservation of fluid volume and the representation of subgrid structures.

Next, we present a finite-volume algorithm for solving the incompressible Euler equations in two and three dimensions in the presence of a flow-driven free surface and a dynamic rigid body. In this development, the chief concerns are efficiency, scalability, and extensibility (to higher-order and truly conservative methods). These priorities informed a number of important choices: The air phase is substituted by a pressure boundary condition in order to greatly reduce the size of the computational domain, a cut-cell finite-volume approach is chosen in order to minimize fluid volume loss and open the door to higher-order methods, and adaptive mesh refinement (AMR) is employed to focus computational effort and make large-scale 3D simulations possible. This algorithm is shown to produce robust and accurate results that are well-suited for the study of ocean waves and the development of wave energy conversion (WEC) devices.