Mathematical Modeling of the Calcium Binding Proteins Calmodulin and Troponin C Undergraduate Research Thesis Presented in partial fulfillment of the requirements for graduation with honors research distinction in the undergraduate colleges of The Ohio State University by Garrett Hauck The Ohio State University 2021 Project Advisor: Professor Jonathan Davis, Department of Physiology and Cell Biology Abstract Cellular calcium either directly controls, or at a minimum modulates nearly, if not, all human functions. As such, the structural motif that forms the calcium binding pocket (EF-hand) is one of the most common protein elements found in Mammalia. The calcium binding properties of the EF-hands appear to be “tuned” to receive and respond to different calcium signals and can vary by over 6 orders of magnitude. Our laboratory is attempting to understand the rules that govern calcium binding to proteins, so that we might be able to engineer these proteins as gene therapies to treat diseases. Calmodulin is a small, switch-like, calcium binding protein that plays multiple roles in all human cells depending on what protein system it is attached. We are using rational design, super computers as well as evolutionary clues to help us engineer calmodulins with specific functionalities. Using a simple mathematical model to describe the calcium binding events of calmodulin and available steady-state and kinetic data, we determined the exact rate parameters that govern this interaction for wild type calmodulin and other variants (such as plant calmodulins, disease mutants, and engineered calmodulins). Troponin C is the “cousin” of calmodulin and is very important in the calcium regulation of muscle contraction. Using a previously created mathematical model for the activation of the cardiac troponin complex, we modified the rate parameters to effectively simulate the calcium binding data for the slow skeletal isoform and characterized the effects of a potential small molecule therapeutic. This information better helps us identify what we want to alter in order to achieve a particular outcome (in terms of engineering novel calmodulins) and may help explain the different responses between cardiac and slow skeletal troponin. ii Table of Contents Abstract……………………………………………………………………………………………ii Chapter 1…………………………………………………………………………………………..1 Properties of Calmodulin………………………………………………………………….3 Calmodulin Variants………………………………………………………………………5 Previous Calmodulin Models……………………………………………………………...7 Troponin C and the Troponin Complex………………………………………………….11 Previous Troponin Models……………………………………………………………….15 Chapter 2 – Methods……………………………………………………………………………..17 Protein Expression & Purification……………………………………………………….18 Steady-State Calcium Titrations………………………………………………………....18 Kinetic Experiments……………………………………………………………………..19 Model Creation & Simulations…………………………………………………………..21 Chapter 3 – Results……………………………………………………………………………....26 Wild Type Calmodulin – C Domain……………………………………………………..27 Other Calmodulin Variants – C Domain………………………………………………...27 Wild Type and Other Calmodulin Variants – N Domain………………………………..30 Symmetrical Parallel Model of Wild Type C Domain…………………………………..31 iii Asymmetrical Parallel Models of Wild Type C Domain………………………………...32 Slow Skeletal Troponin Model………………………………………………………...…33 Cardiac Troponin Drug Model…………………………………………………………...35 Chapter 4 – Discussion…………………………………………………………………………..37 Series vs. Parallel Models………………………………………………………………..38 Symmetrical vs. Asymmetrical Models………………………………………………….38 Mathematical Considerations…………………………………………………………….40 Slow Skeletal Troponin Insights…………………………………………………………42 Troponin Drug Model Takeaways……………………………………………………….43 Conclusion & Future Directions…………………………………………………………44 References………………………………………………………………………………………..46 Figures and Tables…………………………………………………………………………….....50 iv Chapter 1 In nearly every branch of science, from the physical sciences to the social sciences, the concept of modeling is incredibly important. Whether the subject is molecular biology, quantum physics, abnormal psychology, or criminology, a model will always be of use to aid in the description of systems and their study. In addition to simplifying a system of interest, models can be used to perform experiments that may not be feasible for a number of reasons – too costly, too risky, or too difficult. Researchers or others may also opt to use a model system to anticipate the impacts of an action or experimental modification. Perhaps a demolition crew would like to model the impact of their explosives on a building in an empty field before tearing down a building in the heart of a city, or perhaps a biochemist would like to anticipate whether an experiment with costly reagents is worth pursuing before spending the money. In any sense, a model can be beneficial in practically any scenario, and it is clear that models for biochemical systems can aid in the understanding of their molecular interactions. When it comes to modeling biochemical interactions, there are several approaches that can be taken. One of the most frequently used approaches is modeling a system with molecular structures. A molecular structure can be obtained in many different ways – there are the experimental approaches of X-ray crystallography, nuclear magnetic resonance, and cryogenic electron microscopy, and a structure can also be predicted using the plethora of homology modeling software that now exists. If the structure of a protein or nucleic acid is known or predicted, researchers can use the model of the macromolecule to make predictions about interactions, investigate the potential impacts of mutations, and visualize potential or real conformational changes. These structural models are very powerful, allowing the scientist to view an extraordinarily small system on a computer screen. In addition to structural models, models can 1 be made from a mathematical standpoint to simulate the kinetics of a system, which can be used to describe binding events in a different fashion than structural models. The advantages of this approach are plentiful and distinct from a molecular structure. Firstly, the system of interest can be expanded to include more than a structural model. A mathematical model can involve several macromolecules and/or several small molecules or ions and can simulate several different binding events. Second, a mathematical model can allow other questions to be asked or other aspects of a system to be looked at. A mathematical model uses microscopic properties of individual molecules to describe binding events that cannot be investigated through structure. The affinity for a target cannot be determined by looking at a structure (although it can be estimated or predicted relative to similar interactions), but it can be determined using the specifics of a mathematical model. Finally, mathematical models are more efficient than most experiments that involve structural models. The software required is cheaper, there is less demand in terms of computing power, and the experiments simply do not consumer as much time. With the same resources, more work can be accomplished in one day with a mathematical model than most experiments that a structural model is used for. It is this concept of mathematical modeling that this project is centered around. In the context of the calcium binding protein calmodulin (CaM), the protein-ion binding events are useful to model to obtain a deeper understanding of the microscopic mechanisms that govern the properties of this interaction. Similarly, the calcium binding protein troponin C (TnC), part of the troponin complex (Tn), and its interaction with troponin I (TnI), can be effectively modeled to understand the specifics of the binding events. Both of these proteins are important for function and regulation of the contraction/relaxation cycle in muscle, and hence have serious implications 2 in several disease states, such as arrythmia and various cardiomyopathies. For these reasons, it is important to understand them with as much depth as possible. Properties of Calmodulin As a protein, CaM is arguably one of the most influential proteins to be characterized and researched. CaM is essential for mammalian life, so much so that there are multiple copies of the CaM gene in certain organisms to prevent lethal recombination events involving the CaM gene (Fischer et al., 1988). CaM is a calcium-binding protein that allows for signals communicated by changing calcium levels to be translated into specific cellular and biochemical events (Chin & Means, 2000). CaM has hundreds of downstream binding targets, which can include a plethora of different protein classes, examples including Myosin Light Chain Kinase (MLCK), Calcineurin (CaN), Nitric Oxide Synthase (NOS), and the Ryanodine Receptor (RyR) (Vogel, 1994). CaM is a small protein of 148 amino acid residues, consisting of two domains which create a dumbbell shape. The two domains are connected via a flexible linker, allowing for a large amount of flexibility within the CaM structure, hence providing the opportunity for the vast array of downstream targets for CaM. Each domain contains two calcium binding sites, which are part of the EF-hand family, providing four binding motifs in one protein. The EF-hand structural motif consists of a helix-loop-helix pattern, where the loop contains amino acids designed to coordinate a calcium ion. In addition to binding calcium, CaM is known to competitively bind magnesium ions, competing with up to 1 mM free magnesium