ABSTRACT

1. CO Mediated Sickle Cell Polymer Melting Crystal Bennett1, Oala Omer2, Taylor Rosemond3, & Phillip Thomas4 Mentors: Dr. Liping Liu5and Dr. Catherine D. White6 1,3,5Department of Mathematics, North Carolina A&T State University, Greensboro, NC 2Department of Chemistry, North Carolina A&T State University, Greensboro, NC 4,6Department of Biology, North Carolina A&T State University, Greensboro, NC GUI’s • GOALS • Verify the models’ work numerically using several different computational tools and numerical methods • Produce GUI using computational tools. • Be able to describe the different numerical schemes used to describe the model. ABSTRACT Sickle cell anemia is a disorder caused by a mutation in DNA that replaces the nucleic acid Glutamic with Valine. This replacement causes a change in the characteristics of hemoglobin that allows the monomers, the simplest units of chemically binding molecules, to stick together. These chains of monomers, called polymers, distort the shape and properties of the red blood cell. The malformed cells do not efficiently pass through capillaries or transport oxygen to the body's tissues. In order to make these cells more effective, the polymers must be broken apart. The process of breaking polymers apart is called melting. In the referenced study, the melting was induced by immersing the polymers in a buffer solution containing carbon monoxide. The mathematical model of this process was produced in a separate study. The purpose of this paper is to analyze and reproduce the current mathematical model using various computational and numerical tools. BACKGROUND Sickle cell anemia, is a homozygous recessive genetic disorder. Carriers of sickle cell must inherit the disease from both parents. Sickle cell is prevalent in persons of African and/or Mediterranean descent. These are areas that are heavily plagued with malaria. Normal hemoglobin (HbA) allows oxygen to be circulated throughout the body. As HbA moves from a oxygenated state to a deoxygenated state the hemoglobin molecules are free moving and the healthy cell is soft and flexible. However, sickle cells move from being oxygenated to being deoxygenated the defective hemoglobin ( HbS) sticks together creating polymers and the cells become brittle. The polymers distort the shape of the cell, causing them to resemble ‘sickles’. The rigid, distorted cells often become logged in the capillaries, inhibiting other red blood cells from transporting oxygen to the tissue and muscles in the body. The body recognizes these deformed red blood cells and attacks them, shortening their lifespan. The lack of oxygen these blockages create cause sickle cell patients to experience severe pain and can also cause long term damage to organs that are deprived of oxygen. To find a treatment for this debilitating disease, studies have focused on inducing polymer melting which would convert hemoglobin polymers into hemoglobin monomers, allowing sickle cells to return to a relatively normal state. The studies used as the basis for this project focused on carbon monoxide (CO) mediated polymer melting. The main experiment focuses on the melting of sickle cell hemoglobin polymers induced by rapid dilution using a stopped flow apparatus. Polymer melting in the absence of any hemoglobin ligand was compared to melting when the diluting buffer was saturated with carbon monoxide (CO). Since hemoglobin molecules have a stronger affinity for CO than they do for themselves, this is an effective method for dissolving the polymers into hemoglobin monomers. • WHAT’S THE PROBLEM • Defective hemoglobin ( HbS) moves from being oxygenated to being deoxygenated and sticks together creating polymers and the cells become brittle. • Polymers distort the shape of the cell, causing them to resemble ‘sickles’. • Rigid, distorted cells often become logged in the capillaries, inhibiting other red blood cells from transporting oxygen to the tissue and muscles in the body. MATERIALS AND METHODS Figure 7. Gui’s developed by Matlab, Excel, and Simbiology Figure 2. Dynamical System showing the flow of monomers as they become polymers, oxygenated polymers, and then oxygenated monomers • Variables and Constants of Model • Cmd: molar concentration of deoxyHbS in the solution phase. • Cpd: molar concentration of deoxyHbS cells in the polymer phase. • Cmco: molar concentration of CO-bound HbS in the solution phase. • Cpco: molar concentration of HbS polymer fibers. • CO]: molar concentration of carbon monoxide. • Km: CO binding (ligation) rate for solution phase HbS. • Kd: Dissociation (melting) rate constant of deoxyHbS cells. • Kco: CO-mediated dissociation rate constant • Kp: binding rate of CO to CO - bound polymers • Cs: Solubility concentration of phosphate buffer • Csco: Solubility concentration of CO in buffer Figure 3. The three possible states of the dynamical system including: CO Free, CO saturation, and a General CO mediated state. Figure 8. Graph of CO Bound monomers using various step sizes. Numerical Schemes Used Three states of Dynamical System Figure 8. Graphs comparing molar concentrations at various Kp values for CPCO and CPD. • Forward Euler • Backward Euler • Runge-Kutta (2nd and 4th order) • Non-Standard Finite Difference method • CO Free – when there is no CO present in the system there is no oxygenation taking place and so the monomers will become polymers and remain deoxygenated. • CO Saturation – enough CO is put into the system where no deoxygenating is present and so all oxygenated polymers will become oxygenated monomers. • General CO Case – an arbitrary amount of CO is present and both oxygenation and deoxygenating take place. All four populations are present in the model • CONCLUSIONS/FUTURE RESEARCH • Analyzed Mathematical Model • Sensitivity Analysis • Programmed Methods to Verify Claims • Investigated the effects of parameter changes • Investigate the effect temperature has on CO mediated melting • Modify the existing model to include temperature • Analyze the results the new model yields Figure 5.Scheme for Non Standard Finite Difference Method REFERENCES CITED Aroutiounian, S. Kh, J. G. Louderback, S. K. Ballas, and D. B. Kim-Shapiro. "Evidence for Carbon Monoxide Binding to Sickle Cell Polymers during Melting." Biophysical Chemistry 181st ser. 91.167 (2001): 167-81. Print. Chen, Mingxiang, Dominic P. Clemence, and Gregory Gibson. "Analysis of Numerical Schemes of a Mathematical Model for Sickle Cell Depolymerization." Applied Mathematics and Compution 216 (2010): 1489-500. Print. Daniel-Jones, Nikki, Mingxiang Chen, Dominic P. Clemence, and Gregory Gibson. "A Mathematical Model for Sickle Cell Depolymerization: Dynamical Properties and Numerical Experiments." International Journal of Qualitative Theory of Differential Equation and Applications2 (2008): 183-200. Print. Figure 1. Illustration of how pain crisis occur by outlining the polymer formation due to a lack of oxygen in the red blood cell. Figure 4. System discretized using Forward Euler. Figure 6. Discretization using NSFD