ABSTRACTSickle 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.

BACKGROUNDSickle 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.

CO Mediated Sickle Cell Polymer Melting

Crystal Bennett1, Oala Omer2, Taylor Rosemond3, & Phillip Thomas4

Mentors: Dr. Liping Liu5 and Dr. Catherine D. White6

1,3,5Department of Mathematics, North Carolina A&T State University, Greensboro, NC2Department of Chemistry, North Carolina A&T State University, Greensboro, NC

4,6Department of Biology, North Carolina A&T State University, Greensboro, NC

REFERENCES CITEDAroutiounian, 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.

MATERIALS AND METHODS

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

GUI’sWHAT’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.

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.

Variables and Constants of Model• Cmd: molar concentration of deoxy HbS in the solution

phase.• Cpd: molar concentration of deoxy HbS 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 deoxy HbS

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 2. Dynamical System showing the flow of monomers as they become polymers, oxygenated polymers, and then oxygenated monomers

Figure 3. The three possible states of the dynamical system including: CO Free, CO saturation, and a General CO mediated state.

Figure 7. Gui’s developed by Matlab, Excel, and Simbiology

• Forward Euler• Backward Euler• Runge-Kutta (2nd and 4th order)• Non-Standard Finite Difference method

Numerical Schemes Used

Figure 6. Discretization using NSFD

Figure 1. Illustration of how pain crisis occur by outlining the polymer formation due to a lack of oxygen in the red blood cell.

• 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

Three states of Dynamical System

Figure 4. System discretized using Forward Euler.

Figure 5. Scheme for Non Standard Finite Difference Method

Figure 8. Graph of CO Bound monomers using various step sizes.

Figure 8. Graphs comparing molar concentrations at various Kp values for CPCO and CPD.