Mathematical Model for Diabetes in Non-Obese Diabetic Mice Valerie Vidal MTH 496: Senior Project Advisor: Shawn D. Ryan, Dept. of Mathematics Spring 2018
Abstract In the United States, the number of people who suffer from diabetes is currently on the rise. In order to develop a cure or treatment for diabetes, researchers need to understand how this disease develops in the body. For type 1 diabetics, the body targets, attacks, and kills healthy, insulinproducing beta-cells. In the paper "Modeling Cyclic Waves of Circulating T Cells in Autoimmune Diabetes", the authors Joseph Mahaffy and Leah Edelstein-Keshet develop a model for the development and behavior of T cells in non-obese diabetic mice. Mice and humans have similar diabetic tendencies, so the study of these mice can aid in the production of a model for how T cells react in type 1 diabetic humans. Mahaffay and Edlestein-Keshet created their model using several assumptions, one of which was that effector T-cells are terminal. Although this was useful in simplifying the model, when considering the actual behavior of T-cells, some effector cells become memory cells instead of dying off. This paper takes Mahaffy's and Edlestein- Keshet's reasearch and explores what would happen if one were to consider a model that includes a factor to represent the effector cells that become memory cells.
Table of Contents 1. Introduction 1 2. Background Information 2 3. Modified Model 5 4. Conclusions/Further Research 7 5. References..9 6. Appendix 10
Introduction According to the American Diabetes Association, out of the population of individuals who suffer from diabetes, 5% have Type 1. This is still a significant amount of people with around 1.25 million Americans who currently have it and an estimated 40,000 people who will be newly diagnosed every year in the United States [6]. Those who have Type 1 diabetes will likely suffer from hyperglycemia (high blood glucose) and/or hypoglycemia (low blood glucose). It is important for individuals with these symptoms to monitor them and use preventative measures to maintain a healthy blood glucose level. According to an analysis published by Nova Science Publishers, Inc., an active and fit way of life can reduce blood glucose level and increase insulin sensitivity, primarily in the skeletal muscles in diabetic patients, which leads to a reduced need for insulin [4]. Along with this prescribed diet and exercise, in order to decrease the cases and effects of diabetes, it is crucial to continue the effort in understanding how Type 1 diabetes develops in patients. In order to study the development of diabetes, many researchers rely on the study of mice. Mice have immune systems similar to humans [2] making it possible to develop models of cellular behavior that closely mimics that of humans. Trudeau et al. used non-obese diabetic (NOD) mice to study how T-cells react. NOD mice are mice that either are Type 1 diabetic or have a high chance of becoming Type 1 diabetic. Type 1 diabetics suffer from an autoimmune disease that causes their T-cells to attack β-cells which produce insulin to regulate blood sugar. When a β- cell dies it releases peptide and then the peptide is cleared immediately[1]. In Type 1 diabetics, the peptide isn't cleared as fast and T cells register this extra peptide as a foreign substance. When a T cell encounters a foreign substance it becomes activated and either becomes an effector cell or a memory cell. Effector cells are the cells that attack a foreign substance as if it were a disease. Memory cells are T cells that retain information about the foreign substance but do not attack. If an activated T cell becomes a memory cell, the cell can then multiply into more memory cells or become regular T cells again. If an activated T-cell becomes an effector cell, then it multiplies into more effector cells and attacks the β-cells. After the attack and when the β- cells die, they release the peptide and the process begins again continuing until all of the β-cells have been terminated[5]. A graphical representation of this process is as follows:
Figure 1. Stages of a T-cell attacking β-cells include activation, becoming an effector cell or memory cell and attacking/killing the β-cells [5]. Background Mahaffay and Edlestein-Keshet created a linear diagram of the way that the T-cells behave in the NOD mice. The diagram is as follows:
Figure 2. A linear representation of Figure 1.1. Activated T-cells become either memory or effector cells and the effector cells, attack and kill insulin producing β-cells [1] In this diagram, A represents the activated T-cells, M represents the memory T-cells, and E represents the effector T-cells. B represents β-cells and p represents the peptide produced when a β-cell dies. Mahaffay and Edelstein-Keshet developed a model that determines the amount of T-cells in the body of a NOD mouse. Before developing the model, they made the following assumptions; Consider the T-cells to be in a well-mixed compartment. Consider only the cells behavior after 4-5 weeks. Effector Cells are terminal. Peptide is produced at a linear rate. Once β-cells are gone, the immune response stops. In the mice (and humans) T-cells are in three distinct parts of the body: the blood, the pancreas, and the lymph nodes. In the first assumption, however, they combined the three and consider the T-cells to be in one well-mixed compartment so that they could eliminate the parameters of the rate at which T cells flow between these parts of the body. This allows for a spatially homogeneous model. The authors make the second assumption because in the first few weeks of life the immune system is still developing and recognizing diseases and healthy substances. The third assumption is that once an activated T cell becomes an effector cell, it cannot become a memory cell or an activated T-cell again; it dies as an effector cell. The last assumption is assuming that once there are no more β-cells to attack, the T-cells stop attacking and no longer produce more effector cells or memory cells to target the β-cells. This last assumptions is a shut off mechanism for the model.
Joseph Mahaffy and Leah Edelstein-Keshet created a mathematical model to represent the growth and decay of the different types of T-cells. The model that Mahaffay and Edelstein- Keshet developed using these assumptions is the following: Figure 3. Mahaffay and Edelstein-Keshet s 5 equation model for the rates of change of number of Activated T-cells, Memory T-cells, Effector T-cells, β-cells, and the amount of peptide. [1] In this model, A represents activated T-cells, M represents memory cells, E represents effector cells, p represents peptide and B represents the β-cells. The other parameters are included in the following table as well as the values used in Mahaffay s and Edelstein-Keshet s paper.
Figure 4. The parameters and the values used in the models for T-cells in Modeling Cyclic Waves of Circulating T-cells in Autoimmune Diabetes In this model, the change in the amount of peptide and β-cells are included. However, since the change in these two occurs over the course of days and the change in the other types of cells occurs over the course of a few hours, the authors considered them to be slow variables and were able to reduce the model to a 3-equation model instead of 5. The reduced model can be written as,: Figure 5. Mahaffay and Edelstein-Keshet s 3 equation model for the rates of change of number of Activated T-cells, Memory T-cells, Effector T-cells.
In the Mahaffay and Edlestein-Keshet model, the original model and graphs were recreated using Matlab: Figure 6. Graph of the change in Activated, Effector, and Memory T-cells as well as β-cells created using the code for the 3 equation model found in Modeling Cyclic Waves of Circulating T-cells in Autoimmune Diabetes [1] In this figure, it is clear that there is a spike in the number of activated T-cells and that there is a sudden decrease in β-cells around time t=30. In the original paper, Mahaffay and Edlestein- Keshet included a graph that spiked several times causing the β-cells to die out. Modified Model Although the Mahaffay and Edlestein-Keshet model creates an understanding of the behavior of T-cells, it could represent biology more closely by including a term for when effector cells become memory cells. One of the assumptions in Mahaffay and Edelstein-Keshet s model was that once effector cells attacked β-cells they died off. However, in actual biological circumstances, effector cells can also become memory cells. Adding this additional term to the
equation stabilizes the model. In the second equation of the Mahaffay and Edlestein-Keshet model, a new term of γδee can be added to represent the effector cells that become memory cells. The parameter γ represents the proportion of effector cells that become memory cells and δe and E represent the same parameters of the decay rate of effector cells and effector cells, respectively. The code used in Matlab to simulate the model is included in the appendix. The value for gamma was then altered to see the effect of having a high, medium, and low proportion of effector cells becoming memory cells. When gamma was set to 0.1, it produced the following graph, There is an oscillation most clearly in the activated T-cell from time t=0 until around time t=160 when all of the β-cells have died off. However, the spike in activated T-cells is less dramatic then in the original model. The other major difference between this modified and the original is that the amount of memory cells spikes much higher in this modified model. Logically, this makes sense because in the new model effector cells are becoming memory cells, so there are more ways to produce memory cells. Taking a look at when the value for gamma was set to 0.5 and 0.9, respectively, and produced the following: There is still some oscillation in the first graph where gamma was set to 0.5, but in the second graph where gamma was set to 0.9, the curve of the activated cells seems to be much smoother. This suggests that the addition of this new term stabilizes the model. The question is then, why did Mahaffay and Edelstein-Keshet ignore this extra term in their original model. In a paper written by Kaech et al, the authors explain that after an autoimmune
response, only 5-10% of activated effector cells become memory cells [2]. If one were to run the code for the new model using values between 0.05 and.1 for gamma, the following outputs would be produced: In the above figure, the graphs represent when gamma is 0.09, 0.075 and 0.05, respectively. These graphs suggest that the smaller the gamma value is (the fewer effector cells that become memory cells), the more oscillation occurs. These graphs also represent the range value given in Kaech et al for the present of effector cells that become memory cells. Since there is still significant oscillation in the actual biological range for the cells, that could be one of the reasons that the new term was left out of the original model. However, since the addition of the new term still shows the steadying of the oscillations when gamma is greater, this knowledge could be used in finding ways to cure or manage Type 1 diabetes. Conclusion/Further Research In the original model, the assumptions made by Mahaffay and Edelstein Keshet simplified the model. When the model is modified to include when effector cells become memory cells instead of being terminal, it appears that the larger the value for gamma is, the more stable the model becomes. In biology the value for gamma is low, and it doesn t have a huge impact on the final results; however, the gamma term still affects the oscillation of the model. These oscillatory behaviors of the T-cells is what is believed to cause Type 1 diabetes because each time that the T-cells spike, the amount of β-cells drops drastically. If researchers can find a way to raise the gamma term and eliminate these oscillations, they could be one step closer to finding a cure for diabetes. For further research, one could look at the other assumptions that Mahaffay and Edelstein Keshet used. For example, one could include rate at which T cells flow between these parts of the body since in real life, the T-cells move between three parts of the body instead of staying in one well-mixed compartment. This addition would also help to create a better understanding of how exactly T-cells develop and interact with β-cells in the human body.
References 1. Joseph, M. & Leah, E. (2007). Modeling cyclic waves of circulating T-cells in autoimmune diabetes. SIAM Journal On Applied Mathematics, (4), 915. doi:10.1137/060661144 2. Kaech, S., Wherry, E., Konieczny, B., Ahmed, R., Tan, J., & Surh, C. (2003). Selective expression of the interleukin 7 receptor identifies effector CD8 T-cells that give rise to long-lived memory cells. Nature Immunology, 4(12), 1191-1198. doi:10.1038/ni1009 3. Kaech, S. M., Wherry, E. J., & Ahmed, R. (2002). Effector and memory T-cells differentiation: implications for vaccine development. Nature Reviews: Immunology, 2(4), 251-262. 4. Pearce, Z. (2014). Type 1 diabetes: causes, treatment and potential complications. New York: Nova Science Publishers, Inc. 5. Trudeau, J. D., Kelly-Smith, C., Verchere, C. B., Elliott, J. F., Dutz, J.P., Finegood, D. T., & Tan, R. (2003). Prediction of spontaneous autoimmune diabetes in NOD mice by quantification of autoreactive T-cells in peripheral blood. The Journal Of Clinical Investigation, 111(2), 217-223. (2018). Type 1 Diabetes. American Diabetes Association. Retrieved from www.diabetes.org/diabetes-basics/type-1/?loc=util-header_type1
1. Code for Modified Model Matlab Graphs clear Appendix dt =.01; t = 0:dt:400; A(1) = 0.5; B(1) = 1; E(1) = 1; M(1) = 0; a1 = 2; %n a2 = 2; %k1 a3 = 3; %m a4 = 0.7; %a a5 = 1.0; %k2 a6 =.02; %sigma a7 = 20;%alpha a8 = 1; %beta + deltaa a9 = 1; %epsilon a10 = 1;%beta 2^m1 a11 =.01; %deltam a12 =.1; %beta2m^m2 a13 = 0.3;%deltaE a14 = 50;%R a15 = 1;%deltap a16 =.1;%scale a17 =.14;%K a18 =.0; %gamma for i = 1:size(t,2)-1 y4 = a14*e(i)*b(i)/a15; f1 = y4^(a1)/(a2^(a1)+y4^(a1)); f2 = a4*a5^(a3)/(a5^(a3)+y4^(a3)); A(i+1) = A(i) + dt*(f1*(a6+a7*m(i))-a8*a(i)-a9*a(i)*a(i)); M(i+1) = M(i) + dt*(a10*f2*a(i)-f1*a16*a7*m(i)-a11*m(i)+a18*a13*e(i)); E(i+1) = E(i) + dt*(a12*(1-f2)*a(i)-a13*e(i)); B(i+1) = B(i) + dt*(-a17*e(i)*b(i)); end plot(t,a,'r',t,m,'k--',t,e,'b',t,b,'g') legend({'a(t)','m(t)','e(t)','b(t)'},'fontsize',16,'location','northwest')