A Mathematical Study of Germinal Center Formation

Transcription

1 A Mathematical Study of Germinal Center Formation Samantha Erwin Adviser: Dr. Stanca Ciupe Virginia Tech October 1, 2014 Samantha Erwin Modeling Germinal Center Formation 1/19

2 1 Biology 2 The Model 3 Results 4 Current Work Samantha Erwin Modeling Germinal Center Formation 2/19

3 Biology Long term goal: Develop mathematical models of immune responses to chronic infections. Currently, developing mathematical model of immune responses in non chronic infections. Recently experimentalist discovered T follicular helper cells play a role in adaptive immune responses. Germinal center formations is believed to be dependent on T follicular helper cell and B cell interactions. Samantha Erwin Modeling Germinal Center Formation 3/19

4 Biology Image of T follicular helper cell migration, development, and B cell interactions in extra follicular and germinal centers. [ Weinstein, J. S., Hernandez, S. G., and Craft, J. T cells that promote B-cell maturation in systemic autoimmunity. Immunological Reviews, 247: , 2012.] Samantha Erwin Modeling Germinal Center Formation 4/19

5 The Model We first look at the host-pathogen dynamics leading to successful antibody response capable of clearing an infection. dn dh dg db 0 db i db n dp = s N d N N α N VN, = α N VN d H H γhb 0, = βγhb 0 d G G ηg = d 0 B 0 σb 0 G, n B i, i=0 = 2ασB i 1 G σb i G d i B i, = 2ασB n 1 G d i B n κb n, = κb n. Samantha Erwin Modeling Germinal Center Formation 5/19

6 The Model Our goal for this work is to determine the dynamical evolution of the total B cells in the germinal centers. B T = n i=0 B i for healthy and HIV chronically infected individuals who do and do not control the virus. We have started this work with germinal center formation during non-chronic disease. Samantha Erwin Modeling Germinal Center Formation 6/19

7 Known Parameters Name Value Units d N 0.01 per day d H 0.01 per day d G 0.01 per day d 0...d N 0.8 per day α N ml/(virus x day) s N 10 4 cells per ml V 10 4 copes per ml κ 1.2 per day α ml/(cell x day) β 1.97 ml/(cell x day) η ml/(cell x day) Samantha Erwin Modeling Germinal Center Formation 7/19

8 Initial Conditions Cells Initial Condition Units N s N d H cells per ml H 0 cells per ml G 0 cells per ml B 0 3 cells per ml B i 0 cells per ml B i 0 cells per ml B n 0 cells per ml P 0 cells per ml Samantha Erwin Modeling Germinal Center Formation 8/19

9 Numerical Results Data was gathered from Hollowood & Macartney. They used young, pathogen free mice and measured splenic germinal center cell proliferation responses to a T-dependent antigen. The total number of B cells in a germinal center, B T, versus time (in days). t B T In natural infection germinal center B cells undergo 5 10 steps of somatic hypermutations maturation, or n = 5 to n = 10 in our model. Samantha Erwin Modeling Germinal Center Formation 9/19

10 Data Fitting For our results we used n = 8 and fit the parameters σ and γ which represent B cells maturation rate and T FH cells recruitment inside the germinal centers. Parameter Best Fit Description γ γhb 0 σ σb i G 10 4 B T, n=8 Collected Data Total Cells per Germinal Center Days Samantha Erwin Modeling Germinal Center Formation 10/19

11 Data Fitting All parts of the model N 10 6 H G P B 0 B 2 B 4 B 6 B n B t Populations Total B Cells in Germinal Center Days Days Samantha Erwin Modeling Germinal Center Formation 11/19

12 Germinal Center Formation for Non-Chronic Infection Clone distribution By the time the germinal center becomes extinct, almost all B cells have the highest degree of somatic hypermutation. This results holds even when η = 0, suggesting that B cells do not compete for T FH cells. 1 Distribution of clones at t=10 1 Distribution of clones at t=20 1 Distribution of clones at t= clone number clone number clone number B i /B T B i /B T B i /B T Samantha Erwin Modeling Germinal Center Formation 12/19

13 Chronic Infections In HIV patients highly mutated, broadly neutralizing antibodies are formed and an increase in steps of B cells somatic hypermutations occur. We assumed that all parameters are as in the non-chronic case and used n=50 in our model. We predict that B T grows to 10 6 cells and the germinal centers take longer than 30 days to end B T, n=50 Collected Data 10 5 Total Cells per Germinal Center Days Samantha Erwin Modeling Germinal Center Formation 13/19

14 Chronic Infections All parts of the model Populations N H G P Total B Cells in Germinal Center B 0 B 20 B 30 B 40 B n B t Days Days Samantha Erwin Modeling Germinal Center Formation 14/19

15 The Model We first look at the host-pathogen dynamics leading to successful antibody response capable of clearing an infection. dn dh dg db 0 db i db n dp = s N d N N α N VN, = α N VN d H H γhb 0, = βγhb 0 d G G ηg = d 0 B 0 σb 0 G, n B i, i=0 = 2ασB i 1 G σb i G d i B i, = 2ασB n 1 G d i B n κb n, = κb n. Samantha Erwin Modeling Germinal Center Formation 15/19

16 Chronic vs. Nonchronic Infection B, n=8 T 10 6 B, n=50 T 10 3 Collected Data G, n=8 G, n=50 Total Cells per Germinal Center Populations Days Days Samantha Erwin Modeling Germinal Center Formation 16/19

17 Germinal Center Formation for Chronic Infection 0.12 B cell competition for T FH cells is important in this scenario, and it leads to a decrease in T FH cell numbers at the peak of B T. As a consequence, B cells of highest somatic hypermutation allowed by the model are not reached. Our future goals are to determine the factors that allow for the emergence and dominance of the high affinity clones. Distribution of clones at t= Distribution of clones at t= Distribution of clones at t= clone number 0.06 clone number 0.06 clone number B i /B T B i /B T B i /B T Samantha Erwin Modeling Germinal Center Formation 17/19

18 Mutating Virus dn dh dg db 0 db i db n dv 0 dv i dv n dp i = s N d N N α N Vi N = α N Vi N d H H γhb 0 = βγhb 0 d G G ηg B i = σb 0 GV 0 d 0 B 0 κ 0 B 0 = 2ασB i 1 V i G σb i V i G d i B i k i B i = 2ασB n 1 V ng d nb n κ nb n = d V V 0 µ 0 B 0 V 0 = p i 1 V i 1 d V V i p 0 V 0 µ i B i V i = p n 1 V n 1 d V V n p i V i µ nb nv n = κ i B i d pp i Samantha Erwin Modeling Germinal Center Formation 18/19

19 Thank you Thank you! Samantha Erwin Modeling Germinal Center Formation 19/19