Biology The Model Results Current Work
A Mathematical Study ofGerminal Center Formation
Samantha Erwin
Adviser: Dr. Stanca CiupeVirginia Tech
October 1, 2014
Samantha Erwin Modeling Germinal Center Formation 1/19
Biology The Model Results Current Work
1 Biology
2 The Model
3 Results
4 Current Work
Samantha Erwin Modeling Germinal Center Formation 2/19
Biology The Model Results Current Work
Biology
Long term goal: Develop mathematical models of immuneresponses to chronic infections.Currently, developing mathematical model of immuneresponses in non chronic infections.Recently experimentalist discovered T follicular helper cellsplay a role in adaptive immune responses.Germinal center formations is believed to be dependent onT follicular helper cell and B cell interactions.
Samantha Erwin Modeling Germinal Center Formation 3/19
Biology The Model Results Current Work
Biology
Image of T follicular helper cell migration, development, and Bcell interactions in extra follicular and germinal centers.[ Weinstein, J. S., Hernandez, S. G., and Craft, J. T cells that promote B-cell maturation in systemicautoimmunity. Immunological Reviews, 247: 160-171, 2012.]
Samantha Erwin Modeling Germinal Center Formation 4/19
Biology The Model Results Current Work
The ModelWe first look at the host-pathogen dynamics leading to successfulantibody response capable of clearing an infection.
dNdt
= sN − dNN − αNVN,
dHdt
= αNVN − dHH − γHB0,
dGdt
= βγHB0 − dGG − ηGn∑
i=0
Bi ,
dB0
dt= −d0B0 − σB0G,
dBi
dt= 2ασBi−1G − σBiG − diBi ,
dBn
dt= 2ασBn−1G − diBn − κBn,
dPdt
= κBn.
Samantha Erwin Modeling Germinal Center Formation 5/19
Biology The Model Results Current Work
The Model
Our goal for this work is to determine the dynamicalevolution of the total B cells in the germinal centers.
BT =n∑
i=0
Bi
for healthy and HIV chronically infected individuals who doand do not control the virus. We have started this workwith germinal center formation during non-chronic disease.
Samantha Erwin Modeling Germinal Center Formation 6/19
Biology The Model Results Current Work
Known Parameters
Name Value UnitsdN 0.01 per daydH 0.01 per daydG 0.01 per dayd0.....dN 0.8 per dayαN 5 × 10−10 mL/(virus x day)sN 104 cells per mLV 104 copes per mLκ 1.2 per dayα 10.05 mL/(cell x day)β 1.97 mL/(cell x day)η 4 × 10−7 mL/(cell x day)
Samantha Erwin Modeling Germinal Center Formation 7/19
Biology The Model Results Current Work
Initial Conditions
Cells Initial Condition UnitsN sN
dHcells per mL
H 0 cells per mLG 0 cells per mLB0 3 cells per mLBi 0 cells per mLBi 0 cells per mLBn 0 cells per mLP 0 cells per mL
Samantha Erwin Modeling Germinal Center Formation 8/19
Biology The Model Results Current Work
Numerical Results
Data was gathered from Hollowood & Macartney. Theyused young, pathogen free mice and measured splenicgerminal center cell proliferation responses to aT-dependent antigen.The total number of B cells in a germinal center, BT ,versus time (in days).
t 4 7 9 14 21BT 0 356 1149 498 49
In natural infection germinal center B cells undergo 5 − 10steps of somatic hypermutations maturation, or n = 5 ton = 10 in our model.
Samantha Erwin Modeling Germinal Center Formation 9/19
Biology The Model Results Current Work
Data Fitting
For our results we used n = 8 and fit the parameters σ andγ which represent B cells maturation rate and TFH cellsrecruitment inside the germinal centers.
Parameter Best Fit Descriptionγ 5.47843 γHB0σ 0.00386 σBiG
0 5 10 15 20 25 30100
101
102
103
104
Days
Tota
l Cel
ls p
er G
erm
inal
Cen
ter
BT, n=8Collected Data
Samantha Erwin Modeling Germinal Center Formation 10/19
Biology The Model Results Current Work
Data Fitting
All parts of the model
0 5 10 15 20 25 3010−1
100
101
102
103
104
105
106
Days
Popu
latio
ns
NHGP
0 5 10 15 20 25 3010−1
100
101
102
103
104
Days
Tota
l B C
ells
in G
erm
inal
Cen
ter
B0B2B4B6BnBt
Samantha Erwin Modeling Germinal Center Formation 11/19
Biology The Model Results Current Work
Germinal Center Formation for Non-Chronic Infection
Clone distributionBy the time the germinal center becomes extinct, almost allB cells have the highest degree of somatic hypermutation.This results holds even when η = 0, suggesting that B cellsdo not compete for TFH cells.
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bi/BT
clon
e nu
mbe
r
Distribution of clones at t=10
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bi/BT
clon
e nu
mbe
r
Distribution of clones at t=20
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bi/BT
clon
e nu
mbe
r
Distribution of clones at t=30
Samantha Erwin Modeling Germinal Center Formation 12/19
Biology The Model Results Current Work
Chronic Infections
In HIV patients highly mutated, broadly neutralizingantibodies are formed and an increase in steps of B cellssomatic hypermutations occur. We assumed that allparameters are as in the non-chronic case and used n=50in our model.We predict that BT grows to 106 cells and the germinalcenters take longer than 30 days to end.
0 5 10 15 20 25 30100
101
102
103
104
105
106
Days
Tota
l Cel
ls p
er G
erm
inal
Cen
ter
BT, n=50Collected Data
Samantha Erwin Modeling Germinal Center Formation 13/19
Biology The Model Results Current Work
Chronic Infections
All parts of the model
0 5 10 15 20 25 3010−1
100
101
102
103
104
105
106
Days
Popu
latio
ns
NHGP
0 5 10 15 20 25 3010−1
100
101
102
103
104
105
106
Days
Tota
l B C
ells
in G
erm
inal
Cen
ter
B0B20B30B40BnBt
Samantha Erwin Modeling Germinal Center Formation 14/19
Biology The Model Results Current Work
The ModelWe first look at the host-pathogen dynamics leading to successfulantibody response capable of clearing an infection.
dNdt
= sN − dNN − αNVN,
dHdt
= αNVN − dHH − γHB0,
dGdt
= βγHB0 − dGG − ηGn∑
i=0
Bi ,
dB0
dt= −d0B0 − σB0G,
dBi
dt= 2ασBi−1G − σBiG − diBi ,
dBn
dt= 2ασBn−1G − diBn − κBn,
dPdt
= κBn.
Samantha Erwin Modeling Germinal Center Formation 15/19
Biology The Model Results Current Work
Chronic vs. Nonchronic Infection
0 5 10 15 20 25 3010
0
101
102
103
104
105
106
Days
Tota
l C
ells
per
Germ
inal C
ente
r
0 5 10 15 20 25 3010
−1
100
101
102
103
Days
Popula
tions
BT, n=8
BT, n=50
Collected Data
G, n=8
G, n=50
Samantha Erwin Modeling Germinal Center Formation 16/19
Biology The Model Results Current Work
Germinal Center Formation for Chronic Infection
B cell competition for TFH cells is important in thisscenario, and it leads to a decrease in TFH cell numbers atthe peak of BT .As a consequence, B cells of highest somatichypermutation allowed by the model are not reached.Our future goals are to determine the factors that allow forthe emergence and dominance of the high affinity clones.
0 5 10 15 20 25 30 35 40 45 500
0.02
0.04
0.06
0.08
0.1
0.12
Bi/BT
clon
e nu
mbe
r
Distribution of clones at t=10
0 5 10 15 20 25 30 35 40 45 500
0.02
0.04
0.06
0.08
0.1
0.12
Bi/BT
clon
e nu
mbe
r
Distribution of clones at t=20
0 5 10 15 20 25 30 35 40 45 500
0.02
0.04
0.06
0.08
0.1
0.12
Bi/BT
clon
e nu
mbe
r
Distribution of clones at t=30
Samantha Erwin Modeling Germinal Center Formation 17/19
Biology The Model Results Current Work
Mutating Virus
dN
dt= sN − dN N − αN
∑Vi N
dH
dt= αN
∑Vi N − dH H − γHB0
dG
dt= βγHB0 − dGG − ηG
∑Bi
dB0
dt= −σB0GV0 − d0B0 − κ0B0
dBi
dt= 2ασBi−1Vi G − σBi Vi G − di Bi − ki Bi
dBn
dt= 2ασBn−1VnG − dnBn − κnBn
dV0
dt= −dV V0 − µ0B0V0
dVi
dt= pi−1Vi−1 − dV Vi − p0V0 − µi Bi Vi
dVn
dt= pn−1Vn−1 − dV Vn − pi Vi − µnBnVn
dPi
dt= κi Bi − dpPi
Samantha Erwin Modeling Germinal Center Formation 18/19