a mathematical study of germinal center formation · biologythe modelresultscurrent work a...

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


1 Biology

2 The Model

3 Results

4 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.



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.]



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.



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.



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)



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



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.



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



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



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


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




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



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



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.



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



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


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


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




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



Thank you

Thank you!


a mathematical study of germinal center formation · biologythe modelresultscurrent work a...

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