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June  2017, 14(3): 655-671. doi: 10.3934/mbe.2017037

Germinal center dynamics during acute and chronic infection

460 McBryde Hall, Virginia Tech, Blacksburg, VA 24061, USA

* Corresponding author: Stanca M. Ciupe

Received  February 16, 2016 Accepted  October 12, 2016 Published  December 2016

The ability of the immune system to clear pathogens is limited during chronic virus infections where potent long-lived plasma and memory B-cells are produced only after germinal center B-cells undergo many rounds of somatic hypermutations. In this paper, we investigate the mechanisms of germinal center B-cell formation by developing mathematical models for the dynamics of B-cell somatic hypermutations. We use the models to determine how B-cell selection and competition for T follicular helper cells and antigen influences the size and composition of germinal centers in acute and chronic infections. We predict that the T follicular helper cells are a limiting resource in driving large numbers of somatic hypermutations and present possible mechanisms that can revert this limitation in the presence of non-mutating and mutating antigen.

Citation: Samantha Erwin, Stanca M. Ciupe. Germinal center dynamics during acute and chronic infection. Mathematical Biosciences & Engineering, 2017, 14 (3) : 655-671. doi: 10.3934/mbe.2017037
References:
[1]

C. AllenT. Okada and J. Cyster, Germinal-center organization and cellular dynamcs, Immunity, 27 (2007), 190-202. Google Scholar

[2]

B. AsquithC. DebacqA. FlorinsN. GilletT. Sanchez-AlcarazA. Mosley and L. Willems, Quantifying lymphocyte kinetics in vivo using carboxyfluorescein diacetate succinimidyl ester (CFSE), Proc Biol Sci, 273 (2006), 1165-1171. Google Scholar

[3]

D. Burton and J. Mascola, Antibody responses to envelope glycoproteins in HIV-1 infection, Nat Immunol, 16 (2015), 571-576. doi: 10.1038/ni.3158. Google Scholar

[4]

R. CubasJ. MuddA. SavoyeM. Perreau and J. van Grevenynghe, Inadequate T follicular cell help impairs B cell immunity during HIV infection, Nat Med, 19 (2013), 494-499. doi: 10.1038/nm.3109. Google Scholar

[5]

R. De Boer and A. Perelson, Quantifying T lymphocyte turnover, J Theor Biol, 327 (2013), 45-87. doi: 10.1016/j.jtbi.2012.12.025. Google Scholar

[6]

A. HauserM. Shlomchik and A. Haberman, In vivo imaging studies shed light on germinal-centre development, Nat Rev Immunol, 7 (2007), 499-504. doi: 10.1038/nri2120. Google Scholar

[7]

B. Haynes, New approaches to HIV vaccine development, Curr Opin Immunol, 35 (2015), 39-47. doi: 10.1016/j.coi.2015.05.007. Google Scholar

[8]

P. HodgkinJ. Lee and A. Lyons, B cell differentiation and isotype switching is related to division cycle number, J Exp Med, 184 (1996), 277-281. doi: 10.1084/jem.184.1.277. Google Scholar

[9]

K. Hollowood and J. Macartney, Cell kinetics of the germinal center reaction -a stathmokinetic study, Eur J Immunol, 22 (1992), 261-266. doi: 10.1002/eji.1830220138. Google Scholar

[10]

T. Kepler and A. Perelson, Cyclic re-entry of germinal center B cells and the efficiency of affinity maturations, Immunol Today, 14 (1993), 412-415. doi: 10.1016/0167-5699(93)90145-B. Google Scholar

[11]

C. Kesmir and R. De Boer, A mathematical model on germinal center kinetics and termination, J Immunol, 163 (1999), 2463-2469. Google Scholar

[12]

C. Kesmir and R. de Boer, A spatial model of germinal center reactions: Cellular adhesion based sorting of B cells results in efficient affinity maturation, J Theor Biol, 222 (2003), 9-22. doi: 10.1016/S0022-5193(03)00010-9. Google Scholar

[13]

F. KroeseA. WubbenaH. Seijen and P. Nieuwenhuis, Germinal centers develop oligoclonally, Eur J Immunol, 17 (1987), 1069-1072. doi: 10.1002/eji.1830170726. Google Scholar

[14]

R. KuppersM. ZhaoM. Hansmann and K. Rajewsky, Tracing B cell development in human germinal centers by molecular analysis of single cells picked from histological sections, Embo J, 12 (1993), 4955-4967. Google Scholar

[15]

P. Kwong and J. Mascola, Human antibodies that neutralize HIV-1: Identification, structures, and B cell ontogenies, Immunity, 37 (2012), 412-425. doi: 10.1016/j.immuni.2012.08.012. Google Scholar

[16]

V. L and M. Diaz, Autoreactivity in HIV-1 broadly neutralizing antibodies: Implications for their function and induction by vaccination, Curr Opin HIV AIDS, 9 (2014), 224-234. Google Scholar

[17]

H. LeeE. HawkinsM. ZandT. Mosmann and H. Wu, Interpreting CFSE obtained division histories of B cells in vitro with Smith-Martin and cyton type models, Bull Math Biol, 71 (2009), 1649-1670. doi: 10.1007/s11538-009-9418-6. Google Scholar

[18]

M. LindqvistJ. van LunzenD. SoghoianB. Kuhl and S. Ranasinghe, Expansion of HIV-specific T follicular helper cells in chronic HIV infection, J Clin Invest, 122 (2012), 3271-3280. Google Scholar

[19]

I. MacLennan, Germinal centers, Annu Rev Immunol., 12 (1994), 117-139. doi: 10.1146/annurev.iy.12.040194.001001. Google Scholar

[20]

M. Meyer-HermannE. MohrN. PelletierY. ZhangG. Victoria and K. Toellner, A theory of germinal center B cell selection, division, and exit, Cell Reports, 2 (2012), 162-174. doi: 10.1016/j.celrep.2012.05.010. Google Scholar

[21]

M. Meyer-HermannM. Figge and K. Toellner, Germinal centres seen through the mathematical eye: B-cell models on the catwalk, Trends in Immunol, 30 (2009), 157-164. doi: 10.1016/j.it.2009.01.005. Google Scholar

[22]

M. Meyer-Hermann and P. Maini, Cutting edge: Back to one-way germinal centers, J Immunol, 174 (2005), 2489-2493. doi: 10.4049/jimmunol.174.5.2489. Google Scholar

[23]

M. Meyer-HermannP. Maini and D. Iber, An analysis of B cell selection mechanisms in germinal centers, Math Med Biol, 23 (2006), 255-277. doi: 10.1093/imammb/dql012. Google Scholar

[24]

H. MiaoX. JinA. Perelson and H. Wu, Evaluation of multitype mathematical models for CFSE-labeling experiment data, Bull Math Biol, 74 (2012), 300-326. doi: 10.1007/s11538-011-9668-y. Google Scholar

[25]

I. MikellD. SatherS. KalamsM. AltfeldG. Alter and L. Stamatatos, Characteristics of the earliest cross-neutralizing antibody response to HIV-1, PLoS Pathog, 7 (2011), 1-15. doi: 10.1371/journal.ppat.1001251. Google Scholar

[26]

M. MoodyR. ZhangE. WalterC. Woods and G. Ginsburg, H3N2 influenza infection elicits more cross-reactive and less clonally expanded anti-hemagglutinin antibodies than influenza vaccination, PLoS One, 6 (2011), e25797, 14pp. doi: 10.1371/journal.pone.0025797. Google Scholar

[27]

J. Moreira and J. Faro, Modelling two possible mechanisms for the regulation of the germinal center dynamics, J Immunol, 177 (2006), 3705-3710. doi: 10.4049/jimmunol.177.6.3705. Google Scholar

[28]

M. Oprea and A. Perelson, Somatic mutation leads to efficient affinity maturation when centrocytes recycle back to centroblasts, J Immunol, 158 (1997), 5155-5162. Google Scholar

[29]

M. OpreaE. van Nimwegen and A. Perelson, Dynamics of one-pass germinal center models: Implications for affinity maturation, Bull Math Biol, 62 (2000), 121-153. doi: 10.1006/bulm.1999.0144. Google Scholar

[30]

S. PallikkuthA. Parmigiani and S. Pahwa, The role of interleukin-21 in HIV infection, Cytokine growth factor rev, 23 (2012), 173-180. doi: 10.1016/j.cytogfr.2012.05.004. Google Scholar

[31]

M. PerreauA.-L. SavoyeE. De CrignisJ. Corpataux and R. Cubas, Follicular helper T cells serve as the major CD4 T cell compartment for HIV-1 infection, replication, and production, J Exp Med, 210 (2013), 143-156. doi: 10.1084/jem.20121932. Google Scholar

[32]

J. PublicoverA. GaggarS. NishimuraC. Van Horn and A. Goodsell, Age-dependent hepatic lymphoid organization directs successful immunity to hepatitis B, J Clin Invest, 123 (2013), 3728-3739. Google Scholar

[33]

J. PublicoverA. GoodsellS. NishimuraS. Vilarinho and Z. Wang, IL-21 is pivotal in determining age-dependent effectiveness of immune responses in a mouse model of human hepatitis B, J Clin Invest, 121 (2011), 1154-1162. Google Scholar

[34]

M. RadmacherG. Kelsoe and T. Kepler, Predicted and inferred waiting times for key mutations in the germinal centre reaction: evidence for stochasticity in selection, Immunol and Cell Bio, 76 (1998), 373-381. doi: 10.1046/j.1440-1711.1998.00753.x. Google Scholar

[35]

T. SchwickertG. VictoriaD. FooksmanA. Kamphorst and M. Mugnier, A dynamic T cell-limited checkpoint regulates affinity-dependent B cell entry into the germinal center, J Exp Med, 208 (2011), 1243-1252. doi: 10.1084/jem.20102477. Google Scholar

[36]

Z. ShulmanA. GitlinS. TargM. Jankovic and G. Pasqual, T follicular helper cell dynamics in germinal centers, Science, 341 (2013), 673-677. doi: 10.1126/science.1241680. Google Scholar

[37]

G. Siskind and B. Benacerraf, Cell selection by antigen in the immune response, Adv. Immunol., 10 (1969), 1-50. doi: 10.1016/S0065-2776(08)60414-9. Google Scholar

[38]

M. StaffordL. CoreyY. CaoE. DaarD. Ho and A. Perelson, Modeling plasma virus concentration during primary HIV infection, J theor Biol, 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076. Google Scholar

[39]

L. Stamatatos, HIV vaccine design: The neutralizing antibody conundrum, Curr Opin Immunol, 24 (2012), 316-323. doi: 10.1016/j.coi.2012.04.006. Google Scholar

[40]

L. StamatatosL. MorrisD. Burton and J. Mascola, Neutralizing antibodies generated during natural HIV-1 infection: Good news for an HIV-1 vaccine?, Nat Med, 15 (2009), 866-870. doi: 10.1038/nm.1949. Google Scholar

[41]

L. VerkoczyG. KelsoeM. Moody and B. Haynes, Role of immune mechanisms in induction of HIV-1 broadly neutralizing antibodies, Curr Opin Immunol, 23 (2011), 383-390. doi: 10.1016/j.coi.2011.04.003. Google Scholar

[42]

G. Victora and L. Mesin, Clonal and cellular dynamics in germinal centers, Curr Opin Immunol, 28 (2014), 90-96. doi: 10.1016/j.coi.2014.02.010. Google Scholar

[43]

C. Vinuesa, HIV and T follicular helper cells: A dangerous relationship, J Clin Invest, 122 (2012), 3059-3062. Google Scholar

[44]

C. VinuesaI. Sanz and M. Cook, Dysregulation of germinal centres in autoimmune disease, Nat Rev Immunol, 9 (2009), 845-857. doi: 10.1038/nri2637. Google Scholar

[45]

J. WeinsteinS. Hernandez and J. Craft, T cells that promote B-cell maturation in systemic autoimmunity, Immunol Rev, 247 (2012), 160-171. doi: 10.1111/j.1600-065X.2012.01122.x. Google Scholar

[46]

I. WollenbergA. Agua-DoceA. HernandezC. Almeida and V. Oliveira, Regulation of the germinal center reaction by Foxp3+ follicular regulatory T cells, J Immunol, 187 (2011), 4553-4560. doi: 10.4049/jimmunol.1101328. Google Scholar

[47]

X. WuT. ZhouJ. ZhuB. Zhang and I. Georgiev, Focused evolution of HIV-1 neutralizing antibodies revealed by structures and deep sequencing, Science, 333 (2011), 1593-1602. doi: 10.1126/science.1207532. Google Scholar

[48]

X. ZhangS. IngA. FraserM. ChenO. Khan and J. Zakem, Follicular helper T cells: New insights into mechanisms of autoimmune diseases, Ochsner J, 13 (2013), 131-139. Google Scholar

show all references

References:
[1]

C. AllenT. Okada and J. Cyster, Germinal-center organization and cellular dynamcs, Immunity, 27 (2007), 190-202. Google Scholar

[2]

B. AsquithC. DebacqA. FlorinsN. GilletT. Sanchez-AlcarazA. Mosley and L. Willems, Quantifying lymphocyte kinetics in vivo using carboxyfluorescein diacetate succinimidyl ester (CFSE), Proc Biol Sci, 273 (2006), 1165-1171. Google Scholar

[3]

D. Burton and J. Mascola, Antibody responses to envelope glycoproteins in HIV-1 infection, Nat Immunol, 16 (2015), 571-576. doi: 10.1038/ni.3158. Google Scholar

[4]

R. CubasJ. MuddA. SavoyeM. Perreau and J. van Grevenynghe, Inadequate T follicular cell help impairs B cell immunity during HIV infection, Nat Med, 19 (2013), 494-499. doi: 10.1038/nm.3109. Google Scholar

[5]

R. De Boer and A. Perelson, Quantifying T lymphocyte turnover, J Theor Biol, 327 (2013), 45-87. doi: 10.1016/j.jtbi.2012.12.025. Google Scholar

[6]

A. HauserM. Shlomchik and A. Haberman, In vivo imaging studies shed light on germinal-centre development, Nat Rev Immunol, 7 (2007), 499-504. doi: 10.1038/nri2120. Google Scholar

[7]

B. Haynes, New approaches to HIV vaccine development, Curr Opin Immunol, 35 (2015), 39-47. doi: 10.1016/j.coi.2015.05.007. Google Scholar

[8]

P. HodgkinJ. Lee and A. Lyons, B cell differentiation and isotype switching is related to division cycle number, J Exp Med, 184 (1996), 277-281. doi: 10.1084/jem.184.1.277. Google Scholar

[9]

K. Hollowood and J. Macartney, Cell kinetics of the germinal center reaction -a stathmokinetic study, Eur J Immunol, 22 (1992), 261-266. doi: 10.1002/eji.1830220138. Google Scholar

[10]

T. Kepler and A. Perelson, Cyclic re-entry of germinal center B cells and the efficiency of affinity maturations, Immunol Today, 14 (1993), 412-415. doi: 10.1016/0167-5699(93)90145-B. Google Scholar

[11]

C. Kesmir and R. De Boer, A mathematical model on germinal center kinetics and termination, J Immunol, 163 (1999), 2463-2469. Google Scholar

[12]

C. Kesmir and R. de Boer, A spatial model of germinal center reactions: Cellular adhesion based sorting of B cells results in efficient affinity maturation, J Theor Biol, 222 (2003), 9-22. doi: 10.1016/S0022-5193(03)00010-9. Google Scholar

[13]

F. KroeseA. WubbenaH. Seijen and P. Nieuwenhuis, Germinal centers develop oligoclonally, Eur J Immunol, 17 (1987), 1069-1072. doi: 10.1002/eji.1830170726. Google Scholar

[14]

R. KuppersM. ZhaoM. Hansmann and K. Rajewsky, Tracing B cell development in human germinal centers by molecular analysis of single cells picked from histological sections, Embo J, 12 (1993), 4955-4967. Google Scholar

[15]

P. Kwong and J. Mascola, Human antibodies that neutralize HIV-1: Identification, structures, and B cell ontogenies, Immunity, 37 (2012), 412-425. doi: 10.1016/j.immuni.2012.08.012. Google Scholar

[16]

V. L and M. Diaz, Autoreactivity in HIV-1 broadly neutralizing antibodies: Implications for their function and induction by vaccination, Curr Opin HIV AIDS, 9 (2014), 224-234. Google Scholar

[17]

H. LeeE. HawkinsM. ZandT. Mosmann and H. Wu, Interpreting CFSE obtained division histories of B cells in vitro with Smith-Martin and cyton type models, Bull Math Biol, 71 (2009), 1649-1670. doi: 10.1007/s11538-009-9418-6. Google Scholar

[18]

M. LindqvistJ. van LunzenD. SoghoianB. Kuhl and S. Ranasinghe, Expansion of HIV-specific T follicular helper cells in chronic HIV infection, J Clin Invest, 122 (2012), 3271-3280. Google Scholar

[19]

I. MacLennan, Germinal centers, Annu Rev Immunol., 12 (1994), 117-139. doi: 10.1146/annurev.iy.12.040194.001001. Google Scholar

[20]

M. Meyer-HermannE. MohrN. PelletierY. ZhangG. Victoria and K. Toellner, A theory of germinal center B cell selection, division, and exit, Cell Reports, 2 (2012), 162-174. doi: 10.1016/j.celrep.2012.05.010. Google Scholar

[21]

M. Meyer-HermannM. Figge and K. Toellner, Germinal centres seen through the mathematical eye: B-cell models on the catwalk, Trends in Immunol, 30 (2009), 157-164. doi: 10.1016/j.it.2009.01.005. Google Scholar

[22]

M. Meyer-Hermann and P. Maini, Cutting edge: Back to one-way germinal centers, J Immunol, 174 (2005), 2489-2493. doi: 10.4049/jimmunol.174.5.2489. Google Scholar

[23]

M. Meyer-HermannP. Maini and D. Iber, An analysis of B cell selection mechanisms in germinal centers, Math Med Biol, 23 (2006), 255-277. doi: 10.1093/imammb/dql012. Google Scholar

[24]

H. MiaoX. JinA. Perelson and H. Wu, Evaluation of multitype mathematical models for CFSE-labeling experiment data, Bull Math Biol, 74 (2012), 300-326. doi: 10.1007/s11538-011-9668-y. Google Scholar

[25]

I. MikellD. SatherS. KalamsM. AltfeldG. Alter and L. Stamatatos, Characteristics of the earliest cross-neutralizing antibody response to HIV-1, PLoS Pathog, 7 (2011), 1-15. doi: 10.1371/journal.ppat.1001251. Google Scholar

[26]

M. MoodyR. ZhangE. WalterC. Woods and G. Ginsburg, H3N2 influenza infection elicits more cross-reactive and less clonally expanded anti-hemagglutinin antibodies than influenza vaccination, PLoS One, 6 (2011), e25797, 14pp. doi: 10.1371/journal.pone.0025797. Google Scholar

[27]

J. Moreira and J. Faro, Modelling two possible mechanisms for the regulation of the germinal center dynamics, J Immunol, 177 (2006), 3705-3710. doi: 10.4049/jimmunol.177.6.3705. Google Scholar

[28]

M. Oprea and A. Perelson, Somatic mutation leads to efficient affinity maturation when centrocytes recycle back to centroblasts, J Immunol, 158 (1997), 5155-5162. Google Scholar

[29]

M. OpreaE. van Nimwegen and A. Perelson, Dynamics of one-pass germinal center models: Implications for affinity maturation, Bull Math Biol, 62 (2000), 121-153. doi: 10.1006/bulm.1999.0144. Google Scholar

[30]

S. PallikkuthA. Parmigiani and S. Pahwa, The role of interleukin-21 in HIV infection, Cytokine growth factor rev, 23 (2012), 173-180. doi: 10.1016/j.cytogfr.2012.05.004. Google Scholar

[31]

M. PerreauA.-L. SavoyeE. De CrignisJ. Corpataux and R. Cubas, Follicular helper T cells serve as the major CD4 T cell compartment for HIV-1 infection, replication, and production, J Exp Med, 210 (2013), 143-156. doi: 10.1084/jem.20121932. Google Scholar

[32]

J. PublicoverA. GaggarS. NishimuraC. Van Horn and A. Goodsell, Age-dependent hepatic lymphoid organization directs successful immunity to hepatitis B, J Clin Invest, 123 (2013), 3728-3739. Google Scholar

[33]

J. PublicoverA. GoodsellS. NishimuraS. Vilarinho and Z. Wang, IL-21 is pivotal in determining age-dependent effectiveness of immune responses in a mouse model of human hepatitis B, J Clin Invest, 121 (2011), 1154-1162. Google Scholar

[34]

M. RadmacherG. Kelsoe and T. Kepler, Predicted and inferred waiting times for key mutations in the germinal centre reaction: evidence for stochasticity in selection, Immunol and Cell Bio, 76 (1998), 373-381. doi: 10.1046/j.1440-1711.1998.00753.x. Google Scholar

[35]

T. SchwickertG. VictoriaD. FooksmanA. Kamphorst and M. Mugnier, A dynamic T cell-limited checkpoint regulates affinity-dependent B cell entry into the germinal center, J Exp Med, 208 (2011), 1243-1252. doi: 10.1084/jem.20102477. Google Scholar

[36]

Z. ShulmanA. GitlinS. TargM. Jankovic and G. Pasqual, T follicular helper cell dynamics in germinal centers, Science, 341 (2013), 673-677. doi: 10.1126/science.1241680. Google Scholar

[37]

G. Siskind and B. Benacerraf, Cell selection by antigen in the immune response, Adv. Immunol., 10 (1969), 1-50. doi: 10.1016/S0065-2776(08)60414-9. Google Scholar

[38]

M. StaffordL. CoreyY. CaoE. DaarD. Ho and A. Perelson, Modeling plasma virus concentration during primary HIV infection, J theor Biol, 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076. Google Scholar

[39]

L. Stamatatos, HIV vaccine design: The neutralizing antibody conundrum, Curr Opin Immunol, 24 (2012), 316-323. doi: 10.1016/j.coi.2012.04.006. Google Scholar

[40]

L. StamatatosL. MorrisD. Burton and J. Mascola, Neutralizing antibodies generated during natural HIV-1 infection: Good news for an HIV-1 vaccine?, Nat Med, 15 (2009), 866-870. doi: 10.1038/nm.1949. Google Scholar

[41]

L. VerkoczyG. KelsoeM. Moody and B. Haynes, Role of immune mechanisms in induction of HIV-1 broadly neutralizing antibodies, Curr Opin Immunol, 23 (2011), 383-390. doi: 10.1016/j.coi.2011.04.003. Google Scholar

[42]

G. Victora and L. Mesin, Clonal and cellular dynamics in germinal centers, Curr Opin Immunol, 28 (2014), 90-96. doi: 10.1016/j.coi.2014.02.010. Google Scholar

[43]

C. Vinuesa, HIV and T follicular helper cells: A dangerous relationship, J Clin Invest, 122 (2012), 3059-3062. Google Scholar

[44]

C. VinuesaI. Sanz and M. Cook, Dysregulation of germinal centres in autoimmune disease, Nat Rev Immunol, 9 (2009), 845-857. doi: 10.1038/nri2637. Google Scholar

[45]

J. WeinsteinS. Hernandez and J. Craft, T cells that promote B-cell maturation in systemic autoimmunity, Immunol Rev, 247 (2012), 160-171. doi: 10.1111/j.1600-065X.2012.01122.x. Google Scholar

[46]

I. WollenbergA. Agua-DoceA. HernandezC. Almeida and V. Oliveira, Regulation of the germinal center reaction by Foxp3+ follicular regulatory T cells, J Immunol, 187 (2011), 4553-4560. doi: 10.4049/jimmunol.1101328. Google Scholar

[47]

X. WuT. ZhouJ. ZhuB. Zhang and I. Georgiev, Focused evolution of HIV-1 neutralizing antibodies revealed by structures and deep sequencing, Science, 333 (2011), 1593-1602. doi: 10.1126/science.1207532. Google Scholar

[48]

X. ZhangS. IngA. FraserM. ChenO. Khan and J. Zakem, Follicular helper T cells: New insights into mechanisms of autoimmune diseases, Ochsner J, 13 (2013), 131-139. Google Scholar

Figure 1.  Dynamics of model (2) applied to an acute infection
Figure 2.  Sensitivity Analysis
Figure 3.  Comparison of model (1)'s dynamics when $n$, $\alpha$ and $\sigma$ are varied
Figure 4.  B clone distribution in acute and chronic infections
Figure 5.  Comparison of model (3)'s dynamics when $f$ is varied
Figure 6.  Virus strains and B clone dynamics for slow mutating virus
Figure 7.  Models (3)-(4)'s dynamics
Table 1.  Variables and fixed parameter values.
NameValueUnitsDescriptionCitation
$s_N$$10^4$cells per ml per dayNaive CD4 T-cell recruitment rate[38]
$d_N$$0.01$per dayNaive CD4 T-cell death rate[38]
$\alpha_N$$1.8\times 10^{-11}$ml per day per cellPre-Tfh cell production rate
$d_H$$0.01$per dayPre-Tfh cell death rate[38]
$d_G$$0.01$per dayTfh cell death rate[38]
$d$$0.8$per dayB-cell death rate[11]
$\kappa$$1.2$per dayPlasma cells production rate
$\gamma$$2$per cell per dayPre-Tfh cell differentiation rate[36]
$\mu$$2$per cell per dayAntigen removal rate
$\eta$$10^{-5}$per cell per dayTfh competition rate
$N(0)$$10^6$cells per mlInitial amount of CD4 T cells[38]
$H(0)$0cells per mlInitial amount of Pre-Tfh cells
$G(0)$0cells per mlInitial amount of Tfh cells
$B_0(0)$3cellsInitial amount of B-cells[13,11]
$B_i(0)$0cellsInitial amount of B-cell clones
$P(0)$0cellsInitial amount of plasma cells
$V(0)$$2\times 10^8$per mlInitial amount of non-mutating antigen[9]
NameValueUnitsDescriptionCitation
$s_N$$10^4$cells per ml per dayNaive CD4 T-cell recruitment rate[38]
$d_N$$0.01$per dayNaive CD4 T-cell death rate[38]
$\alpha_N$$1.8\times 10^{-11}$ml per day per cellPre-Tfh cell production rate
$d_H$$0.01$per dayPre-Tfh cell death rate[38]
$d_G$$0.01$per dayTfh cell death rate[38]
$d$$0.8$per dayB-cell death rate[11]
$\kappa$$1.2$per dayPlasma cells production rate
$\gamma$$2$per cell per dayPre-Tfh cell differentiation rate[36]
$\mu$$2$per cell per dayAntigen removal rate
$\eta$$10^{-5}$per cell per dayTfh competition rate
$N(0)$$10^6$cells per mlInitial amount of CD4 T cells[38]
$H(0)$0cells per mlInitial amount of Pre-Tfh cells
$G(0)$0cells per mlInitial amount of Tfh cells
$B_0(0)$3cellsInitial amount of B-cells[13,11]
$B_i(0)$0cellsInitial amount of B-cell clones
$P(0)$0cellsInitial amount of plasma cells
$V(0)$$2\times 10^8$per mlInitial amount of non-mutating antigen[9]
Table 2.  Parameter estimates and confidence intervals
Name Units Value Description Confidence Intervals
$\alpha$ 27.469 B-cell offspring production rate [14.015 40.924]
$\sigma$ ml per cell per day $1.1 \times 10^{-5}$ Affinity maturation rate [4.8 $\times 10^{-6}$ 1.7 $\times 10^{-5}$]
Name Units Value Description Confidence Intervals
$\alpha$ 27.469 B-cell offspring production rate [14.015 40.924]
$\sigma$ ml per cell per day $1.1 \times 10^{-5}$ Affinity maturation rate [4.8 $\times 10^{-6}$ 1.7 $\times 10^{-5}$]
[1]

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