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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. 

[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. 

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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. 

[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.

[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. 

[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.

[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. 

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[43]

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

[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.

[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.

[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.

[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.

[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. 

show all references

References:
[1]

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

[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. 

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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. 

[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.

[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. 

[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.

[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. 

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[43]

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

[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.

[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.

[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.

[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.

[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. 

Figure 1.  Dynamics of model (2) applied to an acute infection

(a) $B_t$ as given by (2) versus data (•); (b) B-cells that underwent different levels of somatic hypermutations and plasma cells, for $\mu=2$.; (c) Pre-Tfh cells $H$ per ml and Tfh cells $G$ per ml; and (d) Antigen per ml; for $\mu=2$ per cell per day (solid lines), $\mu=2\times 10^{-2}$ per cell per day (dashed lines) and $\mu=2\times 10^{-4}$ per cell per day (dotted lines). The dashed-dotted line is the antigen limit of detection of $3\times 10^{-4}$ sRBC per ml. The other parameter values are given in Tables 1 and 2.

Figure 2.  Sensitivity Analysis

$B_t$ and $G$ (grey lines) and the semi-relative sensitivity curves, $q \frac{\partial B_t}{\partial q} $and $q \frac{\partial G}{\partial q}$, over time for $q=\alpha$ (solid lines), $q=\gamma$ (dotted lines) and $q=\sigma$ (dashed lines).

Figure 3.  Comparison of model (1)'s dynamics when $n$, $\alpha$ and $\sigma$ are varied

(a) $B_t$ as given by (2) versus data (•); and (b) $G$ per ml as given by model (1) for $n=8$ (solid lines); $n=50$, $\alpha=27.5$, $\sigma=1.1\times 10^{-5}$ ml per cell per day (dashed lines); and $n=50$, $\alpha=1.6$, $\sigma=\times 10^{-3}$ ml per cell per day (dotted lines). The other parameters are given in Tables 1 and 2.

Figure 4.  B clone distribution in acute and chronic infections

Clonal distribution $B_i/B_t$ for $0\leq i \leq n$, $t=10$, $t=20$, $t=30$ days following infection for $n=8$ (top row) and $n=50$ (second row). Note that for $n=8$ the germinal center contains the B clone with the highest level of somatic hypermutation $B_8$, while for $n=50$ case the germinal center is dominated by middle clones and the $B_{50}$ clone is absent. We then show two mechanisms to achieve the $B_{50}$ clone: (third row) $\eta=0$ and (bottom row) $\alpha=1.6$, $\sigma=10^{-3}$ ml per cell per day, $\eta=10^{-5}$ per cell per day. $B_t$ is given by (2), and the other parameters are given in Tables 1 and 2. In both situations clone $B_{50}$ dominates the germinal center B-cell population $20$ days following infection.

Figure 5.  Comparison of model (3)'s dynamics when $f$ is varied

(a) $B_t$ and $P$, (b) $G$ per ml, and (c) $V_t=\sum_{i=0}^{n-1} V_i$ as given by (3) for $n=8$ and $f=0.9$ (solid lines); $f=0.1$ (dashed lines); $f=0.01$ dotted lines. The other parameters are given in Tables 1, 2, $\alpha_N^\phi=3.6 \times 10^{-6}$ and $V_0^\phi=10^3$. Note that $P$ for $f=0.01$ is negligible.

Figure 6.  Virus strains and B clone dynamics for slow mutating virus

(a) $V_i$ per ml, (b) $B_i$ as given by (3) for $n=8$ and $f=0.01$. The other parameters are given in Tables 1, 2, $\alpha_N^\phi=3.6 \times 10^{-6}$ and $V_0^\phi=10^3$.

Figure 7.  Models (3)-(4)'s dynamics

(a) $B_t$, (b) $G$ per ml, and (c) $V_t=\sum_{i=0}^{n-1} V_i$ as given by (3) and (4) for $n=8$, $f=0.01$ and $r=0$ (solid lines); $r=0.75$ (dashed lines); and $r=0.8$ (dotted lines); for $n=50$, $f=0.1$ and $r=0.75$ (dashed-dotted line). The other parameters are given in Tables 1, 2, $\alpha_N^\phi=3.6 \times 10^{-6}$ and $V_0^\phi=10^3$.

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 $\times 10^{-5}$]
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