A Mathematical Model for Germinal Centre Kinetics and Affinity Maturation
Catherine
Lloyd
Bioengineering Institute, University of Auckland
Model Status
This is the original unchecked version of the model imported from the previous
CellML model repository, 24-Jan-2006.
Model Structure
Affinity maturation refers to an increase in the affinity of antibodies for the antigen produced during an immune response. This is achieved by mutation of the genes encoding the antibody, followed by selection for the B cells which express B cell receptors with the highest affinity for antigen. Both gene mutation and subsequent protein selection occur in the germinal centres of secondary lymphoid organs. Upon immunisation (infection), the antigen is concentrated in the secondary lymphoid organs, where a small subset of all B cells recognise it. After successful presentation of the antigen fragments to T cells, B cells enter the blast state. After about three days, the B blasts differentiate into centroblasts, and then into centrocytes. Centrocytes express B cell receptors on their surface and they are believed to be subject to selection by antigen in that they need to bind, internalise and present antigen to T cells in order to prevent their under going apoptosis. Centrocytes that successfully interact with T cells (selected centrocytes), can either differentiate into memory cells, antibody forming cells (AFCs), or they can revert back to the centroblast state (see below).
Shortly after the first differentiation of centroblasts into centrocytes, somatic hypermutation starts to act on the centroblasts and changes the affinity of the B cell receptors for the antigen. About 53% of the mutations are silent, and have no affect on affinity. About 28% of the mutations are fatal, leading to B cell apoptosis. The remaining 19% of mutations either increase or decrease affinity. Selection is therefore needed to select for the mutations leading to an increase in affinity and driving affinity maturation.
The molecular mechanisms underlying these processes are not yet clearly understood, and as yet there is no experimental system available to investigate this. A mathematical model is therefore useful in investigating hypotheses and comparing their simulation results with available experimental data. In 2002, Iber and Maini published a mathematical model for germinal centre kinetics and affinity maturation. They began with a simple model of the primed primary immune response (see below), and then in this they embedded the model of affinity maturation. Model simulations showed that antigen masking by antibodies can drive affinity maturation and provide a stabilising feedback mechanism. Iber and Maini proposed that the selection probability of centrocytes and the recycling probability of selected centrocytes vary over time. They also show that the efficiency of affinity maturation is highest if clones with a very high affinity antigen leave the germinal centre for either the memory or the plasma cell pool. It is further shown that the termination of somatic hypermutation for several days before the end of the germinal centre immune reaction is beneficial for affinity maturation.
The complete original paper reference is cited below:
A Mathematical Model for Germinal Centre Kinetics and Affinity Maturation, Dagmar Iber and Philip K. Maini, 2002,
Journal of Theoretical Biology
, 219, 153-175. (A PDF version of the article is available for Journal Members on the Journal of Theoretical Biology website.) PubMed ID: 12413873
reaction schematic for the model
A schematic diagram showing the model scheme for the primed primary immune response.
$\mathrm{rho}=\frac{24.0}{6.5}\ln 2.0d=12.0\ln 2.0$
$\mathrm{pr}=0.6+0.2\frac{\mathrm{Ag}^{n}}{\mathrm{Ag}^{n}+\mathrm{Ag\_min}^{n}}\mathrm{pr1}=0.95\mathrm{pr2}=0.7\mathrm{eta}\mathrm{eta}=\begin{cases}\frac{\mathrm{K\_AgAb}}{\mathrm{C1}+\mathrm{C2}} & \text{if $\frac{\mathrm{K\_AgAb}}{\mathrm{C1}+\mathrm{C2}}< 1.0$}\\ 1.0 & \text{otherwise}\end{cases}$
$\frac{d \mathrm{Ag}}{d \mathrm{time}}=-(ukC+u\mathrm{Cs}(1.0-k))\mathrm{Ag}-\mathrm{k\_on}\mathrm{Ag}\mathrm{Ab}$
$\frac{d \mathrm{Ab}}{d \mathrm{time}}=\mathrm{beta}\mathrm{AFC}-\mathrm{k\_on}\mathrm{Ag}\mathrm{Ab}$
$\frac{d \mathrm{K\_AgAb}}{d \mathrm{time}}=\mathrm{k\_on}\mathrm{Ag}\mathrm{Ab}\mathrm{theta}=0.3\frac{\mathrm{Ag}}{\mathrm{C1}+\mathrm{C2}}\frac{\mathrm{Ag}}{\mathrm{Ag}+\mathrm{K\_AgAb}}$
$s=\mathrm{sc}-h+f\frac{\mathrm{Ag\_min}^{n}}{\mathrm{Ag\_min}^{n}+\mathrm{Ag}^{n}}\mathrm{sc}=\frac{\mathrm{delta\_c}}{d(2.0\mathrm{pr}m-1.0)+\mathrm{delta\_c}}\mathrm{s1}=0.01+0.09\mathrm{zeta}\mathrm{s2}=0.1+0.85\mathrm{zeta}\mathrm{zeta}=\frac{\mathrm{Ag}}{\mathrm{C1}+\mathrm{C2}}$
$\frac{d B}{d \mathrm{time}}=\mathrm{rho}B\frac{d \mathrm{B1}}{d \mathrm{time}}=\mathrm{pr1}\mathrm{rho}\mathrm{Cs1}-\mathrm{rho}\mathrm{B1}\frac{d \mathrm{B2}}{d \mathrm{time}}=\mathrm{pr2}\mathrm{rho}\mathrm{Cs2}-\mathrm{rho}\mathrm{B2}$
$\frac{d C}{d \mathrm{time}}=2.0\mathrm{rho}mB-\mathrm{mu}C\frac{d \mathrm{C1}}{d \mathrm{time}}=2.0\mathrm{rho}m(\mathrm{M11}+\mathrm{B1})(\mathrm{M12}+\mathrm{B2})-\mathrm{mu1}\mathrm{C1}\frac{d \mathrm{C2}}{d \mathrm{time}}=2.0\mathrm{rho}m(\mathrm{M21}+\mathrm{B1})(\mathrm{M22}+\mathrm{B2})-\mathrm{mu2}\mathrm{C2}\mathrm{mu}=ds+\mathrm{delta\_c}(1.0-s)\mathrm{mu1}=d\mathrm{s1}+\mathrm{delta\_c}(1.0-\mathrm{s1})\mathrm{mu2}=d\mathrm{s2}+\mathrm{delta\_c}(1.0-\mathrm{s2})\mathrm{delta\_c}=1.5\ln 2.0\mathrm{M11}=1.0-\mathrm{M21}\mathrm{M12}=1.0-\mathrm{M22}$
$\frac{d \mathrm{Cs}}{d \mathrm{time}}=dsC-\mathrm{rho}\mathrm{Cs}\frac{d \mathrm{Cs1}}{d \mathrm{time}}=d\mathrm{s1}\mathrm{C1}-\mathrm{rho}\mathrm{Cs1}\frac{d \mathrm{Cs2}}{d \mathrm{time}}=d\mathrm{s2}\mathrm{C2}-\mathrm{rho}\mathrm{Cs2}$
$\frac{d M}{d \mathrm{time}}=(1.0-\mathrm{theta})\mathrm{rho}(1.0-\mathrm{pr})\mathrm{Cs}\frac{d \mathrm{M1}}{d \mathrm{time}}=(1.0-\mathrm{pr1})\mathrm{Cs1}\frac{d \mathrm{M2}}{d \mathrm{time}}=(1.0-\mathrm{pr2})\mathrm{Cs2}$
$\frac{d \mathrm{AFC}}{d \mathrm{time}}=\mathrm{theta}\mathrm{rho}(1.0-\mathrm{pr})\mathrm{Cs}$
germinal centre kinetics
immunology
gene regulation
Journal of Theoretical Biology
keyword
2002-12-06
2002
This is the CellML description of Iber and Maini's 2002 mathematical
model for germinal centre kinetics and affinity maturation.
The University of Auckland
The Bioengineering Institute
Dagmar
Iber
Iber and Maini's 2002 mathematical model for germinal centre kinetics
and affinity maturation.
Catherine
Lloyd
May
A Mathematical Model for Germinal Centre Kinetics and Affinity
Maturation
219
153
175
c.lloyd@auckland.ac.nz
The University of Auckland, Bioengineering Institute
Philip
Maini
K
Catherine Lloyd