Affinity Maturation in Germinal Centers Strongly Determined by Rate of Dissociation

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UNIVERSITY OF AMSTERDAM

BACHELOR THESIS

Affinity maturation in germinal centers strongly

determined by rate of dissociation

Author:

Harry Thavaganeshan

Supervisor:

dr. ir. Huub Hoefsloot

A Thesis Presented for the Degree of

Bachelor of Science

in the

Biomedical Sciences

University of Amsterdam

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Abstract

Germinal centers are immunological hot spots that stimulate the generation of matured B cells. These cells are of paramount importance with respect to the immune system and battling pathogens. There is still much to discover within the topic of germinal centers and one of the interesting points is the process of affinity maturation, the process that simulates a micro evolution of B cells to create highly affinitive B cells to combat the pathogens. In this paper, the impact of the rate of association as opposed to dissociation is being investigated to determine the most impactful of the two with respect to the amount of activated B cells. Information on the rates could prove vital to better examination of germinal centers by creation of more accurate artificial doubles. By modelling the dynamics in germinal centers using models by Faro et al. (2019) and implementing multiple B cells as well as affinity maturation and the option to modulate the rates per B cells, different situations could be created to determine the impact of both rates. From the models, it can be concluded that the rate of dissociation has a higher impact on the dynamics in germinal centers than the rate of association. Interestingly, a decrease of antigens makes the impact of changing the rate of dissociation even more extreme. Also, no clear relation was seen between affinity and the amount of maturated B cells. As the outcomes of this research cannot be strongly verified due to lack of information on rates with respect to germinal center output, more models should be made on the centers focused on both rates as this information could prove vital to better creation of artificial models of germinal centers and agents that influence the dynamics in these centers.

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Introduction

Immune system

The human immune system protects the host from pathogens. The immune system can be divided into two systems, one of which is the innate immune system that supports pattern recognition and direct reactions. The other system is the adaptive immune system that works through recognition of antigens after which specialized B cells can be developed (Murphy et al., 2013).

The B cells must bind to the antigens that result from the invasion. The process is characterized by the antigen binding to the receptors of the B cells, after which a microevolution of B cells takes place, known as affinity maturation. This process starts with extreme generation of B cells and through apoptotic selection, B cells with a high affinity towards an antigen are selected. The B cell will secrete compounds that alert the immune system to send a spectrum of compounds to signal and destroy the invader (Clark & Ledbetter, 1994). The alert marks all invaders which use the antigen that the B cell is highly affinitive for, therefore acting as a local beacon (Murphy et al., 2013). As a pathogen can spread throughout the whole body, it takes more than one B cell to ensure elimination of all antigens.

Germinal centers

Researching the maturation of antibodies and the development of B cells has become popular in the last decades as the current advances in immunology also demand a thorough understanding of these

processes. Germinal centers are characterized by the maturation and development of antibodies and B cells respectively. The centers consist of B cells, stromal cells and CD4 T-cells, the latter population is also known as the follicle helper T cell population (Murphy et al., 2013). Germinal centers are activated in response to the immune system requesting an increase in B cells, which can be due to multiple reasons such as a bacterium or virus entering a human body. This activation then leads to the use of antigens, follicle dendritic cells and B cells to make specialized B cells. These B cells undergo a selection process resulting in B cell populations with high affinities towards the antigen (Victoria & Nussenzweig, 2012). This selection process is preceded by a period of clonal diversification, which is characterized by the creation of mutant B cell receptors that each have a varying affinity towards the antigen (Shlomchik et

al., 2019). The affinity is determined by the affinity of the B cell receptors towards the antigens.

An important process in these germinal centers is the process of affinity maturation, which is

characterized by the maturation of the activated B cells through selection based on affinity. The process relies on antigen presentation by follicular dendritic cells to the B cells, which then will be stimulated to divide extremely fast (Murphy et al., 2013). During this division an increase in the number of mutations in the B cells can be found. This variety of mutations is important to the B cells, as the variance will lead to the cells developing a different affinity towards the antibody (Bannard & Cyster, 2017). Germinal centers, when matured, are divided in dark and light zones. While the dark zones support the division and clonal diversification of the B cells, the light zone is responsible for the selection procedure, which happens under control of T cells and follicle dendritic cells that secrete signals necessary for the mutated B cells to survive (MacLennan, 1994). Since there is not enough signaling for every B cell to survive, the B cells with the highest affinity towards the antigens will survive, thus it can be concluded that this procedure can be perceived as a selection procedure determined by affinity. The process results in

various mutant B cell populations with high affinity towards the antigen and it is believed that these cells are required for maintaining long term antibody production as well as memory cells after the first three weeks of development (MacLennan, 1994).

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Points of interest

A point of interest can be found when delving deep into the fundaments of affinity maturation. The affinity of a B cell is determined by both association and dissociation of FDC bound antigens to B cells if it is assumed that the fundamental process of affinity maturation is based on general binding kinetics (Oprea & Perelson, 1997). The point of interest lies within the systematic increase of affinity as it is still vague whether the affinity rises through increase of association or decrease in dissociation. An increase in association of the antigen to the substrate would point out that it is the urge of the antigen wanting to bind the B cell, that raises affinity. A decrease in dissociation would mean that the urge of the antigen to let go of the B cell lessens, therefore increasing affinity of the B cell towards the antibody.

Into the unknown

In this research, the hypothesis whether affinity maturation is the result of an increase in association or a decrease in dissociation will be tested. This will be done through simulation of a base model and

adapting it to the various situations with respect to affinity maturation.

First, the base model is constructed based on literature references, after which the possibility of having multiple activated B cells shall be integrated. From here, the model can be used both as a fundament for further research as well as a control mechanism for this further research. The different situations of affinity maturation will then be modelled within the base model for comparison. This research tests the differences between antibody association and dissociation in relation to affinity maturation, and every step during simulation will be checked by comparison to the base model to ensure correct dynamics.

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Materials and methods

Models

Some fundamental processes that take place in the body on cellular level are hard to research due to limitations caused by the current technology. A solution to this is simulating the processes as far as possible through computational modeling. This allows the possibility of manipulating the environments created in the model, therefore making it possible to research how processes change under the influence of changing parameters. However, modelling a process also means that the model will only describe the situations created by the coherence of assumptions that were defined before construction of the model, thus a good balance must be found between the amount of assumptions and the margin of error. Models can be a great way of researching uncommon or rare situations within an environment. This way

situations that remain relatively unknown can be researched, which could lead to relevant hypotheses with respect to process understanding and even problem solving.

Goal

The goal of this research is to investigate the impact of both the rate of association (Kon) and dissociation

(Koff), and the differences between each rate in relation to activated B cell outputs. This was done by

analysis of three models, each taking certain assumptions into account. After which relevant parameters from these models were recombined into one model that was used for the main analysis.

Bioinformatics

For modelling R (v3.6.2) was used in combination with the ODE-solver (deSolve). The original ode function from deSolve was used and the modelling was done on a Lenovo yoga laptop. Scripts of the three base models and the finalized model are provided in the appendix (page 20-50).

Method

To investigate the process of affinity maturation, the three different models in Faro et al. (2019) were replicated and checked for functionality. The models describe the dynamics in germinal centers in the following way. At first the antigens (Af) and B cells (B) meet, after which the antigens bind to the B cells (Ba). This is then followed by detachment of the antigen which leaves the B cell in a stimulated state (Be). After this, these stimulated B cells bind to T cells (T) and form B-T cell conjugates (Tb), which then detach from each other. What is left, are activated B (Bd) and T cells (Td) which both return to their original populations.

There are multiple ways to expand this standard model. Faro et al. (2019) show three ways to expand the model (Figure 1). Model 1 eliminates the assumption of constant supply of antigens as it incorporates depletion of non bound antigens through implementation of antigen consumption.

Model 2 expands on this by adding the involvement of FDCs (F) that present the antigen and also make B cell maturation (Bm) possible through activation by antigen and B cell (Bam).

Model 3 adds a new pathway that involves matured T cells (Tm) that also make maturation of B cells possible through a similar stimulated B cell to activated B & T cell (Bdm & Tdm respectively) route by forming conjugates (Tbm) to activate each other. The activated T cells then go back into the matured T cell population while the activated matured B cells will become matured B cells that can act as plasma cells or long living B cells (Bm). A clear parameter explanation can be found in the appendix (page 16).

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7 Checking the model in this research was done by

replication of the exact parameters by Faro et al. (2019), followed by comparison of graphs. Models 1 and 3 were reproducible without any margin of error as opposed to model 2, which, when plotted, seemed to deviate significantly from the reproduced model 2, therefore the decision was made to eliminate model 2 in this research.

Following the successful recreation of models 1 and 3, the goal was ultimately to add new parameters, routes and competition in the model. Modelling competition was prioritized as it plays a big role in germinal centers. Since validating an enhanced model without reference material is impossible, the choice was made to implement competition firstly. After this, validation of the model could be done by diagram comparison, since the graphs should look similar to the normal model in case the competition differential equations are set to not express any competition.

Another key aspect unconsidered in the models from Faro et al. (2019) is the process of affinity maturation which is characterized by extreme B cell division and mutation. This is followed by a selection process by T cells during which the B cells with the highest affinity to an antigen will survive. During this process cells with low affinity are left to die due to apoptosis that can only be stopped by signals coming from the aforementioned T cells. The T cells only signal the cells with the highest affinity. A germinal center can thus be considered a microenvironment in which evolution takes place. After modelling competition, expansion of the model through adding alternative routes and parameters was done. For this, literature research was conducted to find implementable sources with respect to germinal centers. At first, this research was focused on finding missing fundaments of germinal centers in the models. After implementation of these missing parts, the focus is shifted towards new

developments in germinal center research, which can then be simulated in the model. The model’s utility to analyse current dynamics in germinal centers can then shift to prediction of the dynamics taking the implemented developments into account.

A danger to this type of modelling is the lack of both validation as well as falsification due to non-existence of any reference model, therefore the results of this analysis should be taken carefully into consideration as the predictions cannot be confirmed yet. However, it does increase the power of analysis if literature is used as reference material for the predictions as validation of predictions does support the functionality of the model. The model will ultimately focus on generating hypotheses that can be tested in future research.

Figure 1. A schematic overview of the components of each model The base model is indicated by the black lines. Model 1 adds antigen depletion and is shown by the red delta symbol. Model 2 features a pathway to maturated B cells through FDC interaction indicated by the green arrows. Model 3 is characterized by a pathway through matured T cell selection to matured B cell output, indicated by the light blue arrows. The coloured pathways are to be read as if combined with the black pathway.

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8 The newly added processes were competition through creation of 16 cell lines instead of 1 B-cell line, which was created by changing the one amount of B cells to 16 equal amounts of B cells by using vectors. After this, the rate of association (c1) and dissociation (a1) were changed for every cell line to enable the process of affinity maturation and the ability to investigate the impact of the rate of association and dissociation (Figure 1). Different situations were then created by modulation of the rates. These situations were each created to set up a “ruling out”-system that enables to discriminate between the impact of the rate of association as well as the rate of dissociation, thus taking the affinity into account.

Affinity maturation

As discussed earlier, there are different concepts of affinity maturation. The process can be described as an evolutionary process and as the process of affinity-increase resembles substrate binding as described in enzyme kinetics, the equation that defines affinity would look like the following:

𝐴𝑓𝑓𝑖𝑛𝑖𝑡𝑦 ~ 𝐾𝑜𝑛

𝐾𝑜𝑓𝑓 , Kon being the rate of association (a1) and Koff being the rate of dissociation (c1) The question that rises is whether an increase in affinity is the result of an increase in association or a decrease in dissociation. It could be the result of both as well. There are multiple possibilities; firstly it must be considered that there are three states for every K, namely an increase in K, a decrease in K or K remaining neutral. When this setup is used, the relevant possibilities are the ones where the affinity rises, which are:

1. Kon increases a lot, Koff increases little

2. Kon increases, Koff remains constant

3. Kon increases, Koff decreases

4. Kon remains constant, Koff decreases

5. Kon decreases a little, Koff decreases a lot

To investigate the effects of Kon and Koff, various situations were designed to compare the effects of both

parameters on output cell and activated B cell populations. The situations had to be designed in a way that enables a ‘ruling-out’ system that can be used to determine the parameter impact by comparison of the situations. One situation should contain a constant affinity as it can be used as a control factor. The others should all focus on increasing affinity. One should only have an increasing Kon and another should

only have a decreasing Koff. Lastly, a situation including an increasing Kon as well as a decreasing Koff at

the same time was included to investigate the significance of affinity as a parameter as well, since affinity would rise the most in this situation.

The situations were the following:

• Situation 1: Kon increases, Koff decreases, Aff increases extremely • Situation 2: Kon and Koff increase proportionally, Aff remains constant • Situation 3: Kon increases, Koff remains same, Aff increases slowly • Situation 4: Kon remains same, Koff decreases, Aff increases mildly

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9 It should be mentioned that the formula assumes that the B cells before (B) and after (Ba) association are considered the same. In the models of Faro et al.(2019) a distinction between these B cells is made (Figure 1). An overview of the model dynamics used in this research can be found in the appendix (page 17).

The T cell maturation pathway from model 3 that makes use of the antigen signaled B cells (Be) could also have the effect of taking up more of these B cells during this part of the model. This could influence the impact of changing the constants as well since they share the component of the antigen signaled B cell (Be), therefore both the basic cycle as well as the T cell maturation pathway are directly affected by changes of the a1 (Koff) factor and indirectly by changes of the c1 (Kon) factor.

The values of Kon, Koff and affinity in the different situations can be found in Table 1. All steps taken for

Kon and Koff in the situations will differ 0.03c or 0.03a from the previous step. Affinity was calculated by

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Results

As shown in Figure 2 situations 1 and 4 show a lower production of output cells and activated B cells, whereas situation 3 and 2 show a high production of the same cells, this difference is very small in case of an endless supply of antigen however. Different amounts of antigen were used as the amount of antigens are directly related to the rate of association and the rate of

dissociation. The standard amount of antigen defined by Faro et al. (2019) was 3000.

Firstly, antigen amounts three times as big and three times as small were taken. After this, the choice was made to implement amounts ten times bigger and smaller. Antigen amounts of 300, 1000, 3000, 9000, 30000 were used in this research to simulate antigen deficiency (Ag < 3000) and surplus (Ag > 3000).

When looking at the graphs produced with an antigen amount of 3000, the effects of affinity changes are hard to notice. The amount of activated B cells totals up to approximately 5500. When the amount of antigen is increased, the graphs show no changes in output-cell amounts.

Lowering the amount of antigen by two thirds seemed to simulate an even greater effect since the difference between situation 4 & 1 as opposed to situation 3 and 2 rose. The difference between 4 and 1 or 3 and 2 mainly remained the same though. Lowering it by ten times resulted in extreme differences that did not resemble the original dynamics, therefore it is not

Figure 2. Activated B cell amounts in relation to antigen amounts.

All graphs portray the relation between activated B cell amounts and the amount of antigen. Each graph contains the different situations; situation 1 being a disproportional increase in affinity, situation 2 being a constant affinity, situation 3 being an increase in Kon and situation 4 being a decrease in Koff. On the left side, the graphs with antigen amounts of 300, 1000 and 3000 are shown from top to bottom. On the right side, graphs with an antigen amount of 9000 and 30 000 can be found.

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11 clear whether this is due to model instability or proper dynamic changes. Therefore the choice was made not to take this result into consideration as a reasonable outcome.

Following the outcome of the Ag1000 graph, it can be stated that Koff has a greater impact on the dynamics in germinal centers than Kon, since in both situation 1 and 4 there is a decrease of Koff. In contrary to the hypothesis that reducing Koff would lead to increased output cells, the opposite is true. Reducing Koff leads to a decrease in total activated B cells. To ensure that this statement can be validated, situation 3 and 4, which contain a constant Koff and constant

Kon respectively, were compared to

similar situations with less

difference between each cell Kon or

Koff. As an antigen amount equal to

1000 was the most stable type of model that clearly showed the differences, this was chosen to introduce the two new situations (Figure 3).

Situation 3 and 4 were used once again and compared to less extreme ranges of Kon/Koff values.

Situation 3 and 4 has steps of 0.03c or 0.03a between each different value (Table 1), whereas the new situations have a difference of 0.02 and 0.01 c or a. (Figure 3).

The results from the comparison of the steps support the statement that Koff has a stronger effect on

activated B cell output as opposed to Kon. In situation 3, where only Kon is variable, the results show no

clear difference between the change of steps. In situation 4, a clear difference can be seen between each step. A positive correlation between Koff and activated B cell output could be proposed. This means that

the higher the rate of dissociation, the higher the amount of activated B cells, which does not fall in line with the formula of affinity. No correlation could be proposed for situation 3.

Figure 3. Overview of different layers of situation 3 and 4.

The upper two graphs show the amount of activated B cells at an amount of 1000 antigens. The different diagrams show 0.01, 0.02 and 0.03 K-steps in situation 3 and 4. The graph on the lower left shows the two upper graphs recombined. Situation 3 shows no changes, situation 4 shows a clear distinction between graphs, where the steps correlate negatively with the total of cells.

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Discussion

Conclusion

To determine whether the rate of association or the rate of dissociation is more dominant in changing dynamics of germinal centers, various situations were designed to investigate how the changes would relate to the amount of activated B cells. The results of comparing the situations led to the conclusion that Koff, the rate of dissociation of antigens from the B cells, has more effect on the amount of

activated B cells when changed than Kon. This effect becomes even greater when the amount of antigen is lowered, simulating a deficiency of antigens in a germinal center.

Evaluation

Since this research relies heavily on Faro et al. (2019), all validation procedures are based on the graphs that were produced in their research. This could mean that in case of parameter errors, these would be consistent in both their models and the models in this research. In addition, the differential equations considering competition were validated using the same graphs, from which it can be concluded that these were consistent.

However, the same validation does not apply to implementation of the newly introduced parameters, since no reference model was available to compare the found data for yet. This means that none of the predictions can be applied for real world germinal center dynamics, since possibility for both validation and falsification is needed to set up a conclusion based on a finding or prediction. Since the dynamics do resemble the models from Faro et al. (2019) and these were based on confirmed researches, the model can be considered at least accurate with respect to Faro et al.

A study conducted by Rosenfeld et al. (2017) did show a more significant effect of alternating the rate of dissociation as opposed to the rate of association of antibodies, however, this was a very specific study focused on a specific antibody with high affinity to ricin. The use of the T cell maturation pathway profits from changes of Koff directly as well, which could also have influenced the outcome of this

research. If possible, future studies to confirm that this would not only apply to a single protein and a T cell maturation pathway, would fall in line with the conclusion of this research, thus validating the outcome that the rate of dissociation has a stronger effect on germinal center dynamics than the rate of association.

In case the rate of dissociation does indeed have more effect on germinal center dynamics, this would mean that it would be favorable to modulate this rate instead of the rate of association when engineering immunological agents that have an effect on germinal centers if dynamics are to be quickly changed. In case more control is needed, the agents could focus on the rate of association as this could provide more control since alternating this rate will bring smaller changes to the dynamics. This means Kon should be used to bring small changes as alternating the rate will affect the dynamics slower, hence providing more control over the situations as it decreases the chance of sudden major changes. In addition, when

designing artificial germinal centers, the distinction between the effects of the different rates can be implemented to recreate a more accurate germinal center.

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13 Interestingly, antigen deficiency showed a positive correlation between activated B output cells and Koff rates. The lower the antigen, the more the difference shows. Since antigen deficiency can be present in humans, medication focused on increasing Koff would be able to compensate for the deficiency and boost

the amount of output B cells to improve the immunological response. As this finding does contradict the formula of affinity that states an inversely proportional relationship between Koff and affinity, more

research should be done on this particular finding to validate it.

Recommendations

There is still debate as to how the whole process of affinity maturation unfolds in reality. Due to the chaotic environment in a germinal center, it would be logical to think that the process of affinity maturation would be chaotic as well, therefore being characterized by parallel processing. This would mean that the antigen presentation, T cell approval and division evolution would all happen at the same time (Iber & Maini, 2002). However, another hypothesis is that the process would happen sequentially, which would then be characterized by sequential processing. Arguments for this are that the B cells need the antigen presentation by the follicle dendritic cells before they move into the light zone, which means that antigen presentation would be a rate-limiting step, therefore indicating a sequential process (Berek

et al., 1991). It would be interesting to implement the various types of affinity maturation in the future

to see whether this changes the effects of the parameter changes in the same situations.

The main weakness of this research is the lack of comparison material, hence why it is of importance to support the creation of more models that explore the effects of association and dissociation of antigens on B cells. As all of the models in this research are based on Faro et al. (2019), this also means that in case of inaccuracies in their models, the models in this research will also suffer from the same inaccuracies. Apart from the need of elimination of these inaccuracies in case they are present, another argument for making more models is the fact that not all possible parameters have been included in the current models. Expansion of the models with state of the art knowledge in the future is advisable to generate even more accurate results.

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References

Anderson, S. M., Khalil, A., Uduman, M., Hershberg, U., Louzoun, Y., Haberman, A. M., … Shlomchik, M. J. (2009). Taking Advantage: High-Affinity B Cells in the Germinal Center Have Lower Death Rates, but Similar Rates of Division, Compared to Low-Affinity Cells. The Journal of Immunology.

https://doi.org/10.4049/jimmunol.0902452

Bannard, O., & Cyster, J. G. (2017). Germinal centers: programmed for affinity maturation and antibody diversification. Current opinion in immunology, 45, 21-30.

Berek, C., Berger, A., & Apel, M. (1991). Maturation of the immune response in germinal centers. Cell, 67(6), 1121-1129.

Clark, E. A., & Ledbetter, J. A. (1994). How B and T cells talk to each other. Nature, 367(6462), 425-428.

De Boer, R. J., Oprea, M., Antia, R., Murali-Krishna, K., Ahmed, R., & Perelson, A. S. (2001). Recruitment Times, Proliferation, and Apoptosis Rates during the CD8+ T-Cell Response to Lymphocytic Choriomeningitis Virus. Journal of Virology. https://doi.org/10.1128/jvi.75.22.10663-10669.2001

Faro, J., von Haeften, B., Gardner, R., & Faro, E. (2019). A sensitivity analysis comparison of three models for the dynamics of germinal centers. Frontiers in Immunology, 10(AUG). https://doi.org/10.3389/fimmu.2019.02038 Giorgino, T. (2009). Computing and visualizing dynamic time warping alignments in R: The dtw package. Journal of Statistical Software. https://doi.org/10.18637/jss.v031.i07

Iber, D., & Maini, P. K. (2002). A mathematical model for germinal centre kinetics and affinity maturation. K. Murphy; C. Janeway, P. T. M. walport. (2013). Janeway’s Immunology. Journal of Chemical Information and Modeling. https://doi.org/10.1017/CBO9781107415324.004

MacLennan, I. C. M. (1994). Germinal Centers. Annual Review of Immunology, 12(1), 117–139.

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Oprea, M., & Perelson, A. S. (1997). Somatic mutation leads to efficient affinity maturation when centrocytes recycle back to centroblasts. The Journal of Immunology, 158(11), 5155-5162.

Rosenfeld, R., Alcalay, R., Mechaly, A., Lapidoth, G., Epstein, E., Kronman, C., ... & Mazor, O. (2017). Improved antibody-based ricin neutralization by affinity maturation is correlated with slower off-rate values. Protein

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Appendices

Summary of immunological terms

*all of the following processes were explained using Janeway’s Immunology (K. Murphy; C. Janeway, 2013).

There are various cells involved in the immunity system. B cells are important for immunity since they secrete antibodies that are able to fight off antigens. Tfh cells are helper T cells which, when signaled by other B cells, construct and maintain germinal centers. These germinal centers will then serve as a space that stimulates B cell proliferation and differentiation. Matured B cells originating from these germinal centers can either be memory cells which are stored for future intrusion of the same antigen, so that an immediate reaction can take place, or the B cells can turn into long living plasma cells which make strong antibodies against antigens.

Follicular dendritic cells (FDCs) are not dendritic cells and belong to the immune system. These cells are necessary for generating germinal centers as they indirectly aid in organisation of lymphoid cells,

therefore enabling germinal center construction. FDCs can also trap antigens and they also enable B cells to become memory B cells as the requirement for this is, that B cells need to bind the antigen that the FDC presents.

Germinal centers can be seen as systems with their own behaviour and sequence of interactions. Upon forming a germinal center, differentiated B cells and FDCs interact during which the FDC provides the B cells with an antigen. These antigens are then presented to early Tfh cells which are positioned on the border between the light and dark zone. The dark zone is responsible for B cell proliferation as well as clonal diversity generation, while the light zone will serve for competition in the later stages occurring in germinal centers. Firstly, the B cells (in the dark zone) focus on dividing to generate a prevalent

population after which they deliberately mutate their own DNA, therefore stimulating clonal diversity. This leads to an increased efficiency in finding strong antibodies as statistically more options are made available to persist the antigen.

The B cells that are now maturing will move to the light zone after which antibody expression follows. These cells will then be signaled to go into apoptosis after which they will have to get signals from Tfh cells to stop the apoptosis. Since not all B cells will get these signals, it creates a competitional

environment for the B cells in which clonal evolution can take place, since the most adapted will survive. After survival these B cells will go back to the dark zone again to repeat the process and that will lead to even more mutations and, therefore, a greater chance of creating an even better antibody. After this whole process the cells can either become memory cells by restarting the whole cycle from the beginning again or they can become long living plasma cells which will be used to fight off antigens.

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Overview Absolute Affinity Values

Situation 1 Situation 2 Situation 3 Situation 4

Cell Kon Koff Aff Kon Koff Aff Kon Koff Aff Kon Koff Aff

B1 19.13 19.43 0.98 19.13 19.43 0.98 19.13 19.43 0.98 19.13 19.43 0.98 B2 19.70 18.85 1.05 19.70 20.01 0.98 19.70 19.43 1.01 19.13 18.85 1.02 B3 20.28 18.26 1.11 20.28 20.60 0.98 20.28 19.43 1.04 19.13 18.26 1.05 B4 20.85 17.68 1.18 20.85 21.18 0.98 20.85 19.43 1.07 19.13 17.68 1.08 B5 21.43 17.10 1.25 21.43 21.76 0.98 21.43 19.43 1.10 19.13 17.10 1.12 B6 22.00 16.52 1.33 22.00 22.34 0.98 22.00 19.43 1.13 19.13 16.52 1.16 B7 22.57 15.93 1.42 22.57 22.93 0.98 22.57 19.43 1.16 19.13 15.93 1.20 B8 23.15 15.35 1.51 23.15 23.51 0.98 23.15 19.43 1.19 19.13 15.35 1.25 B9 23.72 14.77 1.61 23.72 24.09 0.98 23.72 19.43 1.22 19.13 14.77 1.30 B10 24.30 14.18 1.71 24.30 24.68 0.98 24.30 19.43 1.25 19.13 14.18 1.35 B11 24.87 13.60 1.83 24.87 25.26 0.98 24.87 19.43 1.28 19.13 13.60 1.41 B12 25.44 13.02 1.95 25.44 25.84 0.98 25.44 19.43 1.31 19.13 13.02 1.47 B13 26.02 12.44 2.09 26.02 26.42 0.98 26.02 19.43 1.34 19.13 12.44 1.54 B14 26.59 11.85 2.24 26.59 27.01 0.98 26.59 19.43 1.37 19.13 11.85 1.61 B15 27.16 11.27 2.41 27.16 27.59 0.98 27.16 19.43 1.40 19.13 11.27 1.70 B16 27.74 10.69 2.60 27.74 28.17 0.98 27.74 19.43 1.43 19.13 10.69 1.79

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Germinal Center Overview

1

2

3

Suppl. Figure 1. An simplified overview of B cell maturation in germinal centers.

The figure only shows B cells to prevent confusion between the various cell types, hence why this is an oversimplified schematic. It starts with a B-cell that has just been exposed to an antigen delivered by an FDC (1). This occurrence initiates the clonal expansion of the B-cells in the dark zone (moon) which is characterized by proliferation as well as mutation of B-cells (2). B-cells that have almost matured will move to the light zone (sun) in which the process of clonal evolution starts as they enter the apoptotic state in this zone. To prevent death, the B-cells will need to receive survival signals by the FDC and Tfh cells. Since these survival signals are antibody-affinity dependent, only the most adapted will survive (3), after which they can return into the cycle to create an even better cell type (4).

4

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20

Final Model (incl. mutations, affinity maturation and situations)

library(deSolve)

vraag3 <- function() { ## PARAMETERS

tspan <- seq(0,30, by = 1/24)

## KON INCREASES; KOFF DECREASES OUT PROPORTION (KON/KOFF INCREASES OVER TIME)

#factorenc <- seq(1,1.30,by = 0.02) #factorena <- rev(seq(0.70,1, by = 0.02))

## KON AND KOFF INCREASE IN PROPORTION (KON/KOFF = CONSTANT) #factorenc <- seq(1,1.30,by = 0.02)

#factorena <- seq(1,1.30,by = 0.02)

## KON INCREASES (KON/KOFF INCREASES OVER TIME) #factorenc <- seq(1,1.30,by = 0.02)

#factorena <- c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

## KOFF DECREASES (KON/KOFF INCREASES OVER TIME) factorenc <- c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) factorena <- rev(seq(0.70,1,by = 0.02)) c1 <- factorenc * 19.13 c2 <- 16.77 a1 <- factorena * 19.43 a2 <- 14.29 p1 <- 2.25 p2 <- 2.70

(21)

21 db <- 2.03 dt <- 0.39 p1m <- 1.16 p2m <- 1.63 dbm <- 1.20 Kt = 315.18 Kb = 12632.48 mu2 <- 0.10

# ANTIGEN AMOUNT (NORMALLY: 3000) #Af <- 3000 #Af <- 2000 Af <- 1000 #Af <- 500 # Y WAARDEN i <- 100/16 B <- c(i,i,i,i,i,i,i,i,i,i,i,i,i,i,i,i) Ba <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) Be <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) T1 <- 10 Tb <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) Td <- 0 Bd <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) Bm <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) Tm <- 0 Tbm <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) Tdm <- 0 Bdm <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(22)

22 # MUTATIONS OFF #mutability = c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) # MUTATIONS ON mutability = c(0.05, 0.1, 0.15, 0.20, 0.02, 0.04, 0.08, 0.16, 0.01, 0.02, 0.04, 0.06, 0.03, 0.06, 0.12, 0.24 ) mt1 = runif(16,0.0,0.01) mt2 = runif(16,0.0,0.01) mt3 = runif(16,0.0,0.01) mt4 = runif(16,0.0,0.01) mt5 = runif(16,0.0,0.01) mt6 = runif(16,0.0,0.01) mt7 = runif(16,0.0,0.01) mt8 = runif(16,0.0,0.01) mt9 = runif(16,0.0,0.01) mt10 = runif(16,0.0,0.01) mt11 = runif(16,0.0,0.01) mt12 = runif(16,0.0,0.01) mt13 = runif(16,0.0,0.01) mt14 = runif(16,0.0,0.01) mt15 = runif(16,0.0,0.01) mt16 = runif(16,0.0,0.01) mutstore = rbind(mt1,mt2,mt3,mt4,mt5,mt6,mt7,mt8,mt9,mt10,mt11,mt12,mt13,mt14,mt15,mt16) mutations = apply(mutstore, 2, sum)

mutations1 = c(0.04, 0.08, 0.09, 0.07, 0.09, 0.08, 0.07, 0.04, 0.05, 0.07, 0.08, 0.05, 0.02, 0.01, 0.08, 0.07)

# DEZE FACTOREN WORDEN BIJ DE RESPECTIEVELIJKE CELHOEVEELHEDEN OPGETELD EN DE MUTABILITY FACTOR WORDT ERAF GEHAALD

print("The following mutation rates were used (from B1 (L) to B16 (R)): "); print(mutations[1:4]); print(mutations[5:8]); print(mutations[9:12]); print(mutations[13:16])

(23)

23

statevar = c(Af, B, Ba, Be, T1, Tb, Td, Bd, Bm, Tm, Tbm, Tdm, Bdm)

par1 <- c(c1, c2, a1, a2, p1, p2, db, dt, Kb, Kt, mu2, p1m, p2m, dbm, mutations, mutability ) out <- ode(statevar, tspan, opgave3, par1)

matplot(out, type="l", ylab="Number of cells ", xlab="Time (hours)", main= "Germinal center, model 3" )

return(out) }

opgave3 <- function(t, y, parms) { # PARAMETERS c1 <- parms[1:16] c2 <- parms[17] a1 <- parms[18:33] a2 <- parms[34] p1 <- parms[35] p2 <- parms[36] db <- parms[37] dt <- parms[38] Kb <- parms[39] Kt <- parms[40] mu2 <- parms[41] p1m <- parms[42] p2m <- parms[43] dbm <- parms[44] mutations <- parms[45:60]

(24)

24 mutability<- parms[61:76] # Y WAARDEN Af <- y[1] B <- y[2:17] Ba <- y[18:33] Be <- y[34:49] T1 <- y[50] Tb <- y[51:66] Td <- y[67] Bd <- y[68:83] Bm <- y[84:99] Tm <- y[100] Tbm <- y[101:116] Tdm <- y[117] Bdm <- y[118:133] # INTERACTIEPARAMETERS

Bt = sum(B) + sum(Ba) + sum(Be) + sum(Tb) + sum(Bd) + sum(Bm) + sum(Tbm) + sum(Bdm) Tt = T1 + sum(Tb) + Td + sum(Tbm) + Tm + Tdm n = 1 alfaT = Kt/(Kt+Tt) alfaB = (2**n - 1) * Kb/(Kb+Bt) m2 <- (mu2 * sum(Tb))/(T1+sum(Tb)+Td) # ALLE DIFFERENTIAALVERGELIJKINGEN dAfdt = -1*sum(c1*B)*Af + sum(a1*Ba)

dBdt = -1*db * B - c1 * B * Af + p1*(1+alfaB) * Bd dBmdt = -1 * dbm * Bm + p1m *(1+alfaB)*Bdm dBadt = c1 * B * Af - a1 * Ba

(25)

25 dBedt = -1*c2*Be*(T1 + Tm) + a1*Ba - db*Be

dTdt = -1*c2*sum(Be)*T1 + p2*(1-m2)*(1+alfaT) * Td - dt*T1 dTbdt = c2*Be*T1-a2*Tb

dTddt = a2 * sum(Tb) - p2 * Td

dBddt = a2 * Tb - p1 * Bd - mutability*Bd + mutations*Bd*mutability

dTmdt = -1*c2 * sum(Be) * Tm + p2*m2*(1+alfaT) * Td + p2m*(1+alfaT)*Tdm - dt*Tm dTbmdt = c2 * Be * Tm - a2 * Tbm

dTdmdt = a2 * sum(Tbm) - p2m * Tdm dBdmdt = a2 * Tbm - p1m * Bdm

# ALLE JUISTE DIFFERENTIAALOUTPUTS return(list(c(af = dAfdt,

dBdt[1], dBdt[2], dBdt[3], dBdt[4],dBdt[5], dBdt[6], dBdt[7], dBdt[8],dBdt[9], dBdt[10], dBdt[11], dBdt[12],dBdt[13], dBdt[14], dBdt[15], dBdt[16],

dBadt[1], dBadt[2], dBadt[3], dBadt[4],dBadt[5], dBadt[6], dBadt[7], dBadt[8],dBadt[9], dBadt[10], dBadt[11], dBadt[12],dBadt[13], dBadt[14], dBadt[15], dBadt[16],

dBedt[1], dBedt[2], dBedt[3], dBedt[4],dBedt[5], dBedt[6], dBedt[7], dBedt[8],dBedt[9], dBedt[10], dBedt[11], dBedt[12],dBedt[13], dBedt[14], dBedt[15], dBedt[16],

dTdt, dTbdt[1], dTbdt[2], dTbdt[3], dTbdt[4],dTbdt[5], dTbdt[6], dTbdt[7], dTbdt[8],dTbdt[9], dTbdt[10], dTbdt[11], dTbdt[12],dTbdt[13], dTbdt[14], dTbdt[15], dTbdt[16], dTddt, dBddt[1], dBddt[2], dBddt[3], dBddt[4],dBddt[5], dBddt[6], dBddt[7], dBddt[8],dBddt[9], dBddt[10], dBddt[11], dBddt[12],dBddt[13], dBddt[14], dBddt[15], dBddt[16], dBmdt[1], dBmdt[2], dBmdt[3], dBmdt[4],dBmdt[5], dBmdt[6], dBmdt[7], dBmdt[8],dBmdt[9], dBmdt[10], dBmdt[11], dBmdt[12],dBmdt[13], dBmdt[14], dBmdt[15], dBmdt[16], dTmdt, dTbmdt[1], dTbmdt[2], dTbmdt[3], dTbmdt[4],dTbmdt[5], dTbmdt[6], dTbmdt[7], dTbmdt[8],dTbmdt[9], dTbmdt[10], dTbmdt[11], dTbmdt[12],dTbmdt[13], dTbmdt[14], dTbmdt[15], dTbmdt[16], dTdmdt,

(26)

26 dBdmdt[1], dBdmdt[2], dBdmdt[3], dBdmdt[4],dBdmdt[5], dBdmdt[6], dBdmdt[7],

dBdmdt[8],dBdmdt[9], dBdmdt[10], dBdmdt[11], dBdmdt[12],dBdmdt[13], dBdmdt[14], dBdmdt[15], dBdmdt[16]))) # the function returns a new value of dR/dt

}

# DATAGENERATIE EN AANMAKEN VAN EEN KOLOM MET DE OPGETELDE WAARDEN Data = vraag3()

dAf <- cbind(Data[,c(1,2)])

dB <- cbind(Data[,c(1,3:18)], rowSums(Data[,3:18])) dBa <- cbind(Data[,c(1,19:34)], rowSums(Data[,19:34])) dBe <- cbind(Data[,c(1,35:50)], rowSums(Data[,35:50])) dT1 <- cbind(Data[,c(1,51)]) dTb <- cbind(Data[,c(1,52:67)], rowSums(Data[,52:67])) dTd <- cbind(Data[,c(1,68)]) dBd <- cbind(Data[,c(1,69:84)], rowSums(Data[,69:84])) dBm <- cbind(Data[,c(1,85:100)], rowSums(Data[,85:100])) dTm <- cbind(Data[,c(1,101)]) dTbm <- cbind(Data[,c(1,102:117)], rowSums(Data[,102:117])) dTdm <- cbind(Data[,c(1,118)]) dBdm <- cbind(Data[,c(1,119:134)], rowSums(Data[,119:134]))

## DISTINCTIE VAN DE DIVERSE SITUATIES DOOR OPSLAG IN NIEUWE VARIABELE #UltimateData1 <- cbind(dAf, dB[,18], dBa[,18], dBe[,18], dT1[,-1], dTb[,18], dTd[,-1], dBd[,18], dBm[,18], dTm[,-1], dTbm[,18], dTdm[,-1], dBdm[,18])

#UltimateData2 <- cbind(dAf, dB[,18], dBa[,18], dBe[,18], dT1[,-1], dTb[,18], dTd[,-1], dBd[,18], dBm[,18], dTm[,-1], dTbm[,18], dTdm[,-1], dBdm[,18])

#UltimateData3 <- cbind(dAf, dB[,18], dBa[,18], dBe[,18], dT1[,-1], dTb[,18], dTd[,-1], dBd[,18], dBm[,18], dTm[,-1], dTbm[,18], dTdm[,-1], dBdm[,18])

UltimateData4 <- cbind(dAf, dB[,18], dBa[,18], dBe[,18], dT1[,-1], dTb[,18], dTd[,-1], dBd[,18], dBm[,18], dTm[,-1], dTbm[,18], dTdm[,-1], dBdm[,18])

(27)

27 # PLOTTEN VAN ALLE MAXWAARDEN

matplot(UltimateData[,-1], type = "l", ylab="Number of cells ", xlab="Time (hours)", main= "Germinal center New Model", col = c(2,3,4,5,6,7,8,1,2,3,4,5,6), lty = c(1,1,1,1,1,1,1,1,2,2,2,2,2)); legend("topleft", ncol = 2, legend = c("Af","B","Ba","Be","T","Tb","Td","Bd","Bm","Tm","Tbm","Tdm", "Bdm"), col = c(2,3,4,5,6,7,8,1,2,3,4,5,6), lty = c(1,1,1,1,1,1,1,1,2,2,2,2,2), cex = 0.7)

# VERGELIJKING VAN OUD MODEL EN NIEUW MODEL OM OVEREENKOMSTEN TE VINDEN (eerst andere model (Model3.R) inladen s.v.p.)

Oudmodel = vraag4()

#NewModel1 = matplot(UltimateData1[,-1], type = "l", ylab="Number of cells ", xlab="Time (hours)", main= "Germinal center Situation 1", col = c(2,3,4,5,6,7,8,1,2,3,4,5,6), lty = c(1,1,1,1,1,1,1,1,2,2,2,2,2)); legend("topleft", ncol = 2, legend = c("Af","B","Ba","Be","T","Tb","Td","Bd","Bm","Tm","Tbm","Tdm", "Bdm"), col = c(2,3,4,5,6,7,8,1,2,3,4,5,6), lty = c(1,1,1,1,1,1,1,1,2,2,2,2,2), cex = 0.7)

NewModel4 = matplot(UltimateData4[,-1], type = "l", ylab="Number of cells ", xlab="Time (hours)", main= "Germinal center Situation 4", col = c(2,3,4,5,6,7,8,1,2,3,4,5,6), lty = c(1,1,1,1,1,1,1,1,2,2,2,2,2)); legend("topleft", ncol = 2, legend = c("Af","B","Ba","Be","T","Tb","Td","Bd","Bm","Tm","Tbm","Tdm", "Bdm"), col = c(2,3,4,5,6,7,8,1,2,3,4,5,6), lty = c(1,1,1,1,1,1,1,1,2,2,2,2,2), cex = 0.7)

# ALLE BELANGRIJKE OUTPUTPARAMETERS #dBd1 <- cbind(Data[,c(1,69:84)] , rowSums(Data[,69:84 ])) #dBd2 <- cbind(Data[,c(1,69:84)] , rowSums(Data[,69:84 ])) #dBd3 <- cbind(Data[,c(1,69:84)] , rowSums(Data[,69:84 ])) #dBd4 <- cbind(Data[,c(1,69:84)] , rowSums(Data[,69:84 ])) #dBm1 <- cbind(Data[,c(1,85:100)], rowSums(Data[,85:100])) #dBm2 <- cbind(Data[,c(1,85:100)], rowSums(Data[,85:100])) #dBm3 <- cbind(Data[,c(1,85:100)], rowSums(Data[,85:100])) #dBm4 <- cbind(Data[,c(1,85:100)], rowSums(Data[,85:100]))

(28)

28 # CODE VOOR DE TOEKOMST OM ONDERLINGE VERGELIJKINGEN TE MAKEN (nog niet relevant)

matplot(dB [,c(-1,-18)], type = "l", ylab="Number of cells ", xlab="Time (hours)", main= "Germinal center New Model B" , col = c(1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8), lty = c(1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2)); legend("topright", legend = c(paste0("cell", 1:16 )),col = c(1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8), lty = c(1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2), ncol = 2, cex = 0.7 )

matplot(dBa [,c(-1,-18)], type = "l", ylab="Number of cells ", xlab="Time (hours)", main= "Germinal center New Model Ba" , col = c(1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8), lty = c(1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2)); legend("topleft" , legend = c(paste0("cell", 1:16 )),col = c(1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8), lty =