OPEN Clinical & Translational Immunology (2017) 6, e131; doi:10.1038/cti.2016.

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THEORETICAL ARTICLE

A generalized quantitative antibody homeostasis

model: antigen saturation, natural antibodies
and a quantitative antibody network
Jzsef Prechl1,2

In a pair of articles, we present a generalized quantitative model for the homeostatic function of clonal humoral immune system.
In this second paper, we describe how antibody production controls the saturation of antigens and the network of antibody
interactions that emerges in the epitome space with the establishment of the immune system. Efcient control of antigens, be it
self or foreign, requires the maintenance of antibody concentrations that saturate antigen to relevant levels. Simple calculations
suggest that the observed diverse recognition of antigens by natural antibodies is only possible by cross-reactivity whereby
particular clones of antibodies bind to diverse targets and shared recognition of particular antigens by multiple antibody
clones contribute to the maintenance of antigen control. We also argue that natural antibodies are none else than the result of
thymus-independent responses against immunological self. We interpret and explain antibody production and function in a
virtual molecular interaction space and as a network of interactions. Indeed, the general quantitative (GQM) model we propose is
in agreement with earlier models, conrms some assumptions and presumably provides the theoretical basis for the construction
of a real antibody network using the sequence and interaction database data.
Clinical & Translational Immunology (2017) 6, e131; doi:10.1038/cti.2016.90; published online 17 February 2017

THE GQM APPLIED TO ANTIBODY HOMEOSTASIS apparent afnity to indicate that factors like multiple binding sites
By denition, antigens are molecules recognized by antibodies. Most modulate the observed strength of the interaction.
denitions however fail to further elaborate what exactly is meant by Assuming that antibodies are produced with the intent of regulating
recognition. The strength of the interaction between the antigen- antigen availability, best control over antigen concentration is achieved
binding site (paratope) of an antibody and the antibody-binding site when the concentration of antibody is close to the KD (Figure 1). In
(epitope) of the antigen is characterized by afnity, kinetics of our map, this zone for a range of [Ab] and KD values is dened by a
association and dissociation, and binding energy. Antibodies often line, where [Ab] = KD, which is the line representing 50% saturation of
recognize more than one target. Immunological assays usually require the antigen (Figure 1). By lowering or increasing antibody production,
the titration of the antibody, which is the identication of lowest the host can release or capture antigens, and likewise by changing the
concentration that binds to the nominal target but does not bind to efciency of Ab binding, the host can modulate antigen saturation
others. This is quite logical for antibodies intentionally produced in (Figure 1). Various immunological mechanisms are responsible for
animals, but how we dene the target of an antibody in vivo? By removing antibodyantigen complexes, called immune complexes,
changing the concentration of antigen and antibody, saturation of any from the circulation.
can be achieved even when afnity of the interaction is low. The The range of [Ab] values we will be using in our model reect
absolute and relative concentration of antigens and antibodies does actual immunoglobulin concentrations in blood plasma, and start
matter and our GQM attempts to reveal antibody function by around the concentration that a single plasma cell could achieve by
addressing these factors. continuous secretion of antibody. The range of KD values includes
The general equation dening equilibrium dissociation constant KD: afnity constants usually observed1 for antibodyantigen interactions
(of KD = 10 610 10 M) but extends to both lower and higher values
AbAg to provide exibility for interpreting apparent afnities. Please note
KD
AbAg that these are exactly the same dimensions, which we use in our
tells us that KD = [Ab] when [Ag] = [AbAg]. That is when antigen is accompanying sister paper on B-cell development.2 Let us now analyze
half saturated, free antibody concentration is equal to KD. For the sake various characteristic immune responses in the order of increasing
of simplicity, we will regard [Ab] as the concentration of paratope and antibodyantigen interaction afnity. We will consider a single uidic
[Ag] as the concentration of epitope and we shall use the term compartment, the blood plasma for this theoretical framework,

1
R&D Laboratory, Diagnosticum zrt, Budapest, Hungary and 2MTA-ELTE Immunology Research Group, at Etvs Lornd University, Budapest, Hungary
Correspondence: Dr J Prechl, R&D Laboratory, Diagnosticum zrt, Attila ut 126, Budapest 1047, Hungary.
E-mail: jprechl@diagnosticum.hu
Received 26 August 2016; revised 15 November 2016; accepted 24 November 2016
 A quantitative antibody homeostasis model II
J Prechl
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Figure 1 Outlines of the GQM for regulation of antibody production. (a) Antibodies will saturate antigen by increasing their concentration or by increasing
apparent afnity. (b) Low concentrations of low-afnity antibodies do not bind antigen at relevant extent, antigen concentrations can be best controlled at
around 50% saturation, while elimination is achieved by increasing saturation further. (c) ASC produce antibodies to increase concentration, while prior
differentiation account for increased afnity or ability to remove antigen. Immune complexes are removed by different cells and silent or proinammatory
events.

however, with proper adjustments the model can be possibly extended As we have discussed in our accompanying article, our GQM
to include the extracellular space and mucosal surfacessites of key assumes that B1I cells develop from immature B cells in blood, as a
importance for immunological action. result of continuous BCR signaling triggered by blood-borne antigens.
Their precursors, immature B cells were selected based on their
NATURAL ANTIBODIES AND TI ANTIBODY RESPONSES polyreactive property, tested by self-antigen displayed on the devel-
Can low-afnity antibodies mediate any relevant biological effect at all? oping cells. Polyreactivity against common, shared molecular targets
For an antibody with KD = 10 6 M a concentration of 10 6 M should allows clusters of antibodies to cooperatively bind to highly abundant
be reached for substantial binding to its target, which is quite close to self-antigens, reaching relevant fractional saturation despite the
the total immunoglobulin concentration in plasma (Figure 2). relatively low afnity of the interactions. Any particular antibody
Obviously, no single antibody can dominate to such an extent can belong to several different such clusters, thereby increasing the
(except for pathological antibodies in disease, like monoclonal concentration of antibodies against that particular antigen. In other
gammopathies). Multiple antibody-binding sites on the antigen words, by shared, distributed recognition several different antigenic
increase the apparent afnity of the interaction by avidity effects, targets can be saturated by the same amount of antibody. Cross-
but not to the value required here. Most likely, a combination of these reactivity increases the apparent clonal diversity because clusters of the
effects, large cumulative concentration achieved by a large number of same network respond to different targets. Removal of cellular debris,
cross-reacting antibody producing B-cell clones and avidity might which contains a huge number of different molecules may become
confer effector functions to low-afnity antibodies. possible this way by a limited number of specicities and antibodies.
Natural antibodies are low-afnity antibodies, constitutively pro- Polyspecic, low-afnity, high off-rate interactions in the plasma result
duced by B1I cell populations that are relatively well characterized in in dynamic short-lived contacts, these natural antibodies acting like a
the mouse3 but only recently described in humans.4 Natural antibodies lubricant rather than like cement. On the other hand, molecular aging,
are polyreactive or polyspecic, binding to structurally different self often presenting as aggregation, results in polymerization of the target,
and microbial targets as well,5,6 these targeted epitopes being mostly increasing the avidity of interactions with natural antibodies and
non-protein molecules. The afnity of natural antibodies to mono- aiding removal. This removal process is continuous and silent. Natural
valent glycan has been determined to be in the range of 10 4106 M.7 IgM xes C1q, which in turn is captured and taken up by phagocytes
While mostly, of the IgM class,8 functionally similar antibodies throughout the body, utilizing different C1q receptors.1316 Natural
belonging to the IgA and IgG classes are also found. Besides providing antibodies may also cover and seal leaks in the endothelium,
immediate protection against invading microbial agents, natural promoting regeneration and healing.17
antibodies are known to play key roles in the clearance of self In addition to B1I cells, MZ B cells also contribute to the fast
molecules, thereby contributing to homeostatic control, suppressing production of antibodies upon challenge.18 These two populations are
inammation and autoimmunity.9,10 In humans, B1I cells represent the cells responsible for thymus-independent (TI) responses. Is there,
from less than 1 to 9% of the circulating B cells.11 Calculating with an then, a difference between natural antibodies and TI antibody
average 4 109 white blood cells per liter of blood, 5% B cells in white responses? We believe there is not. Natural antibodies are dened as
blood cells, and an average 5% of B1I of all the B cells, we arrive at an being produced in the absence of a known antigenic stimulus. This is
averaged 5 107 B1 cells in 5 l of blood, capable of a dominant possibly a wrong interpretation of the events. Accepting that B-cell
contribution to the plasma antibody pool. The number of circulating development involves selection of low-afnity, polyspecic self-
B1I cells indeed shows correlation with serum IgM levels.12 reactivity, the antigenic stimulus is self. Natural IgM antibodies are

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help. B cells with increased afnity to the eliciting antigen emerge in

germinal center reactions, giving rise to antibody-secreting effector
cells (ASC) and memory B cells. Our GQM postulates that antibody
secretion is an effort to decrease [Ag] and return B cells to their
comfort BCR signaling zone.2 These ASC leave the germinal center
and circulate in blood, and are found in blood-rich compartments of
the bone marrow and spleen, some settling and forming long-lived
plasma cells in suitable microenvironments.23 Antibody secretion by
ASC contributes to the pool of antibodies in blood. Along with the
increase in afnity against the target antigen, antibodies lose their
polyreactivity, forming less exible but more precisely tting hyper-
variable loops to contact antigen. Repeated exposure to the same
antigen (booster injections or hypervaccination) recruits memory B
cells, which have an already high afnity, making further steps along
the afnity axis in our map (Figure 2) until limits are reached.24
Increasing the afnity means that less and less [Ab] is required for
saturating the antigen, which translates into less and less ASC required
for eliciting elimination of antigen. This is the idea behind most of our
prophylactic vaccines.25 Our map suggests that even with high-afnity
binding of 10 10 M a huge number of ASC clones are needed for
Figure 2 Balance of Ab and Ag achieved by different humoral immune conferring such protection. The situation drastically changes of course,
responses. The epitope-antibody interaction landscape as dened by free if we look at local concentrations instead of plasma concentration,
antibody concentration and afnity. Second signals required by B cells for where even a single ASC can generate adequate amounts of antibody.
becoming antibody-secreting cells are listed next to the type of immune Afnity maturation is usually accompanied by class switch, so the
response. The range of total serum immunoglobulin concentration and the
heavy chain class of high-afnity antibodies changes from IgM to IgA,
concentration achieved by a single ASC clone are indicated. Bm, memory
B-cell; btk, Brutons tyrosine kinase; NKT, NK T cells; PRR, pattern IgG and IgE. This has very important bearing on the way how
recognition receptors; TD, thymus dependen; TFH, follicular helper T cells; immune complexes forming from these antibodies are eliminated.
Th, helper T-cell; TI, thymus independent. Complement activation and FcR-mediated effector function properties
of these antibody classes can be signicantly different from those of
also present in germ-free mice,19 where the only template for antibody natural antibodies.2628 Increased afnity, mostly accompanied by
production is self. decreased dissociation rates,24 lends rigidity to the forming immune
Natural antibodies are born silently and are eliminated silently complexes, also making them more prone to elimination. Saturation
because no inammatory events accompany either induction of by high-afnity antibody means close to continuous presence of
antibody production or removal of forming immune complexes. They bound antibody, which is poised to lead to interaction of the complex
function to aid the catabolism of cellular debris, of aging proteins, with other components of the immune system such as cells bearing Ig
of endothelial cells and contribute to the molecular and cellular receptors or complement proteins. All characteristics of a TD response
regeneration of blood and blood vessels. Since the same type of cells imply that the immune system does not accept the targeted antigen as
produce natural antibodies and TI antibodies, we consider these part of the immunological self and it concerts its efforts to reject the
antibodies essentially the same, especially for IgM. Whereas the antigen.
continuous presence of stimulatory concentrations of blood-borne From our perspective, an immunodominant B-cell epitope of a TD
antigen drives antibody production by B1 cells, TI responses are response will be one which evades recognition and clearance by
elicited by further increasing Ab production and recruitment of natural antibodies (or preformed induced antibodies), yet is
natural memory cells, the MZ B cells. This can be triggered by recognized by the naive B cells and especially their afnity matured
administration of a proper antigen: pattern recognition receptor progeny. Simply put, an epitope that is the least like self, and its
ligands for TI-1,20 highly repetitive microbial and viral motifs for matching antibody is the most efcient at selectively removing the
TI-2,21 and recruitment of myeloid cell help for TI-322 (Figure 2). antigen bearing the epitope, is likely to initiate a response leading to
Second signals provided by microbial products and cytokines can afnity maturation, B-cell and antibody memory. This epitope will be
induce class switching, leading to the generation of IgA and IgG one that is extruded from the self-epitope space, as discussed in the
(mainly IgG1 and IgG2) antibodies. Overall, TI responses are the next section.
manifestation of antibody-secreting effector function of B cells selected The properties of the BCR and antibodies are summarized in
in the bone marrow and primed in the periphery for controlling Figure 4, layers representing stages of development and antigen
abundant self and non-self antigens. In short, antibodies produced by recognition. Immunological self is a part of the antigenome recognized
TI responses constitute the natural antibody repertoire (Figure 3). as part of the host. Extended immunological self incorporates
embryonic or genetic self and the molecular environment recognized
TD ANTIBODY RESPONSES by natural antibodies. Catabolism of the repertoire of molecules in the
The afnity of a given antibody to a given antigen can only be extended self or immunological self is provided for by natural
increased by changing its structure and sequence. While structural antibodies. Accepted environment is the border of immunological
changes, such as oligomerization (IgM pentamer 4hexamer, IgA self, with enhanced removal of the forming immune complexes.
monomer 4dimer conversions) play role in TI responses, template- Rejected environment is recognized by high-afnity antibodies, which
driven maturation of afnity by sequence modication requires T-cell elicit strong effector functions and attempt to eliminate the antigen.

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 A quantitative antibody homeostasis model II
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represented as a point in this space. This will be the epitope that is

recognized with the highest afnity by that particular BCR. Let us call
this the cognate epitope of the antibody. Because developing fetal
B cells only come into contact with self, this will be a self epitope in
the epitope space of all epitopes, the epitome. Please note that the
exact identity of cognate antigen can change, as the immune system
encounters new epitopes, with potentially higher binding afnity. The
antibodies these cells (or their effector forms) secrete will bind to this
cognate epitope but also to closely related epitopes, even though with
lower afnity. To characterize the collection of epitopes in the
neighboring epitope space to which these antibodies bind, we can
now use a sphere. The amount of antibody available for binding to
epitopes can be symbolized as [Ab]/KD, an expression called antigen-
binding capacity. So we can draw a sphere with a radius [Ab]/KD
(Figure 5). These points, representing B cells with identical antigen-
binding properties, and spheres representing the capacity of antibodies
to bind to the contained epitopes, will be the nodes in our network.
These points and spheres are connected by distances depicting
cross-reactivity of the antibodies. We can characterize epitope
similarity, or in other words, antibody cross-reactivity, by averaging
the differences in afnity between cognate antigen and the other
antigen (Figure 5). As B cells are developing more and more points,
Figure 3 Thymus-independent- and thymus-dependent antibody responses in spheres appear in the epitope space. Closely related antibodies, such
the interaction landscape. Relative concentration of epitopes, antibodies and the as progeny of a pre-BI cell, with identical heavy chains, will be
afnity of the interactions dene natural and thymus-dependent antibodies. close to each other and farther away from other clusters with a
different heavy chain.
It is important to note that there is neither absolute denition nor
intrinsic property for a self or foreign epitope. Rather, the developing
immune system carves out a space that will dene the network
of interactions dening self. We can call this space the extended
immunological self (Figure 4). As IgM production increases, these
molecules can present as epitopes themselves, seeding new interactions
and leading to the development of anti-idiotype antibodies.29 The
resulting closely knit network denes the space containing epitopes
regarded as self. Once lymphoid organs are structured and the host is
born, foreign molecules enter and various kinds of antibody responses
develop. TI responses increase Ab concentration without changing
KD, thus the position of the node does not change only the size.
TD responses also decrease KD, changing the position of a node in the
epitope space and its distance to related clones. Huge increases in
afnity along with loss of polyspecicity result in the extrusion of the
node from the network (Figure 6).
Continuous unregulated production of antibodies by long-lived
plasma cells lends rigidity to the interaction space: these interactions
are hardwired into the system, as such cells are x posts in the
Figure 4 Layers of interactions of antibodies with immunological self and
non-self. mutBCR, BCR with somatic hypermutation. interaction space. Whereas the developing immune system is quite
exible and dynamic, the immune system of an adult has lots of these
Using the dimensions [Ag], [Ab] and KD, we have been able to hardwired posts and possesses less adaptability. Ideally, this more
draw a raw general map of clonal humoral immune responses, rigid interaction space provides protection from pathogens. A badly
hardwired system, such as a chronic autoimmune condition, is very
positioning B cells at different stages of development and antibodies
difcult to heal because of this rigidity, however.30
produced by them on these maps. With these quantitative descriptors,
The fact that an antigen reaches and stimulates a B-cell, reects that
we can now try to design a network that models immune function.
it has not been masked by antibodies from being recognized by that
clone. Additionally, uptake via the BCR can be accompanied by the
INTERPRETING THE HUMORAL IMMUNE RESPONSE AS A uptake of lipids or proteins recognized by helper T cells. These two
NETWORK PHENOMENON events warn the immune system that a molecule that needs to be
Let us imagine a space of epitopes. In this space, certain coordinates eliminated breached the network barrier. The production of
identify interactions with the immune system. The space is uid, high-afnity antibodies means that a new node (connected to other
epitopes can change their positions. The developing immune system clones with the same specicity but lower afnity) appears in the
seeds this space with interactions by generating antibodies that bind to respective antigenic space, ling the space with an increased node size,
epitopes. Each formed B-cell with an antigen receptor (BCR) is which reects increased [Ab]free/KD value.

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Figure 5 Denitions of the antibody network. (a) Antibody Abi reacts with epitope A and cross-reacts with epitope B. Abj reacts with epitope A and cross-
reacts with epitope B. Distances, expressed as negative logarithm of KD are averaged to get the distance in the epitope space, providing an edge in the
network. (b) Free antibody concentration divided by KD reects antigen-binding capacity. This value is displayed as the radius of the node. Changes in
antigen concentration will trigger changes in node radius. New antigens entering the system may react with higher afnity with a given antibody and thereby
change its position in the epitope space.

Figure 6 Simplied schematic representation of the antibody-epitome interaction network. Only nodes of the network are shown, edges are not drawn for the
sake of simplicity but are represented as internodal distances. Blue nodes are natural antibody interactions, purple nodes are interactions with the accepted
environment, red nodes are TD interactions in the epitope space. Green area represents B cells in the quiescent state, not producing antibodies but
presenting themselves in the epitome space. Immunological self is a superconnected central component, while epitopes marked for elimination are extruded
and disconnected.

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With the denitions we have built up of the immunological self. The reactivity networks generated by
Bransburg-Zabary et al.37 by antigen microarray analysis are probably
1. B cells respond to [Ag] by developing into ASC the experimental observations closest to our model. In these experi-
2. B cells respond to [Ag] by changing KD ments, correlation analysis of antibody reactivity between individuals
3. [Ab] is regulated to reach KD for control of [Ag] revealed the presence of connected specicities. The deconvolution of
4. Network node size is [Ab]/KD polyclonal antibody reactivity within individuals will be the key to
5. i-j internodal distance is |KDi-KDj| generate the experimental data directly supporting our model.
6. In a network with N = number of B cells with distinct antigen Quantitative, descriptive approaches to antibody networks currently
binding properties nodes rely on genetic information obtained by new-generation sequencing
technologies.3840 These can now generate antibody sequence
we have outlined a dynamic, weighted network model of the repertoires with immense depth and ne resolution. While it is
humoral immune system, shown in a simplied two-dimensional possible to sequence the B-cell repertoire of a whole zebra sh,41 we
representation in Figure 6. Because TI responses utilize polyclonal cannot do the same with a human B-cell repertoire. There will be a
responses, antibody production against a given epitope region is
sampling bias depending on where the B cells are obtained from: bone
always shared, [Ab]/KD is always o1. For oligoclonal or monoclonal
marrow, blood, tonsils, lymph nodes, spleen. Additionally, sequencing
TD responses, this value may exceed 1. The model shows the space
cannot provide antibody concentrations, even less, so free antibody
lling property with multi-sized spheres, the core representing
concentrations or antigen concentrations in the sampled organism.
immunological selfa giant superconnected component of the
Immunomics approaches will need to utilize the databases on
network of antibodies.
antibodyantigen interactions that have already been compiled
CONCLUDING REMARKS (the immune epitope database, allergome database). The development
We have introduced homeostatic antibody model based on the of novel immunological methods is required for further formal proof
assumption that the clonal humoral immune system seeks after an of this theory and for its application in various elds of immunology.
equilibrium between antibody and antigen. This requires that the By combining genetic, immunologic and immunoinformatics data,
membrane-bound form of antibody, the BCR, regulates the fate of we expect that a true bioinformatics modeling of the complete human
cells that produce antibodies, as shown in our previous article. In this antibody network, its dynamics, disease-associated changes will be
article, we argued that both the afnity and the concentration of the possible in the near future.
antibodies produced in the host are tuned for silently degrading,
carefully removing or aggressively eliminating antigen. CONFLICT OF INTEREST
An important message of this approach is that molecular abundance The author declares no conict of interest.
is the dening factor for immunological self. Everything abundantly
present in blood and in the bone marrow during the development of ACKNOWLEDGEMENTS
the system is self immunologically. Immunological homeostasis is This paper is dedicated to all immunology theorists, past and present,
about controlled catabolism and regeneration of self. Self is silently who advanced the eld of immune network theories. I wish to thank Kroly
Liliom for consulting us on enzyme kinetics and Anita Orosz and Krisztin
eliminated when ages and aggregates. Molecules binding with high
Papp for working with me on methodological approaches to quantitative
afnity to cells screened for low-afnity self-binding and accompanied
immunomics. I thank my family and my colleagues for their patience and
by danger signals are regarded to be eliminated. In this sense,
understanding in the moments when I was too immersed in thinking. We
low-afnity self-recognition is necessary not to avoid binding to self received Grant K109683 from the National Research, Development and
but rather to set a point of reference in the epitope space. Innovation Ofce of Hungary (http://nkh.gov.hu/english) supported this
There have been several different approaches and theories to work.
provide general, perhaps, mathematical explanations for the complex-
ity, the functioning of the immune system. The idiotype network
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