Calculating the parameters and boundary conditions for the antigen independent pathway for the origin of primer antibody.
Links to the Ig-genesis program and math details.
The requirement for primer antibody arises from the following reasoning:
1 - There is a significant portion of the pathogenic universe to which the innate defense mechanisms are blind. This is best seen in individuals that for one reason or another have an impaired "adaptive" immune system (e.g. RAG-/- mutation) but an intact innate (germline-encoded) defense system. These individuals succumb to infection by organisms not recognized by the innate defense system.
2 - The first step in the initiation of an immune response is the uptake and processing of antigen by Antigen Presenting Cells (APC) which have innate (germline-encoded) receptors that recognize a portion of the pathogenic universe, as well as Fc receptors that recognize antigen-antibody complexes, [Ag-Ab]n, the formation of which is dependent on the immune system. As these latter permit processing of antigen not recognized by the innate system (i.e., antigen seen uniquely by the somatically generated recognitive paratopic repertoire of the immune system), the question arises, "where does this 'primer' antibody come from?" The induction of protective levels of secreted antibodies requires the prior induction of effector T-helpers that, in turn, is dependent on the processing by APC of [Ag-Ab]n - complexes to peptides. Where does the antibody necessary to prime this uptake originate? This is the "primer question."
3 - The "primer" antibody must be maintained at an effective concentration and have been purged of anti-self reactivity. The Ig-genesis program calculates the resulting B cells and antibodies in an immune system from any choice of rate constants. The state of the system is calculated and displayed at intervals or steps that can be translated into time or any derivative function.
Under the persistence/transience model (1,2) for the Self/Nonself discrimination all antigen-responsive cells, in this case B cells, are born without effector activity. Naive B cells have two pathways open to them, tolerance (inactivation) and induction (activation). There is a period during development where only self-antigen (S) is present and effector T-helpers are absent. During this time window interaction with antigen results in inactivation and this process is maintained thoughout life as long as the self-antigen persists.
We use the symbol "i" to describe the initial state of cells when they are incapable of expressing effector activity upon antigen binding. The iB cells which do not encounter antigen undergo an antigen-independent differntiation to eB. As self-antigen is present when iB cells arise and persists throughout life the anti-self reactivity is purged. Interaction with antigen blocks the antigen-independent conversion of iB to eB; the only way to induce an aB cell to eB is by interaction with eTh, which results in activation, proliferation, and induction to eB. Antigen drives i-state cells to an anticipatory state (aB), which in the absence of further signals collapses into cell death. Eventually eB cells die out (k4), and antibodies decay (k8), resulting in a requirement for a continuous resupply of anti-nonself eB cells that produce the "primer" antibodies. A steady state production of "primer" antibody from totality of the native repertoire is achieved.
The antigen-independent differentiation of iB to eB, permits an effective uptake concentration of anti-nonself "primer" antibody and a negligible or non-debilitating concentration of anti-self antibody. The rate constants k1 thru k6 relate to cell type conversions. The rate constants k7 and k8 correspond to antibody production and decay respectively, and do not affect cell conversions.
There is a steady state N = (k1xG) of newly arriving iB per unit time from the bone marrow. These cells are a mixture of anti-self and anti-nonself such that SI is the proportion that are anti-self and (1 - SI) is the proportion anti-nonself. There is, therefore, a steady state production of iB anti-self, iBs in number, that is equal to SIxN and of iB anti-nonself, iBns in number, that is equal to (1 - SI)xN. We consider here the steady state levels of each component in the absence of nonself and in the presence of self. Cells that are iB anti-self interact with self and are blocked (eventually die through the k6 pathway) from entering the antigen-independent pathway to effectors. The various pathways and their rate constants, k, are diagrammed.
The rate equations were derived and summed over time. This computer program allows the user to vary the rate constants (k), the size of the Protecton (T) (3), the repertoire size (R), and the probability that an iB cell will be anti-self (SI). The rate constant, k per unit time, is related to the half-life of the cell, k = (ln 2) / (t1/2).
The rate constant k7 may be expressed in any units desired to make Abs and Abns meaningful. For example k7 can be inserted as ng / ml of Ig produced by a single eB cell per unit time, or as absolute numbers of Ig molecules produced by a single eB cell per unit time. The resulting Abs and Abns values will maintain the same units as those given for k7. By contrast k8, is the proportion of the Ig that will decay per unit time, or its half-life.
The output of the calculation is the steady state level of iBs, aBs, eBs, Abs, iBns, eBns, Abns, and new cells from the bone marrow N = k1xG in the absence of nonself. Four of these outputs are also available per antigenic determinant (d), namely these are eBs/d, eBns/d, Abs/d, and Abns/d. The decision as to whether the choice of parameters are acceptable depends on the boundary conditions chosen to permit an effective response to nonself and a negligible response to self.
A set of default input values are provided to illustrate what appears to be an acceptable output. The choice of SI, and rate constants are currently based on Th-genesis model (4,5), although more fine-tuning will be performed for Ig-genesis in the near future.
1 - Cohn, M. (1992) The Self-Nonself Discrimination: Reconstructing a Cabbage from a Sauerkraut. Res. Immunol. 143:323-334.
2 - Langman, R. E., editor (2000). "Self-Nonself Discrimination Revisited." Seminars in Immunology 12: pgs. 344.
3 - Cohn, M. and R. E. Langman (1990). "The Protecton: The unit of humoral immunity selected by evolution." Immunol. Reviews 115: 7-142.
4 - Cohn M., Langman R.E., and Mata J.J. (2002) A computerized model for the self - non-self discrimination at the level of the Th (Th-genesis). I. The origin of 'primer' effector Th cells. Int'l. International Immunol. 14:1105-1112.
5 - Langman, R.E., Mata, J.J., Cohn, M. (2002) A computerized model for the self-nonself discrimination at the level of the T-helper (Th-genesis). II. The behavior of the system upon encounter with nonself antigens. International Immunol. 15:593-609