Quantification of Factor H Mediated Self vs. Non-self Discrimination by Mathematical Modeling

Abstract

The complement system is part of the innate immune system and plays an important role in the host defense against infectious pathogens. One of the main effects is the opsonization of foreign invaders and subsequent uptake by phagocytosis. Due to the continuous default basal level of active complement molecules, a tight regulation is required to protect the body's own cells (self cells) from opsonization and from complement damage. A major complement regulator is Factor H, which is recruited from the fluid phase and attaches to cell surfaces where it effectively controls complement activation. Besides self cells, pathogens also have the ability to bind Factor H; they can thus escape opsonization and phagocytosis causing severe infections. In order to advance our understanding of the opsonization process at a quantitative level, we developed a mathematical model for the dynamics of the complement system-termed DynaCoSys model-that is based on ordinary differential equations for cell surface-bound molecules and on partial differential equations for concentration profiles of the fluid phase molecules in the environment of cells. This hybrid differential equation approach allows to model the complement cascade focusing on the role of active C3b in the fluid phase and on the cell surface as well as on its inactivation on the cell surface. The DynaCoSys model enables us to quantitatively predict the conditions under which Factor H mediated complement evasion occurs. Furthermore, investigating the quantitative impact of model parameters by a sensitivity analysis, we identify the driving processes of complement activation and regulation in both the self and non-self regime. The two regimes are defined by a critical Factor H concentration on the cell surface and we use the model to investigate the differential impact of complement model parameters on this threshold value. The dynamic modeling on the surface of pathogens are further relevant to understand pathophysiological situations where Factor H mutants and defective Factor H binding to target surfaces results in pathophysiology such as renal and retinal disease. In the future, this DynaCoSys model will be extended to also enable evaluating treatment strategies of complement-related diseases.

Keywords: complement regulator factor H; complement system; hybrid differential equation approach; immune evasion; mathematical modeling; non-self recognition; self-tolerance.

Figure 1

Hybrid differential equation model of the complement system. Complement activation can be divided into five parts: (i) activation, (ii) opsonization, (iii) stabilization, (iv) amplification, and (v) regulation. (A) The model focuses on the dynamics of the central component C3b: Active C3b in the fluid phase, C3b^f, results from cleavage of precursor molecule C3^f. The interaction of the fluid phase molecule C3b^f with the cell surface is modeled by the interaction with free surface binding sites $B_{C3b,free}^s$ and binding sites $B_{C3b}^s$ that are occupied with molecules C3b^s on the surface. C3b^f that does not bind to the cell surface gets inactivated via a Factor H mediated inhibition process, or gets stabilized by water molecules and is no longer able to bind to the cell surface. Surface-bound C3b^s can form C3-convertase molecules—C3b^sBb and C3b^sBbP–that cleave C3^f molecules to C3b^f molecules in the vicinity of the cell surface. C3b^s can be inactivated via an inhibition process that is mediated by surface-bound Factor H, whose concentration depends on the concentration of binding sites on the cell surface $B_{fH,max}^s$. (B) The lifetime of active C3b^f is short such that, depending on the distance from the cell surface, the fraction of molecules that reach the cell surface is small; for example, only 1% at a distance of 196 nm within a simple decay model.

Figure 2

Steady states of the molecules C3b^f and C3a^f in the fluid phase for varying surface-bound Factor H concentrations. (A) Radial steady state concentration profile of the fluid-phase C3b^f concentration. (B) Radial steady state concentration profile of the fluid-phase C3a^f concentration. Mind the different scales in (A,B).

Figure 3

Summary of the numerical integration of 1,000 simulations for varied fH^s concentrations. The time intervals indicate how much time passes until a certain amount of C3b^s is attached to the cell surface. Two typical molecular concentration dynamics are shown for small fH^s concentrations (top left) and high fH^s concentrations (top right).

Figure 4

Proportion of C3b molecules at the cell surface. Based on the dominating types of C3b molecules, we distinguish the two extreme regimes for non-self recognition and self-tolerance that are separated by a transition regime where complement evasion takes place.

Figure 5

Sensitivity analysis of the model parameters for the steady state of surface-bound C3b molecules. The upper plot shows the relative concentration of complement molecules on the cell surface. The lower plots for selected Factor H concentrations give the relative local sensitivity of the complement molecules with respect to a parameter p that is individually varied. Positive sensitivities correlate with increasing molecule concentration. The non-self regime is sensitive to the rates r_dec and $r_{inhib}^s$ (see a,b). The transition region (c) is dominated by $r_{inhib}^s$ and r_amp and the self-regime is dominated by the parameters A, r_dec, $r_{inhib}^s$, and r_amp (see d,e).

Figure 6

Relative opsonization in the steady state for varied system parameters as a function of the relative Factor H binding site concentration. The white line indicates the DynaCoSys model for standard parameter values. (A) Variation of the spontaneous C3b^f activation A, (B) variation of the fH^f concentration, (C) variation of the fB^f concentration, and (D) variation of the relative concentration of complement molecules in the fluid phase.

Figure 7

Relative opsonization in the steady state for varied system parameters as a function of the relative Factor H binding site concentration. The white line indicates the DynaCoSys model for standard parameter values. (A) Variation of the dissociation constant for Factor H and binding sites on the cell surface. (B) Variation of the reaction rate for fH^s binding to C3b^s molecules.