The DynaCoSys model is a hybrid differential equation model that describes the complement activation of the alternative pathway. The dynamics of the surface bound molecules are described by a set of ordinary equations and the dynamics of the key fluid phase molecules are described by partial differential equations. The non-linear model includes all features of the complement system: activation, opsonization, stabilization, amplification and regulation. The DynaCoSys model is used in our studies for analyzing the complement activation mediated by Factor H and researching the complement immune evasion of *candida albicans* in the complement immune evasion study.

##### 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 $C_3b$: Active $C_3b$ in the fluid phase, $C_3b^f$, results from cleavage of precursor molecule $C_3^f$. The interaction of the fluid phase molecule $C_3b^f$ with the cell surface is modeled by the interaction with free surface binding sites $B^s_{C_3b,free}$ and binding sites BsC3b that are occupied with molecules $C^3b_s$ on the surface. $C_3b^f$ 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 $C_3b^s$ can form $C_3$-convertase molecules—$C_3b^sBb$ and $C_3b^sBbP$–that cleave $C_3^f$ molecules to $C_3b^f$ molecules in the vicinity of the cell surface. $C_3b^s$ 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 $B^s_{fH,max}$. (B) The lifetime of active $C_3b^f$ 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.