In fact, these parameters represent global effective responses to a large complex network of molecular and cellular www.selleckchem.com/products/AZD2281(Olaparib).html processes that are associated with the bacteria natural growth and with the capture and killing of the bacteria by neutrophils. A calibration of these parameters with extensive experimental data and with the corresponding molecular markers may be important for identifying the main effect of these cellular processes on the bacteria-neutrophil population dynamics in health and in disease. This model is non-linear�Cit takes into account the saturation effects that appear at high concentrations. If these effects are neglected, and the influx is set to zero (so ), we arrive at the linear model: , (see e.g. [11]). This linear model has, for all only a single equilibrium at the origin.

This fixed point is unstable for small and stable when (as in [11]). Thus, the transient bacterial dynamics is always independent of the initial bacterial concentration: if (respectively ), the bacterial population grows exponentially without bound (respectively shrinks exponentially to zero). Moreover, the exponential rate of transient growth/decay of the bacterial population depends only on the neutrophil concentration: it is independent of the initial positive concentration of the bacteria. Notice that this linear model fails to satisfy assumptions A1 and A3. One may postulate that since saturation occurs only at high concentrations (near ), the linear model will adequately describe the dynamics at smaller concentrations.

Next, we show that near this postulate is false: the dynamics in the non-linear and linear models are substantially different, even when is small. Results Two Possibilities: Robust Dynamics vs. Bistable Dynamics Mathematical analysis of Eq. (1) shows that depending on the parameters, one of exactly two types of behaviors, called hereafter type I and II, can occur (a similar statement can be made for all models satisfying assumptions A1�CA4.). In the type I parameter regime, the model has no critical dependence on neutrophil concentration . That is, for all levels of neutrophils there is a single stable equilibrium point (EP) which depends gradually on : for low values it corresponds to the high concentration point associated with the maximal capacity branch�Cthe branch of stable equilibria that emanates from the point , where is the maximal capacity state of the natural bacterial dynamics.

As the neutrophil level increases, this EP gradually lowers till, for sufficiently high neutrophil level (), it reaches the origin (see Fig. 1a and Models section; for simplicity of presentation, we consider here the zero influx and show in the Bacterial Influx section that the results are only slightly modified for small bacterial influx, see also AV-951 Fig. 2).