To quantify the dynamics of immune systems, we first identify the components of the immune systems. We can use populations of the different cell types as the variables or the components of the immune systems. In this way, the dynamics of the immune systems can be described by the following ordinary differential equations [1]:


dx/dt = F(x)

Where x={x1,x2,x3,...} represents the populations of 1, 2, 3, …nth cell types and  F(x)={F1(x), F2(x), F3(x), ...Fn(x)} represents the driving force or the interactions of the other components (cell types) to the specific cell types (1, 2, 3, …nth cell types).

A state of the immune system can be defined by a combination of cell type populations at certain values, . In other words, a state of the system is determined by one set of combinations of cell type populations with certain values while another state is determined by another set of combinations of cell populations with different values. If each cell type can have M possible population values, then there are M^N states in the entire state space of immune system. In fact, many of our concerned biological functional states such as immune active states, immune inactive states, disease states, etc. all should be contained in the states of the system. The dynamics of the immune systems can be seen as the temporal evolution from one state to another. This approach can give us a description of the immune system dynamics. It falls into the general nonlinear dynamics category [2].


A conventional treatment of the above evolution dynamics in nonlinear dynamics is to find the fixed points and explore its local stability [2]. These can give us some hints about the states where the cell type populations would like to be clustered and accumulated to. These states might correspond to the biological functional states such as immune active states, disease states, etc. For biological relevant questions, here we are particularly interested in how one can go from one state to another, for example, how do we go from disease state to immune active state or vice versa. Understanding of these issues will help us to design prevention and curing strategies for diseases. However, the conventional local stability analysis can’t provide such a global connection between local stable states [2].

Recently, inspired by the global picture of Waddington landscape for development and differentiation [3], the complex nonlinear system dynamics have been shown to be globally determined by the two driving forces [4,5,6]. One is related to the probability landscape of the system and the other is the associated curl probability flux. The probability landscape quantifies the probability of each state while the curl probability flux quantifies the flow around the states. The flux is a quantitative measure of detailed balance breaking: a non-equilibrium signature of energy, material or information exchange from the environment to the system. While the probability landscape can identify the higher probability states and correlate to the biological functional states, the connections among those functional states are quantified by both the landscape and the flux. The flux can make the path from one state to another to deviate from the naively expected steepest descent path of saddle path. Furthermore, the forward and backward paths are different due to the presence of the flux. In estimating the rate of the state switching processes, conventionally the kinetic rate is determined by the barrier from one state to the landscape saddle. However,  the optimal paths do not go through the saddle points due to the presence of the curl flux, the barrier has to be measured at the new saddle on the optimal path rather than the landscape saddle. This can be illustrated clearly as the following figure. 



Fig. 1. Two and three dimensional Illustration of the potential landscape with the optimal paths between basins of attraction S and S’ (purple and green lines with arrows) and the steepest descent gradient path (white line). represents the saddle point and  represents the saddle point along the dominant path” (from [6])

Through the landscape and flux approach, we can identify the global driving force of the immune systems as the landscape and flux. We see their effects in determining the probability of states, finding the location and depth of the basins of attraction of the system for describing the function and stability (for example, disease or immune active state basins), quantifying the process and speed of biological functional state switching (switching between immune active and disease or vice versa). In this way, we can identify the immune active and disease states through landscape local minimum, quantify the stability by basin depths and barrier for the disease and immune active states and explore the switching between them. Furthermore, since the landscape topography and flux in determining the functions and global stability of the functional states are globally quantified, we can explore which underlying cell type interactions or cell types these global measures are sensitive to.  Through such global sensitivity analysis, the key cell types and cell type interactions can be identified which are critical to the stabilities and switching of the disease to immune active states or vice versa. This provides the hopes and chances of designing new preventing and new strategy from the global systems perspectives.

[1] Carmen Molina-Paris and Grant Lythe. Mathematical Models and Immune Cell Biology. Springer, New York. (2011).

[2] E. A. Jackson, Perspectives of Nonlinear Dynamics. Cambridge University Press. (1990)

[3] C.H. Waddington. The Strategy of the Genes. Allen &Unwin, London. (1957).

[4] J. Wang, L. Xu, E. K. Wang. Potential landscape and flux framework of nonequilibrium networks: robustness, dissipation, and coherence of biochemical oscillations. Proc. Natl. Acad. Sci. USA , 105: 12271-12276. (2008).

[5] J. Wang, K. Zhang, L. Xu, E.K. Wang. Quantifying the Waddington landscape and biological paths for development and differentiation. Proc. Natl. Acad. Sci. USA. 108(20):8257-8262(2011).

[6] J. Wang,  Landscape and flux theory of non-equilibrium dynamical systems with application to biology, Advances in Physics, 64:1, 1-137. (2015).