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]:
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].
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).