Biological applications of bifurcation theory

Biological applications of bifurcation theory provide a framework for understanding the behavior of biological networks modeled as dynamical systems. In the context of a biological system, bifurcation theory describes how small changes in an input parameter can cause a bifurcation or qualitative change in the behavior of the system. The ability to make dramatic change in system output is often essential to organism function, and bifurcations are therefore ubiquitous in biological networks such as the switches of the cell cycle. Biological applications of bifurcation theory provide a framework for understanding the behavior of biological networks modeled as dynamical systems. In the context of a biological system, bifurcation theory describes how small changes in an input parameter can cause a bifurcation or qualitative change in the behavior of the system. The ability to make dramatic change in system output is often essential to organism function, and bifurcations are therefore ubiquitous in biological networks such as the switches of the cell cycle. Biological networks originate from evolution and therefore have less standardized components and potentially more complex interactions than many networks intentionally created by humans such as electrical networks. At the cellular level, components of a network can include a large variety of proteins, many of which differ between organisms. Network interactions occur when one or more proteins affect the function of another through transcription, translation, translocation, or phosphorylation. All these interactions either activate or inhibit the action of the target protein in some way. While humans build networks with some concern for efficiency and simplicity, biological networks are often adapted from others and exhibit redundancy and great complexity. Therefore, it is impossible to predict quantitative behavior of a biological network from knowledge of its organization. Similarly, it is impossible to describe its organization purely from its behavior, though behavior can indicate the presence of certain network motifs. However, with knowledge of network interactions and a set of parameters for the proteins and protein interactions (usually obtained through empirical research), it is often possible to construct a model of the network as a dynamical system. In general, for n proteins, the dynamical system takes the following form where x is typically protein concentration: These systems are often very difficult to solve, so modeling of networks as a linear dynamical systems is easier. Linear systems contain no products between xs and are always solvable.They have the following form for all i: Unfortunately, biological systems are often nonlinear and therefore need nonlinear models. Despite the great potential complexity and diversity of biological networks, all first-order network behavior generalizes to one of four possible I/O motifs: hyperbolic or Michaelis–Menten, ultra-sensitive, bistable, and bistable irreversible (a bistability where negative and therefore biologically impossible input is needed to return from a state of high output).