Editing Gene regulatory network (section)

=== Coupled ordinary differential equations ===
{{for|an example of modelling of the cell cycle with ODEs|cellular model}}

It is common to model such a network with a set of coupled [[ordinary differential equation]]s (ODEs) or [[Stochastic differential equation|SDE]]s, describing the reaction kinetics of the constituent parts. Suppose that our regulatory network has <math>N</math> nodes, and let <math>S_1(t),S_2(t), \ldots, S_N(t)</math> represent the concentrations of the <math>N</math> corresponding substances at time <math>t</math>. Then the temporal evolution of the system can be described approximately by

: <math> \frac{dS_j}{dt} = f_j \left (S_1,S_2, \ldots, S_N \right) </math>

where the functions <math> f_j </math> express the dependence of <math>S_j</math> on the concentrations of other substances present in the cell. The functions <math>f_j</math> are ultimately derived from basic [[rate equation|principles of chemical kinetics]] or simple expressions derived from these e.g. [[Michaelis–Menten]] enzymatic kinetics. Hence, the functional forms of the <math>f_j</math> are usually chosen as low-order [[polynomials]] or [[Hill equation (biochemistry)|Hill function]]s that serve as an [[ansatz]] for the real molecular dynamics. Such models are then studied using the mathematics of [[dynamical system|nonlinear dynamics]]. System-specific information, like [[reaction rate]] constants and sensitivities, are encoded as constant parameters.<ref>{{cite journal | vauthors = Chu D, Zabet NR, Mitavskiy B | title = Models of transcription factor binding: sensitivity of activation functions to model assumptions | journal = Journal of Theoretical Biology | volume = 257 | issue = 3 | pages = 419–429 | date = April 2009 | pmid = 19121637 | doi = 10.1016/j.jtbi.2008.11.026 | bibcode = 2009JThBi.257..419C | s2cid = 12809260 | url = https://kar.kent.ac.uk/24077/1/myzabetpaper.pdf }}</ref>

By solving for the [[Fixed point (mathematics)|fixed point]] of the system:

: <math> \frac{dS_j}{dt} = 0 </math>

for all <math>j</math>, one obtains (possibly several) concentration profiles of proteins and mRNAs that are theoretically sustainable (though not necessarily [[stability (mathematics)|stable]]). [[Steady state]]s of kinetic equations thus correspond to potential cell types, and [[Oscillation|oscillatory]] solutions to the above equation to naturally cyclic cell types. Mathematical stability of these [[attractor]]s can usually be characterized by the sign of higher derivatives at critical points, and then correspond to [[Steady state (biochemistry)|biochemical stability]] of the concentration profile. [[Critical point (mathematics)|Critical point]]s and [[Bifurcation theory|bifurcation]]s in the equations correspond to critical cell states in which small state or parameter perturbations could switch the system between one of several stable differentiation fates. Trajectories correspond to the unfolding of biological pathways and transients of the equations to short-term biological events. For a more mathematical discussion, see the articles on [[nonlinearity]], [[dynamical systems]], [[bifurcation theory]], and [[chaos theory]].