Continuous dynamical system (Systems of ODEs), Dynamical systems on networks.

https://crnt.osu.edu/

Based on the theory of Chemical kinetics

Mass-action dynamical system

Can write as a dynamical system

$\dot{\mathbf{x}} = S \cdot \mathbf{v}$

where $S$ is the stoichiometrix matrix, and $v$ is the flux vector.

This assumes the Law of mass-action. Mass action kinetics does not always capture correct dynamics, because of more complicated constraints and stochastic dynamics.. In more, general models $v$ has a more complicated form.

Steady state

$\dot{x}=Sv_s=0$. $v_s$ has to be in the null space of $S$.

Conservation relations

Vectors that satisfy $\mathbf{w}^T \dot{\mathbf{x}} = \mathbf{w}^T \mathbf{S} \mathbf{v}=0$.

$w$ lives in the null space of $S^T$

Flux analysis (stochiochemistry)

Because the S matrices are very often fat matrices, the null space is and thus the space of stationary state fluxes is often relatively high dimensional. This is useful for design (Synthetic biology).

$\alpha v_1 + \beta v_2 + ... \geq 0$ is a constraint on the fluxes. This defines a cone (a special case of the polyhedra found in Linear programming)! Hm I can only see it being a cone if there are conditions like $\alpha \geq 0$ otherwise, it's just the intersection of the positive part of the full space and the null subspace.

Sensitivity analysis

Generalized Mass Action/S-Systems

Used in Biochemistry, and Chemistry.