104 A Practical Guide to Cancer Systems Biology
- Mathematical fundamentals
2.1. Introduction to differential equations
A differential equation is a mathematical equation that contains derivatives.
Basically, the differential equation expresses the rate of change of a quantity
(the state variable) as a function of time or the state variable itself. For
example,
dm
dt=f(m, t) (9.1)wheremis the state variable,tdenotes the time,f(m, t) represents a function
of time and state variablem,anddmdt indicates the rate of change of state
variablem. In systems biology, the state variables are usually represented
by the expression level of genes, proteins, or phosphoproteins.
Given the differential equation, we always want to know the solution
of the differential equation, that is, the quantity of state variable over
time. When solving the differential equation, we need to have the initial
condition of the state variable, i.e., the value of the state variable at any
given time. With the differential equation and the initial condition of the
state variable, calculus techniques can be applied to analytically solve the
differential equation. However, not all differential equations can be solved
analytically by calculus techniques. In this case, the Euler method and
the Runge–Kutta method are two popular ways to numerically solve the
differential equations,^2 which are useful for computational simulations.
2.2. From differential equations to discrete dynamic models
Traditionally, differential equations are used to describe continuous-time
systems. Although biological systems are mostly continuous-time systems,
the experimental data generated from the biological systems such as
transcriptomics, quantitative proteomics, and phosphoproteomics data are
always discrete-time. In this case, it is not appropriate to fit discrete-time
experimental data to differential equations. Consequently, the differential
equations need to be modified as the discrete dynamic models. Based on
the definition of the derivative, the differential equation in (9.1) can be
expressed as
m(t+Δt)−m(t)
Δt=f(m, t) (9.2)