## Introduction

In physics, the equations of motion are thought of and referred to as the ‘governing equations’, and can be solved to compute trajectories of objects in space and time when subjected to various forces. The closest we have come to governing equations in biology are systems of ordinary differential equations (ODEs) which arise when applying mass action kinetics to biochemical equations. For example, the widely known Michaelis–Menten and Hill functions arise when describing the kinetics of enzymes (Hill, 1910; Murray, 2005). Models of gene regulatory networks described as systems of nonlinear ODEs where transcription factor regulation is described with Hill functions are also ubiquitous in biology (Karlebach & Shamir, 2008). Nonlinear ODE systems have been used to model a large number of mechanisms in plants, including the *Arabidopsis* circadian clock (Locke *et al.*, 2005a,b, 2006; Zeilinger *et al.*, 2006; Pokhilko *et al.*, 2010), carbohydrate metabolism (Nägele *et al.*, 2010) and photosynthesis (Poolman *et al.*, 2000).

Modelling biological mechanisms with nonlinear systems enables a detailed description and analysis of the underlying biochemical interactions. For investigating the role of specific regulators (e.g. transcription factors, proteases, F-box proteins, etc.), this approach is particularly appropriate, as it is straightforward to simulate the effect of mutations and compare the dynamics with experimental measurements. However, this comes at a cost. The number of model variables and kinetic parameters can quickly grow as more details are represented within the model. Often the kinetic rate parameters are not known, and when they are, the uncertainty associated with those values may be high, as they may be measured in a different context to those desired, for example, different temperature, pH, reagents, *in vitro* vs *in vivo*. Taking these considerations into account, the level of abstraction for a model of a biological system should be largely determined by the level of detailed biochemical understanding *a priori*, and also the quantity and relevance of experimental measurements available for directing model construction.

When the level of biochemical understanding is limited, and few experimental observations exist for constituent components, mechanistic models may be of limited utility, as it is difficult to know the precise functional forms and the underlying topology of the regulatory network. In such cases, it can be more appropriate to describe the dynamics of biological processes with black-box models, which seek to describe the measured outputs signals, potentially driven by variable input signals. In the following sections, I describe first the use of mechanistic models in dissecting the *Arabidopsis* circadian clock network, then progress onto an example of using black-box models to understand high-level properties of circadian networks, discussing along the way the breakthroughs that have arisen, and the relative advantages/disadvantages of each approach (see Table 1 for a summary).

Factor | Mechanistic model | Black-box model |
---|---|---|

Physical interpretation | Intuitive. Mechanisms are explicitly represented by equations or rules. | Very limited. Internal variables are hidden and generic. |

Model simulation | Variable. Depending on the level of mechanistic detail desired. Model reduction techniques enable good efficiency. | Fast. |

Parameter estimation | Difficult and idiosyncratic. Many algorithms exist, but often unreliable. | For linear time-invariant (LTI) systems, parameter identification is efficient and mostly reliable. Nonlinear black-box models suffer analogous to nonlinear mechanistic models. |

Model analysis | For ordinary differential equation (ODE) models, stability and bifurcation analysis tools exist. Frequency response analysis not straightforward. Many analyses rely on linearisation. | For LTI systems, frequency response analysis is straightforward and very informative. |