Table 1: Different mathematical approaches used in computational systems biology.

Approach DescriptionAdvantages and disadvantagesRecent application in toxicology

Ordinary differential equations (ODES)Is the most routinely used approach to modeling biological systems, treats biological systems as a series of reaction based equations, and assumes that reactions are continuous and deterministic in nature Advantages
Generally not computationally intense (fast)
Use is well understood
Cannot model spatial dynamics
Stochasticity not considered
Model of steroidogenesis in H295R cells: role of oxysterols and cell proliferation to improve predictability of biochemical response to endocrine active chemicalmetyrapone [59]

Stochastic reaction networksDeals with stochasticity by representing reactions as discrete random molecular collisions. Important when molecule numbers are low Disadvantages
Molecular fluctuations. Computationally this approach is implemented using the Gillespie algorithm/one of its variants
Cellular systems are inherently stochastic and such models help to deal with this
Computationally inefficient
A systems-based computational model for dose-response comparisons of two modes of action hypotheses for ethanol-induced neurodevelopmental toxicity [60]

Bayesian networksBayesian networks are a type of probabilistic network graphs, where each node within the graph represents a variable. Nodes can be discrete or continuous and are connected to a probability density function, which is dependent on the values of the inputs to the nodesAdvantages
Useful for representing variability within biological systems and also toxicant risk estimates
Constrained when it comes to modeling feedback
A Bayesian approach to probabilistic ecological risk assessment: risk comparison of nine toxic substances in Tokyo surface waters [61]

Petri netsPetri nets are a directed bipartite graph, with two types of nodes, called places and transitions. Places and transitions are connected via arrows/arcs. Each place contains a number of tokens which is a kin to a discrete number of biochemical molecules. A Petri net functions by input-output firing at the “transitions” within the network. The “firing” of transitions is a kin to a biochemical reaction taking place. There are many different variants of Petri net, for example, colored, hybrid, continuous, and stochasticAdvantages
Petri nets are intuitive
A limitation from a biological point of view is that they are restricted to small network modeling
Derived biochemical kinetics (is this a bit blunt-need further explanation?)
An enhanced Petri-net model to predict synergistic effects of pairwise drug combinations from gene microarray data [62]

Boolean networks Boolean networks are comprised of nodes that can either be in an “on” or “off” state. The dynamics of the model are acted out by a series of time steps, with the state of each Boolean variable being updated at each time step. Similar to Petri nets, Boolean models are regularly employed to examine gene regulatory networksAdvantages
Utility lies in its ability to represent genetic networks via “on”/”off” responses
Boolean networks are limited as this network does not deal with biological mechanisms or biochemical kinetics
Simulating quantitative cellular responses using asynchronous threshold Boolean network ensembles [63]

Partial differential equations Partial differential equations (PDEs) are multivariable functions with partial derivatives. Not as ubiquitous as ODE modelsAdvantages
Can deal with both spatial and temporal dependencies
A disadvantage of PDE models is that they can be computationally intensive and thus slow
A Simple model for assessment of antitoxin antibodies [64]

Agent based modelsA rule based method that employs clusters of independent agents whose behaviour is underpinned by simple rules. These agents are capable of interacting with one another through space and time Advantages
Very useful for representing spatial and temporal aspects of biological systems
The principal disadvantage of this approach is the challenges associated with studying the interconnectivity between the agent rules and the dynamics of the biological system
A computational model predicting disruption of blood vessel development [65]