
Approach  Description  Advantages and disadvantages  Recent 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 Disadvantages 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 networks  Deals 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  Advantages Cellular systems are inherently stochastic and such models help to deal with this Disadvantages Computationally inefficient  A systemsbased computational model for doseresponse comparisons of two modes of action hypotheses for ethanolinduced neurodevelopmental toxicity [60] 

Bayesian networks  Bayesian 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 nodes  Advantages Useful for representing variability within biological systems and also toxicant risk estimates Disadvantages 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 nets  Petri 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 inputoutput 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 stochastic  Advantages Petri nets are intuitive Disadvantages A limitation from a biological point of view is that they are restricted to small network modeling Derived biochemical kinetics (is this a bit bluntneed further explanation?)  An enhanced Petrinet 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 networks  Advantages Utility lies in its ability to represent genetic networks via “on”/”off” responses Disadvantages 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 models  Advantages Can deal with both spatial and temporal dependencies Disadvantages 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 models  A 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 Disadvantages 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] 
