# Deterministic vs. Stochastic Models

# The Two Equations Used to Model Gene Expression

**Determinisic**

A deterministic equation uses a rate equation to describe the transcription and translation of genes. Deterministic equations are characterized as behaving predictably; more specifically a single input will consistently produce the same output. Returning to one of the Collins graphs, the blue lines represent the deterministic model for protein production and the red line represents a corresponding stochastic model (figure 1).

Figure 1 was obtained at http://www.ncbi.nlm.nih.gov/sites/entrez?db=pubmed&uid=15883588&cmd=showdetailview&indexed=google permission pending

Figure 1 displays a stochastic function superimposed on a corresponding deterministic function.

**Stochastic**

Stochastic models take into account the "randomness" of transcription and translation by utilizing variables for the formation and decay of single molecules and multi-component complexes.

**Figure 2**

Figure 2 was obtained at http://www.pnas.org/cgi/content/abstract/0608451104v1 permission pending

**Figure 3**

Figure 3 was obtained at http://www.pnas.org/cgi/content/abstract/99/20/12795 permission pending

Above are two visual representation of stochastic models (Figure 2,3). In the first stochastic model (Figure 2), nodes represent states of molecular complexes while arrows represent binding of transcriptional, translational and degradation proteins. The second stochastic model (Figure 3) also depicts transcription and translation factors, but also represents competition for binding to the leader region mRNA between degradosomes and proteins that induce translation (MRu represents the leader region of mRNA).