Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets

Abstract

An integrated understanding of molecular and developmental biology must consider the large number of molecular species involved and the low concentrations of many species in vivo. Quantitative stochastic models of molecular interaction networks can be expressed as stochastic Petri nets (SPNs), a mathematical formalism developed in computer science. Existing software can be used to define molecular interaction networks as SPNs and solve such models for the probability distributions of molecular species. This approach allows biologists to focus on the content of models and their interpretation, rather than their implementation. The standardized format of SPNs also facilitates the replication, extension, and transfer of models between researchers. A simple chemical system is presented to demonstrate the link between stochastic models of molecular interactions and SPNs. The approach is illustrated with examples of models of genetic and biochemical phenomena where the ULTRASAN package is used to present results from numerical analysis and the outcome of simulations.

Figure 1

SPN representation of the dimerization reaction 2R ⇌ R₂. The transition t₊ represents dimerization, with weight function w_t+ = c₊m_monomer(m_monomer − 1), where c₊ is a constant. The transition t₋ represents dissociation, with weight function w_t− = c₋m_dimer. When no initial marking is specified for a place, it is assumed to be zero. Thus, the initial marking is M₀ = {n_m, 0}.

Figure 2

SPN representation of a simple model of protein synthesis. The SPN contains three places: p₁ = inactive gene, p₂ = active gene, and p₃ = protein. The four transitions and their respective rate parameters are activation (λ), inactivation (μ), synthesis (ν), and degradation (δ). The dot inside place inactive gene represents a single token (a single copy of the gene). Thus, the initial marking of the SPN is M₀ = {1, 0, 0}. Other molecules in this system, such as RNA polymerase, are assumed to be in constant concentration and are not explicitly represented.

Figure 3

Distribution of protein number in simple model of gene product synthesis. The distribution was generated from steady-state numerical analysis using the same parameter values as Table 2.

Figure 4

SPN representation of plasmid ColE1 replication system. The reactions represented and the notation used are from fig. 1 of Brendel and Perelson (20). Plasmid DNA occurs in free form (D), or in complexes with RNA II (D_II^s, D_II^l, and D_p), with RNA II and RNA I (D_c^∗ and D_c), or with RNA II, RNA I, and Rom protein (D_M). Replication occurs when primed plasmid DNA (D_p) is converted to free DNA (D). Free RNA I and Rom protein are represented by places R_I and M, respectively. Initial marking of free plasmid is 1. (Modified with permission from ref. .)

Figure 5

Distribution of plasmid copy number in a single bacterium during first generation with 80-min doubling time.

Figure 6

Plasmid copy number during 10 generations starting from 1 or 100 plasmids with 80-min doubling time. Mean and SD of plasmid number is measured every 20 min and at the beginning and end of each generation.