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StoJ is a polyadic, asynchronous stochastic pi calculus with input join and no summation. It is intended for experimentation with different styles of modelling molecular processes, just like the stochastic pi calculus of Priami, Regev, Shapiro and Silverman.

This article discusses some of the issues in stochastic-pi based biological modelling and hints at some of the rationale for Stoj. You can find a short tutorial on Stoj syntax and can experiment with our current implementation using a web interface.

Rates are associated with a join, not with a channel. We implemented Gillespie’s “first reaction” method, as it seemed simplest. Despite its inefficiency it’s performing reasonably well, although we suspect we have Moore’s “law” to thank for that more than anything else.

We used OCaml as the implementation language—the parsing tools are quite smooth and good to work with—and gnuplot to graph the output of each simulation run.

Here’s an example model in StoJ. I duplicated the rates and the overall design of the model from the code archive at the BioSPI website. This model is discussed in Aviv Regev’s paper “Representation and simulation of molecular pathways in the stochastic pi-calculus.” The implementation of the model below gives the same output graphs as Regev’s model gives, which is a nice confirmation of the model and of the StoJ system itself.

// Circadian clock model
// Based on Regev et al

  DNA_a, DNA_r,
   !MRNA_a, !MRNA_r,
   !protein_a, !protein_r,
 DNA_a_promoted, DNA_r_promoted,
 !complex_ar . (

// Start state.
DNA_a<> | DNA_r<>

// Basal rate transcription.
| rec L. DNA_a() ->[4]        (DNA_a<> | MRNA_a<> | L)
| rec L. DNA_r() ->[0.001]   (DNA_r<> | MRNA_r<> | L)

// MRNA degradation.
| rec L. MRNA_a() ->[1.0]    (L)
| rec L. MRNA_r() ->[0.02]    (L)

// Translation.
| rec L. MRNA_a() ->[1.0]  (MRNA_a<> | protein_a<> | L)
| rec L. MRNA_r() ->[0.1]    (MRNA_r<> | protein_r<> | L)

// Protein degradation.
| rec L. protein_a() ->[0.1]  (L)
| rec L. protein_r() ->[0.01] (L)

// A/R Complex formation.
| rec L. protein_a() & protein_r() ->[100.0] (complex_ar<> | L)

// A/R Complex degradation.
| rec L. complex_ar() ->[0.1] (protein_r<> | L)
| rec L. complex_ar() ->[0.01]    (protein_a<> | L)

// Activator binding/unbinding to A promoter, and enhanced transcription
| rec L. protein_a() & DNA_a() ->[10] (DNA_a_promoted<> | L)
| rec L. DNA_a_promoted() ->[10] (protein_a<> | DNA_a<> | L)
| rec L. DNA_a_promoted() ->[40]  (DNA_a_promoted<> | MRNA_a<> | L)

// Activator binding/unbinding to R promoter, and enhanced transcription
| rec L. protein_a() & DNA_r() ->[10]   (DNA_r_promoted<> | L)
| rec L. DNA_r_promoted() ->[100]    (protein_a<> | DNA_r<> | L)
| rec L. DNA_r_promoted() ->[2]       (DNA_r_promoted<> | MRNA_r<> | L)


Here are a couple of graphs produced by the simulator from the program above—first a graph of reagent concentrations varying with time, and then a graph of Repressor (R) molecule concentration against Activator (A) molecule concentration, showing the bistable nature of the whole system. If you take a look at the graphs in Regev’s paper, you’ll see that the spikes in reagent concentration occur in the same places with the same frequency in our model as in Regev’s.

Concentrations against time

R vs A

We also implemented a simple Java applet/application that uses a (human-assisted) mass-spring layout to draw a graph of the (statically apparent) dataflow in the StoJ program it takes as input. It still needs a bit of refining—the graphs/programs tend to be quite densely connected in places, and not only is mass-spring layout poor at densely-connected graphs, especially with cycles, but it doesn’t identify any potential clustering in the graph either. Clustering (subgraphs) might turn out to be important for visualising dataflow in larger models.

Here’s a simple chemical equilibrium model in StoJ:

// A simple H + Cl <--> HCl model,
// similar to that in Andrew Phillips' slides
// with numbers taken from his slides.
// See
// page 16.

new !h, !cl, !hcl . (
  rec ASSOC.  h() & cl() ->[100] (hcl<> | ASSOC)
| rec DISSOC. hcl() ->[10] (h<> | cl<> | DISSOC)
| h<> * 100 | cl<> * 100

And here’s the display generated by the Java program visualisation application. To see it in motion, and experiment with dragging the nodes around using the mouse, visit the web interface page, and click the “Visualise program” button.

Simple chemical equilibrium visualisation

As you can imagine, for larger models the display gets cluttered quite quickly. A note on implementation though – we found Java2D to be surprisingly pleasant to work with. The built-in support for double-buffering of JComponents along with the various classes of Shape helped us quickly develop the application.

This page is based on an earlier article I wrote when ran the first few experiments with the system.

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