Stochastic Join?

mikeb and I were discussing the papers given at BioConcur the other day, and we were puzzling out the differences between the standard way of modelling biochemical and biological reactions (including genetic interference reactions) and our “alternative” modelling style. I suppose the “alternative” style must have been tried before, and that it must display some obvious weakness we haven’t spotted yet.

Here's a comparison of the two styles:

Join and Rates

We've been thinking about using a stochastic π-variant with a join operator. Having join available allows the expression of state machines to be much more direct than in a joinless stochastic π. We started modelling the lambda-switch to fit the model from this BioConcur paper, and saw that it seems more natural to attach the reaction rate to each individual join rather than to each channel as BioSPI does.

Bisimulation

The standard model has comm meaning interaction; we have comm meaning movement in some sense: atoms (mass) move from one locus (molecular species) to another, or a molecule undergoes conformational change, which is a kind of motion from one state or class or species to another. This means that bisimulation in the alternative model has to do with movement rather than interaction — which seems more fine-grained, but perhaps less natural.

Uncertainty in Biological Simulation

We also thought about a few properties biological simulation systems ought to have. One important one is a notion of uncertainty. Few of the current systems are honest about their limits — only one of the papers at BioConcur displayed error bars on any of its graphs! The systems in use all generate probable trajectories through state-space, and this is I assume implicitly to be understood by the reader of the various program outputs, but it would be useful to have the uncertainty in the results reflected in the various displays that these systems produce.

Besides the uncertainties that come from stochastic modelling and Monte-Carlo-type approaches in general, all the inputs to the systems (eg. rates) are purely empirical data, with associated uncertainties, and these uncertainties are not propagated through the system. Not only do the outputs not show the variation in results due to random sampling, but the outputs do not explore the limits of the accuracy of the inputs, either.

Reagent concentration vs. Gillespie vs. Modelling reagent state

The alternative model deals well with high reagent concentrations, since a high concentration is modelled as a large number of messages outstanding on a channel. The Gillespie scheduling algorithms are O(n)-time in the number of enabled reactions, not in the concentrations of the reagents. The standard model suffers here because each molecule is modelled as a process — that is, as an enabled reaction. In a naïve implementation, this makes the scheduling complexity O(n)-time (or at best, O(log(n)) time if the Gibson-Bruck algorithm is used) in the number of molecules in the system.

However, modelling molecules as processes seems to provide one advantage that doesn't easily carry over into the alternative model: molecule-processes can involve internal state that any interacting molecule need not be concerned with. For example, enzyme-private channels with different rates associated with them might be used to model the different reactivities of phosphorylated and unphosphorylated states of the enzyme.

Perhaps a hybrid of the two models will be workable; after all, even in the alternative model, you have the freedom of the restriction operator available for implementing complex behaviours. The alternative model allows some common patterns to be expressed more simply than the standard joinless model though, it seems.

We're still thinking about ways of keeping the molecules-as-messages approach while allowing reactions to specialise on molecule-message state or to ignore it, as appropriate.

Compartments

BioSPI 3.0 took the standard stochastic-π implementation and added BioAmbients to it (see the 2003 paper by Regev, Panina, Silverman, Cardelli and Shapiro). Any stochastic-join will probably benefit from being able to model compartments similarly.

The simple, ambientless approach suffers some of the same problems as SBML is facing with regard to recognising the similarity of species across compartments. Some way of factoring out reactions/processes/etc common across all compartments should be useful.

Biological Systems Element Standard Model "Alternative" Model
MoleculeProcessMessage
Molecule species-Channel
Interaction CapabilityChannelProcess
InteractionCommunicationMovement (and computation)
Modification State and/or channel change State and/or channel change
Reaction Rate attached to channel attached to prefix