Surface-Volume Reactions (Example: IP3 Model)

The simulation script described in this chapter is available at STEPS_Example repository.

In the previous chapter we used objects of type steps.model.Reac to represent a reaction taking place inside a volume. In this chapter we consider another type of kinetic reaction represented by the steps.model.SReac class (associated with the steps.model.Surfsys container) which defines a reaction taking place on a surface (or patch) connecting two compartments (arbitrarily naming one of them the “inner” compartment, and the other one the “outer” compartment). Reactants and products can therefore be freely moving around in a volume or embedded in a surface. Therefore, it is necessary to firstly specify the location of the reactant and product species.

Note: Surface reactions are designed to represent reactions where one reactant is embedded in a membrane, but in fact if all reactants and products belong to the same compartment and none appear on a patch it will behave exactly like the equivalent Reac object.

The stoichiometry of the surface reaction is specified by the following lists:

• Species on the left hand side of the reaction (the reactants):

  • Species on the surface (slhs).
  • Species in the “outer” compartment (olhs).
  • or Species in the “inner” compartment (ilhs).

• Species on the right hand side of the reaction (the products):

  • Species on the surface (srhs).
  • Species in the “outer” compartment (orhs).
  • Species in the “inner” compartment (irhs).

Note: Reactant species cannot belong to different compartments, so attempting to create an SReac object with both olhs and ilhs will result in an error.

To become familiar with these objects we will build a simplified version of the inositol 1,4,5-trisphosphate (IP $_{3}$) model (described in Doi T, et al,Inositol 1,4,5-Triphosphate-Dependent $Ca^{\text{2+}}$ Threshold Dynamics Detect Spike Timing in Cerebellar Purkinje Cells, J Neurosci 2005, 25(4):950-961) in STEPS.

In the IP3 receptor model, reactions (i.e. receptor binding of IP3 andcalcium molecules) take place on the membrane separating the endoplasmicreticulum (ER) and the cytosol. Therefore, we will use Surface Reaction objects to describe the reactions.

In the figure below we can see a schematic diagram of the states and transitions in the model. We see that our reactions take place on the membrane between the cytosol and the Endoplasmic Reticulum (ER) and therefore must be described by an SReac object, with each “binding” reaction described by a second order surface reaction and each “unbinding” reaction by a first order surface reaction.

We will go through the Python code to build this model in STEPS, but providing only brief descriptions of operations we are familiar with from the previous chapter.

Model specification

Model container

First we need to import the steps.model package and create a steps.model.Model container object named mdl, as we did in previous chapter:

Species

Now we create the species in this model based on the above diagram of the states and transitions, declaring all receptor states as separate steps.model.Spec objects. Recall that all identifier strings must be unique and we should also make sure not to reuse a variable name so that we do not lose any references to the objects we create:

Surface System

Next we create a surface system. The function of a surface system is similar to the volume system used to group steps.model.Reac objects we saw in the tutorial on well mixed models. Basically, surface systems group a set of reaction rules that are described by steps.model.SReac objects. It is often the case that such reactions are modeled as taking place on a membrane surface and not within a volume, although this is actually not a necessity. We need to create an object of type steps.model.Surfsys. The arguments to the class constructor are an identifier string and a reference to the parent steps.model.Model object:

Reactions

Now it is time to specify the reaction stoichiometry, shown in diagram of states. Unlike in Well-Mixed Reaction Systems, the reactions in this model are defined by the surface reaction objects (steps.model.SReac) in which the arguments include information about which compartment or patch the reactants and products belong to. Therefore each reactant and product of a surface reaction may be a chemical species within a volume or one which is bound to a surface.

Therefore surface reaction objects can deal with three types of reactions, classified by the locations of the reactants, and the object is smart enough to know what type of reaction it is so that the solver knows what kind of reaction it is dealing with. The three types of reactions are;

• Volume-Surface reactions. In this case molecules within a volume interact with molecules embedded in a surface and result in products that may reside within in a volume or a surface. The units for the reaction parameters in this case are the same as for ordinary volume reactions, namely: a first order reaction parameter has units $s^{-1}$; a second order reaction parameter has units $\left(M.s\right)^{-1}$; a third order reaction $\left(M^{2}.s\right)^{-1}$; and so on.

• Surface-Surface reactions. In this case the reactants are all embedded in a surface. Quite clearly, the dimensions of the reaction are different from a volume reaction and the reaction parameter is assumed to be two-dimensional. This is an important point because the reaction parameter will be treated differently from a volume-volume or volume-surface interaction. A further complication is that parameters for ordinary volume reactions are based on the litre, where there is no convenient equivalent 2D concentration unit. Surface-surface reaction parameters are based on units of area of square meters. A first order surface-surface reaction parameter is therefore required in units of $s^{-1}$; a second-order surface-surface reaction parameter has units $\left(mol.m^{-2}\right)^{-1}.s^{-1}$; a third-order surface-surface reaction parameter has units $\left(mol.m^{-2}\right)^{-2}.s^{-1}$; and so on. Zero-order surface reactions are not supported because of the ambiguity of interpreting the reaction parameter.

• Volume-Volume reactions. It is possible for a surface reaction to contain reactant species that are all in a volume, in which case the reaction behaves similarly to an ordinary volume reaction (steps.model.Reac), though products may belong to connected volumes or surfaces.

As mentioned previously, to create our surface reaction objects we have to include some information about the location of the reaction: which compartment are the reactants to be found in, and are any molecules embedded in a surface and which of the two compartments that the surface connects are the products injected into? We supply this information to STEPS by labelling our compartments that a patch connects, arbitrarily choosing the labels 'inner' and 'outer'. When the surface reaction's parent surface system object is added to a certain patch, the compartment labelling in the surface reaction stoichiometry will match the compartment labelling in the patch definition. We will come to creating a patch later in this chapter.

So, at this stage we must chose which compartment we will label 'outer' and which we will label 'inner' and make sure to maintain this labelling throughout our definitions, and also in our geometry description. We chose to label the cytosol as the 'outer' compartment and the ER as the 'inner' compartment, so should be very careful that this ties in correctly to our description when we create our steps.geom.Patch object to represent a surface to connect the two compartments.

We will first complete all “forward” binding reactions, recalling that “forward” and “backward” reactions must be declared separately:

We model our calcium flux from the ER to the cytosol simply as a second order reaction. In effect we are saying, when such a reaction takes place, a calcium ion from the ER passes instantaneously through an open receptor to the cytosol.

We can see that it is vital that when we come to describing our geometry that our compartment labelling is maintained. At this level we have specified whether some reactants and products belong to the 'inner' or 'outer' compartment, with the intention to label the cytosol as the 'outer' compartment and the ER as the 'inner' compartment, but we will not actually make that distinction until we come to describing our geometry.

Next, we set all reaction constants' default values (see Doi T, et al,Inositol 1,4,5-Triphosphate-Dependent $Ca^{\text{2+}}$ Threshold Dynamics Detect Spike Timing in Cerebellar Purkinje Cells, J Neurosci 2005, 25(4):950-961). These constants could have been passed to the initializer when we were creating our steps.model.SReac objects, but for clarity we chose to set them here with method setKcst. Since these are volume-surface interactions, we must make sure to supply our values in Molar units as discussed previously in this chapter.

Geometry specification

The next step is to create the geometry for the model. We will chose well-mixed geometry, as in the chapter on well-mixed models, but we now have two compartments which are connected by a surface 'patch'. We create two steps.geom.Comp objects to represent the Endoplasmic Reticulum (which we intend to label the 'inner' compartment) and the cytosol ('outer' compartment), and a steps.geom.Patch object to represent the ER membrane between the ER and cytosol. We then add the stoichiometry we previously defined and grouped in our surface system object to the patch object. Note that any volume-reactions we defined with Reac objects and grouped in Volsyscontainers would be added to the Compartments at this stage.

First we create the two well-mixed compartments. With more than one compartment in the model we must make sure that the identifier strings are be unique amongst all compartments in the geometry container. We create the cytosol compartment with the minimum information (identifier string and reference to container), setting the volume with class method steps.geom.Comp.setVol, but set the volume of the ER during object construction purely to demonstrate the two possible ways to achieve the task:

We now create a steps.geom.Patch object, defining the 'inner' and 'outer' compartments. We wish to label the ER as the 'inner' compartment and the cytosol as the 'outer' compartment, which we achieve by their order to the patch object constructor. The 3rd (required) argument to the constructor (here I am calling the string Id the 1st argument, not the 'zeroth') is a reference to the 'inner' compartment and the 4th (optional) argument is a reference to the 'outer' compartment. It is vital that care is taken in the order of the steps.geom.Comp objects to the constructor, so that the required labelling from or surface reaction definitions is maintained. Note: A Patch must have an inner compartment by convention, but does not require an outer compartment. This is an easy way to remember the order to the constructor; since an inner compartment is always required it must come first to the constructor, and the optional outer compartment comes after. Obviously any surface reaction rules that contain reactants or products in the outer compartment cannot be added to a Patch that doesn't have an outer compartment.

We can check the labelling is as desired after object construction if we like with methods steps.geom.Patch.getOComp and steps.geom.Patch.getIComp. Note: Typically, get functions return references to the object, not the identifier string, so we can use any of the object methods on the returned reference to access information about the object. Here we use method getID, which returns the identifier string of the object.

We also set the surface area of the patch:

Simulation with Wmdirect

Now the model is completed and ready for simulation. To run the simulation, we create a random number generator object, as we did previously in Well-Mixed Reaction Systems:

and use the steps.solver.Wmdirect solver as we also did in Well-Mixed Reaction Systems:

To run the simulation and plot the data, we import modules from numpy and create arrays to store the data, as we also did previously. We also import modules from matplotlib in order to plot the results. This time we also want to plot the standard deviation so we create arrays to store that data too. Here we create variable NITER and assign it the value 100, the number of iterations we wish to run. Note: It is perhaps better to group simulation parameters together at the beginning of the script (such as number of iterations, simulation end time, data collection time step, etc) so we can change a parameter simply by changing one variable, which reduces the amount of typing and reduces the scope for error. This is an approach we will adopt in the next chapter.

At the beginning of the simulation, we reset the solver state and set the initial concentration or count (by “count” we mean the number of molecules) of each species (any species we don't explicitly assign a concentration or count to will be initialized with the default value of zero which was set when we called the reset function) and run the simulation for NITER number of iterations. In the following example code, we record the number of IP3 receptors in open state ('Ropen' in 'memb') and the concentration of calcium in the cytosol ('Ca' in 'cyt'). We include a pylab.plot call within our main loop to plot the number of open receptors for each individual iteration:

We can also calculate the mean and standard deviation of our data using NumPy functions and plot them.