A fundamental component of a model involves a set of interaction rules, or reactions. This is a set of rules that describes the possible evolution for objects or processes.  

Here we have a natural hierarchy: 

Simple reactions are the simple objects interactions

Complex reactions are the interactions between more complex objects (e.g., high level description: NK activates Macrophage)

Basic and complex reactions define the Signaling pathways (e.g. NFkB pathway, Apoptosis pathway etc.)

Due to the limited knowledge regarding reaction rate constants, we distinguish only a few reaction speeds that in the real system correspond to different orders of magnitudes. We limit our discussion here to three speeds (although our simulation program can have any finite number of speeds). 

 We introduce a symbol to express each of them: 

->>> high (fast) rate

->>   low (slow) rate

->     very low (very slow) rate

 Note: it can easily be extended to any finite number of levels.   

Dynamical behavior is mostly due to nonlinearity. Whether we use one metaphor or another, the immune system is about interactions of molecules; some are secreted, some reside on cell surfaces, other participate in signaling that results in changes at the gene level, etc. Gene expression changes are responsible for differentiation (irreversible change), for cytokine secretion, for chemokine expression, and more. Even higher level behavior, such as ‘learning’ or ‘selection’ in this system, can be explained from the basic interaction of organs and cells through molecules. Our approach in addressing the dynamical complexity is to be as close to the biological presentations as possible. We express the dynamics in the immune system in terms of interactions between objects which include binary interactions, thus allowing for nonlinearity which may gives rise to complex dynamical behavior. 

Each regular reaction has a list of (one or two) reactants (left), a list of products (right) and a speed, and is  written as follows:

 <reactants> <speed> <products> 

Examples:

R + L ->>> R:L                                      receptor R binds very fast to ligand L to form a complex R:L

R:L ->> W                                              a complex R:L fast initiates a process W

R:L ->> SignalingX                              a complex R:L fast initiates a signaling X (unknown)

SignalingX -> CytokineX                    signaling X results in slow production of cytokine X

 To deal with secretion of molecules by cells, we also have transporter reactions which use the following syntax:

< name>  ::  <name> @ <location > <speed>  <name> @ <location>.

The first <name> is the name of a transporter, the second it the name of the simple object to be transported, <location>s specify origin and destination, and <speed> is one the symbols ->, ->>, ->>>, etc. Our data structures for objects are trees, therefore, we use for <location> the values ‘Self’,  ‘Parent’,  ‘GrandParent’. The convention we used is that transporters are always in the deeper level of the two <location>s specified. 

Reaction Graph. A set of reactions can be viewed as weighted directed graph with two types of nodes: reaction nodes, and ‘simple objects’ nodes. We consider unary and binary reactions in our formulation, so the number of links coming to a reaction node is at most two. The number of links going out of a node is not limited. This representation is different from what is commonly used in biological signaling diagrams, as it refers to precise mechanisms, not only for their outcome. Up/Down regulation for example can be the result of several different reaction sets.