The flowchart shows how the populations of each of the 8 types of cells affect each other (9 types if one includes G, the thymus). In order to maintain a constant system size, the thymus produces an equal number of cells as those that go to Cell Death.
Example: The change in the Initial self population (Is) is equal to the number of new Initial cells from the thymus [ = + k1 * G * SI ], plus cells converting from Es to Is at a rate k4 [ = + Es * k4 ] , minus Is to Es at a rate k2 [ =  Is * k2 ], minus Is to As at a rate k3 [ =  Is * k3], minus the number of initial cells that die [ =  Is * k5 ].
In equation form, this statement looks like this:
Is' = (k1*G* SI) + (Es*k4)  (Is*k2)  (Is*k3)  (Is*k5);
Now we have a system of equations to calculate the cell population in each
group. It is convenient to think of X' as the (current time step) variables
that we calculate knowing X data (time step 1). The program executes
the equations in the execution order (ExecOrder) as listed in the table.
However, the most meaningful equations are those where the bulk of the cell
are converted; namely, those listed as ExecOrder #6, which are listed first.
Equations in the other ExecOrders are mostly supporting rules. These rules
were seperated out, because some calculations were too complex to be included
as a single rule within the bulk conversion equations (#6). Most of the
detailed complexities are a result of constraints that are placed on
certain variables. These supporting rules include such constraints as
not to exceed the number of APCs, or to normalize outgoing rates from
the autocatalytic rate equations, or to simply prevent cell division
if there isn't enough cells to displace for such a thing to occur,
without exceeding the defined total system size.

Variable  Equation  

Is'  =  Is + ((N)*SI) + (Es_{1}*k4)  (Is*k2)  (Is*k3)  (Is*k5) + ((M)*SI); 

As'  =  As + (Is*k3)  P  Q; 

Es'  =  Es_{1} + (Is*k2)  (Es_{1}*k4) + Q; 

Inse'  =  Inse + ((N)*(1SI)*L) + (Ense_{1}*k4)  (Inse*k2)  (Inse*k3)  (Inse*k5) + (M*(1SI)*L); 

Anse'  =  Anse + (Inse*k3)  U  V; 

Ense'  =  Ense_{1} + (Inse*k2)  (Ense_{1}*k4) + V; 

Insu'  =  Insu_{1} + ((N)*(1SI)*(1L))  (Insu_{1}*k2)  (Insu_{1}*k5) + (Ensu_{1}*k4) + (M*(1SI)*(1L)); 

Ensu'  =  Ensu_{1} + (Insu_{1}*k2)  (Ensu_{1}*k4); 
where: (most are internal variables, with the exception of T and N)  

M  =  Cells Added to the system to initially grow it. Equals 0 once the system reaches Total size (T). 

T'  =  (T + M) = (Is' + As' + Es' + Inse' + Anse' + Ense' + Insu' + Ensu') 

P  =  As*k6; 
Note: P is normalized with Q if ((P+Q) > As)  

Q  =  (As/(R*SI)) * ((Es+Ense*XRxn)/(R*SI)) * k7 * PIEs * (R*SI); 
Note: Q must be <= # APC, and normalized with P if ((P+Q) > As)  

U  =  Anse*k6; 
Note: U is normalized with V if ((U+V) > As)  

V  =  (Anse/(R*(1SI)*L)) * ((Ense+Es*XRxn)/(R*(1SI)*L)) * k7 * PIEnsu * (R*(1SI)*L); 
Note: V must be <= #APC, and normalized with U if ((U+V) > As)  

Es_{1}  =  Es + Effector cells that are now dividing that went through the As to Es (via k7) before. 

Ense_{1}  =  Ense + Effector cells that are now dividing that went though the Anse to Ense (via k7) before. 

Insu_{1}  =  Insu  Es_{1}  Ense_{1} 

N  =  P + (Is*k5) + U + (Inse*k5) + (Insu_{1}*k5) = cells that die = cells that are reborn = k1*G 
Executed when Foreign AG is initially introduced, to convert the % of unengaged cells to the engaged type:  

Inse'  =  Insu'*L'; 

Ense'  =  Ensu'*L'; 

Insu'  =  Insu'  Inse'; 

Ensu'  =  Ensu'  Ense'; 
PIE Calculations:  

PIEs'  =  (1e^{((As'/APC)/(SI*R))}) * (1e^{((Es'/APC)/(SI*R))}) * APC; 

PIEnse'  =  (1e^{((Anse'/APC)/(L'*(1SI)*R))}) * (1e^{((Ense'/APC)/(L'*(1SI)*R))}) * APC; 

PIEnsu'  =  (1e^{((Insu'/APC)/((1L')*(1SI)*R))}) * (1e^{((Ensu'/APC)/((1L')*(1SI)*R))}) * APC; 
Effector / peptide calculations:  

Es'/p  =  Es'/(R*SI); 

Ens'/p  =  Ense'/(R*(1SI)*L'); 

Ensu'/p  =  Ensu'/(R*(1SI)*(1L')); 
Further model notes:
Autocatalytic k7 rate:
The autocatalytic rate k7, which convert Astype to Estype
cells (also true for nonself engaged cells, Anse to Ense)
can use some explanation. First, the simple case without being autocatalytic
or interacting with APC, the number of cells going through this pathway
would simply be As*k7 = # of new Es cells through this pathway.
Adding catalytic feedback from Es makes the equation: As*Es*k7.
This equation implies that any selfspecific peptide on As interacts
with any selfspecific peptide on Es. This isn't the case as they
must be specific to the exact same peptide for them to interact. Thus,
we must divide out the number of peptides for each As and Es
which gives us the following for each peptide: (As/ (R* SI))*
(Es/ (R* SI))* k7. Now, we include the probability
that an AsAPCEs interaction will happen in the same physical
location, namely a Priming Inductive Event (PIEs) would yield, (As/(R*SI))
* (Es/(R*SI)) * k7 * PIEs. For more details
on PIE, see the math details in the ThGenesis Minimal Model. However,
recall that this calculation is in terms of per peptide, so multiply by
the number of self peptides (R*SI) gives the entire amount
of cells converting from As to Es as: (As/(R*SI))
* (Es/(R*SI)) * k7 * PIEs * (R*SI).
k6 vs k7 competition on As & Anse:
Since k6 is large (default value of 0.1386) and k7 is autocatalyic,
and thus effectively changes with respect to the concentration of the anticipatory
and effecotor cells, there is a competition for As (likewise for Anse).
We place a few constraints: The first constraint is that no more than the
number of APC/p*(number of peptides) may go through the k7 pathway.
And second, if the pull from the k6 and k7 pathway exceeds the number of
Anticipatory cells, the two pathways are normalized. The Normalizing Equation
for values A and B, with to a maximum of C is A' = C*A/(A+B) and B' = C*B/(A+B).
Instant conversion of Insu to Inse and Ensu
to Ense cells:
Equations in the execution order of #7 & #8 in the table above result
in instant appearance of a large number of Inse and Ense cells. This conversion
is equal to the nonself antigenic load times the unengaged cells. This
is performed because of the actuality that Insu and Ensu
cells are no different than Inse and Ense cells. Their only
difference is that Inse and Ense cells would react with the
newly introduced foreign antigen. Thus these unengaged cells, would now
engage, and thus are renamed accordingly. This also helps in keeping the
math straight, as nonself engaged becomes symmetrical mathematically to self.
Steady State:
Steady state values can be observed by allowing the system to continue
until the numbers within each cell groups converge on a number and stays
there regardless of additional time steps. It is important to note that
the system is generally allowed sufficient time after the initial system
growth stage to become steady state, before the introduction of the foreign
antigen. After addition of the antigen, the system again will adjust itself
to find a new steady state. Generally, it does. However on occasions the
system will oscillate slightly. This is due to the fact that cell divisions
are timed to be exactly nsteps, and cease to divide when there is no Insu
to displace. Eventually Insu type cells regenerate, and after some delay,
effector cell divisions gets up to speed and take them down again. Thus
causing a small oscillation.
Last Modified: November 1, 2002