Multiple Cytokine Effect Description: General Formulas

Musatova Veronika, Rubina Svetlana, Demin Oleg
October 8, 2021

Case #1: Process has non-zero rate without effector impact

First, we will describe the case when the process has non-zero rate without regulators impact (see Fig.1).

Types of effectors considered:

  • Activators (Ai) accelerate a process;
  • Inhibitors (Ij) slow down a process;
  • Positive/negative modulators of effector Ai (MAik) strengthen/weaken the effect of an effector;
  • Positive/negative modulators of inhibitor Ij (MIjk) is considered to act as MAik and are not described below.

Description of regulation of a cell dynamics process:

1. A cell dynamics process is transition between different states of a cell

2. Control of each particular process is associated with a generalized receptor. Effector binds to receptor and effect is mediated by state of receptor. States of the receptor with bound activators/inhibitors promote faster/slower transition between cell states (effect). Binding of an effector with receptor is assumed to be in equilibrium.

3. There are 2 pairs of binding sites in the receptor: activator binding site and site for binding of activator ’s modifiers; inhibitor binding site and site for binding of inhibitor ’s modifiers. Sites associated with activators are completely independent on those associated with inhibitors. Sites for activators and their modifiers are dependent, i.e., binding of a modifier may lead to accumulation of state of receptor bound with an activator. The same is true for inhibitors.  Activators compete for “activator binding site ”. Inhibitors compete for “inhibitor binding site ”. Modifiers compete for corresponding binding site.  

4. Rate law of a process regulated by multiple effectors:

Let's assume that k(A,I,M) depends on relative concentration of receptor states.

Rtot - total concentration of receptors;
R - concentration of unbound receptors;
Ai - concentration of i-th activator;
I- concentration of j-th inhibitor;

1) All stages of the catalytic cycle can be subdivided into a group of fast reactions and a group of slow reactions.

2) Fast reactions can be considered at quasi-equilibrium in comparison with slow reactions.

3) All concentrations of enzyme states can be expressed in terms of parameters of the catalytic cycle and substrate/product/effector concentrations on the basis of equilibrium relationships valid for fast reactions.

Expressing parameters of receptor-effector binding in terms of "kinetic" parameters:

1) the maximal stimulatory effect of i-th activator / j-th inhibitor:

2) supposing that receptor binding and level of tissue response is in direct proportion:


Notice that it is not reflected in the formulas below if a modifier has stimulatory or inhibitory effect.

Mj - concentration of l-th modifier of j-th inhibitor;

Modifiers of 2nd order ("modifiers of modifier") and 3rd order (a very rare case) have the same construction as 1st order modifiers.

Modifiers of 2nd order

SMh - h-th second modifier; all the other values - in a similar to mentioned above.

Modifiers of 3rd order

TMf - f-th third modifier; all the other values – in a similar to mentioned above.

Case #2: Process has zero rate without effector impact

In this case process has a non-zero rate only if there is an activator bonded to the receptor (see Fig.2). Any effector slowing down or accelerating the process caused by an activator should be interpreted as a modifier.


The equation (2gen) in this case may be simplified:

From the (4gen) and (10gen):

or, supposing that receptor binding and level of tissue response is in direct proportion:

Lets define

With this designation for the caseв„–2 following equations may be used:

Modifiers of 2nd order

Modifiers of 3rd order

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