January 27, 2022

**Introduction**

Mechanistic modeling of immune response is very important to correctly predict response of patients to immuno-therapies in different therapeutic areas. Cell dynamics processes included in a QSP model of immune response can be governed by multiple regulatory molecules. Most part of the data quantifying the regulatory effects is represented by results of *in vitro* experiments demonstrating dependence of the cell dynamics process on dose of individual effector molecule. Combined effect of 2 or more regulatory molecules is less frequently presented in the literature [*PMID: 1358972*], [*PMID: 2104901*], [*PMID: 7523571*]. Different formulas to describe multiple effects of cytokines were proposed in [*PMID: 20491795*], [*PMID: 16986268*], [*PMID: 21305530*], [*PMID: 23841912*], [*biophys2012 poster*], [*ACoP7 poster*]. Some of them are based on empiric considerations but others take into account any mechanistic reasoning.

All the approaches enable to describe multiple effects of cytokines, chemokines and other regulatory molecules on immune cell proliferation, differentiation, migration, apoptosis and cytokine production. Basing on the analysis of the approaches we have chosen those published in [*ACoP7 poster*] and then described it in the note. Below we present derivation of formula describing multiple effects of activators, inhibitors and modifiers.

**Types of effectors considered **

We have considered following types of effectors

- *Activators* (A_{i}) accelerate a process

- *Inhibitors* (I_{k}) slow down a process

- *Modifiers* of effector Ai (M^{Ai}_{j}) strengthen/weaken effect of an effector

**Description of regulation of a cell dynamics process with multiple activators and inhibitors **

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, see Fig 1A, B for schematic representation). Binding of an effector with receptor is assumed to be in equilibrium.

*Figure 1.** Schematic representation of assumptions underlying derivation of formular describing multiple effects of cytokines on cell dynamics processes*

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 (see Fig 2). 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.

*Figure 2.** Schematic representation of sites of generalized receptor for binding of activators, inhibitors and their modifiers.*

4. Rate law of a process regulated by multiple effectors can be presented in following manner (see Fig 1A for schematic representation):

V = k(A,I,S)*[Cell State 1] (1)

5. Function ** k(A,I,S)** describes influence of effectors on the process rate in terms of their concentrations in extracellular medium. Derivation is based on assumption that

where R, R_{tot}, I_{j}°R, R°A_{i}, I_{j}°R°A_{i} are concentrations of free receptor, total receptor, complex of receptor with inhibitor I_{j}, complex of receptor with activator A_{i}, complex of receptor with inhibitor I_{j} and activator A_{i}, correspondently.

are basal rate constant, rate constant corresponding to effect of activator Ai and portions of decrease in basal rate constant corresponding to effect of inhibitor Ij. Values of these parameters should hold following inequalities true:

It is important to stress that in framework of the approach we have assumed that binding of an inhibitor decreases the ability of the receptor-inhibitor complex stimulate the process by the same factor independently on bound activator.

5. Taking into account conservation law for total receptor concentration and applying quasi-equilibrium approach to calculate concentrations of receptor-effector complexes in terms of effector concentration and binding parameters, one obtains:

Substituting equations (4) into expression (2) and expressing parameters describing effector to receptor binding and rate constants in terms of EC_{50}, IC_{50}, E_{max}, I_{max} for corresponding activators and inhibitors we have come to following expression:

where

Equation (5) describes dependence of rate constant of a cell dynamics process in a way illustrated in Fig 3A,B.

*Figure 3.** Qualitative dependence of rate constant of a cell dynamics process on activator (A) and inhibitor (B) concentration under condition when concentrations of all other effector are equal to zero.*

**Description of regulation of a cell dynamics process with 2 activators and 1 modifier**

6. Let us take into account in Eq (5) effect of modifiers. To illustrate details how we do it in framework of proposed approach we consider example of 2 activators (A_{1} and A_{2}) and 1 modifier (M) which is able to influence on effect of both activators but at different extent. Scheme representing the example is shown in Fig 4.

*Figure 4.** Scheme representing interaction of 2 activators and 1 sensitizer.*

For this case dependence of the rate constant k on states of a receptor bound with the 2 activators and 1 sensitizer can be described by following equation:

Here, we assume (and, subsequently, take into account in the equation) that binding of modifier M to receptor does not change rate constant of the cell dynamics process but the binding can either strengthen or weaken effects of activators in different manner and extent. Parameters g_{1} and g_{2} are responsible for the influence of modifier on effect of activators A_{1} and A_{2}, correspondently.

7. Taking into account conservation law for total receptor concentration and applying quasi-equilibrium approach to calculate concentrations of receptor-effector complexes, one obtains following expression:

Expressing parameters of Eq (7) describing effector to receptor binding in terms of EC_{50} for corresponding activators and modifier we have come to following expression:

**Description of regulation of a cell dynamics process with any numbers of activators, inhibitors and modifiers**

In general case of n_{A} activators, nI inhibitors and n_{M,Ai} modifiers for each activator A_{i}, n_{M,Ij} modifiers for each inhibitor I_{j} we have come to equation (5) with following expressions for E_{max}, I_{max}, EC_{50}, IC_{50} in terms of modifiers:

Here, we use following designations

**Conclusion**

The approach proposed in the note allows to successfully describe effect of multiple activators, inhibitors, modifiers on cell dynamics of immune cells.

**References**

[*PMID: 20491795*] Shoda, L., H. Kreuwel, K. Gadkar, Y. Zheng, C. Whiting, M. Atkinson, J. Bluestone, D. Mathis, D. Young and S. Ramanujan (2010). "The Type 1 Diabetes PhysioLab Platform: a validated physiologically based mathematical model of pathogenesis in the non-obese diabetic mouse." Clin Exp Immunol 161(2): 250-267.

[*PMID: 16986268*] Rullmann, J. A., H. Struemper, N. A. Defranoux, S. Ramanujan, C. M. Meeuwisse and A. van Elsas (2005). "Systems biology for battling rheumatoid arthritis: application of the Entelos PhysioLab platform." Syst Biol (Stevenage) 152(4): 256-262.

[*PMID: 21305530*] Meeuwisse, C. M., M. P. van der Linden, T. A. Rullmann, C. F. Allaart, R. Nelissen, T. W. Huizinga, A. Garritsen, R. E. Toes, R. van Schaik and A. H. van der Helm-van Mil (2011). "Identification of CXCL13 as a marker for rheumatoid arthritis outcome using an in silico model of the rheumatic joint." Arthritis Rheum 63(5): 1265-1273.

[*PMID: 23841912*] Schmidt, B. J., F. P. Casey, T. Paterson and J. R. Chan (2013). "Alternate virtual populations elucidate the type I interferon signature predictive of the response to rituximab in rheumatoid arthritis." BMC Bioinformatics 14: 221.

[*biophys2012 poster*] Uvarovsky, A.N., Speshilov, G.I., Metelkin, E.A. (2012) “Modelling of neutrophils dynamics in the presence of survival factors.” p. 141, IV Congress of Russian biophysicists. Symposium II “Physical basics of physiological processes”. Nizhniy Novgorod, 2012. – 206 p. Abstract available at http://www.unn.ru/biophys2012/pages/materials/Vol2.pdf

[*ACoP7 poster*] Demin O., Metelkin E., Lebedeva G., Smirnov S. (2016) “Mechanistic approach to describe multiple effects of regulatory molecules on cell dynamics process in immune response” American Conference om Pharmacometrics. Abstract available at https://isop.memberclicks.net/assets/Legacy_ACOPs/ACoP7/Abstracts/w-21.pdf

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