# Dynamic Effects in Nucleation of Receptor Clusters

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Background

#### 2.2. Mathematical Model

## 3. Results

#### 3.1. The Potential Barrier of New Bond Formation in a Cluster

#### 3.2. The Kinetic Barrier of Bond Formation

#### 3.3. Von Neumann Entropy Approximation

#### 3.4. Critical Size of Receptor Cluster

#### 3.5. Heterogenous Nucleation Efficiency

#### 3.6. Homologous Series in T Cell Activation by Oligomeric MHC

## 4. Discussion

- The formation of supercritical clusters by the mechanism of heterogeneous nucleation;
- The growth of these clusters to a productive state that initiates cell’s intrinsic signaling pathways;
- Signal transmission along the signaling pathway to the cell nucleus, where gene expression takes place.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## Appendix D

## References

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**Figure 1.**Force and energy dependence on the reaction coordinate. (

**a**) The reaction coordinate q is shown for one bond in the cluster. (

**b**) The dependence of bond force on that coordinate is shown. (

**c**) The dependence of bond energy on the coordinate is shown. The extremum of the energy curve corresponds to an intermediate complex and has the energy of potential barrier ${E}_{p}$.

**Figure 2.**Energy and phase diagrams at different levels of dissipation. (

**a**–

**c**) The trajectories of the intermediate complex during the formation of a bond in the coordinate axes q of the oscillations of the entire cluster (abscissa) and the energy E at this coordinate (ordinate). (

**d**,

**e**) The same trajectories but in phase space in the coordinate q (abscissa) and momentum p (ordinate) of the oscillations of the entire cluster. Trajectories that do not overcome the potential barrier are marked in purple. Green—trajectories correspond to a bound state. Yellow—trajectories that lead to the formation of long-lived bonds. Red—trajectories that lead to the formation of short-lived bonds. Diagrams (

**a**,

**d**) correspond to motion without dissipation, (

**b**,

**e**) correspond to weak dissipation, and (

**c**,

**f**) correspond to strong dissipation.

**Figure 3.**Phase diagram accounting for the energy redistribution between modes. (

**a**) The projection of the phase space on the coordinate axis q and momentum p of oscillations of the entire cluster; (

**b**) the axis of energy falling on the remaining degrees of freedom ${E}_{\perp}$. Trajectories that do not lead to bond formation are marked in purple. Red—trajectories that lead to the formation of short-lived bond without energy redistribution. Cyan—trajectories that lead to the formation of long-lived bonds. Magenta—trajectories that do not lead to the formation of long-lived bond after energy redistribution.

**Figure 4.**Phase diagram accounting for the dissipation and the energy redistribution between modes. (

**a**) The projection of the phase space on the coordinate axis q and momentum p of oscillations of the entire cluster; (

**b**) the axis of energy falling on the remaining degrees of freedom ${E}_{\perp}$. Trajectories that do not lead to bond formation are marked in purple. Yellow—trajectories that leaf to the formation of long-lived bond without energy redistribution. Red—trajectories that lead to the formation of short-lived bond without energy redistribution. Cyan—trajectories that lead to the formation of long-lived bonds. Magenta—trajectories that do not lead to the formation of long-lived bond after energy redistribution.

**Figure 5.**Dependence of the von Neumann entropy on the cluster size. The solid line is the interpolation of averaged entropy values for random bond graphs with constraints on the vertex degrees (see Appendix B). The area around solid line is the standard deviation in data. The dashed line is the approximation of the von Neumann entropy in accordance with Equation (10).

**Figure 6.**Dependence of the critical cluster size on the concentration of free receptors. The critical cluster size ${N}_{c}$ hyperbolically depends on logarithm of free receptor concentration ${C}_{r}$. Calculation was made in accordance with Equation (11).

**Figure 7.**Dependence of T cell activation on oligomericity of the Major Histocompatibility Complex (MHC). The abscissa M shows the number of individual MHC in a synthetic MHC oligomer. The ordinate $-\mathrm{lg}{C}_{l}$ shows the minimal MHC oligomer concentration logarithm at which T cell activation occurs. The dashed line shows the “homologous series” predicted by the developed theory. The points with error bars represent the data of the experiments [65].

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**MDPI and ACS Style**

Prikhodko, I.V.; Guria, G.T.
Dynamic Effects in Nucleation of Receptor Clusters. *Entropy* **2021**, *23*, 1245.
https://doi.org/10.3390/e23101245

**AMA Style**

Prikhodko IV, Guria GT.
Dynamic Effects in Nucleation of Receptor Clusters. *Entropy*. 2021; 23(10):1245.
https://doi.org/10.3390/e23101245

**Chicago/Turabian Style**

Prikhodko, Ivan V., and Georgy Th. Guria.
2021. "Dynamic Effects in Nucleation of Receptor Clusters" *Entropy* 23, no. 10: 1245.
https://doi.org/10.3390/e23101245