# Development of a Simple Kinetic Mathematical Model of Aggregation of Particles or Clustering of Receptors

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## Abstract

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## 1. Introduction

## 2. Modeling Approaches

#### 2.1. Aggregation Model

#### 2.1.1. Model Equations

_{i,j}is the rate constant of coagulation of particle with size i with the particle with size j; f

_{i,j}is the rate constant of fragmentation of an aggregate of size i + j into to particles of size i and j.

_{i,j}) and fragmentation (f

_{i,j}) kernels in the terminology of Smoluchowski equations. We assume that there is no difference between the reactions ${A}_{i}+{A}_{j}\text{}\leftrightarrow {A}_{i+j}$ and ${A}_{j}+{A}_{i}\text{}\leftrightarrow {A}_{i+j}$, so kernels are considered to be symmetrical. In the real biological systems, aggregates cannot achieve an infinite size, so we restricted the maximal achievable cluster size to N. The values of a

_{i,j}, f

_{i,j}, and N are chosen based on the following assumptions:

- The mechanism of fragmentation is assumed to be independent of the aggregate size.

#### 2.1.2. Model Parameters

_{1}b represents the initial increase in optical density that is observed in the experiment due to the platelet shape change [33]. We approximate it as $b=\frac{t}{t+lag}\to 1,\text{}{c}_{1}\approx 0.1\text{}$(where t denotes time from activation, lag denotes a lag-time parameter) to achieve precise description of data; however, this parameter rapidly approaches constant value and has no effect on aggregation after the first ten seconds. Lag-time does not have any relation with other constants; it is an independent parameter introduced to describe processes that are not related to aggregation (as platelet shape change). The experimental data of mean aggregate size was also obtained in the LTA assay and was estimated by the following function:

#### 2.1.3. Platelet Aggregation Experiments

#### 2.2. Clustering Model

#### 2.2.1. Model Equations

#### 2.2.2. Model Parameters

#### 2.3. “2-Equation” Model

_{2}is the probability of new aggregate/cluster formation from two single particles, k

_{1}is the probability of another particle attachment to an existing aggregate/cluster, k

_{−1}is the probability of single-particle detachment from an aggregate/cluster, k

_{−2}is the probability of formation of one aggregate/cluster from two existing ones, and k

_{3}is the probability of an aggregate/cluster fragmenting into two. The mean aggregate/cluster size, s, for “2-equation model” is calculated according to the formula:

_{0}denotes the initial concentration of particles. It should be noted here that such simplification of the models violates mass conservation law because the clusters, n, could have variable mass; that is why additional restrictions on s should be introduced in some cases.

_{3}). The main feature that distinguishes the “2-equation model” model from other published models is the description of all population of aggregates/clusters by a single variable.

#### 2.4. Methods for Parameter Estimation, Model Solution, and Comparison of the Models

## 3. Results

#### 3.1. Description of Protein Aggregation Data

_{0}= 1. Figure 2a shows that both models successfully described all sets of data. Parameters for each experimental curve appeared to be equal between both (“Aggregation” and “2-equation”) models (Table 1). Their values for different concentrations of ArgEE are given in Table 1. It appears that all parameter values reduced with the increase in the ArgEE concentration. Furthermore, from the comparison between relative ki-s changes, we can conclude that the suppressive effect of arginine derivatives is mainly caused by a decrease in particle attachment to an existing aggregate (Figure 2b).

#### 3.2. Description of Platelet Aggregation Data

_{−1}= 0.0001, k

_{1}= 0.009, k

_{−2}= 8.0 × 10

^{−7}, k

_{2}= 0.0016, k

_{3}= 0.0068, p

_{0}= 4.8; and for “2-equation model” are k

_{−1}= 0.0061, k

_{1}= 0.0090, k

_{−2}= 8.0 × 10

^{−7}, k

_{2}= 0.0016, k

_{3}= 0.0014, p

_{0}= 4.24. The “Aggregation models” with maximal cluster sizes N

_{max}= 10 and N

_{max}= 100 describe experimental data with the same parameters and the same distribution of particles between aggregates of different sizes (Figure S1). Therefore, we can conclude that (1) the particles are mainly distributed between smaller aggregates, and (2) it is rational to use N

_{max}= 10 to approximate experimental data.

#### 3.3. Additional Restrictions on Parameter Values for “Aggregation Model”

_{−1}equals 1.0 × 10

^{−4}or 4.5 × 10

^{−17}). One way to improve the process of parameter estimation is to introduce additional approach for the description of the same experimental process and thus reduce the amount of variability in the model parameters. For example, the aggregation process can be described by the changes in the solution’s optical density or by the changes in mean aggregate size. We have used both experimental datasets (OD (t) and s (t)) simultaneously to obtain the parameter values for data our own experimental datasets on platelet aggregation (Figure S2 and Table S4). The new parameter values change a little between activation with ADP at 1 μM or 2 μM with the exception of k

_{−2}, which appears to be negligible. The parameters of single platelet attachment to aggregate appear to increase with ADP while the aggregate stability decreases in line with previously obtained results [33].

#### 3.4. Description of Receptor Clustering Data

_{−1}= 0.053, k

_{1}= 6.1 × 10

^{−6}, k

_{−2}= 10.2, k

_{2}= 0.56, p

_{0}= 10

^{4}and for “2-equation model” are k

_{−1}= 6.6 × 10

^{−9}, k

_{1}= 2.1 × 10

^{−4}, k

_{−2}=0.03, k

_{2}= 7.5 × 10

^{−6}, k

_{3}= 1.16, p

_{0}= 10

^{4}. It should be noted that the parameter values between the “Clustering model” and the “2-equation model” are not comparable because they describe probabilities of different processes. The whole set of parameters for other experimental curves is given in Table S2. The description of all sets of experimental data by the “Clustering model” and the “2-equation model” gave equal values of AIC. Therefore, these models describe the data similarly (Table S3). Moreover, both models described the kinetics of single receptors concentration similarly (Figure 4c).

_{1}= 5.4 × 10

^{−5}, k

_{−1}= 6.7 × 10

^{−4}, k

_{2}= 3.4 × 10

^{−5}, k

_{−2}= 6.7 × 10

^{−}

^{4}. The parameter values appear to be significantly lower for this model than for the model of GPVI receptor clustering, reflecting the longer characteristic times of the neutrophil activation compared to platelet activation. The parameters for “2-equation model” were also estimated and are the following: k

_{−1}= 6.8 × 10

^{−10}, k

_{1}= 0.0028, k

_{−2}= 6.5 × 10

^{−4}, k

_{2}= 3.2 × 10

^{−4}, k

_{3}= 0.0061, p

_{0}= 16.4 (Figure S3).

#### 3.5. Additional Restrictions on Parameter Values for “Clustering Model”

_{1}and k

_{2}$~\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r$}\right.$ and these parameters are related to each other. Therefore, we can restrict the parameters values as [59]:

#### 3.6. Features of Receptor Clustering Process Revealed by the Mathematical Models

_{2}/k

_{−2}and k

_{1}/k

_{−1}; those represent dissociation constants of the corresponding reactions (Figure S5). The typical behavior of clusters of different sizes for each type of steady-state is presented in Figure S6. As could be expected, the dissociation rate values affect the steady-state of the system, while the parameter values affect the characteristic times to a steady-state.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schemes of the reactions for the “Aggregation model” (

**a**) and the “Clustering model” (

**b**). (

**a**) The spheres represent single particles, and the merged spheres represent aggregates; the parameters of the reactions are: k

_{2}is the probability of new aggregate formation from two single particles; k

_{1}is the probability of another particle attachment to an existing aggregate; k

_{−1}is the probability of the detachment of single-particle from an aggregate; k

_{−}

_{2}is the probability of formation of one aggregate from two existing ones; k

_{3}is the probability of an aggregate fragmenting into two. (

**b**) The cylinders represent single receptors in the membrane (grey plane); the merged cylinders represent clusters; the parameters of the reactions are: k

_{1}(k

_{−1}) is the probability of attachment (detachment) of single receptor to (from) a cluster; k

_{2}(k

_{−2}) is the probability of attachment (detachment) of a dimer to (from) a cluster.

**Figure 2.**Parameter estimation for the “Aggregation model” and “2-equation model” based on BSA aggregation data. The estimation of model parameters for each model was conducted automatically. For each set of experimental data, parameters of the models were estimated independently. (

**a**) Kinetics of BSA aggregation in the presence of ArgEE at the following concentrations: 50, 100, and 700 mM. Description of experimental data by the “Aggregation model” (red) or the “2-equation model” (blue); parameters of the models appear to be equal and are given in Table 1, as well as correlation of obtained parameters with ArgEE concentration. (

**b**) Relative change between parameters, obtained for 50 and 700 mM of ArgEE.

**Figure 3.**Parameter estimation for the “Aggregation model” and “2-equation model” based on platelet aggregometry data. Estimation of six model parameters and initial platelet concentration for each model was conducted automatically. For each set of experimental data, parameters of the models were estimated independently. Experimental data on platelet aggregation in response to 3 μM of ADP (

**a**) or 2 μM of ADP (

**b**). (

**a**,

**b**) Description of experimental OD-curves (dots) by the “Aggregation model” (red) or the “2-equation model” (blue); parameters of the models are given in Table S1. (

**c**,

**d**) calculated time-course of the mean size of aggregate (

**c**) and concentration of single platelets (

**d**) for models describing experimental data for 3 μM of ADP [40,41].

**Figure 4.**Parameter estimation for the “Clustering model” and “2-equation model” based on platelet GPVI receptor clustering. The estimation of model parameters and initial receptor concentration for each model was conducted automatically. For each set of experimental data, parameters of the models were estimated independently. Experimental data on GPVI receptor clustering on Col-III (

**a**) or III-30 (

**b**). (

**a**,

**b**) Description of the experimental amount of clusters (dots) by the “Clustering model” (red) or the “2-equation model” (blue); parameters of the models are given in Table S2. (

**c**) Calculated time-course of the concentration of single receptors (

**c**) for stimulation with Col-III. (

**d**) The size of predominant clusters in steady state. Typical k

_{2}/k

_{−2}and k

_{1}/k

_{−1}dependence of steady state for N = 15. There are only 3 types of steady states: unclustered, dimers, and full clustering.

**Figure 5.**Description of size distribution of CR3 clusters on human neutrophils by the “Clustering model”. Experimental data [2] are shaded; model data are given in grey. Note that on the right panel, the sum of experimental data columns does not equal 100%. The parameters of the “Clustering model” were: k

_{1}= 5.4 × 10

^{−5}, k

_{−1}= 6.7 × 10

^{−4}, k

_{2}= 3.4 × 10

^{−5}, k

_{−2}= 6.7 × 10

^{−4}. The “Clustering model” adequately predicts the non-steady state at 10 min.

**Table 1.**Automatically assessed model parameters for experimental datasets given on Figure 2a.

Parameter | ArgEE Concentration, mM | Pearson Correlation Coefficient | ||
---|---|---|---|---|

50 | 100 | 700 | ||

k_{1} | 2.3 × 10^{−3} | 4.3 × 10^{−3} | 2.3 × 10^{−3} | −0.69 |

k_{−1} | 1.3 × 10^{−3} | 1.6 × 10^{−4} | 4.5 × 10^{−4} | −0.34 |

k_{2} | 8.6 × 10^{−4} | 7.1 × 10^{−4} | 2.7 × 10^{−4} | −0.98 |

k_{−2} | 0.052 | 0.022 | 1.1 × 10^{−3} | −0.85 |

k_{3} | 0.015 | 5.3 × 10^{−3} | 3.4 × 10^{−4} | −0.80 |

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**MDPI and ACS Style**

Garzon Dasgupta, A.K.; Martyanov, A.A.; Filkova, A.A.; Panteleev, M.A.; Sveshnikova, A.N.
Development of a Simple Kinetic Mathematical Model of Aggregation of Particles or Clustering of Receptors. *Life* **2020**, *10*, 97.
https://doi.org/10.3390/life10060097

**AMA Style**

Garzon Dasgupta AK, Martyanov AA, Filkova AA, Panteleev MA, Sveshnikova AN.
Development of a Simple Kinetic Mathematical Model of Aggregation of Particles or Clustering of Receptors. *Life*. 2020; 10(6):97.
https://doi.org/10.3390/life10060097

**Chicago/Turabian Style**

Garzon Dasgupta, Andrei K., Alexey A. Martyanov, Aleksandra A. Filkova, Mikhail A. Panteleev, and Anastasia N. Sveshnikova.
2020. "Development of a Simple Kinetic Mathematical Model of Aggregation of Particles or Clustering of Receptors" *Life* 10, no. 6: 97.
https://doi.org/10.3390/life10060097