# Modeling of Multivalent Ligand-Receptor Binding Measured by kinITC

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Biomolecular Preliminaries and Notations

^{−1}s

^{−1}. The dissociation takes place according to the unbinding rate ${k}_{\mathrm{off}}$ in units s

^{−1}. Rate constants are numeric representations of the time needed for molecules to associate or dissociate. The dissociation constant ${K}_{d}$ and the association constant ${K}_{a}$ are the ratios of ${k}_{on}$ and ${k}_{\mathrm{off}}$, i.e., ${K}_{d}={\displaystyle \frac{{k}_{\mathrm{off}}}{{k}_{on}}}$ and ${K}_{a}={\displaystyle \frac{{k}_{on}}{{k}_{\mathrm{off}}}}$ [25]. Note that in this paper, we assume a one-to-one stoichiometry between the protein and ligand after the bindings. Although aggregation may occur in some settings, i.e., for molecules with many binding sites, we consider in theory one multivalent ligand interacting with one multivalent receptor only.

^{−1}] instead of [M

^{−1}t

^{−1}]. The rate equation for the change in bivalent complexes is:

## 3. From ITC to kinITC

#### ITC Data Generation for the Study

- Dissociation constant: ${K}_{d}$ in mM
- Receptor concentration at injection time: ${c}_{R}$ in mM
- Ligand concentration at injection time: ${c}_{L}$ in mM
- Complex concentration at injection time: ${c}_{LR}$ in mM
- Receptor concentration at equilibrium: ${c}_{R}$ in mM
- Ligand concentration at equilibrium: ${c}_{L}$ in mM
- Complex concentration at equilibrium: ${c}_{LR}$ in mM
- Wiseman parameter at the beginning of each titration: ${c}_{0}$

- Heat power: $E=\frac{dQ}{dt}$ in $\mu \mathrm{cal}/\mathrm{s}$

## 4. The Mathematical Model

- -
- the off-diagonal entries are positive: ${k}_{ij}\ge 0\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}i\ne j$
- -
- each column sum is zero: ${\sum}_{i}{k}_{ij}=0\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4.pt}{0ex}}j$
- -
- each diagonal element is the sum of the column entries: ${k}_{ii}={\sum}_{i\ne j}{k}_{ij}$.

**1**and

**2**and the receptors

**A**and

**B**.

* | diagonal matrix with the negative column sum such that the column sum is zero |

k_{off} | row vector of length ${n}^{2}$ with entry ${k}_{{\mathrm{off}}_{1}}$ |

${k}_{on}\xb7{c}_{L}$ | column vector of length ${n}^{2}$ with entry ${c}_{L}\xb7{k}_{{on}_{1}}$ |

${X}_{i}$ | sparse matrix of size $(\left(\right)open="("\; close=")">\genfrac{}{}{0pt}{}{n}{i}\xb7n)$ with entries ${k}_{{\mathrm{off}}_{2}}$ to ${k}_{{\mathrm{off}}_{n}}$, respectively. |

The positions of these entries depend on the theoretical order of conformational states. | |

${Y}_{i}$ | sparse matrix of size $(\left(\right)open="("\; close=")">\genfrac{}{}{0pt}{}{n}{i+1}\xb7n)$ with entries ${k}_{o{n}_{2}}$ to ${k}_{o{n}_{n}}$, respectively. |

The positions of the entries are the transpose of the respective ${X}_{i}$ submatrices. |

#### PCCA+

Algorithm 1: Find the optimal overall ${k}_{on}$ curve after clustering |

Require: 14 k_on-ITC and cR values from ITC |

P ← permutation matrix of possible k_on and k_off values |

for all rows of P do |

set up rate matrix K; |

for all titration steps t do |

perform PCCA+ to obtain $Kc$; |

k_on $\leftarrow Kc(2,1)/cR\left(t\right)$; |

end for |

C ← correlation coefficient of k_on-ITC and k_on; |

index $\leftarrow max\left(C\right)$; |

end for |

return P(index), k_on(index); |

## 5. Results and Discussion

#### 5.1. Bivalent Model Fitting

#### Man(1,5)-5

#### 5.2. Trivalent Model Fitting

#### Man(1,3,5)-5

^{th}overall ${k}_{on}$ value was clearly an outlier, probably due to measurement inconsistencies. For completeness, it is depicted in the graph. The correlation coefficient was 0.87. The binding rates were the following: ${k}_{o{n}_{1}}=1000,{k}_{o{n}_{2}}=1000,{k}_{o{n}_{3}}=1000$. All three unbinding rates were ${k}_{{\mathrm{off}}_{i}}=1$. One possible interpretation is that all the fully-bound state was what Whitesides called the kinetic origin of high stability [36].

#### 5.3. Pentavalent Model Fitting

#### Man(1,3,5,7,9)S-9

#### 5.4. Limitations of the Model and Discussion

## 6. Conclusions and Outlook

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Bivalent bindings have seven possible states. The ligand binding sites are equal. The red and blue color refer to the possible binding permutations.

**Figure 2.**Comparison of the computed and experimental [14] ${k}_{on}$ rates for bivalent Man(1,5)-5; see the Supplementary Information (SI) for additional information on experimental data. The theoretical ${k}_{on}$ rates were 100 and one, and those of ${k}_{\mathrm{off}}$ were one and one.

**Figure 3.**Comparison of the computed and experimental [14] ${k}_{on}$ rates for bivalent Man(1,3,5)-5. The ${k}_{on}$ rates were 1000, 1000, and 1000 and for ${k}_{\mathrm{off}}$ were 1, 1, and 1.

**Figure 4.**Comparison of the computed and experimental [14] ${k}_{on}$ rates for pentavalent Man(1,3,5,7,9)S-9; see the Supplementary Information (SI) for additional information on experimental data. The ${k}_{on}$ rates were 1, 1, 10,000, and 10 and for ${k}_{\mathrm{off}}$ were 10,000, 1, 1, and 1.

Valency (N) | Structure | Compound Name |
---|---|---|

bivalent (2) | Man(1,5)-5 | |

trivalent (3) | Man(1,3,5)-5 | |

pentavalent (5) | Man(1,3,5,7,9)S-9 |

**Table 2.**Order of the seven possible states for bivalent bindings: as described in Section 2, $\left[L{R}^{1}\right]$ comprises the sum of the conformational states II, ..., V, and $\left[L{R}^{2}\right]$ comprises the sum of states VI and VII.

State Number | Combination | Number of Bindings |
---|---|---|

I | $\left[\begin{array}{c}1-\\ 2-\end{array}\right]$ | 0 |

II | $\left[\begin{array}{c}1A\\ 2-\end{array}\right]$ | 1 |

III | $\left[\begin{array}{c}1B\\ 2-\end{array}\right]$ | 1 |

IV | $\left[\begin{array}{c}1-\\ 2B\end{array}\right]$ | 1 |

V | $\left[\begin{array}{c}1-\\ 2A\end{array}\right]$ | 1 |

VI | $\left[\begin{array}{c}1A\\ 2B\end{array}\right]$ | 2 |

VII | $\left[\begin{array}{c}1B\\ 2A\end{array}\right]$ | 2 |

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**MDPI and ACS Style**

Erlekam, F.; Igde, S.; Röblitz, S.; Hartmann, L.; Weber, M.
Modeling of Multivalent Ligand-Receptor Binding Measured by kinITC. *Computation* **2019**, *7*, 46.
https://doi.org/10.3390/computation7030046

**AMA Style**

Erlekam F, Igde S, Röblitz S, Hartmann L, Weber M.
Modeling of Multivalent Ligand-Receptor Binding Measured by kinITC. *Computation*. 2019; 7(3):46.
https://doi.org/10.3390/computation7030046

**Chicago/Turabian Style**

Erlekam, Franziska, Sinaida Igde, Susanna Röblitz, Laura Hartmann, and Marcus Weber.
2019. "Modeling of Multivalent Ligand-Receptor Binding Measured by kinITC" *Computation* 7, no. 3: 46.
https://doi.org/10.3390/computation7030046