# Exploring the Limits of the Geometric Copolymerization Model

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Geometric Copolymerization Model

#### 2.2. Monte Carlo Reaction Schemes

^{•}, and a polymer chain ending with X as ∼ X, where X can be one of the monomers A or B, or initiator I. Two types of reactions, initiation and propagation reactions were modeled:

#### 2.3. Datasets and Monte Carlo Parameters

#### 2.4. Log Likelihood Ratio

## 3. Results and Discussion

#### 3.1. Parameter Space Reduction

#### 3.2. Parameter Optimization

#### 3.3. Beyond Living Polymerization

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MS | Mass spectrometry |

ODE | Ordinary differential equation |

LP | Living polymerization |

RLP | Reversible living polymerization |

FRP | Free radical polymerization |

CRP | Controlled radical polymerization |

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**Figure 1.**All possible transitions for copolymer chain lengths $\le 2$. For example, the transition from the initiator state I to the state ${M}_{2,0}^{\mathrm{A}}$ (copolymer chains having two A-monomers and ending in A) corresponds to adding the sequence AA. Note that transitions that add more than two monomers correspond to multiple events. For example, the transition of I to ${M}_{2,1}^{\mathrm{A}}$ corresponds to adding the two sequences BAA and ABA.

**Figure 2.**Filled contours: copolymer fingerprints of $D{P}_{n}=25$ computed by Monte Carlo simulations with no (

**left**) and high applied noise (

**right**). Contours: fingerprints computed by the geometric model using the best parameters computed by the optimization algorithms for each of the fingerprint-generating functions (direct, spline, and ODE).

**Figure 3.**Log likelihood ratios of the results computed by the optimization algorithms as a function of noise. The ratios are averaged over all three algorithms for each fingerprint-generating function (direct, spline, ODE). The higher the ratios, the better the observed data is “explained” by the identified model parameterizations. If the ratio is below zero, the null model achieves a higher likelihood than the geometric model with the given parameterization.

**Figure 4.**Running times of the optimizations averaged over all datasets with degree of polymerization $D{P}_{n}$ = 3, 25, and 45 for each fingerprint-generating function (direct, spline, ODE).

**Figure 5.**Filled contours: copolymer fingerprints of the Monte Carlo simulations of controlled radical polymerization (CRP,

**left**), free radical polymerization (FRP,

**center**), and reversible living polymerization (RLP,

**right**) with the highest used termination and propagation reaction rates of 0.1. Contours: fingerprints computed by the model with the best parameters resulting from the optimizations using the ODE fingerprint-generating function.

**Figure 6.**Log likelihoods (

**left**) and log likelihood ratios (

**right**) of the results from the optimizations using the ODE fingerprint-generating function for the controlled radical polymerization (CRP), free radical polymerization (FRP), and reversible living polymerization (RLP) as a function of termination and depropagation rates.

**Table 1.**Overview of the modeled reactions types for the living polymerization (LP), reversible living polymerization (RLP), free radical polymerization (FRP), and controlled radical polymerization (CRP).

Reaction Type | LP | RLP | FRP | CRP |
---|---|---|---|---|

Initiation | × | × | × | × |

Propagation | × | × | × | × |

Depropagation | × | |||

Termination (Recomb. & Disprop.) | × | × | ||

Initiator Decomposition | × | |||

(De-)Activation | × |

**Table 2.**Overview of the top three optimization algorithms for each fingerprint-generating function, selected based on Supplementary Figures S1–S3. We ranked the results of the algorithms for each dataset based on the log likelihood ratios and counted the ranks.

Algorithm | #Ranks | |||
---|---|---|---|---|

1st | 2nd | 3rd | ||

Direct | Cloning, Information Gain, Aging (CLI) [28] | 4 | 5 | 7 |

Probabilistic Crowding (PC) [29] | 6 | 5 | 5 | |

Restricted Tournament Selection (RTS) [30] | 6 | 6 | 4 | |

Spline | Covariance Matrix Adaptation Evolution Strategy (CMAES) [31] | 3 | 6 | 7 |

Deterministic Crowding (DC) [32] | 3 | 8 | 5 | |

Generalized Extremal Optimization (GEO) [33] | 10 | 2 | 4 | |

ODE | Genetic Algorithm (GA) [34] | 5 | 9 | 2 |

Generalized Extremal Optimization (GEO) [33] | 8 | 0 | 8 | |

Mutation Hill Climber (MHC) [35] | 3 | 7 | 6 |

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**MDPI and ACS Style**

Engler, M.S.; Scheubert, K.; Schubert, U.S.; Böcker, S. Exploring the Limits of the Geometric Copolymerization Model. *Polymers* **2017**, *9*, 101.
https://doi.org/10.3390/polym9030101

**AMA Style**

Engler MS, Scheubert K, Schubert US, Böcker S. Exploring the Limits of the Geometric Copolymerization Model. *Polymers*. 2017; 9(3):101.
https://doi.org/10.3390/polym9030101

**Chicago/Turabian Style**

Engler, Martin S., Kerstin Scheubert, Ulrich S. Schubert, and Sebastian Böcker. 2017. "Exploring the Limits of the Geometric Copolymerization Model" *Polymers* 9, no. 3: 101.
https://doi.org/10.3390/polym9030101