# DNA-Guided Assembly for Fibril Proteins

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### DNA-Guided Assembly of Nanocellulose Meshes

## 3. Results

#### 3.1. Coarse-Grained Computational Modeling of the DNA-Fibril Dynamical System

#### 3.2. Tailored Stochastic Modeling of Assembly Formation and Dynamics

zone 1 | zone 2 | zone 3 | zone 4 |

zone 5 | zone 6 | zone 7 | zone 8 |

zone 9 | zone 10 | zone 11 | zone 12 |

zone 13 | zone 14 | zone 15 | zone 16 |

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Table of Reference Distributions

**Table A1.**List of the fifteen reference distributions with positive support used to test the goodness of fit for the empirical distribution of the trimmed mean gap size.

Distribution | Acronym | Parameters | |
---|---|---|---|

Beta | beta | $f\left(x\right)=\frac{\Gamma (\alpha +\beta )}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{x}^{\alpha -1}{(1-x)}^{\beta -1}$ | $\alpha ,\beta >0$ |

Burr | burr | $f\left(x\right)=\frac{\alpha k{\left(\frac{x}{\beta}\right)}^{\alpha -1}}{\beta {\left[1+{\left(\frac{x}{\beta}\right)}^{\alpha}\right]}^{k+1}}$ | $x\ge 0$, $\alpha ,\beta ,k>0$ |

Dagum | dagum1 | $f\left(x\right)=\frac{\alpha k{\left(\frac{x}{\beta}\right)}^{\alpha k-1}}{\beta {\left[1+{\left(\frac{x}{\beta}\right)}^{\alpha}\right]}^{k+1}}$ | $x\ge 0$, $\alpha ,\beta ,k>0$ |

Gumbel | gumbel | $f\left(x\right)=\frac{1}{\sigma}{e}^{-(\frac{x-\mu}{\sigma}+{e}^{-\frac{x-\mu}{\sigma}})}$ | $\mu \in \mathbb{R}$, $\sigma >0$ |

Logistic | logis | $f\left(x\right)=\frac{{e}^{-\frac{x-\mu}{\sigma}}}{\sigma {\left(1+{e}^{-\frac{x-\mu}{\sigma}}\right)}^{2}}$ | $\mu \in \mathbb{R}$, $\sigma >0$ |

Log-Logistic | llogis | $f\left(x\right)=\frac{\beta {\left(\frac{x}{\alpha}\right)}^{\beta -1}}{\alpha {\left[1+{\left(\frac{x}{\alpha}\right)}^{\beta}\right]}^{2}}$ | $\alpha ,\beta >0$ |

Gamma | gamma | $f\left(x\right)=\frac{1}{\Gamma \left(\alpha \right){\beta}^{\alpha}}{x}^{\alpha -1}{e}^{-\frac{x}{\beta}}$ | $\alpha ,\beta >0$ |

Pareto 1 | pareto1 | $f\left(x\right)=\alpha \frac{{\beta}^{\alpha}}{{x}^{\alpha +1}}$ | $x>\beta $, $\alpha ,\beta >0$ |

Weibull | weibull | $f\left(x\right)=\frac{\alpha}{\beta}{\left(\frac{x}{\beta}\right)}^{\alpha -1}{e}^{-{\left(\frac{x}{\beta}\right)}^{\alpha}},\phantom{\rule{0.166667em}{0ex}}x>0$ | $\alpha ,\beta >0$ |

Inverse Weibull | invweibull | $f\left(x\right)=\alpha {\left(\frac{\beta}{x}\right)}^{\alpha}\frac{{e}^{-{\left(\frac{\beta}{x}\right)}^{\alpha}}}{x},\phantom{\rule{0.166667em}{0ex}}x>0$ | $\alpha ,\beta >0$ |

Normal | norm | $f\left(x\right)=\frac{1}{\sqrt{2\pi}\sigma}{e}^{-\frac{{(x-\mu )}^{2}}{2{\sigma}^{2}}}$ | $\mu \in \mathbb{R}$, $\sigma >0$ |

Log-Normal | lnorm | $f\left(x\right)=\frac{1}{x\sigma \sqrt{2\pi}}{e}^{-\frac{{(logx-\mu )}^{2}}{2{\sigma}^{2}}}$ | $\mu ,\sigma >0$ |

Generalized Extreme Value | gev | $f\left(x\right)=\left\{\begin{array}{cc}\frac{1}{\sigma}{e}^{-{(1+\xi \frac{x-\mu}{\sigma})}^{-\frac{1}{\xi}}}{(1+\xi \frac{x-\mu}{\sigma})}^{-1-\frac{1}{\xi}},\hfill & \xi \ne 0\hfill \\ \frac{1}{\sigma}{e}^{-\frac{x-\mu}{\sigma}-{e}^{-\frac{x-\mu}{\sigma}}},\hfill & \xi =0\hfill \end{array}\right.$ | $\left\{\begin{array}{cc}1+\xi \frac{x-\mu}{\sigma}>0,\hfill & \xi \ne 0\hfill \\ x\in \mathbb{R},\hfill & \xi =0\hfill \end{array}\right.$; $\mu ,\xi \in \mathbb{R}$, $\sigma >0$ |

Nakagami | naka | $f\left(x\right)=\frac{2{m}^{m}}{\Gamma \left(m\right){\Omega}^{m}}{x}^{2m-1}{e}^{-\frac{m}{\Omega}{x}^{2}}$ | $x\ge 0$, $m\ge 0.5$, $\Omega >0$ |

Rayleigh | rayleigh | $f\left(x\right)=\frac{x}{{\sigma}^{2}}{e}^{-\frac{{x}^{2}}{2{\sigma}^{2}}}$ | $x\ge 0$, $\sigma >0$ |

#### Appendix A.2. Table of Top Ranked Fitted Distributions

**Table A2.**Top five fitted distributions according to Anderson–Darling statistic on each of the four central zones and the entire assembly differentiated by the initial distribution of the rod length.

Distribution | Zones | Rank_Dist_1 | Rank_Dist_2 | Rank_Dist_3 | Rank_Dist_4 | Rank_Dist_5 |
---|---|---|---|---|---|---|

Uniform | z6 | gumbel | gev | lnorm | beta | dagum1 |

Uniform | z7 | dagum1 | burr | gumbel | llogis | lnorm |

Uniform | z10 | burr | dagum1 | gev | gumbel | llogis |

Uniform | z11 | gumbel | dagum1 | lnorm | burr | llogis |

Uniform | all | gamma | lnorm | nakagami | burr | normal |

Geometric (RS) | z6 | burr | llogis | dagum1 | gumbel | lnorm |

Geometric (RS) | z7 | gev | dagum1 | burr | gumbel | llogis |

Geometric (RS) | z10 | lnorm | gev | dagum1 | llogis | burr |

Geometric (RS) | z11 | dagum1 | gumbel | gev | burr | llogis |

Geometric (RS) | all | burr | dagum1 | gumbel | llogis | invweibull |

Geometric (LS) | z6 | gev | gumbel | dagum1 | lnorm | burr |

Geometric (LS) | z7 | gev | lnorm | gumbel | gamma | dagum1 |

Geometric (LS) | z10 | burr | gumbel | dagum1 | llogis | lnorm |

Geometric (LS) | z11 | llogis | burr | dagum1 | lnorm | gamma |

Geometric (LS) | all | gamma | lnorm | nakagami | normal | burr |

Binomial | z6 | dagum1 | burr | llogis | gumbel | gev |

Binomial | z7 | gumbel | gev | lnorm | dagum1 | burr |

Binomial | z10 | gumbel | gev | lnorm | dagum1 | burr |

Binomial | z11 | dagum1 | gumbel | lnorm | burr | llogis |

Binomial | all | gumbel | dagum1 | invweibull | burr | lnorm |

Beta-binomial | z6 | gev | lnorm | gamma | dagum1 | llogis |

Beta-binomial | z7 | burr | dagum1 | gev | invweibull | gumbel |

Beta-binomial | z10 | burr | dagum1 | gumbel | llogis | lnorm |

Beta-binomial | z11 | gev | lnorm | gumbel | dagum1 | llogis |

Beta-binomial | all | gev | lnorm | burr | llogis | dagum1 |

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**Figure 1.**DNA origami functionalized by orthogonally aligned specific aptamers, placed on opposite sides, and connected to rod-like structures.

**Figure 2.**Possible origami positioning along a fibril: (

**a**) both origamis are positioned on the same side of the fibril, in which case a minimum (one-origami wide) inter-fibril gap has to be present, and (

**b**) the origamis are positioned on opposite sides of the fibril, in which case, in between two parallel fibrils, there could be almost no space at all.

**Figure 3.**Results of the coarse-grained model: (

**a**) dynamics of the assembly averaged gap size per $O/R$ and rod length variation. Each data point is a median of approximately 30 independent runs. The horizontal axes represents the $O/R$ parameter, and the vertical one is the average size of the hole. (

**b**) The heat map of the 2D matrix representations of the R object interconnections generated within one run instance of the coarse-grained model for the case when $O/R=0.1$ and $l=10$. All assemblies contain 1000 rods.

**Figure 4.**The five initial rod lengths distributions. From left to right: uniform, truncated right skewed geometric (geometric (RS)), truncated left skewed geometric (geometric (LS)), binomial, and beta-binomial.

**Figure 5.**Loess tendency of the trimmed mean gap size for all $O/R$ ratios differentiated by initial distribution of the rod lengths and all models characterized by the minimum gap d allowed in between two consecutive docking positions.

**Figure 6.**Heat maps of 2D representations of the mesh structure for each model and each initial rod length distribution in the case of $O/R=5$ colored with respect to the intersection degree of R objects.

**Figure 7.**Time-dependent evolution of the trimmed mean gap size for the mesh assembly in the case of $d=2$ and $O/R=5$, when the dynamics of the system is frozen once the total number of rods in the structure reaches 1000. The evolution is spitted for each initial distribution of the rod lengths and each of the four central zones.

**Figure 8.**Histogram representation of the trimmed mean gap size for each of the four central zones and the entire assembly along with the top three corresponding fitted distributions.

**Table 1.**Trimmed mean gap size values of 100 independent runs of the in silico experiments in the case of $d=2$ and $O/R=5$ evaluated for each initial rod length distribution, for each of the four central zones, as well as for the entire assembly.

Distribution | z6 | z7 | z10 | z11 | All |
---|---|---|---|---|---|

Uniform | 3.2675 | 3.4093 | 3.2805 | 3.5288 | 3.5151 |

Geometric (RS) | 2.7425 | 3.0276 | 2.1810 | 2.5424 | 2.6354 |

Geometric (LS) | 3.5283 | 3.3386 | 3.4199 | 3.3275 | 3.6453 |

Binomial | 3.1354 | 2.9235 | 3.1980 | 3.0020 | 3.2363 |

Beta-binomial | 3.3983 | 3.4945 | 3.5101 | 3.4127 | 3.5035 |

**Table 2.**The most representative fitted distributions for each initial rod length distribution evaluated by employing a linear weighting scheme on the top five fitted distributions for all four central zones and for the entire structure, as presented in Table A2.

Distribution | Model | O/R Ratio | Fitted Distribution |
---|---|---|---|

Beta-binomial | 2 | 5 | gev |

Binomial | 2 | 5 | gumbel |

Geometric (LS) | 2 | 5 | lnorm |

Geometric (RS) | 2 | 5 | dagum1 |

Uniform | 2 | 5 | gumbel |

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Amărioarei, A.; Spencer, F.; Barad, G.; Gheorghe, A.-M.; Iţcuş, C.; Tuşa, I.; Prelipcean, A.-M.; Păun, A.; Păun, M.; Rodriguez-Paton, A.;
et al. DNA-Guided Assembly for Fibril Proteins. *Mathematics* **2021**, *9*, 404.
https://doi.org/10.3390/math9040404

**AMA Style**

Amărioarei A, Spencer F, Barad G, Gheorghe A-M, Iţcuş C, Tuşa I, Prelipcean A-M, Păun A, Păun M, Rodriguez-Paton A,
et al. DNA-Guided Assembly for Fibril Proteins. *Mathematics*. 2021; 9(4):404.
https://doi.org/10.3390/math9040404

**Chicago/Turabian Style**

Amărioarei, Alexandru, Frankie Spencer, Gefry Barad, Ana-Maria Gheorghe, Corina Iţcuş, Iris Tuşa, Ana-Maria Prelipcean, Andrei Păun, Mihaela Păun, Alfonso Rodriguez-Paton,
and et al. 2021. "DNA-Guided Assembly for Fibril Proteins" *Mathematics* 9, no. 4: 404.
https://doi.org/10.3390/math9040404