# Hierarchical Coarse-Grained Strategy for Macromolecular Self-Assembly: Application to Hepatitis B Virus-Like Particles

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Materials

#### 2.1. Framework Overview

#### 2.2. Reference Structure HBcAg${}_{2}$ Dimer

#### 2.3. Intermolecular Interaction Potential

#### 2.3.1. Molecular Dynamics Simulations

#### 2.3.2. Spatial Descriptors

#### 2.3.3. Multivariant Estimation using Universal Kriging

#### 2.3.4. Molecular Collisions

#### 2.3.5. 2D Example of Kriging and Sampling Algorithm

#### 2.3.6. Biased MD Sampling and Insertion of Empirical Data

#### 2.3.7. Summary and Implementation

- 1.
**Molecular reference structure**of all involved molecules from, e.g., a protein data bank. This reference structure has to be the same as used for the parameterization of the diffusion model [65].- 2.
**Initial interaction potential sampling**using MD and the outlined sampling methodology. (For large interaction spaces proximity sampling might be required for sufficient correlation data.)- 3.
**Trend fitting**in a lower dimensional interaction space of ${\delta}_{m}$ for all potential components.- 4.
**Correlation analysis and sectional variogram fitting**of trend-compensated residual R for all potential components. Identification of potential components with reasonable spatial continuity besides trend (only fulfilled by A-B potential).- 5.
**Grid design**based on interaction distance and memory size constraints.- 6.
**Universal Kriging**for multivariant estimation of interaction potential component residual R.- 7.
**Molecular collision**accounting as a function of molecular overlap and flexibility with increasing interaction potential.- 8.
**Iterative refinement**of field estimate based on estimation variance and extrema (potential minima/maxima, gradient maxima) localization and specification.

#### 2.4. Diffusion Model

#### 2.5. Usage and Implementation within the Molecular Discrete Element Method

#### 2.5.1. Simulation Procedure

#### VLP Binding Agreement and Stability (SP1)

#### VLP Capsid Stability (SP2)

#### VLP Self-Assembly (SP3)

#### 2.5.2. Postprocessing

_{SAS,asymp}(s in Equation (S5)) and time constant ${\tau}_{\mathrm{SAS}}$ (r in Equation (S5))). Additionally, the differentiation between structured and unstructured contacts is used to quantify the assembly quality by the average number of structured and unstructured connections per dimer ${\xi}_{\mathrm{struc}}$ and ${\xi}_{\mathrm{unstruc}}$, respectively. A perfect 120 mer capsid of HBcAg${}_{2}$ is characterized by ${\xi}_{\mathrm{struc}}=4$ and ${\xi}_{\mathrm{unstruc}}$ near zero. The fraction of structured contacts out of all contacts is termed ${\Phi}_{\mathrm{struc}}={\xi}_{\mathrm{struc}}/({\xi}_{\mathrm{struc}}+{\xi}_{\mathrm{unstruc}})$. In addition to a global application, these measures can also be used on a per SAS or per dimer basis.

## 3. Results and Discussion

#### 3.1. HBcAg${}_{2}$ Interaction Potential and VLP Stability

#### 3.1.1. Pure MD-Based Interaction Potential

#### MD Data

#### Convergence

#### Resulting Field

#### 3.1.2. Biased MD Interaction Potential

#### 3.1.3. MD-Based Interaction Potential with Empirical Data

_{bind,center}= −1400 kJ/mol, U

_{bind,outer}= −1000 kJ/mol, and ${r}_{\mathrm{bind}}=1.0$ nm. As it can be seen in Figure 10, in comparison to the pure MD-based potential (Figure 8) the inserted virtual data points create new minima at the binding locations, but do not affect the remaining overall potential. This is important as remaining characteristics, such as the potential barrier at ${\delta}_{m}\phantom{\rule{0.166667em}{0ex}}\approx 1.5$ nm, are kept and consequently knowledge from MD and empirical data are merged.

#### 3.2. VLP Self-Assembly

#### 3.2.1. Assembly Properties

#### 3.2.2. Assembly Kinetics

#### 3.2.3. Assembly Pathways

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

_{2}reference structure.

## Conflicts of Interest

## Abbreviations

AA-MD | All-Atom Molecular Dynamics |

ANN | Artificial Neural Networks |

BD | Brownian Dynamics |

BLUE | Best Linear Unbiased Estimate |

CG-MD | Coarse-Grained Molecular Dynamics |

CPU | Central Processing Unit |

DEM | Discrete Element Method |

DOF | Degree of Freedom |

DPD | Dissipative Particle Dynamics |

FF | Force Field |

GPR | Gaussian Process Regression |

GPU | Graphics Processing Unit |

HBcAg | Hepatitis B Core Antigen |

HBcAg${}_{2}$ | HBcAg Dimer |

HBV | Hepatitis B Virus |

LD | Langevin Dynamics |

MDEM | Molecular Discrete Element Method |

MD | Molecular Dynamics |

NPT | Isothermal-Isobaric Ensemble |

PBC | Periodic Boundary Conditions |

PDB | Protein Data Bank |

PME | Particle Mesh Ewald |

PW | Polarizable Water |

QM/MM | Quantum Mechanics/Molecular Mechanics |

RMS | Root-Mean-Square |

RMSD | Root-Mean-Square Distance |

SAS | Self-Assembled Structure |

SI | Supplementary Information |

SPX | Simulation Procedure X |

SVD | Singular Value Decomposition |

UK | Universal Kriging |

VLP | Virus-Like Particles |

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**Figure 2.**Effect of macromolecule abstraction as anisotropic beads with interaction potential on computational complexity (n is number of atoms, neglecting solvent and ions). Note that a single interaction of HBcAg${}_{2}$ is equivalent to $n=9432$ and further increased by the solvent atoms ($n\approx {10}^{5}$).

**Figure 3.**Framework overview with a focus on intermolecular interaction. For details on the diffusion model see ref. [65]. Blue indicates MD simulations, greyed-out regions are optional components, dotted lines indicated usage of related functionality (i.e., MD simulation is performed), dashed lines indicate information exchange.

**Figure 4.**2D Universal Kriging example after 8 iterations with 10 samples per iteration and 20 initial samples (100 samples in total). For variogram determination the entire truth field was provided to ensure sufficient statistics. Test field possesses no units.

**Figure 5.**2D Universal Kriging example after initial random sampling (20 samples), two resampling iterations of 10 samples each (40 samples total), and six resampling iterations of 10 samples each (80 samples total). Test field possesses no units.

**Figure 6.**Visualization of trimer equilibrium conformations for various interaction potentials after equilibration (SP1): capsid reference conformation (

**a**), pure MD-based potential (

**b**) (Section 3.1.1), biased MD-based potential (

**c**) (Section 3.1.2), with empirical data (

**d**) (Section 3.1.3).

**Figure 7.**Convergence of the iterative resampling procedure for potential changes (

**left**, ${U}_{i}-{U}_{i-1}$) and variance development (

**right**). Figure adapted with permission from Ref. [66]. Copyright 2022, Springer.

**Figure 8.**Visualizations of the potential field based on pure MD-based sampling strategy. (

**a**) Grid average and standard deviation binned over minimum distance. (

**b**) X-Y cross-section minimum over all remaining dimensions. (

**c**) 3D minimum over orientations. (

**d**) 3D mean over orientations. (

**a**,

**b**) adapted from with permission from Ref. [66]. Copyright 2022, Springer.

**Figure 9.**Visualization of two interacting HBcAg${}_{2}$ obtained from the biased MD simulation with lowest potential A-B (side view left, top view right).

**Figure 10.**Visualizations of the potential field based on MD with inserted empirical data. (

**a**) Average and standard deviation binned over minimum distance in grid. (

**b**) X-Y cross-section minimum over all remaining dimensions. (

**c**) 3D minimum over orientations. (

**d**) 3D mean over orientations.

**Figure 11.**Self-assembly of VLPs during 5 ms simulated using MDEM with SP3 simulation protocol (box of 1 μm

^{3}, protein concentration of 5 μM). The size of assemblies formed (${N}_{\mathrm{SAS}}$) is depicted using the designed color scheme and the backbone carbon atom representation.

**Figure 12.**Visualizations of VLP self-assembly using simulation protocol SP3. Colors indicate structure size by number of dimers (${N}_{\mathrm{SAS}}$) and backbone carbon atoms are visualized. Structure 50E exceeds scale with 221 and red structure at top left of (

**d**) contains 233 dimers.

**Figure 13.**Magnified capsids marked in Figure 12 using visualization of dimers as spheres with orientation arrows (x-axis red, y green, z blue). Numbers behind identifier indicate ${N}_{\mathrm{SAS}}$ of structure. Colors match original coloring scheme according to ${N}_{\mathrm{SAS}}$ in Figure 12.

**Figure 14.**Final distribution of numbered size versus diameter of gyration (averaged over last ten saving steps).

**Figure 15.**Histogram of self-assembled structures by number of constituting HBcAg

_{2}(N

_{SAS}) over time. τ

_{SAS}for concentrations in increasing order: 2.91, 1.55, 0.39, 0.21 ms. N

_{SAS,asymp}for concentrations in increasing order: 83.9, 90.7, 103.8, 109.5.

**Figure 16.**Self-assembly by bi-directional transitions between size classes normalized by total number of dimers (major ticks represent unit arrow thickness, i.e., every dimer makes this transition on average). Starting at class 10, the size denotes the class range between −4 to +5 relative to the noted value; 206 incorporates all sizes equal to or larger than 206. Colors provide contrast only. See Section 2.5.2 for further specifications.

**Table 1.**Binding locations between HBcAg${}_{2}$ from the reference capsid (see Figure 1). Positions with respect to the body frame of the reference (molecule A) on x-, y-, z-axis are in nanometer, while angles $\alpha ,\beta ,\gamma $ are in radian.

# | x | y | z | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ |
---|---|---|---|---|---|---|

1 | −2.74 | −0.74 | −3.10 | −0.48 | 0.98 | −0.32 |

2 | 1.47 | −0.91 | −4.14 | −0.88 | −1.05 | 0.67 |

3 | −3.01 | −0.70 | −3.08 | −2.72 | −1.05 | 3.03 |

4 | −0.65 | −0.77 | 4.25 | 2.72 | 0.92 | 2.76 |

**Table 2.**Anisotropic translational (${D}_{\mathrm{t}}$) and rotational (${D}_{\mathrm{r}}$) diffusion coefficients for HBcAg${}_{2}$ at 293 K and 150 mM NaCl used for MDEM (marked in light orange in Figure 3).

${\mathit{D}}_{\mathbf{t}}$ [$\mathsf{\mu}$m${}^{2}$ s${}^{-1}$] | ${\mathit{D}}_{\mathbf{r}}$ [Mrad${}^{2}$ s${}^{-1}$] | ||||
---|---|---|---|---|---|

x | y | z | $\mathbf{\alpha}$ | $\mathbf{\beta}$ | $\mathbf{\gamma}$ |

87.69 | 72.27 | 71.48 | 12.05 | 7.46 | 7.00 |

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**MDPI and ACS Style**

Depta, P.N.; Dosta, M.; Wenzel, W.; Kozlowska, M.; Heinrich, S.
Hierarchical Coarse-Grained Strategy for Macromolecular Self-Assembly: Application to Hepatitis B Virus-Like Particles. *Int. J. Mol. Sci.* **2022**, *23*, 14699.
https://doi.org/10.3390/ijms232314699

**AMA Style**

Depta PN, Dosta M, Wenzel W, Kozlowska M, Heinrich S.
Hierarchical Coarse-Grained Strategy for Macromolecular Self-Assembly: Application to Hepatitis B Virus-Like Particles. *International Journal of Molecular Sciences*. 2022; 23(23):14699.
https://doi.org/10.3390/ijms232314699

**Chicago/Turabian Style**

Depta, Philipp Nicolas, Maksym Dosta, Wolfgang Wenzel, Mariana Kozlowska, and Stefan Heinrich.
2022. "Hierarchical Coarse-Grained Strategy for Macromolecular Self-Assembly: Application to Hepatitis B Virus-Like Particles" *International Journal of Molecular Sciences* 23, no. 23: 14699.
https://doi.org/10.3390/ijms232314699