# IGLOO: An Iterative Global Exploration and Local Optimization Algorithm to Find Diverse Low-Energy Conformations of Flexible Molecules

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Global Optimization Algorithms

#### 2.1.1. Basin-Hopping (BH)

`StoppingCriteria`is satisfied (see Section 2.2): (line 3) a relatively large-amplitude perturbation is applied to the current conformation q, aiming to escape from local basins; (line 4) a new conformation ${q}_{\mathrm{new}}$ is generated by local energy minimization from the perturbed conformation ${q}^{\prime}$; (line 5) the transition to the new local minimum is accepted or rejected based on the usual

`MetropolisTest`[38]. Following this stochastic acceptance test, the new conformation ${q}_{\mathrm{new}}$ is accepted if its energy is lower than or equal to the one of the previous conformation q. Otherwise, it is accepted with a probability that decreases exponentially with the positive energy difference between the two configurations. The implementation of BH used for the comparative analysis in this work follows a multistart procedure, as presented in Algorithm 2. Aiming to cover the space more globally, it performs several rounds of the monotonic-BH algorithm, starting from randomly sampled conformations generated by the function

`SampleRoot`. Note that we successfully applied this implementation of BH in previous work [39].

Algorithm 1: Monotonic-Basin-Hopping |

Algorithm 2: Multistart-Basin-Hopping |

#### 2.1.2. Hybrid-BH-RRT (HYBRID)

`SampleRoot`. The number of initial configurations is defined in the parameters P and tested inside the function

`MaxNumberRoots`. Then, the algorithm follows the main steps of RRT: (line 6) a conformation ${q}_{\mathrm{rand}}$ is randomly sampled in the whole conformational space; (line 7) the nearest conformation ${q}_{\mathrm{near}}$ to ${q}_{\mathrm{rand}}$ among the already explored minima contained in $\mathcal{T}$ is selected; (line 8) the basic tree expansion step in the original RRT algorithm is replaced by an execution of the monotonic-BH algorithm starting from the selected ${q}_{\mathrm{near}}$. Since the probability of a conformation ${q}_{\mathrm{near}}$ to be selected for expansion is proportional to the volume of its associated Voronoi cell (i.e., the subset of points in the conformational space closer to ${q}_{\mathrm{near}}$ than to any other conformation contained in the tree), the tree tends to grow towards the least explored regions of the space. Note that the parameters used for the monotonic-BH within the HYBRID algorithm should ensure that the number of iterations of perturbation and local minimization stages is relatively small compared to the number of iterations within the multistart-BH, so that the efficiency of the exploration via the coupling of the BH with the RRT algorithm will be enhanced.

#### 2.1.3. Iterative Global Exploration and Local Optimization (IGLOO)

`TRRT-Exploration`(line 7), which implements a multi-tree variant of the RRT algorithm that considers an energy threshold as a constraint for the exploration. This algorithm will be described in Section 2.2. Then, local energy minimization is performed from each of the conformations corresponding to the nodes of these trees (lines 8–10). The resulting set of local minima is filtered to eliminate conformations that are too similar, and to reduce the number of local minima taken into account for the following steps (line 11). Note that although the

`TRRT-Exploration`generates conformations with a good dispersion in $\mathcal{C}$, the subsequent local minimization usually produces clusters of conformations. The

`UpdateParameters`function (line 12) adapts the energy threshold, the exploration step size, and the maximum number of roots for the next iteration of the algorithm. The three main steps of IGLOO—global exploration, local minimization, and filtering—are iterated until one of the conditions in the function

`StoppingCriteria`is satisfied. To illustrate the behavior of IGLOO, Figure A1 provided in the appendix section presents results obtained along several iterations of the algorithm to find low-energy conformations of a simple system that has frequently been used in theoretical works: the alanine dipeptide.

Algorithm 3: Hybrid-BH-RRT |

Algorithm 4: IGLOO |

#### 2.2. Implementation Details

`SampleRoot`: This function randomly samples a conformation q from the domain defined for the conformational space variables $\mathcal{C}$. This conformation is locally minimized using the

`LocalMinimization`function described below. If the set $\mathcal{T}$ already contains other local minima, the distance between the new sample and the previous ones is computed, and the new sample is rejected if the minimum value of these distances is below a given threshold defined in the set of parameters P. In that case, the process is repeated until the new sample is sufficiently far from all the others. This strategy is inspired from the Poisson disk sampling process [40], and aims to guarantee a good dispersion of the points used to initialize the exploration.

`LargeAmplitudePerturbation`: As is generally done in Monte Carlo-type methods, this “random move” is not applied to all the variables simultaneously, but to a subset of them at each iteration. In the implementation used in this work, the

`LargeAmplitude- Perturbation`is applied to a single variable at a time (we tested random moves applied to several variables simultaneously but the results did not improve for the test systems presented below). More precisely, these large-amplitude moves are applied only to a subset of the conformational space variables considered to be the most important ones, whereas the

`LocalMinimization`function, explained below, operates on all the variables. Section 2.3 will explain the main and secondary variables for the type of molecules considered in this work. The bounds for the amplitude of the perturbation (i.e., the minimum and maximum step size) are defined in the set of parameters P. A random value between these bounds is sampled at each iteration.

`LocalMinimization`: The local minimization is based on a simple Monte Carlo method at very low temperature. Although this type of stochastic algorithm is in general less computationally efficient than deterministic gradient-based approaches, it is also less sensitive to local traps. Moreover, it does not require the computation of the derivatives of the objective function, which can be difficult and expensive in some cases. The algorithm iteratively applies small random perturbations to the current conformation, which are accepted or rejected based on the Metropolis test (as for the BH algorithm). As the temperature parameter used in this test is very low, the probability of accepting a locally perturbed conformation that increases energy is also very low, but not zero, allowing minimization to escape from small energy basins.

`StoppingCriteria`function explained below, the considered criteria are a maximum number of iterations, a limited computing time, or estimated convergence, based on a maximum number of consecutive rejections of Metropolis test.

`TRRT-Exploration`: The exploration algorithm implemented in this work combines ideas of the Transition-based RRT (TRRT) [34] and the threshold algorithm [8]. The pseudo-code is presented in Algorithm 5. The algorithm constructs several exploration trees starting from the given set of roots, aiming to cover the reachable regions of the conformational space. These trees are constructed by iterating the following steps:

- (line 3) One of the trees ${\mathcal{T}}_{i}$ is randomly selected.
- (line 4) A conformation ${q}_{\mathrm{rand}}$ is randomly sampled in a domain containing the selected tree. In practice, a convex polytope in $\mathcal{C}$ is computed for each tree, and updated when the tree grows. This polytope is enlarged in proportion to the size of the exploration step, and a conformation is randomly sampled within it.
- (line 5) The nearest conformation to ${q}_{\mathrm{rand}}$, ${q}_{\mathrm{near}}$, stored in the nodes of the selected tree ${\mathcal{T}}_{i}$ is chosen for extension.
- (line 6) A new conformation ${q}_{\mathrm{new}}$ is created by extending ${q}_{\mathrm{near}}$ in the direction of ${q}_{\mathrm{rand}}$. This extension is implemented by moving along the interpolated path between ${q}_{\mathrm{near}}$ and ${q}_{\mathrm{rand}}$ of a given step size, provided in the set of parameters P.
- (lines 7–8) A new node containing ${q}_{\mathrm{new}}$ will be added to the corresponding tree if it satisfies a set of constraints. More precisely, the function
`ValidConformation`tests the absence of clashes between non-bonded atoms and that the potential energy of the conformation is below a given threshold (defined in the set of parameters P). The function`AddNode`creates a new node and performs the merging of two trees when ${q}_{\mathrm{new}}$ can be connected to a neighboring node in another tree.

`StoppingCriteria`based on a maximum number of iterations or on a limited computing time are satisfied.

Algorithm 5: TRRT-Exploration |

`FilterConformations`: The aim of this function is to reduce the number of local minima from which the next iteration of the IGLOO algorithm is initialized.

`StoppingCriteria`: Several types of conditions can be considered to determine the end of the iterative process performed by the algorithms. They are based on: (i) a maximum number of iterations; (ii) a limited computing time; (iii) estimated convergence, based on the evolution of the lowest energy value (i.e., the global optimum). All these conditions are evaluated, and the first one to be satisfied stops the algorithm.

#### 2.3. Molecular Model and Energy Function

#### 2.4. Software Availability

## 3. Results and Discussion

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Illustrative Example of IGLOO’s Performance on the Alanine Dipeptide

**Figure A1.**Two-dimensional projections, using the $\mathsf{\Phi}$ and $\mathsf{\Psi}$ dihedral angles, of explored conformations of the alanine dipeptide after four iterations (

**a**–

**d**) of the IGLOO algorithm. Each sampled conformations is depicted with a point, and it is connected to its parent node in the tree by an edge. Each point and edge is colored on a gradient from gray to black, providing information on the order in which the conformations were sampled. The first conformations sampled are in light gray, the last in black. Once locally minimized, conformations are represented by white circles. Nodes selected as roots for the next iteration, which correspond to representative conformations in the different basins, are depicted by red crosses. (

**a**) Extensive exploration of the potential energy surface with an energy threshold allowing the global exploration and the crossing of high-energy barriers. (

**b**) Reduction of the exploration step, inducing closer proximity of the nodes, and of the energy threshold, revealing lower barriers. (

**c**) Further reduction of the exploration step and energy threshold allows the achievement of a resolution enabling basin separation. (

**d**) The threshold value reached limits the exploration to low-energy basins.

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**Figure 1.**Evolution of the Met-enkephalin (top) and df-c-Myb (bottom) lowest energy (in kcal/mol) for BH (red), HYBRID (blue) and IGLOO (green) algorithms as a function of CPU time (in s). The solid lines represent the average value over ten runs of each algorithm. The variability of their performance is illustrated by the colored area between the first and third quartile (dashed lines). Outliers are represented by red circles (BH), blue diamonds (HYBRID), and green crosses (IGLOO).

**Figure 2.**Met-enkephalin lowest-energy structure found by BH (red), HYBRID (blue) and IGLOO (green) and associated energy (in kcal/mol) and CPU time (in s). Residue numbers are depicted in white.

Energy |

CPU time |

−224.40 |

1,519,327 |

−224.01 |

158,039 |

−224.81 |

64,927 |

**Figure 3.**df-c-Myb HPN1, HPN2 and HLX lowest-energy structures found by BH (red), HYBRID (blue) and IGLOO (green) and associated energy (in kcal/mol) and CPU time (in s). Residue numbers are depicted in white.

Energy |

CPU time |

−604.93 |

14,152 |

−615.27 |

34,578 |

−622.45 |

131,471 |

Energy |

CPU time |

−622.37 |

58,509 |

−620.28 |

20,102 |

−623.78 |

110,712 |

Energy |

CPU time |

−632.15 |

311,772 |

−620.26 |

80,916 |

−631.01 |

133,136 |

**Figure 4.**Distribution of minimized conformations of df-c-Myb on a two-dimensional projection at three different CPU times (in s) for BH (red), HYBRID (blue), and IGLOO (green). The two dimensions are defined by the ${}^{4}$C$\alpha {-}^{7}$C$\alpha $ and ${}^{2}$C$\alpha {-}^{5}$C$\alpha $ interatomic distances (in Å). HPN1, HPN2 and HLX regions are depicted by oblique hatches, horizontal hatches and grids, respectively. On each thumbnail, the dark dots correspond to the minima already located during the CPU time corresponding to the previous thumbnail.

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## Share and Cite

**MDPI and ACS Style**

Margerit, W.; Charpentier, A.; Maugis-Rabusseau, C.; Schön, J.C.; Tarrat, N.; Cortés, J.
IGLOO: An Iterative Global Exploration and Local Optimization Algorithm to Find Diverse Low-Energy Conformations of Flexible Molecules. *Algorithms* **2023**, *16*, 476.
https://doi.org/10.3390/a16100476

**AMA Style**

Margerit W, Charpentier A, Maugis-Rabusseau C, Schön JC, Tarrat N, Cortés J.
IGLOO: An Iterative Global Exploration and Local Optimization Algorithm to Find Diverse Low-Energy Conformations of Flexible Molecules. *Algorithms*. 2023; 16(10):476.
https://doi.org/10.3390/a16100476

**Chicago/Turabian Style**

Margerit, William, Antoine Charpentier, Cathy Maugis-Rabusseau, Johann Christian Schön, Nathalie Tarrat, and Juan Cortés.
2023. "IGLOO: An Iterative Global Exploration and Local Optimization Algorithm to Find Diverse Low-Energy Conformations of Flexible Molecules" *Algorithms* 16, no. 10: 476.
https://doi.org/10.3390/a16100476