 Methodology
 Open access
 Published:
Hamiltonian diversity: effectively measuring molecular diversity by shortest Hamiltonian circuits
Journal of Cheminformatics volume 16, Article number: 94 (2024)
Abstract
In recent years, significant advancements have been made in molecular generation algorithms aimed at facilitating drug development, and molecular diversity holds paramount importance within the realm of molecular generation. Nonetheless, the effective quantification of molecular diversity remains an elusive challenge, as extant metrics exemplified by Richness and Internal Diversity fall short in concurrently encapsulating the two main aspects of such diversity: quantity and dissimilarity. To address this quandary, we propose Hamiltonian diversity, a novel molecular diversity metric predicated upon the shortest Hamiltonian circuit. This metric embodies both aspects of molecular diversity in principle, and we implement its calculation with high efficiency and accuracy. Furthermore, through empirical experiments we demonstrate the high consistency of Hamiltonian diversity with realworld chemical diversity, and substantiate its effects in promoting diversity of molecular generation algorithms. Our implementation of Hamiltonian diversity in Python is available at: https://github.com/HXYfighter/HamDiv.
Scientific contribution
We propose a more rational molecular diversity metric for the community of cheminformatics and drug development. This metric can be applied to evaluation of existing molecular generation methods and enhancing drug design algorithms.
Introduction
Thanks to the tremendous development in computational tools and machine learning algorithms, computeraided drug discovery (CADD) has grown considerably in recent years, which can significantly shorten the time of the drug development process [1,2,3]. De novo molecular design algorithms [4,5,6,7,8,9,10,11,12,13] can generate candidate molecules with desirable in silico chemical and biological property scores [14, 15], which can provide meaningful inspirations for downstream preclinical studies and clinical trials.
However, due to the gap between in silico scores and in vivo behaviors of molecules [16], pharmacologists expect the upstream algorithms to provide as diverse a collection of drug candidates as possible, which can increase the probability of them eventually designing a drug to market [17]. Moreover, diverse drug compounds may assist in addressing drug resistance and side effects. Therefore, for molecule generation methods, the diversity of generated candidates is one of their pivotal aspects of performance.
For molecular diversity, besides a large size of the molecular set, we also expect the molecules to be dissimilar to each other, since similar structures cannot provide much inspiration for pharmacologists. Richness [18] and IntDiv (internal diversity) [19], currently the two most widely used molecular diversity metrics in CADD, respectively measure the quantity and dissimilarity of molecules, which are two fundamental aspects of molecular diversity. Nevertheless, as illustrated in Fig. 1, higher Richness may coincide with lower IntDiv, making the comparison of molecular diversity controversial. Hence, we need a more comprehensive metric to assess molecular diversity that reflects both the quantity and dissimilarity of a molecular set.
To address this challenge, we propose Hamiltonian diversity, a novel metric for molecular diversity, in this paper. Specifically, our contributions include:

We review existing molecular diversity metrics theoretically and formulate two general principles of an ideal molecular diversity metric: monotonicity and dissimilarity. None of the existing metrics satisfy both principles simultaneously.

We propose a new Hamiltonian diversity based on the shortest Hamiltonian circuit, which adheres to both principles of molecular diversity metrics. We also provide an intuitive explanation and efficient implementation for the novel metric.

We demonstrate the high consistency between Hamiltonian diversity and realworld chemical diversity. When incorporated into existing molecule generation algorithms, Hamiltonian diversity also helps promote the diversity of generated molecules.
Existing molecular diversity metrics
The druglike chemical space \({\mathcal {S}}\) is vast, with an estimated \(10^{33}\) synthesizable molecular structures [20], and it cannot be described in dimensions. Therefore, a metric function is needed in drug design to evaluate the span of a molecular set in \({\mathcal {S}}\), that is, the molecular diversity.
A molecular diversity metric \(\varphi\) is a function that maps a set of molecules \({\mathcal {M}}\subseteq {\mathcal {S}}\) to a nonnegative real number which reflects the diversity of the set:
where \({\mathcal {P}}(\cdot )\) denotes the power set. In particular, \(\varphi (\emptyset )=0\).
Some existing metrics for molecular diversity meet the above definition, and they can be divided into two categories: referencebased and distancebased.
Referencebased metrics
Referencebased metrics intuitively compare a molecular set \({\mathcal {M}}\) with a reference set \({\mathcal {R}}\):
where \({\mathbb {I}}\) is the indicator function. \({\mathcal {R}}\) can be either a set of molecules or a set of molecular fragments, as a result of which we use the above formulation which is applicable to different kinds of reference sets.
Richness [18] is a widely used referencebase metric where \({\mathcal {R}}={\mathcal {S}}\), counting the number of unique molecules in a set, i.e., \({\mathcal {M}}\).
Distancebased metrics
On the other hand, distancebased metrics quantify molecular diversity based on pairwise distances among molecules instead of depending on given reference sets:
where f is a function to be defined, and d is a distance metric between a pair of molecules:
The Tanimoto distance [21], denoted as \(d_T\), between extendedconnectivity fingerprints (ECFP) [22] of small molecules is a widely used metric function and is considered the most appropriate choice for the distance metric in the chemical space [23, 24]:
where \(\varvec{x},\varvec{y}\in \{0,1\}^n\) are ndimensional binary fingerprints of two molecules \(x,y\in {\mathcal {S}}\).
It is worth noting that our subsequent definition and analysis also apply to other molecular distance functions besides the Tanimoto distance, including Fréchet ChemNet distance [25], VAEdissimilarity [26] and GRMSD [27]. The Tanimoto distance is adopted as the default distance function because it is mathematically a metric function, which is theoretically essential for the properties of Hamiltonian diversity. as stated in Section 3.3.
Among distancebased metrics, IntDiv [19], which measures the average pairwise distance of a molecular set, is currently the most popular one for the evaluation of molecule generation approaches [1, 14]. The recently proposed Circles [28] is defined by the maximum number of mutually exclusive circles of radiuses t that can fit into a molecular set \({\mathcal {M}}\) as neighborhoods, with a subset of it \({\mathcal {C}}\subseteq {\mathcal {M}}\) as the circle centers.
Principles of molecular diversity metrics
The objectives of a molecule generation algorithm include achieving both high Richness and high IntDiv. However, these two metrics are orthogonal to each other, meaning they represent distinct aspects of the molecular sets. Therefore, it is desirable to have an ideal molecular diversity metric that combines both aspects into a single value. With a given distance metric d, we formulate these two aspects as two principles, respectively:
Principle 1 (Monotonicity). A good molecular diversity metric \(\varphi\) should be monotonic, i.e., for a molecule \(x\in {\mathcal {S}}\) and a molecular set \({\mathcal {M}}\subseteq {\mathcal {S}}\), it holds that
and the equality holds if the distance between x and a molecule in \({\mathcal {M}}\) is 0^{Footnote 1}:
Principle 2 (Dissimilarity). A good molecular diversity metric \(\varphi\) should be positively correlated with molecular distances, i.e., for any three molecules \(x,y,z\in {\mathcal {S}}\), it holds that
To ensure the validity of a molecular diversity metric [29], it should satisfy both principles.
As a contrast, the design of Circles [28] follows another set of principles for molecular diversity metrics that diverge from our principles in several aspects: (1) their monotonicity principle lacks the inclusion of the equality condition; (2) their dissimilarity principle merely requires a nonstrict positive correlation (\(\ge\)), contrasting with our stricter criterion; (3) they propose a subadditivity principle, derived from the additivity property of outer measures, but we do not accept this principle since no practical insights are provided to support their introduction of “sub”. These differences lead to the possibility that the results of the principled analysis of metrics in [28] may differ from ours.
Table 1 provides an overview of existing molecular diversity metrics, demonstrating that none of them fully adhere to both monotonicity and dissimilarity principles. Consequently, the development of a new metric is imperative to tackle this key challenge in drug design.
Hamiltonian diversity
In this section, we present a novel metric for quantifying molecular diversity, Hamiltonian diversity, based on identifying the shortest Hamiltonian circuit within the chemical distribution of a molecular set.
Definition
We first formulate a molecular set \({\mathcal {M}}=\{m_1,m_2,...,m_n\} (n\ge 2)\) as an undirected complete graph \(K_n({\mathcal {M}})=(\{m_i\},\{d_{ij}\})\), \(i,j=1,2,...,n,\ i\ne j\). In this graph, each vertex represents a molecule \(m_{i}\), and the edge weight between vertices i and j is determined by the pairwise molecular distance: \(d_{ij}=d(m_i,m_j)\).
Hamiltonian circuit, a crucial concept in graph theory, is a directed cycle that visits each vertex in a graph exactly once. The Hamiltonian diversity, denoted as HamDiv, is defined as the length of the shortest Hamiltonian circuit in the complete graph \(K_n({\mathcal {M}})\).
Definition (Hamiltonian Diversity).
where the last constraint ensures the solution is a single circuit rather than the union of smaller circuits. Here, we adopt the widely recognized DantzigFulkersonJohnson formulation of the Traveling Salesperson Problem (TSP) [31] to establish a mathematically rigorous definition of Hamiltonian diversity.
In particular, if \({\mathcal {M}}=\{x,y\}\), i.e., \({\mathcal {M}}=2\), \(\texttt {HamDiv}({\mathcal {M}})=2d(x,y)\). If \({\mathcal {M}}=0\) or 1, \(\texttt {HamDiv}({\mathcal {M}})=0\).
Explanation
The Hamiltonian diversity adopts the Hamiltonian circuit in a complete graph to measure the diversity of a molecular set, which factors in each molecule equally. Moreover, this approach utilizes insights related to the exploration process within the chemical space, with the shortest Hamiltonian circuit representing the minimal cost of “traveling across” all molecules in a set. Therefore, by employing Hamiltonian diversity, we can not only effectively evaluate the molecular diversity but also obtain interpretability by quantifying each molecule’s contribution to the overall diversity.
It is worth emphasizing that with a given molecular distance metric d, Hamiltonian diversity is hyperparameterfree. In contrast to Circles, which requires a predefined hyperparameter t that can significantly affect diversity values, HamDiv provides a fixed measurement that facilitates fair and uncontroversial evaluations.
Advantages
Hamiltonian diversity adheres to both the monotonicity and dissimilarity principles, and the correctness of the dissimilarity principle is evident since \(\texttt {HamDiv}(\{x,y\})=2d(x,y)\).
The monotonicity principle is fulfilled when the molecular distance is a metric function, meaning that it satisfies the triangle inequality:
Proof: For a molecule \(x\in {\mathcal {S}}\) and a molecular set \({\mathcal {M}}\subseteq {\mathcal {S}}\), we can denote the two molecules near x in the shortest Hamiltonian circuit of \({\mathcal {M}}\cup \{x\}\) as \(x_{1}\) and \(x_{+1}\). Then we have:
A Hamiltonian circuit can be constructed for \({\mathcal {M}}\) with the edge between \(x_{1}\) and \(x_{+1}\), and other edges are all identical to those in the shortest Hamiltonian circuit of \({\mathcal {M}}\cup \{x\}\). This Hamiltonian circuit is not longer than the shortest Hamiltonian circuit of \({\mathcal {M}}\cup \{x\}\), and also not shorter than the shortest Hamiltonian circuit of \({\mathcal {M}}\). Hence, we have:
And Hamiltonian diversity also evidently satisfies the equality condition.
In summary, Hamiltonian diversity is, in principle, an ideal metric of molecular diversity. From this perspective, Hamiltonian diversity has advantages over all the existing diversity metrics in Table 1.
Figure 2 and Table 2 demonstrate an example of Hamiltonian diversity and compare it with Richness and IntDiv. The comparison between {A,B,C,D,E} and {A,B,C,E} shows that IntDiv does not satisfy monotonicity, while the comparison between {A,B,C,D} and {A,B,C,E} shows the defects of Richness in dissimilarity. By contrast, HamDiv meets both principles of diversity metrics in practice.
Efficient implementation
Following previous choices of the molecular distance metric [19, 28], we calculate the Hamiltonian diversity using the Tanimoto distance between ECFPs of molecules, which satisfies the triangle inequality [32, 33].
The calculation of the Hamiltonian diversity is essentially the solution of the Traveling Salesman Problem (TSP), which is a classic NPhard problem in combinatorial optimization. Numerous algorithms have been devised to tackle the TSP in a complete graph, encompassing exact, approximation, and heuristic methods [34, 35]. To strike a balance between accuracy and efficiency in diversity implementation, we run several popular algorithms for the TSP on molecular sets of different sizes, and compare the results and time costs to determine which algorithm is the best for implementing the Hamiltonian diversity.
Molecular sets
We choose the following realworld drug datasets to test different implementations of Hamiltonian diversity:

1.
the 739 inhibitors against the cJun Nterminal kinase 3 (JNK3) target [36];

2.
the 2664 inhibitors against the Glycogen synthase kinase 3 beta (GSK3\(\beta\)) target [36];

3.
the 7218 inhibitors against the Dopamine Receptor D2 (DRD2) target [4].
And smaller sets of sizes 10, 15, 20, and 200 are randomly selected from the JNK3 actives.
Algorithms for solving the TSP
We separately implement Hamiltonian diversity using the following popular algorithms for solving the TSP whose inputs are pairwise distance matrices of the molecular set.

1.
Dynamic programming: also called the BellmanHeldKarp algorithm, giving the exact solution of the TSP with time complexity of \(O(2^nn^2)\) [37];

2.
Christofides algorithm: providing an approximate solution within a factor of 3/2 of the optimal solution length, with a time complexity of \(O(n^3\log n)\) [38];

3.
Greedy algorithm: providing an approximate solution with a time complexity of \(O(n^2\log n)\) (usually as a fast baseline);

4.
Simulated annealing algorithm: a metaheuristic algorithm giving an approximate solution [39];

5.
Threshold accepting algorithm: another metaheuristic algorithm giving an approximate solution [40];

6.
2opt Local Search algorithm: a classic heuristic algorithm for the TSP [41].
As shown in Table 3, we report the results and time costs for calculating the Hamiltonian diversity of different molecular sets using different algorithms. The results show that when the molecular set grows above 20, the time consumption of the precise algorithm becomes unacceptable. When the set size is less than 1000, 2opt local search is the most accurate approximation algorithm and can complete the calculation in a few minutes. When the set size exceeds 1000, the performance of 2opt local search is severely compromised within a limited running time, and the greedy algorithm is most acceptable in this case. As a consequence, we implement the calculation of Hamiltonian diversity as follows:

When \({\mathcal {M}}\le 20\), using dynamic programming;

When \(20<{\mathcal {M}}\le 1000\), using 2opt local search algorithm;

When \({\mathcal {M}}>1000\), using greedy algorithm.
Discussions
Richness and internal diversity are currently the two most commonly used metrics of molecular diversity. Compared with them, Hamiltonian diversity has advantages in principle, but there may be inaccuracies in the calculation. So, is there a better alternative in this regard?
First, the calculation of the recently proposed Circles actually also corresponds to the solution of an NPhard problem, and it is implemented merely using simple greedy algorithms without analysis of accuracies.
In addition, one might want to replace the Hamiltonian circuit with a minimal spanning tree, which can be efficiently solved. But in fact, the molecular diversity metric constructed using a minimum spanning tree does not satisfy the monotonicity principle, because adding one molecule may result in a minimum spanning tree with a smaller total weight.
Above all, since precise molecular diversity values are not required for practical drug design, Hamiltonian diversity is a good metric with superiority in principle.
Experiments
Correlations with biological functionality
We conduct an empirical study comparing molecular diversity metrics by analyzing the correlations between these metrics and a gold standard of biological functionality following the settings presented in [28].
The analysis utilizes the BioActivity dataset [42], which consists of 10,000 compound samples with bioactivity labels sourced from the ChEMBL database [43]. These labels belong to 50 different bioactivity classes, each containing 200 samples. The number of label types covered by a subset of the BioActivity dataset represents its biological functional diversity, which should be reflected by an ideal diversity metric. Therefore, the number of bioactivity labels in a subset is recognized as a proxy gold standard (GS) of molecular diversity.
In order to assess the empirical validity of various molecular diversity metrics, we perform a random sampling of subsets from the BioActivity dataset. The subsets are of fixed sizes, specifically \(n=50, 200, 1000\). Subsequently, we calculate correlations (Spearman’s correlation coefficient) between the molecular diversity metrics and the GS. A better diversity metric should have a higher correlation to the GS. The Tanimoto distance between ECFPs of molecules is employed to calculate all the distancebased diversity metrics.
As shown in Fig. 3, Hamiltonian diversity exhibits a better correlation with the GS compared to all other metrics for \(n=50\) and \(n=200\). When \(n=1000\), although slightly lower than Circles(0.6) and Circles(0.7), Hamiltonian diversity still achieves a high correlation with the GS (\(>0.9\)). These results suggest that Hamiltonian diversity has higher consistency with realworld chemical diversity of molecules than other existing metrics.
Applying Hamiltonian diversity to molecule generation
To demonstrate the potential of using Hamiltonian diversity in realworld drug discovery for promoting molecular diversity, we incorporate it into a reinforcement learning algorithm for molecule generation to encourage diverse exploration in the chemical space.
Scenarios
To simulate practical drug discovery, we consider the following property predictors (also known as oracles): (1) QED (Quantitative Estimate of Druglikeness) [44] and SA (Synthetic Accessibility) [45], two commonly used oracles in CADD; (2) JNK3 (cJun Nterminal kinase3) and GSK3\(\beta\) (glycogen synthase kinase3 beta), two protein targets related to Alzheimer’s disease, whose evaluators are random forest models based on Morgan fingerprints [36, 46]. We employ the oracles implemented by [47].
Moreover, we establish three multiobjective molecule generation settings (with constraints) by combinations of these oracles following [8]:

(a)
JNK3\(\ge 0.5\) and QED\(\ge 0.7\) and SA\(\le 2.5\)

(b)
GSK3\(\beta \ge 0.5\) and QED\(\ge 0.7\) and SA\(\le 2.5\)

(c)
JNK3\(\ge 0.5\) and GSK3\(\beta \ge 0.5\) and QED\(\ge 0.7\) and SA\(\le 2.5\)
Under each setting, the generated compounds meeting the constraints are considered desirable drug candidates, and we aim to assess the diversity of this molecular set.
Algorithms
We use the initial version of Reinvent [4, 48] as a baseline, which is a competitive deep reinforcement learningbased approach for goaldirected molecule generation [49]. The algorithm uses a combination of recurrent neural networks (RNN) and reinforcement learning (RL) to iteratively optimize the generation of molecules toward the desired properties. The loss function for updating the RNN agent in each RL step is:
where x denotes a molecule in a SMILES string form, \(P(x)_M\) refers to the probability of the model M generating x, and \(s(x)\in [0,1]\) is the predicted property score of x and \(\sigma\) is a coefficient. This function encourages the agent to generate molecules with higher property scores. However, in practice, the agent is likely to converge to a certain local optimum in the chemical space, resulting in the generation of structures with low diversity.
An existing variant of Reinvent [17, 50] enhances molecular diversity by penalizing identical scaffolds in s(x). The scaffolds of molecules generated in each step are stored in a “scaffold memory”, and if the scaffold of a newly generated molecule is the same as that of more than k previous molecules, s(x) is set to 0, where k is an integral coefficient.
Inspired by Hamiltonian diversity, we encourage the agent to explore the chemical space more diversely by adding a term on the scoring function:
where \({\mathcal {M}}\) is a memory of molecules generated in previous steps. The minimum distance of a newly discovered molecule from a set of previously generated molecules is corresponding to the increment in each step of the greedy algorithm for solving TSP in HamDiv. The calculation time of the additional term increases with the number of molecules in \({\mathcal {M}}\), but it is not computationally expensive in general, because the Tanimoto distances between molecules can be calculated at a speed of \(10^{6}\) pairs per second. Furthermore, only one natural way to integrate HamDiv into Reinvent is provided here, and we look forward to the exploration of better applications of HamDiv in molecular generation algorithms.
Evaluation details
For each scenario, we apply virtual screening on the ExCAPEDB [51] database and the generated molecular set of each of the three algorithm to obtain the desirable molecules satisfying all the constraints. We use Richness, BM (number of unique BemisMurcko scaffolds), IntDiv and HamDiv as the metrics for molecular diversity, and report their values on those sets of desirable molecules. The Tanimoto distance between ECFPs of molecules is employed to calculate IntDiv and HamDiv.
For hyperparameters, we set \(\sigma =1000,\sigma _1=0.1\), and all the algorithms run 2000 steps with a batch size of 128. All the experiments are conducted on a single NVIDIA A100 GPU and each run cost less than 12 hours.
Experimental results
As shown in Table 4, our algorithm designed with Hamiltonian diversity demonstrates a greater capacity for generating diverse molecules in all three scenarios compared with the two existing methods, especially with higher Hamiltonian diversity values. In addition, our algorithm also performs the best in terms of “scaffold richness” shown by BM.
Moreover, in the second scenario, Reinvent performs better than “Reinvent + scaffold memory”, which is the opposite of the other two scenarios. This is because the second optimization objective is relatively easy, and the diversity constraint added to the baseline will have a large negative effect of “filtering out candidates”. This phenomenon suggests that the “scaffold memory” penalty may be excessive for a simple task, whereas HamDiv does not have this problem.
In summary, the above results suggest the potential benefits of utilizing Hamiltonian diversity in the design of molecule generation algorithms.
Conclusion
In this paper, we review existing metrics for molecular diversity with a principled analysis. Then, we define and implement the Hamiltonian diversity (HamDiv) based on the shortest Hamiltonian circuit and demonstrate its empirical effectiveness through two experiments related to realworld drug discovery. The key advantages of this new molecular diversity metric include:

1.
Hamiltonian diversity satisfies both two principles of molecular diversity metrics: monotonicity and dissimilarity. This fundamentally guarantees its higher effectiveness than all previous metrics.

2.
Hamiltonian diversity can be interpreted intuitively by directly quantifying the effect of each molecule.

3.
Hamiltonian diversity has high consistency with realworld chemical diversity, which reflects its high empirical value.

4.
Hamiltonian diversity can assist in enhancing the diversity of molecule generation, which has a good application prospect in drug discovery.
Availability of data and materials
Our implementation of Hamiltonian diversity in Python is available at: https://github.com/HXYfighter/HamDiv.
Notes
A distance of 0 between two molecules usually indicates that they are almost identical, but they are possible to be slightly different. For example, two slightly different compounds may correspond to the same fingerprint, so the Tanimoto distance \(d_T\) between them is 0.
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Funding
Q. Yao is supported by National Natural Science Foundation of China (under Grant No.92270106) and Beijing Natural Science Foundation (under Grant No.4242039).
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Hu, X., Liu, G., Yao, Q. et al. Hamiltonian diversity: effectively measuring molecular diversity by shortest Hamiltonian circuits. J Cheminform 16, 94 (2024). https://doi.org/10.1186/s13321024008834
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DOI: https://doi.org/10.1186/s13321024008834