Quantum Approximated Graph Cutting: A Rapid Replacement for T-REMD?

Determining an optimal protein configuration for the employment of protein binding analysis as completed by Temperature based Replica Exchange Molecular Dynamics (T-REMD) is an important process in the accurate depiction of a protein’s behavior in different solvent environments, especially when determining a protein’s top binding sites for use in protein-ligand and protein-protein docking studies. However, the completion of this analysis, which pushes out top binding sites through configurational changes, is an polynomial-state computational problem that can take multiple hours to compute, even on the fastest supercomputers. In this study, we aim to determine if graph cutting provide approximated solutions the MaxCut problem can be used as a method to provide similar results to T-REMD in the determination of top binding sites of Surfactant Protein A (SP-A) for binding analysis. Additionally, we utilize a quantum-hybrid algorithm within Iff Technology’s Polar+ package using an actual quantum processor unit (QPU), an implementation of Polar+ using an emulated QPU, or Quantum Abstract Machine (QAM), on a large scale classical computing device, and an implementation of a classical MaxCut algorithm on a supercomputer in order to determine the types of advantages that can be gained through using a quantum computing device for this problem, or even using quantum algorithms on a classical device. This study found that Polar+ provides a dramatic speedup over a classical implementation of a MaxCut approximation algorithm or the use of GROMACS T-REMD, and produces viable results, in both its QPU and QAM implementations. However, the lack of direct configurational changes carried out onto the structure of SP-A after the use of graph cutting methods produces different final binding results than those produced by GROMACS T-REMD. Thus, further work needs to be completed into translating quantum-based probabilities into configurational changes based on a variety of noise conditions to better determine the accuracy advantage that quantum algorithms and quantum devices can provide in the near future.

Determining the most optimal protein configurations for binding models is one of the 2 most important tasks to calculate before completing protein-protein, protein-ligand, or 3 further inter-protein analysis. The complex surface characteristics of proteins, even in 4 their crystallographic form without solvent interactions as found within Protein Data 5 Bank (PDB) formatted files, means that the complete computation of binding 6 characteristics of entire structures can be anywhere from infeasible [29] to impossible on 7 1/18 classical computers [7] without model simplification or cutting the number of residues 8 involved in modeling based on experimental results. While determining the most 9 optimal residues for binding models is a straightforward process if experimental data 10 exists for binding kinetics of that protein, it is difficult to complete intuitively for 11 proteins of which there is little known experimental binding behavior, as chemical 12 formation kinetics is completed in non-deterministic, polynomial (NP) time on a 13 classical device [7]. 14 To solve this problem, a multitude of bioinformaticians have turned to tools that can 15 complete molecular dynamics simulations of the proteins undergoing thermal relaxation 16 in order to more effectively reveal their best binding sites. While tools such as 17 Rosetta [14] provide this functionality in the form of ab initio modeling of protein 18 energies and create rigid protein models, Schrodinger [21] and GROMACS' 19 Temperature-Replica Exchange Molecular Dynamics (T-REMD) [38] complete this task 20 by completing iterative molecular dynamics simulations of each of the protein 21 characteristics' interactions. 22 However, the completion of these approximated computational models can be quite 23 time consuming and computationally complex themselves [10]. With a maximum 24 GROMACS performance of 304 ns/day achieved across 600 physical cores with 2 GPUs 25 used per core of access [5], the scale of computational resources needed for molecular 26 dynamics simulations for even multi-hour levels of full protein analysis means that 27 GROMACS is still quite impractical to use at scale when analyzing full interactomes or 28 virions [27]. 29 Consequently, there remains a need to complete the protein binding site analysis 30 before docking at the precision level of molecular dynamics with much faster speeds and 31 with lower cost computational systems to dramatically increase the ability to 32 computationally model full interactomes in order to effectively model protein-protein 33 and protein-ligand binding at scale. 34 Topological Analysis of Protein Surfaces for Binding Site 35 Analysis 36 Recently, there has been a growing amount of work focused on finding the minimal and 37 maximal points of protein models through the analysis of protein shapes, or topology, 38 before adding electrostatically-or electrodynamically-based equations for full 39 pre-binding analysis in order to screen for top sites prior to compute-intensive force 40 calculation. The completion of this task on classical computers with clever, third-degree 41 polynomial topological algorithms [6] or by quantum annealing devices that straddle the 42 line between highly optimized classical computing devices and fully quantum 43 computational devices within a first-degree polynomial complexity class [25] has led to a 44 new paradigm within biological modeling that is focused more on topological feature 45 distinction.

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Due to the realization of the benefits that could be gained by completing topological 47 models to the overall time of computation, it seemed that the abstraction could be 48 taken one step further through the implementation of Maximum Cut, or MaxCut, 49 graphing algorithms [17]. Not only would these algorithms be able to effectively find the 50 minima and maxima on protein surfaces, but they could actively apply a cost function 51 across the interaction points between the different atoms within the protein to more 52 effectively pick the best binding site with regards to electrodynamics and electrostatics 53 as well. But, while there are potential estimations of a potential Maximum Cut that 54 exist in various algorithmic forms, such as Greedy Cut [39] and the 55 Goemans-Williamson algorithm [19], it has not been formally solved at scale, making it 56 potentially infeasible for protein structure analysis. 57

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Quantum Computing in Improving MaxCut's Odds 58 Quantum processors are devices that use the effects of quantum mechanics for methods 59 of information transfer among bit-like devices. Due to this ability for the bits 60 themselves to hold multiple states, quantum computers seem like ideal solver devices for 61 molecular dynamics calculations, as they could represent every electron within a protein. 62 However, current quantum processors are still in their infancy ; the largest 63 superconducting quantum processor that exists as per the writing of this paper is the 64 Google Bristlecone device, with 72 qubits [8]. As 1 qubit is essentially 1 extra large 65 electron [22], it would take hundreds of qubits to model even the smallest of proteins. 66 However, due to their ability to provide superposition and entanglement features on 67 bits, quantum computers have the potential to take exponential scale problems and turn 68 them into polynomial or even log scale problems [4]. As such, they could be devices that 69 can potentially provide massive improvements to a MaxCut problem. Particularly, the 70 quantum approximate optimization algorithm (QAOA) has been found to be a potential 71 solution to this problem [15]. This hybrid quantum-classical algorithm combines the 72 quantum computer's ability to effectively solve exponential problems with a classical 73 cost function to determine the best cuts within a set of quantum bits, and as such, 74 could provide benefits with few qubits if cleverly designed to scale effectively with 75 classical devices. As such, QAOA has been seen to have biological uses in the past, with 76 the team at ProteinQure using QAOA to effectively fold small proteins [16]. Thus, 77 QAOA seems to be a good candidate for potentially replacing T-REMD. For the past eight months, Iff Technologies has been working with academic and startup 81 clients to test its software, Polar+ [33]. Polar+'s protein pruning tool is an 82 implementation of the QAOA algorithm that includes further bioinformatics 83 contextualization. It has been shown to improve the binding outcomes of protein-ligand 84 interactions [34], However, it has not been directly compared to GROMACS T-REMD 85 and Goemans-Williamson implementations at scale until this point.

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In this paper, Polar+'s protein pruning tool running on a quantum computer 87 connected to a 1-node classical computer is compared in performance to an instance of 88 Polar+'s protein pruning tool running on a quantum abstract machine running on the  The crystallographic coordinates for the SP-A protein structure were determined using 105 Uniprot [13]. Through analysis of this database, the model 5FFR [20] was determined 106 to be the most all-encompassing structural model of SP-A, covering 147 of its amino 107 acids at a resolution of 2.20Å. However, this model did include phosphocholine ligands, 108 which could interfere with direct interrogation of the amino acids that make up the 109 structure of SP-A [20]. Thus, the phosphocholine was removed from the 5FFR model 110 before binding site analysis.

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For the completion of the T-REMD analysis, GROMACS 5.0.4 was utilized to create a 113 suitable solvent environment, along with a set of temperature and pressure controls, in 114 order to most accurately determine the protein configuration in a binding environment, 115 as completed in similar studies [35]. Being a molecular dynamics software, GROMACS 116 completes sets of multi-axial nearest neighbor calculations for a set of forces for 117 coordinate position and velocities across a number of time steps [9]. First, forces for 118 each molecule within a solute and solvent are calculated using a prescribed set of forces 119 unique to different solvent environments. As this study aims to understand a protein in 120 a neutral solvent environment, three different force models were used: the Assisted  Being the oldest force field used, AMBER [11] has the simplest form, with total 125 potential energy for a macromolecule following a summation between bond energy as an 126 ideal spring, geometrical energy from each angle within the covalent bonding between 127 atoms, torsioning due to bond order, and intra-atomic forces represented as a van der 128 Waals force added to an electrostatic force, wherein f ij represents the Fourier 129 transformation, ij represents the well depth of the atom's location, and other constants 130 represent their respective parts. This study used AMBER99sb.  CHARMM [24] is a force-field model that aims to take OPLS further through the 139 addition of an impropers and a Urey -Bradley term, which intend to improve upon the 140 torsional modeling of the atomic interactions through the accounting of bending and 141 non-binding interactions between atoms in the 1,3 positions of an organic molecule due 142 to proximity of electrostatic forces, respectively. This study used CHARMM36.
Equation 3. CHARMM force field formula adapted from Mackerell et al. for electrical potential across protein [23] The following system has been utilized for the benchmarking of the GROMACS 144 software: 145 SP-A protein in water bulk consisting of ∼62,000 atoms (number of water molecules 146 ∼20,000) in 6.5x6.5x6.5 nm 3 cell.

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The testing runs were done using the following parameters: 2fs timestep, PME 148 electrostatics, and van der Waals forces truncated at 1.2 nm with corresponding 149 pressure and temperature control. The benchmark runs were typically for 10000 steps 150 (20ps) with/without writing output any trajectory and coordinate files (Note that with 151 no write trajectories and confout slightly increases the performance). For our tests, we 152 use the "-pin on" and "-dlb yes" GROMACS flags, where "-pin on" stops the kernel 153 from moving processes between cores by locking the cores, and allow dynamic load 154 balancing to automatically run when the load imbalance is 5% or more, which is 155 important for inhomogeneous systems. Note that for optimal performance, we also try 156 5/18 mdrun -resethway and -maxh=0.05 options, which corrects the benching results. After 157 these first test runs, the force fields for SP-A were taken into consideration for a total of 158 10 ns, or 5,000,000 time steps, in order to be able to obtain reasonable accuracy with 159 regards to the interaction of SP-A within a water model.    through the use of helium based cooling chambers [31]. Using the qubits and their angular rotations themselves, which is determined after 208 running the algorithm over a set of test runs for a particular use case and determining 209 which results provide the most problem-specific information, as the nodes within the 210 graphs that MaxCut must be performed on, the total energy of each of these operators 211 simply needs to be decomposed to determine to which node the maximum cut value At the end of this process, basis states representing different qubits being cut from 217 the graph were provided at different probability levels. These basis states, which were 218 binary numbers in order to the qubit and flip state of that qubit, were contextualized to 219 refer to the qubit that needed to be cut from the graph; 1s were taken to be qubits that 220 were part of the new graph, and 0s were taken to be qubits that were eliminated. To Equation 7. Calculation of scalar energetic from the aggregation of shape complementarity(SC) for the larger protein, the receptor (Rsc), and the smaller protein, the ligand (Lsc), as well as the desolvation energy for derived through the summation of each contact potential atomic contact point's desolvation energy obtained experimentally from Zhang et al. [40], wherein α and β are scaling ratios used to ensure that S sc and S elec are of the same scale. With regards to the actual implementation for each of the force fields, only 48 cores 272 were available for running 5,000,000 time steps. As such, the performance obtained was 273 less than that observed in the test runs. This led to a fastest observed run time of    Produced SP-A models from each software

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The newly configured models from GROMACS and cut models from Polar+ and     The reason for this increase in speed is the ability of the quantum processor to 317 utilize superpositions of each state and entanglement between states to produce strong 318 cut probabilities nearly instantaneously for multiple atoms, while classical processing 319 devices must complete these tasks serially for each atom. This allows for a slimmer 320 algorithmic approximation that does not require multi-axial force calculations or higher 321 order functions. More importantly, the steps completed by GROMACS to properly 322 characterize the forces for each atom are additive amongst the atoms and then additive 323 amongst each pair of atoms, creating a computational complexity of O(n 2 ) for each 324 atom [10]. But, this still does not compare to the steps that need to be completed for 325 Goemans-Williamson, which, while having a lower dimensionality requirement, has a 326 13/18 complexity of O(n 2 logn) [28] due to its arccos term. Thus, QAOA is naturally the 327 fastest algorithmic implementation, with an expected performance in the O(logn) 328 regime [15]. 329 Surprisingly, however, the QAM-based Polar+ implementation running on a 330 medium-scale classical device was able to outperform the QPU based implementation. 331 While this means that, in this interpretation, the QPU does not outperform a classical 332 computing device for this task, it does mean that the procedural generation of quantum 333 states on a classical device can provide algorithmic speedups to algorithms within the 334 Bounded Quantum Polynomical, or BQP, regime, in which problems are of greater 335 complexity than those that can be solved within deterministic Polynomial (P) time but 336 lesser complexity than those that require Non-Polynomial deterministic time (NP) [26], 337 thus providing further evidence of a quantum algorithm use case that can, if coupled 338 with an exponential time problem on a quantum device, can provide a definite quantum 339 advantage as raised by Ge and Dunjko [18]. steps were to be used for all atoms. However, the number of time steps was capped at 347 5,000,000 steps, which allowed for a P-time implementation.

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Binding Accuracy

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In Table 9, one can see the comparison of main results across all the discussed models. 350 Based on this table, the following points can be ascertained: glycosylation site, is involved in binding from SP-A's side, which can allow for 360 glycosylated bond formation. From literature, it is seen that the C-type lectin domain is 361 responsible for carbohydrate recognition (see Figure 1) and NAG residues provide the 362 glycosylation of Spike protein that are mainly present on the S2 segment, leaving the 363 RBD open for binding to receptor [30]. Therefore, while the binding model produced by 364 graph cutting methods make more sense when looking at the exact binding interaction 365 from the perspective of SP-A, where there is a NAG residue interacting with an 366 N-linked glycosylation site, the binding models produced by T-REMD make more sense 367 from the perspective of SARS-CoV-2, as strongest binding is more likely to take place 368 within the RBD. Spike. However, due to the shape complementarity term of the ZDOCK algorithm (Ssc) 376 creating a scalar based upon the degree of path matching between receptor and ligand 377 14/18 structures and a lack of structural elements in models produced through graph cutting, 378 the Ssc component for all Polar+/Goemans-Williamson models will be much lower than 379 those produced by GROMACS, and cannot be directly compared. binding, which would make the overall RMSD lower across graph-cut models.

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Through this study, it was found that the operation of a quantum abstract machine 399 (QAM) application of the Quantum Approximate Optimization Algorithm (QAOA) 400 aimed at determining graph cuts within a protein structure in order to find top binding 401 site candidates produced the same top binding sites and was faster than a QPU based 402 version of itself, as well as faster than the Goemans-Williamson algorithm operated on 403 the Bridges supercomputer, and the GROMACS T-REMD system on the JUWELS and 404 Bridges supercomputer due to its superior computational complexity. Moreover, when 405 used to provide binding results using the ZDOCK software, it produced results of 406 chemical merit. However, the models produced by GROMACS for which the same 407 binding test was completed had noticeably different binding results and configurations 408 after this binding site determination step, binding directly to the SARS-CoV-2 Spike S1. 409 For this reason, it is difficult to determine which model is superior without the 410 completion of a biochemical assay to determine the actual method of interaction 411 between the SARS-CoV-2 Spike and SP-A. 412 But, the computational improvements that quantum algorithms can provide in 413 graph cutting is definitely promising in the quest to improve the computational time of 414 complex analysis prior to protein-protein binding. Therefore, QAMs running quantum 415 algorithms do seem to be an interesting area to investigate towards improvements of 416 molecular dynamics simulations, especially with the addition of quantum specific state 417 analysis through the determination of base and excitation energies using the variational 418 quantum eigensolver (VQE) or similar [37] to provide improvements to quantum 419 mechanics based models that can be completed in GROMACS today. 420 Lastly, this study did little to parameterize the qubits for the different thermal 421 environments, which could have provided at par or higher accuracy in the type of 422 modeling generated through graph cutting. Through the introduction of thermal 423 relaxation models [32] to each bond site, the identification of qubits that best represent 424 the electrostatic environment of the atom, the QPU can provide models that can