Computational design of helical artificial metallopeptides: from sequence to activity in Pd-peptide systems

Abstract

Artificial metallopeptides hold immense potential to combine enzymatic activity with the versatility of organometallic catalysts. However, computational de novo design is largely limited to theozyme models that may neglect second-sphere atomic structure, overlook hydrogen-bonding networks, and ignore metal-induced conformational selection. We overcome these limitations for the case of helical metallopeptides and metal-containing helical motifs by proposing a DFT-based bottom-up methodology applied to the design of Pd-binding (Met-X)_n sequences (X = Ala, Val, Ile). Line group symmetry theory is employed to accelerate the calculations by leveraging helical monoperiodicity for computational efficiency. The methodology (a) reproduces the geometric parameters of α-poly-Ala with near-experimental accuracy; (b) to the best of our knowledge, provides the first evidence that the α → π-transition may manifest as a first-order phase transition; (c) identifies (Met-Ala)_n π-helices as preferred matrices for canonical Pd(II) Suzuki coupling intermediates. In contrast, Pd incorporation in the α-helical matrix poses significant challenges, as shown by relaxed potential energy scans. From the periodic π-helix, we extract a cluster containing over 250 atoms and model it in aqueous solution at the ωB97X-V/def2-TZVP-gCP//B97-3c level to obtain reliable energetics for the free energy profile of the key oxidative addition step. The profile featured a low activation barrier and exergonic product formation, with reaction energy falling within the optimal window and barriers lower than those reported for bis-phosphine Pd(0) complexes. This methodology offers an efficient strategy for the de novo design of helical peptides and motifs and environmentally benign bioinorganic catalysts, from sequence to the reactivity of the metal center.

---

1. Introduction

Artificial metalloenzymes (ArMs) are engineered proteins or peptides that incorporate metal ions to catalyze chemical reactions, mimicking the function of natural metalloenzymes and/or organometallic catalysts. Modern ArMs contain a broad range of metals that are bound directly to amino acid side chains (Zn,¹ Ni,² V,³ even Ce⁴) or are introduced through an artificial cofactor. In the latter case, this approach allows the use of metals that are rare or absent in natural enzymes, such as Ni,⁵ Ru,^6,7 Rh,^7,8 Os,^9,10 and Ir,^11,12 as well as biocompatible Fe,^13–15 Mn,¹⁶ and Cu.¹² Harnessing this potential for organic synthesis could deliver immense benefits,^17–20 and recent advances have yielded ArMs that catalyze amine²¹ and sulfide oxidation,^22,23 hydroxylation,²⁴ epoxidation,^25–28 and the hydrolysis of esters and amides.²⁹ Computational de novo design can streamline this effort by guiding the search space, reducing cost, and elucidating reaction mechanisms, thereby accelerating innovation.

However, computational de novo design of ArMs should address challenges arising from both proteomics and organometallic chemistry, many of which remain unresolved. On the proteomics side, uncertain protonation states near the metal site, exchange between low-energy helical conformations that reshape second-sphere contacts, and long-range electrostatics from the periodic hydrogen-bond network all influence structure and reactivity in ways that truncated cluster models systematically miss. On the metal side, coordination-number changes, as well as hemilabile ligands and their possible dissociation, leading to aggregation that competes with catalysis, complicate modeling.^30,31 Accurately modeling bond formation and bond cleavage is sometimes beyond the practical domain of conventional force field methods,³² or requires re-parameterization,³³ while many machine learning interatomic potentials either lack the corresponding data in their training sets or require fine-tuning,^34,35 which motivates the use of quantum chemical methods. Yet even modern semiempirical and tight-binding approaches frequently fail to capture the energetics of relevant chemical reactions.^36,37 Consequently, for each system, one must jointly choose an appropriate DFT method and the relevant biomolecular environment. Cluster models of the catalytic center, which are overly restrictive size-wise, can yield misleading results.³⁸ At the same time, models that are too detailed become computationally prohibitive. These constraints motivate frameworks that retain DFT accuracy and generalizability while representing the full biomolecular environment.

In this work, we aim to preserve quantum-chemical accuracy while maintaining computational tractability and without constraining the active-site model to the theozyme. Conventional molecular models are constrained by the theozyme size, whereas a line group theory-based approach can emulate infinite monoperiodic (metallo)polymers and nanostructures with helical symmetry. This methodology has yielded accurate predictions of structures and properties for helical polymers and nanotubes.^39,40 It applies to any system with monomers aligned along a helical axis without crystalline periodicity, irrespective of composition.^41,42 Section 4 discusses the application of line group theory to monoperiodic systems. By treating the model systems as helical metallopolymers, we model regular peptides and metallopeptides at DFT-level accuracy, circumvent size limitations, and explore extended systems that serve as proxies for ArM scaffolds.

This conceptual model, which treats metal–peptide systems as metal centers embedded in a polymer matrix having a screw axis, is directly relevant to experimentally realized Pd-containing peptide-based and other polymer-based materials. For example, helically chiral Pd-bearing polymer matrices active in asymmetric Suzuki–Miyaura transformations have been reported.⁴³ Site-specific protein–protein conjugates can be formed by modifying a cysteine residue into the S-aryl-Pd-X derivative, which could covalently bind another cysteine residue on the external protein.⁴⁴ Cooperative catalysis between a lipase active site and a Pd center has been shown to accelerate alkyl–alkyl cross-coupling of 1-bromohexane with B-hexyl-9-BBN in a stable single-atom enzyme–metal complex.⁴⁵ In addition, DFT calculations predicted that several proteinogenic amino acids can tightly bind both Pd(0) and Pd(II) centers that catalyze Suzuki cross-coupling.⁴⁶ Taken together, these precedents motivate a quantum chemical, line group theory-based framework for the design of Pd–peptide systems.

Here, we propose a bottom-up methodology for the design of helical metallopeptides and helical metal-containing motifs based on DFT calculations within the line group formalism. First, we demonstrate that a symmetry-based line group formalism can reproduce the geometries and relative energies of canonical helical secondary structures of proteins with near-experimental accuracy. We then design helical peptide matrices that contain methionine-based Pd binding sites and evaluate these Pd–peptide systems at the DFT level. We evaluate structural rearrangements and stability across the Suzuki coupling cycle and assess reactivity in the key oxidative addition step. The framework enables the engineering of chimeric, metal-containing peptide matrices as models of helical (metal-binding motifs of) ArMs, with desired structures, properties, and compositions; it also yields DFT-quality potential-energy surfaces suitable for downstream applications, including training machine-learning interatomic potentials.

2. Results and discussion

2.1. Bottom-up design methodology: from sequence to 3D structure

The first step in the bottom-up methodology in Fig. 1 is to choose the metal-binding center for the metal of interest and the linker amino acid (AA) residues. We focus on Pd as a truly artificial metal that, at the same time, catalyzes a wide spectrum of homogeneous organic reactions not represented among natural enzymatic transformations, including C–C bond formation reactions used in the synthesis of complex small molecules for pharmaceuticals, agrochemicals, and organic electronic materials.^47–49 Methionine (Met) and selenomethionine (SeMet) were identified as the most promising proteinogenic AAs for strongly coordinating the key Pd(0)/Pd(II) intermediates along the Suzuki cross-coupling cycle. Histidine (His) provides somewhat weaker but relatively strong binding, whereas AAs with harder side-chain donors are predicted to bind Pd too weakly, which in turn is expected to promote rapid deactivation of the catalytic center.⁴⁶ Although the –SeCH₃ moiety of SeMet binds Pd(0)/Pd(II) more strongly than its sulfur analogue, direct chemical synthesis of extended poly(SeMet-X) polymers is impeded by the ready oxidation of selenium under ambient conditions. We therefore select Met as the residue that forms the core of the Pd-binding center in the designed metallopeptide.

Fig. 1 Bottom-up design methodology: from the selection of metal-binding residues and spacers to the evaluation of the metallopeptide reactivity.

Since the second step in Fig. 1 involves the design of a peptide sequence that incorporates and stabilizes non-native Pd(0)/Pd(II) centers and primes them for catalysis, we need to optimize intermetal distances between adjacent sites while preventing Pd(0) dissociation, which otherwise promotes rapid aggregation.⁵⁰ To this end, we insert uncharged spacer residues, alanine (Ala), valine (Val), or isoleucine (Ile), between Met donors to tune steric bulk and backbone hydrophobicity. This arrangement serves three purposes: (i) it spatially separates individual metal centers; (ii) it stabilizes the overall helical architecture; and (iii) it increases the hydrophobicity of the matrix, creating hydrophobic regions that can non-covalently bind organic substrates near the catalytic sites, thereby priming the scaffold for activity. In addition, although these hydrophobic residues do not bind Pd(0)/Pd(II) directly (see below), their side chains provide steric shielding reminiscent of common phosphine and N-heterocyclic carbene ligands.^51,52 At the same time, sterically unprotected Gly residues should be avoided due to the undesirable interaction of Pd(II)^53–56 with the polypeptide chain.

Step 2 of the methodology (Fig. 1) is explained in detail in section 2.2. Using line-group symmetry, we assemble helical polypeptides from the selected amino acid residues and, within the same formalism, scan electronic energies over a broad range of backbone torsion angles. This enables a systematic mapping of the geometric parameters and relative energies of the α- and π-helical conformations and provides an estimate of the barrier for the α-to-π-helical transition. Section 4 also includes the results of the DFT method selection and shows close agreement between computed and experimental geometrical parameters for the polyalanine α-helix.

Introducing a single spacer residue between Met residues yielded the most favorable helical conformations. In these constructs, the S–S distances in the adjacent Met residues that define each metal-binding site approached those previously reported for bis-Met Pd(0) complexes,⁴⁶ while the polypeptide backbone remained shielded by bulky nonpolar side chains. Accordingly, we focused subsequent analyses on (Met-X)_n polypeptides, where X denotes Ala, Val, or Ile.

Step 3 in Fig. 1 introduces the metallocenters and evaluates their reactivity within the peptide matrix. We track structural rearrangements of the designed Pd–peptide helices across the canonical steps of the Suzuki cross-coupling cycle, and reactivity in the key oxidative addition step. Step 3 completes the computational workflow and is presented in section 2.3.

Summarizing this subsection, we note that the design of helical (metallo)peptides was carried out in light of the MALEK principle, which states Met, Ala, Leu, Glu, and Lys as the strongest helix-forming residues in aqueous media.⁵⁷ At the same time, for Val and Ile, formation of helical structures in water is unfavorable,⁵⁸ although they do occur in helices within membrane segments, which are hydrophobic environments.⁵⁹ Therefore, spacer preselection should be made with the target medium in mind. Therefore, in section 2.2, we focus on (Met-X)_n polypeptides, where X denotes Ala, Val, or Ile, to model the intrinsic conformational properties of these polypeptides in the helical state using DFT. At the same time, when introducing Pd and evaluating activity, we primarily use water as the medium (characteristic of artificial metalloenzymes and “green” catalysis settings) and include non-polar THF for comparison. This approach allows us to describe the helical framework using DFT and the SMD implicit-solvent model, which treats bulk electrostatics and includes an implicit H-bond contribution via solvent acidity/basicity descriptors (see section 2.3).

2.2. Spatial structure and interconversion of the helices

To select a reliable electronic structure protocol, we benchmarked several widely used DFT protocols on poly(Ala)_n, for which experimental data are available, and chose the best-performing method. With the optimal protocol, the helical pitch (h), the number of residues per turn (Q_aa), H-bond lengths, and other related metrics closely match the reported values for poly(Ala)_n,⁶⁰ as discussed in section 4.

We modeled the geometries of (Met-X)_n in α-helical and π-helical conformations. Fig. 2 summarizes structural and energetic differences between (Met-X)_n helices in the α- and π-conformations and the parameters that govern the α-to-π-transition. For (Met-Ala)_n, the transition from the more densely packed α-helix to the less dense π helix increases the inner radius from 2.31 to 2.84 Å. Concomitantly, the number of hydrogen bonds per helical pitch rises from 3.60 to 4.45. The relationship between h and Q_aa shows that, as Q_aa increases during the transition, h first increases to about 5.6 Å, then abruptly decreases to about 5.0 Å near Q_aa ≈ 4.1. After the formation of new hydrogen bonds characteristic of the π-helix, the pitch increases again. Notably, the h vs. Q_aa trends are only weakly sequence-dependent across X = Ala, Val, and Ile.

Fig. 2 Structural and energetic descriptors of the α-to-π-helix transition. (a) Energy profile (ΔE, top) and helical pitch (h, bottom) as functions of the number of amino acids per turn, Q_aa. (b) Optimized geometries of the α- and π-helices of (Met-Ala)_n; key structural differences are listed below. Radii were measured as the distance between Cα atoms and the screw axis. Element coloring: S is yellow; O is red; N is blue; C is dark grey; H is light gray.

We now analyze the general energetics of the α-to-π-helix transition. A comparison of the top and bottom panels in Fig. 2a shows that the positions of the energy minima do not coincide with the minima in helical pitch. For the α-helix, the energy minimum occurs at h approximately 5.35 Å rather than at the smallest pitch near 5.1 Å. For the π-helix, the energy minimum is at h approximately equal to 5.1 Å, instead of being at the lowest observed pitch of about 5.0 Å. Across all α- and π-helix pairs, the relative energies, ΔE, differ by less than 2.5 kJ mol⁻¹.

While all Met-containing polypeptides considered here undergo the α-to-π-helix transition at Q_aa = 4.18, the corresponding activation barriers span from 20 kJ mol⁻¹ for poly(Met-Ile)_n to 24 kJ mol⁻¹ for poly(Met-Ala)_n (Fig. 2a). This 4 kJ mol⁻¹ spread reflects an unexpected steric acceleration that becomes more pronounced as the helix untwists along the α to π pathway. The transition follows an axial elongation pathway that redistributes the H-bond network; the screw motion intrinsic to the α-to-π conversion entails a concerted sliding of the backbone that positions N–H donors in the helical H-bond network between two C=O acceptors. At the PBE/def2-TZVP level, the associated imaginary frequencies are 27.1i, 22.1i, and 19.6i cm⁻¹ for poly(Met-Ala)_n, poly(Met-Val)_n, and poly(Met-Ile)_n, respectively, based on analysis of the corresponding local H-bond clusters. The supporting ZIP archive includes the coordinates of the PES saddle points corresponding to the N–H donor slides in the local H-bond clusters, with the corresponding animated imaginary modes, together with visualizations of the collective screw-expansion motion.

The computed energy profile displays two basins of comparable depth, while the helical pitch changes abruptly from about 5.5 Å in the α basin to about 5.0 Å in the π basin (Fig. 2a, bottom panel). This discontinuity is accompanied by a notable reorganization of backbone hydrogen bonds (Fig. 2b) and a measurable widening of the helical core. Structural changes of this magnitude cannot arise from infinitesimal fluctuations about a single minimum. In Landau's phenomenological theory,⁶¹ second-order transitions feature a continuous order parameter, whereas first-order transitions exhibit discontinuous jumps in the order parameter, coexisting phases with distinct symmetries, and an energy barrier between them. In this particular case, α-to-π interconversion demonstrates all these characteristics, providing a rationale to see it as a first-order phase transition.

In our formalism, the chains are infinite, so the thermodynamic limit is effectively reached. In this regime, the system could exhibit a latent enthalpy and may show metastability and hysteresis on molecular dynamics timescales, consistent with viewing the α- and π-helices as alternative phases of the same polymer. A more complete group-theoretic analysis that constructs an explicit path in order-parameter space and mathematically devises the relevant symmetry-breaking invariant is suggested in the future, as well as a more complete consideration of lateral packing present in real crystalline systems, for a rigorous assignment of the α-to-π-transformation as a first-order phase transition.

2.3. Pd incorporation and reactivity evaluation

With the poly(Met-X)_n helices identified, we next introduced Pd(0) centers into the peptide matrices. Representative metallopeptide geometries are shown in Table 1. Metal binding elicits different responses in the helical forms. In the α-helical structure, the changes are minimal: Q_aa and h increase slightly, consistent with a small axial elongation. By contrast, the π-helices undergo a more pronounced elongation that better accommodates the metal center (see below). Across X = Ala, Val, and Ile, both Q_aa and h increase, whereas the translational shift per residue, f, remains constant in most cases (see Table 1).

Table 1 Geometries of Pd(0)-bound peptides and the comparison of structural parameters of the poly(Met-X)_n peptides (X = Ala, Val, Ile)

Polymer/parameter	Poly(Met-Ala)		Poly(Met-Val)		Poly(Met-Ile)
	Pd-free	With Pd	Pd-free	With Pd	Pd-free	With Pd
Pd atoms are shown as cerulean spheres. Peptide residues are shown as sticks (Met in grey, X in green). S atoms in Met residues are orange. The helical backbone structure is superimposed.
α-Helical conformation
Q aa	3.61	3.63	3.61	3.61	3.64	3.63
h, Å	5.37	5.39	5.40	5.41	5.39	5.42
f, Å	1.49	1.49	1.49	1.50	1.48	1.50
π-Helical conformation
Q aa	4.43	4.50	4.42	4.50	4.41	4.52
h, Å	5.11	5.18	5.13	5.22	5.11	5.23
f, Å	1.15	1.15	1.16	1.16	1.16	1.16

---

Although all three poly(Met-X)_n Pd-free polymers adopt very similar helical structures, poly(Met-Ala)_n is easier to prepare under standard peptide synthesis protocols,^62,63 exhibits slightly higher structural stability across the α-to-π-transformation (hence slightly more controllable dynamics), and contains fewer atoms, which improves computational tractability. At the same time, selecting sterically unprotected Gly is inadvisable (see section 2.1). Importantly, in the reactivity evaluations below, we target water media. As noted in section 2.1, Met and Ala have a strong propensity to form helical structures in water, whereas helices containing Val and Ile are more typical of hydrophobic (membrane-like) environments. We therefore selected the [Pd(Met-Ala)₂]_n polymer for further investigation.

We next introduced Pd(II) centers that mimic key intermediates in the Suzuki coupling of PhBr with Ph- and vinyl-boronic esters into the peptide matrices. For both α- and π-helical forms, we optimized the geometries and performed relaxed potential energy surface (PES) scans to probe structural rearrangements along the catalytic cycle. Results for the π- and α-helical structures are summarized in Table 2. The π helix is structurally conservative across all intermediates considered, with Q_aa, h, and f changing only marginally. In contrast, the α-helical [Pd(Met-Ala)₂]_n chain shows a larger response, especially in the case of [PhPdBr(Met-Ala)₂]_n, where both Q_aa and h decrease relative to the other intermediates, consistent with local tightening within each helical pitch and the buildup of structural strain.

Table 2 Potential energy surfaces near the minima and structural parameters of the π- and α-helical conformations of [PhPdBr(Met-Ala)₂]_n, [PhPd(OH)(Met-Ala)₂]_n, and [Ph₂Pd(Met-Ala)₂]_n metallopeptides. See Fig. S1 for structures of [PhVinPd(Met-Ala)₂]_n. Element coloring: Pd is cerulean; Br is dark red; S is yellow; O is red; N is blue; C is dark grey; H is light gray

Parameter	[Pd(Met-Ala)2]n	[PhPdBr(Met-Ala)2]n	[PhPd(OH)(Met-Ala)2]n	[Ph2Pd(Met-Ala)2]n	[PhVinPd(Met-Ala)2]n
π-Helical conformation
Q aa	4.50	4.46	4.48	4.46	4.47
h, Å	5.18	5.17	5.18	5.17	5.18
f, Å	1.15	1.16	1.16	1.16	1.16
α-Helical conformation
Q aa	3.63	3.59	3.65	3.66	3.62
h, Å	5.39	5.31	5.40	5.42	5.39
f, Å	1.49	1.48	1.48	1.48	1.49

---

Across all Pd(II)–peptide polymers studied, the α-helical forms are significantly less favorable than their π-helical counterparts (Table 2 and Fig. S1). In the case of [Ph₂Pd(Met-Ala)₂]_n, the relative energy of the α-helical chain, ΔE, is +88.5 kJ mol⁻¹ per monomer. The α-helical conformations of [PhPdBr(Met-Ala)₂]_n and [PhPd(OH)(Met-Ala)₂]_n are +55.1 and +67.8 kJ mol⁻¹ per monomer, relative to the corresponding π-helical conformations. We therefore conclude that the π-helical form is universally preferred for incorporating canonical Pd(II) intermediates of the Suzuki coupling, whereas the α-helical alternative is strongly energetically disfavored.

To estimate the potential of the [Pd(Met-Ala)₂]_n polymer in C–C coupling reactions, we analyzed the key oxidative addition step. Along a relaxed PES scan in the case of the periodic, less favorable α-helical matrix, we found that dissociation of one Met ligand from the [Pd(Met)₂] center (state 1 in Fig. 3a) was required to form the Pd(0)–aryl halide π-complex, [Pd(Met)(η²-PhBr)] (1′). Formation of 1′ is strongly endothermic, ΔE = +109 kJ mol⁻¹, relative to 1. The barrier from 1′ to the oxidative addition transition state TS1 is small, ΔE^‡ = 17 kJ mol⁻¹; however, referenced to 1, it amounts to 126 kJ mol⁻¹ because forming 1′ is highly endothermic. The subsequent formation of the α-helical product, [PhPdBr(Met-Ala)₂]_n (intermediate 2), is exothermic relative to 1′ (ΔE = −84.7 kJ mol⁻¹) while being endothermic relative to 1 (24.3 kJ mol⁻¹).

Fig. 3 (a) Relaxed PES scan of PhBr oxidative addition to the [Pd(Met)₂] center in α-helical [PhPdBr(Met-Ala)₂]_n; (b) free energy profile of PhBr oxidative addition to the [Pd(Met)₂] center in the π-helical [PhPdBr(Met-Ala)₂]_n. See text for details of the computational procedures used in (a) and (b). Note that in the pre-reaction state 1, PhBr is assumed to be non-bonded and is therefore not shown. (Free) Energy profiles include the active center structure, with Met shown schematically for brevity. Element coloring: Pd is cerulean; Br is dark red; S is yellow; O is red; N is blue; C is dark grey; H is light gray.

In summary, whereas the Pd-free polypeptides display two nearly isoenergetic minima separated by barriers below approximately 24.5 kJ mol⁻¹ (Fig. 2), Pd(II) incorporation constrains the helix in the π-conformation (Table 2). The corresponding oxidative addition requires energetically unfavorable Met dissociation, also being endothermic, likely due to strain. These considerations motivate our preferential focus on the more favorable π-helical scaffold below.

To obtain the energetic parameters of the oxidative addition in the π-helical matrix within model water and THF solvents at a higher level of theory, we proceeded as follows. A chain containing four [PhPdBr(Met-Ala)₂]_n units was extracted from the periodic structure, with the terminal atoms capped with Me substituents. One of the inner Pd centers was selected as the reactive center (Fig. 3b). Geometry optimizations of the pre- and post-oxidative addition intermediates, as well as the transition state, were performed at the PBE/def2-SVP (H: def2-TZVP) level. All atoms were constrained except those of the reactive center and the aliphatic side chains of the corresponding two Met residues. The resulting Hessian in the TS system had the target imaginary mode corresponding to C–Br bond cleavage, with frequency 106.15i cm⁻¹. TS coordinates for both the extended helix and the cluster models, along with the mode visualization, are provided in the supporting ZIP archive. Then, we performed single-point energy refinements at the ωB97X-V/def2-TZVP-gCP with SMD solvation (water and THF) and combined these with thermochemical corrections computed at the PBE/def2-SVP (H: def2-TZVP) level to estimate Gibbs free energies of activation and of elementary steps. ωB97X-V was selected because it ranks the best among non-double-hybrid functionals on the comprehensive GMTKN55 benchmark (including thermochemistry, barrier heights, and noncovalent interactions) and has been recommended for metalloenzyme reactions as one of the best options.^64,65

Fig. 3b summarizes the energetic parameters and optimized geometries for the oxidative addition of PhBr to the [Pd(Met)₂] center in the π-helical matrix. Formation of the pre-reaction intermediate from the metallopeptide and PhBr (state 1) yields 1′ with a bridging Br ligand and lies at ΔG = 0.0 and +5.2 kJ mol⁻¹ relative to 1 in water and THF, correspondingly. From 1, oxidative addition proceeds over a relatively low barrier of 64.5 and 73.7 kJ mol⁻¹ in water and THF, respectively. The values reported for typical bisligated phosphine Pd(0) complexes vary from 104 to 122 kJ mol⁻¹.⁶⁶ We took the same bisligated complexes (L = PPh₃, PCy₃, and SPhos; P(t-Bu)₃ did not form the bisligated state) and computed the activation barrier in 1 → TS1 at the ωB97X-V/def2-TZVP-gCP//B97-3c level with SMD solvation. The barriers ranged from 82 to 122 kJ mol⁻¹ in water and from 97 to 136 kJ mol⁻¹ in THF, correspondingly; in all the cases, the PPh₃-ligated complex corresponded to the lowest barrier (all values and structures are provided in the supporting ZIP archive). Finally, the formation of the [PhPdBr(Met)₂] center is strongly exergonic, with ΔG = −118.7 and −111.4 kJ mol⁻¹ in water and THF, respectively, relative to 1. Taken together, these results indicate that the π-helical matrix can accommodate Pd(0)/Pd(II) centers and facilitate the key oxidative addition step relevant to Suzuki and many other cross-coupling reactions.

3. Summary and conclusions

In this work, we present a novel bottom-up computational framework that treats metallopeptides as helical polymers at the DFT level beyond theozyme-sized clusters. The methodology employs line group theory with variable translational parameters to systematically generate α- and π-helical conformations from monomeric peptide units, enabling a comprehensive analysis of polypeptide matrix geometries and their corresponding potential energy surfaces. The presented approach achieves near-experimental geometric accuracy in modeling α-polyalanine chains and offers an efficient strategy for de novo design, with potential applications in training machine learning force fields and exploring other canonical helical structures (e.g., 3₁₀-helices), making it a versatile tool for advancing peptide and bioinorganic catalyst design.

This approach not only facilitates accurate non-empirical modeling of infinite (metallo)peptide systems by exploiting helical symmetry (thereby overcoming the computational limitations of large molecular models, when considered at the DFT level) but also provides a robust theoretical foundation for characterizing structural transitions such as the α-to-π helix transformation. This transition—manifesting itself in the reformation of hydrogen-bond networks and backbone rearrangements—exhibited a 20–24 kJ mol⁻¹ energy barrier and an abrupt helical pitch change (from ∼5.5 Å to ∼5.0 Å), suggesting that such a transformation could manifest itself as a first-order phase transition.

Our results demonstrate the efficacy of this approach in designing [Pd-(Met-X)₂]_n metallopeptides (X = Ala, Val, Ile). We designed the spatial structures and modeled potential energy surfaces near the minima corresponding to the canonical intermediates in the Suzuki cross-coupling reaction, identifying stable helical conformations with Met residue arrangement favoring Pd(0) and Pd(II) binding. Incorporation of Pd(0) and Pd(II) centers revealed that π-helical conformations are energetically more favorable for stabilizing models of the canonical intermediates, especially [PhPdBr(Met-Ala)₂]. In contrast, α-helical structures exhibited strain, requiring energetically costly dissociation of one Met residue for PhBr oxidative addition to proceed, highlighting the superior properties of Met-containing π-helices for catalysis involving the [Pd(Met)₂] center and indicating potential applicability to other helical metallopeptide systems incorporating “softer” transition metals. These findings highlight the critical role of helical conformation in modulating catalytic activity, offering actionable insights for tailoring metallopeptide sequences.

To obtain the kinetic and thermodynamic parameters of PhBr oxidative addition in an aqueous (or THF) medium using a high-accuracy DFT protocol, we generated a cluster model directly extracted from the DFT-optimized periodic π-helix, avoiding a simplistic theozyme model. Indeed, by construction, the cluster inherits second-sphere packing, backbone strain, and hydrophobic shielding from the parent periodic π-helix. The (free) energy profile showed a low barrier of PhBr oxidative addition, well below barriers typical for bis-phosphine Pd(0) complexes.

Such metallopeptides serve as proxies for artificial metalloenzymes due to their ability to mimic the coordination environments and catalytic functionality of natural enzymes while incorporating non-biological metals like Pd for synthetic transformations. Beyond their utility as enzyme mimics, such metal–peptide systems are promising because of their hybrid nature, blending the structural versatility of polypeptides and inherent biodegradability of the peptide matrix with the reactivity of organometallic complexes. The pursuit of such designed Pd–peptide assemblies reflects the growing interest in experimental studies. Metal–peptide frameworks, such as metal-α-helix systems, demonstrate that porous frameworks with characteristic metal sites can be constructed from short peptides.⁶⁷ At the same time, the functional potential of non-helical Pd–peptides has been validated in a biological context: a Pd(II)-stapled β-sheet miniprotein has been shown to catalyze bioorthogonal depropargylation reactions inside living cells.⁶⁸ Pd-containing helical structures have also drawn interest beyond biosystems: the helically chiral polymer ligand PQXphos afforded axially chiral biaryl esters in asymmetric Suzuki coupling with high enantioselectivities, and its helical chirality can be switched by solvent to access either enantiomer.⁴³ Collectively, these works motivate exploration of new structural frameworks, such as the Pd-Met-based helices proposed here. By providing a predictive framework for designing metallopeptides and metallo-helical polymers with tailored reactivity, our work facilitates the engineering of bioinspired catalysts and addresses the pressing need for environmentally benign yet efficient and selective catalytic systems.

4. Computational details

4.1. The mathematical formalism of line-group theory

In line-group theory, the symmetry of polymers is described by equation L = ZP, where L is the symmetry group, Z denotes the group of generalized translations, and P represents the symmetry group of the monomer.⁶⁹ In this context, the group Z may (or may not) contain the translation group as a subgroup. Depending on the combination of Z and P, all line groups are classified into 13 families.⁶⁹

The canonical structures of α and π helices are described by a screw axis. This corresponds to a cyclic group with generator Z = (C_Q|f), where Q is the order of the screw axis and f denotes the magnitude of the shift along the screw axis (with the condition that Q ≥ 1). Technically, Q may either be a rational or an irrational number. As the CRYSTAL17 software used for quantum chemical modeling requires translational periodicity, we will consider only the case in which Q is rational. Only symmetry groups with rational Q include a translation group as a subgroup; they can be expressed within crystallographic notation (q, p, t) and polymer notation (q, r, f).

A rational value of Q can be expressed as the quotient of two co-prime positive integers: Q = q/r. In this case:

(C_Q|f)^q = (C_Q^q|q·f) = (E|t), (1a)

qf = t, (1b)

where t is the translation vector and E is the identity element. Since Q represents the order of the screw axis, the rotation angle can be calculated as:

This implies that before reaching a translationally equivalent position, the helix completes r full turns, while the monomer repeats q times. Consequently, Q = q/r is the number of monomers per turn, r is the number of turns, q is the total number of monomers in the structure. Thus, the height (pitch) of a single turn is expressed as:

These parameters under consideration are presented schematically in Fig. 4. Finally, in our case, a monomer in the chain consists of a single amino acid. Therefore, Q corresponds to the number of amino acid residues per turn (Q_aa).

Fig. 4 Schematic representation of a helical structure, along with its characteristic structural parameters described in eqn (1b), (2), and (3). Monomers are represented by spheres; in (a), the blue spheres indicate monomers belonging to the first turn of the helix; (b) shows rotation angle φ between monomers connected via helical transformation, with spheres indicating distinct positions.

Given the values of q and r, the parameter p can be determined using the following formula:

rp = lq + 1 or rp = 1 mod q, (4)

where l is a positive number. This formula is valid only when the estimated structure is described exclusively by a screw axis. Thus, to define the symmetry group, it is sufficient to know only one set of parameters—either (q, p, t) or (q, r, f)—from which the other can be derived. A more detailed description of line group theory can be found in the dedicated book.⁶⁹ Further discussion of the used mathematical formalism is given in section S2.

Helix transitions between α and π conformations were studied by sequentially optimizing structures across a range of helical axis orders (Q). For each structure, the symmetry-irreducible unit of the preceding monomer and a shift parameter (f) were used to generate the next structure. Geometry optimization was then performed while fixing Q. The helical axis order Q was converted to Q_aa (amino acid residues per turn) using standard helical parameter equations (see SI for details). Structures were optimized for Q∈(1.9, 2.1), which corresponds to φ∈(172°, 189°) and Q_aa∈(3.8, 4.2).

4.2. DFT calculations

We used the CRYSTAL17^70,71 program to perform DFT geometry optimizations and electronic structure calculations. Geometry optimizations were carried out with convergence thresholds of a maximum energy gradient of 0.0003 a.u. (TOLDEG) and a maximum atomic displacement of 0.0012 a.u. (TOLDEX). To ensure accurate electron densities and numerical integration, we enabled XLGRID and employed pruned radial and angular grids with tolerances TOLLDENS = 8 and TOLLGRID = 16. Self-consistent field convergence was enforced with integral accuracy thresholds TOLINTEG = 8 8 8 8 16 and a pseudopotential core-radius tolerance TOLPSEUD = 7. Brillouin-zone sampling used a Monkhorst–Pack grid⁷² generated with SHRINK = 8 16. Exchange–correlation effects were described with the PBE functional (PBEXC).⁷³ CRYSTAL17 performs ab initio calculations on periodic systems using the linear combination of atomic orbitals (LCAO) approximation. Hydrogen atoms were described with the pob-TZVP⁷⁴ basis set to obtain more accurate H-bond properties, while all other elements were described with pob-DZVP-rev2^75,76 to reduce computational workload. This choice follows the benchmarking in Table 3, which compares the performance of PBE and PBE0,⁷⁷ each with and without D3⁷⁸ dispersion corrections, as well as the comparison of full pob-TZVP-based and our protocols included in the SI. Table 3 shows that PBE reproduces structural parameters with accuracy comparable to the hybrid PBE0 while requiring fewer CPU resources. Notably, inclusion of the D3 dispersion correction worsens agreement with experiment for this system.

Table 3 Structural parameters of the polyalanine α-helix, as modeled with different computational protocols

Parameter	PBE	PBE-D3	PBE0	PBE0-D3	Experiment
Q aa	3.60	3.59	3.61	3.59	47/13 ^60,79 (3.62)
h, Å	5.37	5.31	5.36	5.30	5.41 ^80
f, Å	1.49	1.47	1.49	1.47	1.50,^80 1.50 ^60
Hydrogen bond (C=O⋯H–N), Å	2.86	2.80	2.87	2.82	2.82,^79 2.91,^80 2.87 ^60
R(αC), Å	2.30	2.29	2.28	2.30	2.29 ^60,79,80

---

Additional calculations on cluster models were performed using ORCA 5.0.3.⁸¹ Geometry optimizations were conducted using the PBE functional, with the def2-TZVP basis set for hydrogen atoms and the def2-SVP basis set for all other elements.⁸² The def2/J auxiliary basis set⁸³ was employed to enable the RI approximation.⁸⁴ A TightSCF convergence criterion and the DefGrid2 integration grid were used. The same settings were applied for vibrational frequency calculations to obtain thermochemical corrections.

Spin-restricted single-point Hessian calculations to locate saddle points for H-bond sliding were performed at the PBE/def2-TZVP level, using the def2/J auxiliary basis set for the RI approximation. A VeryTightSCF convergence criterion and the DefGrid3 integration grid were employed to ensure high accuracy.

Single-point energy refinements were performed with the ωB97X-V functional,⁸⁵ using the resolution-of-identity with chain-of-spheres exchange approximation,⁸⁶ paired with the def2-TZVP basis set and the def2/J auxiliary basis. A TightSCF convergence criterion was applied with a DefGrid2 integration grid to balance accuracy and computational cost.

For solvation effects, single-point energy calculations were conducted with the SMD model using the B3LYP functional,^87–89 LANL2DZ for Pd and Br,^90,91 and 6-31G* for other elements,^92,93 following the recommended protocol of B3LYP/6-31G* in the original publication⁹⁴ as closely as possible. TightSCF with VerySlowConv and DefGrid2 was enforced to ensure numerical stability. Thermochemical corrections were computed in ORCA using the default settings. A standard-state correction of 7.93 kJ mol⁻¹ was added to the thermochemical corrections to convert from the 1 atm ideal gas reference to the 1 M solution.

Conflicts of interest

There are no conflicts of interest to declare.

Data availability

The data underlying this study are available in the published article and its supplementary information (SI). Supplementary information: the supporting PDF file includes computational details and supplementary figures. The supporting ZIP archive includes the molecular and extended helical structures in the XYZ format, visualizations of imaginary modes, and the .xlsx tables listing computed (free) energies. See DOI: https://doi.org/10.1039/d5qi01794g.

Acknowledgements

The authors thank Natalia Soloveva, an independent designer. Anton V. Domnin and Yaroslav V. Solovev were supported by RSF grant 25-74-30002. Computations by the Moscow-based co-authors were conducted using the shared HPC facilities at Lomonosov Moscow State University.

References

1. M. L. Zastrow and V. L. Pecoraro, Designing Hydrolytic Zinc Metalloenzymes, Biochemistry, 2014, 53(6), 957–978,  DOI:10.1021/bi4016617.
2. K. J. Naughton, R. E. Treviño, P. J. Moore, A. E. Wertz, J. A. Dickson and H. S. Shafaat, In Vivo Assembly of a Genetically Encoded Artificial Metalloenzyme for Hydrogen Production, ACS Synth. Biol., 2021, 10(8), 2116–2120,  DOI:10.1021/acssynbio.1c00177.
3. F. van de Velde, I. W. C. E. Arends and R. A. Sheldon, Vanadium-Catalysed Enantioselective Sulfoxidations: Rational Design of Biocatalytic and Biomimetic Systems, Top. Catal., 2000, 13(3), 259–265,  DOI:10.1023/A:1009094619249.
4. A. S. Klein, F. Leiss-Maier, R. Mühlhofer, B. Boesen, G. Mustafa, H. Kugler and C. Zeymer, A De Novo Metalloenzyme for Cerium Photoredox Catalysis, J. Am. Chem. Soc., 2024, 146(38), 25976–25985,  DOI:10.1021/jacs.4c04618.
5. W. Wang, R. Tachibana, K. Zhang, K. Lau, F. Pojer, T. R. Ward and X. Hu, Artificial Metalloenzymes with Two Catalytic Cofactors for Tandem Abiotic Transformations, Angew. Chem., Int. Ed., 2025, 64(8), e202422783,  DOI:10.1002/anie.202422783.
6. Y. Okamoto, R. Kojima, F. Schwizer, E. Bartolami, T. Heinisch, S. Matile, M. Fussenegger and T. R. Ward, A Cell-Penetrating Artificial Metalloenzyme Regulates a Gene Switch in a Designer Mammalian Cell, Nat. Commun., 2018, 9(1), 1943,  DOI:10.1038/s41467-018-04440-0.
7. D. F. Sauer, Y. Qu, M. a. S. Mertens, J. Schiffels, T. Polen, U. Schwaneberg and J. Okuda, Biohybrid Catalysts for Sequential One-Pot Reactions Based on an Engineered Transmembrane Protein, Catal. Sci. Technol., 2019, 9(4), 942–946,  10.1039/C8CY02236D.
8. S. Morra and A. Pordea, Biocatalyst–Artificial Metalloenzyme Cascade Based on Alcohol Dehydrogenase, Chem. Sci., 2018, 9(38), 7447–7454,  10.1039/C8SC02371A.
9. J. M. Zimbron, T. Heinisch, M. Schmid, D. Hamels, E. S. Nogueira, T. Schirmer and T. R. Ward, A Dual Anchoring Strategy for the Localization and Activation of Artificial Metalloenzymes Based on the Biotin–Streptavidin Technology, J. Am. Chem. Soc., 2013, 135(14), 5384–5388,  DOI:10.1021/ja309974s.
10. T. Kokubo, T. Sugimoto, T. Uchida, S. Tanimoto and M. Okano, The Bovine Serum Albumin–2-Phenylpropane-1,2-Diolatodioxo-Somium(VI) Complex as an Enantioselective Catalyst for Cis-Hydroxylation of Alkenes, J. Chem. Soc., Chem. Commun., 1983, 14, 769–770,  10.1039/C39830000769.
11. J. G. Rebelein, Y. Cotelle, B. Garabedian and T. R. Ward, Chemical Optimization of Whole-Cell Transfer Hydrogenation Using Carbonic Anhydrase as Host Protein, ACS Catal., 2019, 9(5), 4173–4178,  DOI:10.1021/acscatal.9b01006.
12. J. Zhao, J. G. Rebelein, H. Mallin, C. Trindler, M. M. Pellizzoni and T. R. Ward, Genetic Engineering of an Artificial Metalloenzyme for Transfer Hydrogenation of a Self-Immolative Substrate in Escherichia Coli's Periplasm, J. Am. Chem. Soc., 2018, 140(41), 13171–13175,  DOI:10.1021/jacs.8b07189.
13. M. V. Doble, A. G. Jarvis, A. C. C. Ward, J. D. Colburn, J. P. Götze, M. Bühl and P. C. J. Kamer, Artificial Metalloenzymes as Catalysts for Oxidative Lignin Degradation, ACS Sustainable Chem. Eng., 2018, 6(11), 15100–15107,  DOI:10.1021/acssuschemeng.8b03568.
14. S. Shima, D. Chen, T. Xu, M. D. Wodrich, T. Fujishiro, K. M. Schultz, J. Kahnt, K. Ataka and X. Hu, Reconstitution of [Fe]-Hydrogenase Using Model Complexes, Nat. Chem., 2015, 7(12), 995–1002,  DOI:10.1038/nchem.2382.
15. D. J. Raines, J. E. Clarke, E. V. Blagova, E. J. Dodson, K. S. Wilson and A.-K. Duhme-Klair, Redox-Switchable Siderophore Anchor Enables Reversible Artificial Metalloenzyme Assembly, Nat. Catal., 2018, 1(9), 680–688,  DOI:10.1038/s41929-018-0124-3.
16. L. Leone, D. D’Alonzo, V. Balland, G. Zambrano, M. Chino, F. Nastri, O. Maglio, V. Pavone and A. Lombardi, Mn-Mimochrome VI*a: An Artificial Metalloenzyme With Peroxygenase Activity, Front. Chem., 2018, 6, 590,  DOI:10.3389/fchem.2018.00590.
17. C. Van Stappen, Y. Deng, Y. Liu, H. Heidari, J.-X. Wang, Y. Zhou, A. P. Ledray and Y. Lu, Designing Artificial Metalloenzymes by Tuning of the Environment beyond the Primary Coordination Sphere, Chem. Rev., 2022, 122(14), 11974–12045,  DOI:10.1021/acs.chemrev.2c00106.
18. M. Wittwer, U. Markel, J. Schiffels, J. Okuda, D. F. Sauer and U. Schwaneberg, Engineering and Emerging Applications of Artificial Metalloenzymes with Whole Cells, Nat. Catal., 2021, 4(10), 814–827,  DOI:10.1038/s41929-021-00673-3.
19. T. Vornholt, F. Christoffel, M. M. Pellizzoni, S. Panke, T. R. Ward and M. Jeschek, Systematic Engineering of Artificial Metalloenzymes for New-to-Nature Reactions, Sci. Adv., 2021, 7(4), eabe4208,  DOI:10.1126/sciadv.abe4208.
20. F. Schwizer, Y. Okamoto, T. Heinisch, Y. Gu, M. M. Pellizzoni, V. Lebrun, R. Reuter, V. Köhler, J. C. Lewis and T. R. Ward, Artificial Metalloenzymes: Reaction Scope and Optimization Strategies, Chem. Rev., 2018, 118(1), 142–231,  DOI:10.1021/acs.chemrev.7b00014.
21. Y. S. Choi, H. Zhang, J. S. Brunzelle, S. K. Nair and H. Zhao, In Vitro Reconstitution and Crystal Structure of P-Aminobenzoate N-Oxygenase (AurF) Involved in Aureothin Biosynthesis, Proc. Natl. Acad. Sci. U. S. A., 2008, 105(19), 6858–6863,  DOI:10.1073/pnas.0712073105.
22. A. Pordea, M. Creus, J. Panek, C. Duboc, D. Mathis, M. Novic and T. R. Ward, Artificial Metalloenzyme for Enantioselective Sulfoxidation Based on Vanadyl-Loaded Streptavidin, J. Am. Chem. Soc., 2008, 130(25), 8085–8088,  DOI:10.1021/ja8017219.
23. F. van de Velde and L. Könemann, Enantioselective Sulfoxidation Mediated by Vanadium-Incorporated Phytase: A Hydrolase Acting as a Peroxidase, Chem. Commun., 1998, 17, 1891–1892,  10.1039/A804702B.
24. H. C. Kolb, M. S. VanNieuwenhze and K. B. Sharpless, Catalytic Asymmetric Dihydroxylation, Chem. Rev., 1994, 94(8), 2483–2547,  DOI:10.1021/cr00032a009.
25. M. T. Reetz, M. Rentzsch, A. Pletsch, M. Maywald, P. Maiwald, J. J.-P. Peyralans, A. Maichele, Y. Fu, N. Jiao, F. Hollmann, R. Mondière and A. Taglieber, Directed Evolution of Enantioselective Hybrid Catalysts: A Novel Concept in Asymmetric Catalysis, Tetrahedron, 2007, 63(28), 6404–6414,  DOI:10.1016/j.tet.2007.03.177.
26. M. T. Reetz, Directed Evolution of Selective Enzymes and Hybrid Catalysts, Tetrahedron, 2002, 58(32), 6595–6602,  DOI:10.1016/S0040-4020(02)00668-3.
27. A. Fernández-Gacio, A. Codina, J. Fastrez, O. Riant and P. Soumillion, Transforming Carbonic Anhydrase into Epoxide Synthase by Metal Exchange, ChemBioChem, 2006, 7(7), 1013–1016,  DOI:10.1002/cbic.200600127.
28. K. Okrasa and R. J. Kazlauskas, Manganese-Substituted Carbonic Anhydrase as a New Peroxidase, Chem. – Eur. J., 2006, 12(6), 1587–1596,  DOI:10.1002/chem.200501413.
29. J. Chin, M. Banaszczyk, V. Jubian, J. H. Kim and K. Mrejen, Artificial Hydrolytic Metalloenzymes. in Bioorganic Chemistry Frontiers; ed. H. Dugas, Springer, Berlin, Heidelberg, 1991, pp 175–194.  DOI:10.1007/978-3-642-76241-3_5.
30. K. D. Vogiatzis, M. V. Polynski, J. K. Kirkland, J. Townsend, A. Hashemi, C. Liu and E. A. Pidko, Computational Approach to Molecular Catalysis by 3d Transition Metals: Challenges and Opportunities, Chem. Rev., 2019, 119(4), 2453–2523,  DOI:10.1021/acs.chemrev.8b00361.
31. A. Lledós, Computational Organometallic Catalysis: Where We Are, Where We Are Going, Eur. J. Inorg. Chem., 2021, 2021(26), 2547–2555,  DOI:10.1002/ejic.202100330.
32. L. W. Bertels, L. B. Newcomb, M. Alaghemandi, J. R. Green and M. Head-Gordon, Benchmarking the Performance of the ReaxFF Reactive Force Field on Hydrogen Combustion Systems, J. Phys. Chem. A, 2020, 124(27), 5631–5645,  DOI:10.1021/acs.jpca.0c02734.
33. F. Fiesinger, D. Gaissmaier, M. van den Borg, J. Beßner, A. C. T. van Duin and T. Jacob, Development of a Mg/O ReaxFF Potential to Describe the Passivation Processes in Magnesium-Ion Batteries, ChemSusChem, 2023, 16(3), e202201821,  DOI:10.1002/cssc.202201821.
34. M. Radova, W. G. Stark, C. S. Allen, R. J. Maurer and A. P. Bartók, Fine-Tuning Foundation Models of Materials Interatomic Potentials with Frozen Transfer Learning, Npj Comput. Mater., 2025, 11(1), 237,  DOI:10.1038/s41524-025-01727-x.
35. O. T. Unke, S. Chmiela, H. E. Sauceda, M. Gastegger, I. Poltavsky, K. T. Schütt, A. Tkatchenko and K.-R. Müller, Machine Learning Force Fields, Chem. Rev., 2021, 121(16), 10142–10186,  DOI:10.1021/acs.chemrev.0c01111.
36. J. Sirirak, N. Lawan, M. W. V. der Kamp, J. N. Harvey and A. J. Mulholland, Benchmarking Quantum Mechanical Methods for Calculating Reaction Energies of Reactions Catalyzed by Enzymes, PeerJ Phys. Chem., 2020, 2, e8,  DOI:10.7717/peerj-pchem.8.
37. M. Bursch, H. Neugebauer and S. Grimme, Structure Optimisation of Large Transition-Metal Complexes with Extended Tight-Binding Methods, Angew. Chem., 2019, 131(32), 11195–11204,  DOI:10.1002/ange.201904021.
38. E. A. Pidko, Toward the Balance between the Reductionist and Systems Approaches in Computational Catalysis: Model versus Method Accuracy for the Description of Catalytic Systems, ACS Catal., 2017, 7(7), 4230–4234,  DOI:10.1021/acscatal.7b00290.
39. A. V. Domnin, V. V. Porsev and R. A. Evarestov, DFT Modeling of Electronic and Mechanical Properties of Polytwistane Using Line Symmetry Group Theory, Comput. Mater. Sci., 2022, 214, 111704,  DOI:10.1016/j.commatsci.2022.111704.
40. A. V. Domnin, I. E. Mikhailov and R. A. Evarestov, DFT Study of WS2-Based Nanotubes Electronic Properties under Torsion Deformations, Nanomaterials, 2023, 13(19), 2699,  DOI:10.3390/nano13192699.
41. R. A. Evarestov, V. V. Porsev, D. D. Kuruch and P. Yu Cherezova, Torsion and Axial Deformations of Chalcogen Helical Chains (S, Se, Te): First Principles Calculations Using Line Symmetry Groups, Nanomaterials, 2025, 15(7), 505,  DOI:10.3390/nano15070505.
42. V. V. Porsev, A. V. Bandura and R. A. Evarestov, Ab Initio Modeling of Helically Periodic Nanostructures Using CRYSTAL17: A General Algorithm First Applied to Nanohelicenes, Comput. Mater. Sci., 2022, 203, 111063,  DOI:10.1016/j.commatsci.2021.111063.
43. Y. Akai, L. Konnert, T. Yamamoto and M. Suginome, Asymmetric Suzuki–Miyaura Cross-Coupling of 1-Bromo-2-Naphthoates Using the Helically Chiral Polymer Ligand PQXphos, Chem. Commun., 2015, 51(33), 7211–7214,  10.1039/C5CC01074H.
44. H. H. Dhanjee, A. Saebi, I. Buslov, A. R. Loftis, S. L. Buchwald and B. L. Pentelute, Protein–Protein Cross-Coupling via Palladium–Protein Oxidative Addition Complexes from Cysteine Residues, J. Am. Chem. Soc., 2020, 142(20), 9124–9129,  DOI:10.1021/jacs.0c03143.
45. X. Li, Y. Cao, K. Luo, L. Zhang, Y. Bai, J. Xiong, R. N. Zare and J. Ge, Cooperative Catalysis by a Single-Atom Enzyme-Metal Complex, Nat. Commun., 2022, 13(1), 2189,  DOI:10.1038/s41467-022-29900-6.
46. V. V. Petrova, Y. V. Solovev, Y. B. Porozov and M. V. Polynski, Will We Witness Enzymatic or Pd-(Oligo)Peptide Catalysis in Suzuki Cross-Coupling Reactions?, J. Org. Chem., 2024, 89(12), 8478–8485,  DOI:10.1021/acs.joc.4c00409.
47. J. Magano and J. R. Dunetz, Large-Scale Applications of Transition Metal-Catalyzed Couplings for the Synthesis of Pharmaceuticals, Chem. Rev., 2011, 111(3), 2177–2250,  DOI:10.1021/cr100346g.
48. S. Xu, E. H. Kim, A. Wei and E. Negishi, Pd- and Ni-Catalyzed Cross-Coupling Reactions in the Synthesis of Organic Electronic Materials, Sci. Technol. Adv. Mater., 2014, 15(4), 044201,  DOI:10.1088/1468-6996/15/4/044201.
49. P. Devendar, R.-Y. Qu, W.-M. Kang, B. He and G.-F. Yang, Palladium-Catalyzed Cross-Coupling Reactions: A Powerful Tool for the Synthesis of Agrochemicals, J. Agric. Food Chem., 2018, 66(34), 8914–8934,  DOI:10.1021/acs.jafc.8b03792.
50. M. V. Polynski and V. P. Ananikov, Modeling Key Pathways Proposed for the Formation and Evolution of “Cocktail”-Type Systems in Pd-Catalyzed Reactions Involving ArX Reagents, ACS Catal., 2019, 9(5), 3991–4005,  DOI:10.1021/acscatal.9b00207.
51. D. J. Durand and N. Fey, Computational Ligand Descriptors for Catalyst Design, Chem. Rev., 2019, 119(11), 6561–6594,  DOI:10.1021/acs.chemrev.8b00588.
52. H. Clavier and S. P. Nolan, Percent Buried Volume for Phosphine and N-Heterocyclic Carbene Ligands: Steric Properties in Organometallic Chemistry, Chem. Commun., 2010, 46(6), 841–861,  10.1039/B922984A.
53. V. Yeguas, P. Campomanes, R. López, N. Díaz and D. Suárez, Understanding Regioselective Cleavage in Peptide Hydrolysis by a Palladium(II) Aqua Complex: A Theoretical Point of View, J. Phys. Chem. B, 2010, 114(25), 8525–8535,  DOI:10.1021/jp101870j.
54. A. Kumar, X. Zhu, K. Walsh and R. Prabhakar, Theoretical Insights into the Mechanism of Selective Peptide Bond Hydrolysis Catalyzed by [Pd(H₂O)4]2+, Inorg. Chem., 2010, 49(1), 38–46,  DOI:10.1021/ic901071v.
55. N. M. Milović and N. M. Kostić, Palladium(II) Complex as a Sequence-Specific Peptidase: Hydrolytic Cleavage under Mild Conditions of X-Pro Peptide Bonds in X-Pro-Met and X-Pro-His Segments, J. Am. Chem. Soc., 2003, 125(3), 781–788,  DOI:10.1021/ja027408b.
56. N. M. Milović and N. M. Kostić, Palladium(II) Complexes, as Synthetic Peptidases, Regioselectively Cleave the Second Peptide Bond “Upstream” from Methionine and Histidine Side Chains, J. Am. Chem. Soc., 2002, 124(17), 4759–4769,  DOI:10.1021/ja012366x.
57. B. Haimov and S. Srebnik, A Closer Look into the α-Helix Basin, Sci. Rep., 2016, 6(1), 38341,  DOI:10.1038/srep38341.
58. P. C. Lyu, J. C. Sherman, A. Chen and N. R. Kallenbach, Alpha-Helix Stabilization by Natural and Unnatural Amino Acids with Alkyl Side Chains, Proc. Natl. Acad. Sci. U. S. A., 1991, 88(12), 5317–5320,  DOI:10.1073/pnas.88.12.5317.
59. S.-C. Li and C. M. Deber, A Measure of Helical Propensity for Amino Acids in Membrane Environments, Nat. Struct. Biol., 1994, 1(6), 368–373,  DOI:10.1038/nsb0694-368.
60. A. Elliot and B. R. Malcolm, Chain Arrangement and Sense of the α-Helix in Poly-L-Alanine Fibres, Proc. R. Soc. London, Ser. A, 1959 DOI:10.1098/rspa.1959.0004.
61. L. D. Landau and E. M. Lifshitz, Statistical Physics, Butterworth-Heinemann, Oxford, England, 3rd edn, 1996.
62. L. K. Mueller, A. C. Baumruck, H. Zhdanova and A. A. Tietze, Challenges and Perspectives in Chemical Synthesis of Highly Hydrophobic Peptides, Front. Bioeng. Biotechnol., 2020, 8, 162,  DOI:10.3389/fbioe.2020.00162.
63. W. Li, M. T. Jacobsen, C. Park, J. U. Jung, N.-P. Lin, P.-S. Huang, R. A. Lal and D. H.-C. Chou, A Cysteine-Specific Solubilizing Tag Strategy Enables Efficient Chemical Protein Synthesis of Difficult Targets, Chem. Sci., 2024, 15(9), 3214–3222,  10.1039/D3SC06032B.
64. D. A. Wappett and L. Goerigk, Benchmarking Density Functional Theory Methods for Metalloenzyme Reactions: The Introduction of the MME55 Set, J. Chem. Theory Comput., 2023, 19(22), 8365–8383,  DOI:10.1021/acs.jctc.3c00558.
65. L. Goerigk, A. Hansen, C. Bauer, S. Ehrlich, A. Najibi and S. Grimme, A Look at the Density Functional Theory Zoo with the Advanced GMTKN55 Database for General Main Group Thermochemistry, Kinetics and Noncovalent Interactions, Phys. Chem. Chem. Phys., 2017, 19(48), 32184–32215,  10.1039/C7CP04913G.
66. C. L. McMullin, N. Fey and J. N. Harvey, Computed Ligand Effects on the Oxidative Addition of Phenyl Halides to Phosphine Supported Palladium(0) Catalysts, Dalton Trans., 2014, 43(36), 13545–13556,  10.1039/C4DT01758G.
67. R. Richardson-Matthews, K. Velko, B. Bhunia, S. Ghosh, J. Oktawiec, J. S. Brunzelle, V. T. Dang and A. I. Nguyen, Metal-α-Helix Peptide Frameworks, J. Am. Chem. Soc., 2025, 147(20), 17433–17447,  DOI:10.1021/jacs.5c04078.
68. S. Learte-Aymamí, L. Martínez-Castro, C. González-González, M. Condeminas, P. Martin-Malpartida, M. Tomás-Gamasa, S. Baúlde, J. R. Couceiro, J.-D. Maréchal, M. J. Macias, J. L. Mascareñas and M. E. Vázquez, De Novo Engineering of Pd-Metalloproteins and Their Use as Intracellular Catalysts, JACS Au, 2024, 4(7), 2630–2639,  DOI:10.1021/jacsau.4c00379.
69. M. Damnjanović and I. Milošsević, Line Groups in Physics, Lecture Notes in Physics, Springer Berlin Heidelberg, Berlin, Heidelberg, 2010, vol. 801.  DOI:10.1007/978-3-642-11172-3.
70. R. Dovesi, F. Pascale, B. Civalleri, K. Doll, N. M. Harrison, I. Bush, P. D'Arco, Y. Noël, M. Rérat, P. Carbonnière, M. Causà, S. Salustro, V. Lacivita, B. Kirtman, A. M. Ferrari, F. S. Gentile, J. Baima, M. Ferrero, R. Demichelis and M. De La Pierre, The CRYSTAL Code, 1976–2020 and beyond, a Long Story, J. Chem. Phys., 2020, 152(20), 204111,  DOI:10.1063/5.0004892.
71. R. Dovesi, A. Erba, R. Orlando, C. M. Zicovich-Wilson, B. Civalleri, L. Maschio, M. Rérat, S. Casassa, J. Baima, S. Salustro and B. Kirtman, Quantum-Mechanical Condensed Matter Simulations with CRYSTAL, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2018, 8(4), e1360,  DOI:10.1002/wcms.1360.
72. H. J. Monkhorst and J. D. Pack, Special Points for Brillouin-Zone Integrations, Phys. Rev. B, 1976, 13(12), 5188–5192,  DOI:10.1103/PhysRevB.13.5188.
73. J. P. Perdew, K. Burke and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett., 1996, 77(18), 3865–3868,  DOI:10.1103/PhysRevLett.77.3865.
74. M. F. Peintinger, D. V. Oliveira and T. Bredow, Consistent Gaussian Basis Sets of Triple-Zeta Valence with Polarization Quality for Solid-State Calculations, J. Comput. Chem., 2013, 34(6), 451–459,  DOI:10.1002/jcc.23153.
75. D. V. Oliveira, J. Laun, M. F. Peintinger and T. Bredow, BSSE-Correction Scheme for Consistent Gaussian Basis Sets of Double- and Triple-Zeta Valence with Polarization Quality for Solid-State Calculations, J. Comput. Chem., 2019, 40(27), 2364–2376,  DOI:10.1002/jcc.26013.
76. J. Laun, D. V. Oliveira and T. Bredow, Consistent Gaussian Basis Sets of Double- and Triple-Zeta Valence with Polarization Quality of the Fifth Period for Solid-State Calculations, J. Comput. Chem., 2018, 39(19), 1285–1290,  DOI:10.1002/jcc.25195.
77. C. Adamo and V. Barone, Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0 Model, J. Chem. Phys., 1999, 110(13), 6158–6170,  DOI:10.1063/1.478522.
78. S. Grimme, J. Antony, S. Ehrlich and H. Krieg, A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu, J. Chem. Phys., 2010, 132(15), 154104,  DOI:10.1063/1.3382344.
79. S. Arnott and A. J. Wonacott, Atomic Co-Ordinates for an α-Helix: Refinement of the Crystal Structure of α-Poly-l-Alanine, J. Mol. Biol., 1966, 21(2), 371–383,  DOI:10.1016/0022-2836(66)90105-7.
80. L. Brown and I. F. Trotter, X-Ray, Studies of Poly-L-Alanine, Trans. Faraday Soc., 1956, 52(0), 537–548,  10.1039/TF9565200537.
81. F. Neese, The ORCA Program System, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2012, 2(1), 73–78,  DOI:10.1002/wcms.81.
82. F. Weigend and R. Ahlrichs, Balanced Basis Sets of Split Valence, Triple Zeta Valence and Quadruple Zeta Valence Quality for H to Rn: Design and Assessment of Accuracy, Phys. Chem. Chem. Phys., 2005, 7(18), 3297–3305,  10.1039/B508541A.
83. F. Weigend, Accurate Coulomb-Fitting Basis Sets for H to Rn, Phys. Chem. Chem. Phys., 2006, 8(9), 1057–1065,  10.1039/B515623H.
84. F. Neese, An Improvement of the Resolution of the Identity Approximation for the Formation of the Coulomb Matrix, J. Comput. Chem., 2003, 24(14), 1740–1747,  DOI:10.1002/jcc.10318.
85. N. Mardirossian and M. Head-Gordon, ωB97X-V: A 10-Parameter, Range-Separated Hybrid, Generalized Gradient Approximation Density Functional with Nonlocal Correlation, Designed by a Survival-of-the-Fittest Strategy, Phys. Chem. Chem. Phys., 2014, 16(21), 9904–9924,  10.1039/C3CP54374A.
86. F. Neese, F. Wennmohs, A. Hansen and U. Becker, Efficient, Approximate and Parallel Hartree–Fock and Hybrid DFT Calculations. A ‘Chain-of-Spheres’ Algorithm for the Hartree–Fock Exchange, Chem. Phys., 2009, 356(1), 98–109,  DOI:10.1016/j.chemphys.2008.10.036.
87. A. D. Becke, Density–functional Thermochemistry. I. The Effect of the Exchange–only Gradient Correction, J. Chem. Phys., 1992, 96(3), 2155–2160,  DOI:10.1063/1.462066.
88. C. Lee, W. Yang and R. G. Parr, Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density, Phys. Rev. B:Condens. Matter Mater. Phys., 1988, 37(2), 785–789,  DOI:10.1103/PhysRevB.37.785.
89. A. D. Becke, Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior, Phys. Rev. A, 1988, 38(6), 3098–3100,  DOI:10.1103/PhysRevA.38.3098.
90. W. R. Wadt and P. J. Hay, Ab Initio Effective Core Potentials for Molecular Calculations. Potentials for Main Group Elements Na to Bi, J. Chem. Phys., 1985, 82(1), 284–298,  DOI:10.1063/1.448800.
91. P. J. Hay and W. R. Wadt, Ab Initio Effective Core Potentials for Molecular Calculations. Potentials for the Transition Metal Atoms Sc to Hg, J. Chem. Phys., 1985, 82(1), 270–283,  DOI:10.1063/1.448799.
92. M. M. Francl, W. J. Pietro, W. J. Hehre, J. S. Binkley, M. S. Gordon, D. J. DeFrees and J. A. Pople, Self–consistent Molecular Orbital Methods. XXIII. A Polarization–type Basis Set for Second–row Elements, J. Chem. Phys., 1982, 77(7), 3654–3665,  DOI:10.1063/1.444267.
93. P. C. Hariharan and J. A. Pople, The Influence of Polarization Functions on Molecular Orbital Hydrogenation Energies, Theor. Chim. Acta, 1973, 28(3), 213–222,  DOI:10.1007/BF00533485.
94. A. V. Marenich, C. J. Cramer and D. G. Truhlar, Universal Solvation Model Based on Solute Electron Density and on a Continuum Model of the Solvent Defined by the Bulk Dielectric Constant and Atomic Surface Tensions, J. Phys. Chem. B, 2009, 113(18), 6378–6396,  DOI:10.1021/jp810292n.

---