Emergence of fractal geometries in the evolution of a metabolic enzyme

Molecular cloning

The gene encoding CS from S. elongatus PCC 7942 was amplified from genomic DNA by PCR (Q5 High-Fidelity 2× Master Mix, New England Biolabs) and introduced into the pLIC expression vector³⁸ by Gibson cloning (Gibson Assembly Master Mix, New England Biolabs). All other extant and ancestral CS sequences were obtained as gene fragments from Twist Bioscience and introduced into the same expression vector by Gibson cloning. All CS sequences were tagged with a C-terminal polyhistidine tag for purification (tag sequence: LE-HHHHHH-Stop). For single-site mutants and deletions of the CS sequences, KLD enzyme mix (New England Biolabs) was used. Mutagenesis primers were designed with NEBasechanger and used to PCR-amplify the vector encoding for the gene that was to be changed. Resulting PCR products were added to the KLD enzyme mix and subsequently transformed. All cloned genes were verified by Sanger sequencing (Microsynth) before use in experiments.

The DNA sequences of all purified proteins and NCBI identifiers of all extant sequences are presented in Supplementary Table 2.

Protein purification

For heterologous overexpression, the vectors with the gene of interest were transformed into chemically competent Escherichia coli BL21 (DE3) cells. Transformed colonies were used to inoculate expression cultures (500 ml) made from LB medium supplemented with 12.5 g l^–1 lactose (Fisher Chemical). The cultures were incubated overnight at 30 °C and 200 r.p.m. Cells were collected by centrifugation (4,500g, 15 min, 4 °C), resuspended in buffer A (20 mM Tris, 300 mM NaCl and 20 mM imidazole, pH 8) and freshly supplemented with DNAse I (3 units µl^–1, Applichem). The cells were disrupted using a Microfluidizer (Microfluidics) in 3 cycles at 15,000 psi and centrifuged to spin down cell debris and aggregates (30,000g, 30 min, 4 °C). The clarified lysate was loaded with a peristaltic pump (Hei-FLOW 06, Heidolph) on prepacked nickel-NTA columns (5 ml Nuvia IMAC Ni-Charged, Bio-Rad) that were pre-equilibrated with buffer A. The loaded column was first washed with buffer A for 7 column volumes and then with 10% (v/v) buffer B (20 mM Tris, 300 mM NaCl and 500 mM imidazole, pH 8) in buffer A for 7 column volumes. The bound protein was eluted with buffer B and either buffer-exchanged with PD-10 desalting columns (Cytiva) into PBS or 20 mM Tris, 200 mM NaCl, pH 7.5 or further purified by size-exclusion chromatography (SEC). For SEC, the protein was injected on an ENrich SEC 650 column (Bio-Rad) with PBS as the running buffer using a NGC Chromatography System (Bio-Rad). The purity of the proteins was analysed by SDS–PAGE. After either buffer exchange or SEC, the purified proteins were flash-frozen with liquid nitrogen and stored at −20 °C before further use.

Phylogenetic analysis and ancestral sequence reconstruction

Amino acid sequences of 84 CS genes from Cyanobacteria and marine Gammaproteobacteria as the outgroup were collected from the NCBI Reference Sequence database and aligned using MUSCLE (v.3.8.31)³⁹. The maximum likelihood (ML) phylogeny was inferred from the multiple sequence alignment (MSA) using raxML (v.8.2.10)⁴⁰. The LG substitution matrix⁴¹ was used as determined by automatic best-fit model selection as well as fixed base frequencies and a gamma model of rate heterogeneity. The robustness of the ML tree topology was assessed by inferring 100 non-parametric bootstrap trees with raxML, from which Felsenstein’s and transfer bootstrap values were derived using BOOSTER (https://booster.pasteur.fr). Using PhyML (3.0)⁴², we also inferred approximate likelihood-ratio test⁴³ for branches to statistically evaluate branch support in the phylogeny.

Based on the CS tree and the MSA, ancestral sequences were inferred using the codeML package within PAML (v.4.9)⁴⁴. To adjust for gaps and the different lengths of N termini in the CS sequences, their ancestral state was determined using parsimony inference in PAUP (4.0a) based on a binary version of the MSA (1 = amino acid, 0 = gap, no residue). The state assignment for each node in the tree (amino acid or gap) was then applied to the inferred ancestral sequences.

The initial reconstruction of the crucial amino acid substitution q18L was ambiguous in the ancestors ancB and ancA. We determined that this was the case because this L residue is present in the Planktothrix clade and S. elongatus, but not in the Cyanobium/Prochlorococcus clade. Therefore, the L residue was either gained once and then lost along the lineage to Cyanobium/Prochlorococcus or it was gained convergently twice in Planktothrix and S. elongatus. We therefore added the CS sequence from the cyanobacterium Prochlorothrix hollandica to the alignment, which has been stably inferred as a sister group to S. elongatus and the Cyanobium/Prochlorococcus clade by multiple studies^45,46,47. This sequence was previously omitted from analysis as its position on the tree could not be inferred with high support. We manually added a branch to the tree, placing P. hollandica as a sister group to S. elongatus and the Cyanobium/Prochlorococcus clade. Branch lengths were reoptimized using raxML and the ancestral reconstruction repeated using PAML. The results gave high support to the substitution q18L being found in both ancB and ancA (Extended Data Fig. 8b) because P. hollandica also contains the L at position 18. This made the hypothesis of one gain and a subsequent loss in the Cyanobium/Prochlorococcus clade much more probable compared with three independent gains in Planktothrix, P. hollandica and S. elongatus.

MP analysis

Measurements were performed on a OneMP mass photometer (Refeyn). Reusable silicone gaskets (CultureWellTM, CW-50R-1.0, 50-3 mm diameter × 1 mm depth) were set up on a cleaned microscopic cover slip (1.5 H, 24 × 60 mm, Carl Roth) and mounted on the stage of the mass photometer using immersion oil (IMMOIL-F30CC, Olympus). The gasket was filled with 19 µl buffer (PBS or 20 mM Tris, 200 mM NaCl pH 7.5) to focus the instrument. Then, 1 µl of prediluted protein solution (1 µM) was added to the buffer droplet and thoroughly mixed. The final concentration of the proteins during measurement was 50 nM unless stated otherwise. Data were acquired for 60 s at 100 frames per s using AcquireMP (Refeyn, v.1.2.1). The resulting movies were processed and analysed using DiscoverMP (Refeyn, v.2.5.0). The identified protein complexes with corresponding molecular weight were plotted as histograms, and the individual oligomeric state populations appeared as peaks that were fitted by a Gaussian curve (implemented in DiscoverMP). All complexes within the respective Gaussian curve were used to calculate the fraction of CS subunits in each oligomeric state. The instrument was calibrated at least once during each measuring session using either a commercial standard (NativeMark unstained protein standard, Thermo Fisher) or a homemade calibration standard of a protein with known sizes of complexes.

For substrate titrations, the prediluted protein sample (2 µM) was incubated for 10 min with the respective substrate concentration. The same substrate concentration was also included in the buffer in the gasket that was used for focusing. For each substrate concentration, three separate measurements were performed. For pH titrations, the protein sample was diluted into the buffer with the corresponding pH value (20 mM Tris, 200 mM NaCl pH 7–9.5). The dilution factor was at least 200, including predilution and final dilution in the gasket. For each pH value, two separate measurements were performed.

Native mass spectrometry

The purified protein samples were buffer-exchanged into 200 mM ammonium acetate by using centrifugal filter devices (Amicon Ultra) and three successive rounds of concentration and dilution. The concentration of protein was determined by UV absorbance (NanoDrop spectrophotometer, Thermo Fisher) and diluted into aliquots at appropriate monomeric concentrations. Nanoelectrospray was carried out in positive-ion mode on a Q Exactive UHMR mass spectrometer (Thermo Fisher), using gold-coated capillaries prepared in-house and the application of a modest backing pressure (about 0.5 mbar). Sulfur hexafluoride was introduced into the collision ‘HCD’ cell to improve transmission, and the instrument was operated at a resolution of 6,250 (at 200 m/z), with ‘high detector optimization’, and a trapping pressure in the HCD cell set to 4. The rest of the parameters were optimized for each sample, with the following ranges: capillary voltage of 1.2–1.5 kV; capillary temperature of 100–250 °C; in-source trapping from −15 to −150 V; injection times of 50–100 ms; and 1–10 microscans. Mass spectra were deconvolved using UniDec⁴⁸.

Kinetic enzyme assays

For the CS kinetic assays, the colorimetric quantification of thiol groups was used based on 5,5′-dithiobis-(2-nitrobenzoic acid) (DTNB)^49,50. The photospectrometric reactions were carried out in 50 mM Tris pH 7.5, 10 mM KCl, 0.1 mg ml^–1 DTNB and 25 nM protein concentration at 25 °C. To measure K_m values, one substrate was saturated and added to the reaction mix (1 mM oxaloacetate or 0.5 mM acetyl-CoA). The other substrate was varied in concentration and added last to start the reaction. For kinetic measurements at non-saturating substrate concentrations, the protein was diluted only immediately before the reaction start to prevent the disassembly of complexes and added last to the reaction mix. Reaction progress was followed by measuring the appearance of 2-nitro-5-thiobenzoate at 412 nm (extinction coefficient of 14.150 M⁻¹ cm⁻¹) in a plate reader (Infinite M Nano+, Tecan) using Tecan i-control (v.3.9.1). Data analysis and determination of Michaelis–Menten kinetic parameters was done using GraphPad Prism (v.8.4.3). For the kinetic assays with the cys4 variant, the protein was dialysed in a buffer with a glutathione redox system to induce the formation of disulfide bonds of the cysteine residues (50 mM Na₂HPO₄, 150 mM NaCl, 1 mM glutathione and 0.5 mM glutathione disulfide, pH 8). After overnight dialysis, part of the protein sample was used for kinetic assays. The remainder was reduced by incubation with 10 mM dithiothreitol for 3 h at 4 °C and again used for kinetic assays. To exclude additional effects by the treatment itself, the WT SeCS was handled accordingly (dialysis in redox buffer and reduction with dithiothreitol) and measured kinetically for comparison.

Box counting

To quantify fractal scaling, we used a fixed grid scan. The images of the class averages of the 18mer and 54mer assemblies were overlaid with a non-overlapping regular grid (Adobe Illustrator, v.24.0.2). The squares that were needed to fill out the structure were manually counted. This process was repeated for nine different box sizes of the grid (85–17 px). The entire procedure was replicated for three separate grid orientations for both assemblies. Linear regression was performed using GraphPad Prism (v.8.4.3).

Cultivation of S.
elongatus and sample preparation for metabolomics analysis

S. elongatus PCC 7942 was genetically modified to harbour variants of CS by homologous recombination as previously described⁵¹. The standard vector pSyn_6 (Thermo Fisher Scientific) was used as the backbone. A homology cassette was constructed by amplification and extraction of the CS gene and 1,000 bp of the neighbouring homologous regions by PCR from genomic DNA of WT S. elongatus PCC 7942. These were introduced into the pSyn_6 vector that included a spectinomycin-resistance gene to select for transformants. The respective sequence changes of the CS were introduced into this vector (L18Q) to create the corresponding homology cassette. The constructed homology cassettes (WT, L18Q) were transformed into WT S. elongatus PCC 7942 and plated on BG11 plates with 10 µg ml^–1 spectinomycin for selection. Transformants were re-streaked on fresh BG11 plates with spectinomycin, and resulting colonies were analysed for successful integration through the extraction of genomic DNA. All strains were verified by PCR amplification of the introduced cassette (primers were designed to bind outside the introduced DNA region) and Sanger sequencing. All sequences of the homology cassettes are presented in Supplementary Table 3.

S. elongatus PCC 7942 cultures and genetically modified strains were grown in BG11 medium at 30 °C, 100 r.p.m., ambient CO₂ levels and alternating light conditions: 12 h of light (photon flux of 120 μmol m⁻² s⁻¹) and 12 h of darkness. Before the growth experiment, precultures were entrained for 5 days in the circadian conditions to synchronize cells. Then 3 main cultures (50 ml) were set up from 3 independent precultures and inoculated to an OD₇₅₀ of 0.025 or 0.05. Samples for metabolomics analysis were cultivated in specific flasks to facilitate the isolation of culture solution through a syringe valve, which led to slower growth behaviour compared with the standard flasks. The samples were taken at 6 different time points (days 3, 5 and 7) after a light and a dark period.

For recovery experiments under nitrogen deficiency, S. elongatus strains were grown in BG11 medium at full light to an OD₇₅₀of 0.5 in triplicate. The cells were then shifted to medium without a nitrogen source. To do this, the cells were washed twice with BG11 without nitrate and then continuously cultivated in BG11 without nitrogen. The cells underwent chlorosis and fully bleached in the subsequent days. After 14 and 20 days, a serial dilution of the respective cultures was spotted on BG11 agar plates and incubated for 7 further days for recovery.

Sample preparation for metabolomics analysis

The culture volume (1 ml) was taken from the shaking flask through a syringe and immediately quenched in 1 ml 70 % methanol that was precooled in a −80 °C freezer. The sample was mixed and centrifuged (10 min, −10 °C, 13,000g). The supernatant was removed and the pellet was stored at −80 °C until the endometabolome was extracted. At each time point, the cell number and size were measured for each culture using a Coulter counter (Multisizer 4e, Beckman Coulter). The respective biovolume for each cell pellet was then calculated and used to infer a normalized amount of extraction fluid for each sample (extraction fluid = 20,00 × biovolume). All steps of the metabolome extraction were performed on ice and with precooled (−20 °C) reagents. To extract the metabolites, the calculated amount of extraction fluid (50% (v/v) methanol, 50% (v/v) TE buffer pH 7.0) was added to the cell pellets together with the same amount of chloroform. The samples were vortexed and incubated for 2 h at 4 °C while shaking. The phases were then separated by centrifugation (10 min, −10 °C, 13,000g). The upper phase was extracted with a syringe and the same amount of chloroform added again. After mixing, the sample was centrifuged again (10 min, −10 °C, 13,000g) to get remove residual cell fragments and pigments. The upper phase was isolated, added to LC–MS vials and stored at −20 °C until analysis.

Quantification of intracellular metabolites from S.
elongatus by LC–MS/MS

Quantitative determination of acetyl-CoA and citrate was performed using LC–MS/MS. The chromatographic separation was performed on an Agilent Infinity II 1290 HPLC system (Agilent) using a Kinetex EVO C18 column (150 × 2.1 mm, 3 μm particle size, 100 Å pore size, Phenomenex) connected to a guard column of similar specificity (20 × 2.1 mm, 3 μm particle size, Phenomoenex). For acetyl-CoA, a constant flow rate of 0.25 ml min^–1 with mobile phase A being 50 mM ammonium acetate in water at a pH of 8.1 and phase B being 100% methanol at 25 °C was used. The injection volume was 1 µl. The mobile phase profile consisted of the following steps and linear gradients: 0–0.5 min constant at 5% B; 0.5–6.5 min from 5 to 80% B; 6.5–7.5 min constant at 80% B; 7.5–7.6 min from 80 to 5% B; and 7.6 to 10 min constant at 5% B. An Agilent 6470 mass spectrometer (Agilent) was used in positive mode with an electrospray ionization (ESI) source and the following conditions: ESI spray voltage of 4,500 V; nozzle voltage of 1,500 V; sheath gas of 400 °C at 11 l min^–1; nebulizer pressure of 30 psi; and drying gas of 250 °C at 11 l min^–1. The target analyte was identified based on the two specific mass transitions (810.1 → 428 and 810.1 → 302.2) at a collision energy of 35 V and its retention time compared with standards.

For citrate, a constant flow rate of 0.2 ml min^–1 with mobile phase A being 0.1% formic acid in water and phase B being 0.1% formic acid methanol at 25 °C was used. The injection volume was 10 µl. The mobile phase profile consisted of the following steps and linear gradients: 0–5 min constant at 0% B; 5–6 min from 0 to 100% B; 6–8 min constant at 100% B; 8–8.1 min from 100 to 0% B; and 8.1 to 12 min constant at 0% B. An Agilent 6495 ion funnel mass spectrometer (Agilent) was used in negative mode with an ESI source and the following conditions: ESI spray voltage of 2,000 V; nozzle voltage of 500 V; sheath gas of 260 °C at 10 l min^–1; nebulizer pressure of 35 psi; and drying gas of 100 °C at 13 l min^–1. The target analyte was identified based on the two specific mass transitions (191 → 111.1 and 191 → 85.1) at a collision energy of 11 and 14 V and its retention time compared with standards.

Chromatograms were integrated using MassHunter software (Agilent). Absolute concentrations were calculated based on an external calibration curve prepared in sample matrix.

Negative-stain EM

Carbon-coated copper grids (400 mesh) were hydrophilized by glow discharging (PELCO easiGlow, Ted Pella). Next, 5 µl of 450 nM protein suspensions were applied onto the hydrophilized grids and stained with 2% uranyl acetate after a short washing step with double-distilled H₂O. Samples were analysed using a JEOL JEM-2100 transmission electron microscope with an acceleration voltage of 120 kV. A 2k F214 FastScan CCD camera (TVIPS) was used for image acquisition. Alternatively, a JEOL JEM1400 TEM (operated at 80 kV) with a 4k TVIPS TemCam XF416 camera was used. For 2D class averaging, images were taken manually and processed with cisTEM⁵². The following number of particles were averaged: 1,491 particles for 18mers; 200 particles for 54mers; and 186 for 36mers. The 36mer and 54mer particles were isolated from an extended dataset, in which we specifically looked for larger assemblies. The exact percentage of complexes larger than 18mers was difficult to estimate because of very strong preferential orientation. Most particles seemed to have landed not on the face of the triangle but on one of its edges or even one of its tips (Extended Data Fig. 1). To obtain an estimate, another dataset of 150 micrographs without a bias towards larger assemblies was collected. All particles were manually counted for these micrographs and included the assemblies that were laying on their edge and appeared as rectangles. By measuring the edge length, we could assign them to be either a 36mers (30 nm) or 54mers (40 nm). The analysis revealed that under negative-stain TEM conditions (450 nM) approximately 92.8% of detected assemblies were identified as 18mers (1,773 particles), 3.5% as 36mers (66 particles) and 3.8% as 54mers (72 particles). Our estimate of the abundance should still be taken with care and by comparison with our SAXS data, which showed that large complexes only start being reasonably common above 25 µM protein concentration. For the H369R variant of SeCS, a protein concentration of 450 nM was used and 136 particles were averaged to produce the 2D class average of the 18mer shown in Extended Data Fig. 6d.

Crystallography and structure determination

Crystallization was performed using the sitting-drop method at 20 °C in 250 nl drops (Crystal Gryphon, Art Robbins Instruments) consisting of equal parts of protein and precipitation solutions (Swissci 3 Lens Crystallisation Plate). Protein solutions of 250 µM were incubated with 5 mM acetyl-CoA for 10 min at room temperature to induce disassembly into hexamers. The crystallization condition was 0.1 M citrate pH 5.5,and 2.0 M ammonium sulfate. Before data collection, crystals were flash-frozen in liquid nitrogen using a cryo-solution that consisted of motherliquor supplemented with 20% (v/v) glycerol. Data were collected under cryogenic conditions at P13, Deutsches Elektronen-Synchrotron. Data were processed using XDS and scaled with XSCALE⁵³. All structures were determined by molecular replacement with PHASER⁵⁴, manually built in WinCOOT (v.0.9.6)⁵⁵ and refined with PHENIX (v.1.19.2)⁵⁶. The search model for the structure was the hexameric Δ2–6 variant. Images of the structure were generated using PyMOL (v.2.5.2).

Cryo-EM

For cryo-EM sample preparation, 4.5 µl of the protein sample (22.5 µM) was applied to glow-discharged Quantifoil 2/1 grids, blotted for 4 s with force 4 in a Vitrobot Mark III (Thermo Fisher) at 100% humidity and 4 °C, and plunge frozen in liquid ethane, cooled by liquid nitrogen. Cryo-EM data were acquired with a FEI Titan Krios transmission electron microscope (Thermo Fisher) using SerialEM software⁵⁷. Movie frames were recorded at a nominal magnification of ×29,000 using a K3 direct electron detector (Gatan). The total electron dose of about 55 electrons per Ų was distributed over 30 frames at a pixel size of 1.09 Å. Micrographs were recorded in a defocus range from −0.5 to −3.0 μm.

Image processing, classification and refinement

For the SeCS 18mer, all processing steps were carried out in cryoSPARC (v.3.2.0)⁵⁸. A total of 1,408 movies were aligned using the patch motion correction tool, and contrast transfer function (CTF) parameters were determined using the patch CTF tool. An initial set of 10,173 particles were acquired through several rounds of blob picking, 2D classification and template picking for training a Topaz convolutional neural network particle picking model⁵⁹. From all the corrected micrographs, 273,259 particles were extracted in a box size of 350 by 350 pixels at a pixel size of 1.09 Å using the Topaz extract tool together with the trained model. Overall, 224,041 particles were selected for the ab initio reconstruction after removing poor particles through 2D classification. The initial density map was then three-dimensionally (3D) classified and refined using the heterogenous refinement tool, which resulted in three classes. The dominant class (56.7% particles) was subjected to another round of heterogenous refinement, which led to two classes. A 3D non-uniform refinement of the main class (79.8% particles) imposing a C3 symmetry, followed by a local CTF refinement produced a final resolution of 3.93 Å (GSFSC = 0.143), which was used for model building. Local resolution of the density map was calculated with the local resolution estimation tool.

For the ∆2–6 sample, cryo-EM micrographs were processed on the fly using the Focus software package⁶⁰ if they passed the selection criteria (iciness < 1.05, drift 0.4 Å < x < 70 Å, defocus 0.5 µm < x < 5.5 µm, estimated CTF resolution < 6 Å). Micrograph frames were aligned using MotionCor2 (ref. ⁶¹) and the CTF for aligned frames was determined using GCTF⁶². From 5,419 acquired micrographs 1,687,951 particles were picked using the Phosaurus neural network architecture from crYOLO⁶³. Particles were extracted with a pixel box size of 256 scaled down to 96 using RELION (v.3.1)⁶⁴ and underwent several rounds of reference-free 2D classification. Overall, 1,271,457 selected particles (∆2–6) were re-extracted with a box size of 256 and imported into Cryosparc (v.2.3)⁵⁸. For each sample, ab initio models were generated and passed through heterogeneous classification and refinement. Selected particles were re-imported to RELION and underwent several rounds of refinement, CTF-refinement (estimation of anisotropic magnification, fit of per-micrograph defocus and astigmatism and beamtilt estimation) and Bayesian polishing⁶⁵. Final C1 refinement produced models with an estimated resolution of 3.1 Å for ∆2–6 (gold standard FSC analysis of two independent half-sets at the 0.143 cut-off). Local resolution and 3D FSC plots were calculated using RELION and the “Remote 3DFSC Processing Server” web interface⁶⁶, respectively.

For the H369R SeCS 54mer and 18mer, all processing steps were carried out in cryoSPARC (v.4.4.0). In total, 29,126 movies were aligned using the patch motion correction tool, and CTF parameters were determined using the patch CTF tool. Next, 8,583 micrographs of estimated CTF fit ≤ 3.5 Å were selected for subsequent analysis. A Topaz particle picking model was generated by running several rounds of Topaz train and Topaz extract from an initial set of 150 manually picked particles. A total of 95,268 particles were picked using the trained model and extracted in a box size of 1,200 by 1,200 pixels at a pixel size of 0.79 Å. The particles were downsampled to a pixel size of 1.58 Å before 2D classification. 2D classes corresponded to the SeCS 54mer were selected to reconstruct two densities map using the ab initio reconstruction tool. All the extracted particles were re-aligned and 3D classified by running the heterogenous refinement tool using the density map corresponded to an intact 54mer as a reference. The 3D class (18.0% particles) was further refined by non-uniform refinement, which resulted in a final resolution of 5.91 Å (GSFSC = 0.143), which was used for model building. To reconstruct the mutant 18mer, 899,109 particles were picked using a 2D class corresponding to the 18mer as a template and extracted in a box size of 500 by 500 pixels at a pixel size of 0.79 Å. A total of 552,353 particles were selected from 2D classification to generate 3 initial maps. The major class was 3D classified and aligned, followed by a non-uniform refinement to produce the final 18mer density at 3.34 Å (GSFSC = 0.143). Local resolution of the density map was calculated with the local resolution estimation tool, and preferred orientation was assessed using the orientation diagnostics tool.

For 18meric SeCS, initial models were generated separately from their protein sequences using alphaFold⁶⁷ and then fitted as rigid bodies into the density using UCSF Chimera. The model was manually rebuilt using WinCoot (v.0.9.6)⁵⁵. Non-crystallographic symmetry constraints were manually defined in PHENIX (v.1.19.2)⁵⁶ so that each monomer within one hexamer is linked to the two corresponding monomers in the other two hexamers (corresponding to a C3 symmetric refinement of the 18mer). For the Δ2–6 hexamer, a hexameric subunit was extracted from the 18mer model as a starting model for refinement. The model was firstly rigid-body fitted into the density, and manually refined in WinCoot (v.0.9.6)⁵⁵. Both models were subjected to real-space refinements against the respective density maps using phenix.real_space_refine implemented in PHENIX (v.1.19.2)⁵⁶. Images of the structures were generated using PyMOL (v.2.5.2). For the 54mer structure of SeCS H369R, we used the dimers extracted from the WT 18mer structure as our starting model and fitted them as rigid bodies into the density using UCSF Chimera. We then truncated all side chains using pdbtools within PHENIX (v.1.19.2)⁵⁶. The structure was then subjected to one round of real space refinement using default parameters in PHENIX. For the 18mer structure of SeCS H369R, we also used the 18mer SeCS structure as the starting model. Individual dimers were first fitted as rigid bodies into the density using UCSF Chimera. We then subjected the structure to one round of flexible fitting with default parameters, followed by refinement with default parameters using the Namdinator server⁶⁸. The model was then manually rebuilt in WinCoot (v.0.9.6)⁵⁵. In this model, we truncated the substrate lids (residues 220–312, which are not part of the fractal interface) in all chains owing to poorly resolved density in our map, which made it difficult not to introduce register errors during refinement.

SAXS data collection and analysis

SAXS experiments were carried out at the BM29 beamline at the ESRF⁶⁹ using a PILATUS3X 2M photon counting detector (DECTRIS) at a fixed distance of 2,827 m. Protein samples were prepared in 25 mM Tris-HCl buffer pH 7.5 and 200 mM NaCl as a dilution series. Buffer matching was achieved by dialysis and all measurements were carried out at 20 °C. The sample delivery and measurements were performed using a 1 mm diameter quartz capillary, which is part of the BioSAXS automated sample changer unit (Arinax). Before and after each sample measurement, the corresponding buffer was measured and averaged. A total of ten frames (one frame per second) were taken for each sample. All experiments were conducted with the following parameters: beam current of 200 mA; flux of 2.6 × 1,012 photons s^–1 at sample position; wavelength of 1 Å; and estimated beam size of 200 × 200 µm. Processing and analysis of collected SAXS data were performed using ScÅtter IV⁷⁰.The R_g was determined by Guinier approximation. Plotting of the SAXS profiles and Guinier regions used BioXTAS RAW⁷¹.

Construction of atomic models of the 54mers using 18mers

We used the align, translate, and rotate commands within PyMOL (v.2.5.2) to model how a 54mer complex would assemble if the 4.0° and 4.2° dimer rotations and 60° dihedral angle between dimers that are observed in the 18mer structure were applied. The rotation was applied to the connecting dimers of the three 18mer-subcomplexes that built the 54mer. To do this, copies of the hexamers that constitute the 18mer were rotated by 120°, so as to overlay the corner dimers by edge dimers. Two 18mer copies were subsequently connected to the rotated corners by two steps of structural alignment, which placed the residues that should form the third interface 210 Å from each other (Extended Data Fig. 2g).

Calculation of R
_g values

Calculation of R_g values was done using gmx gyrate from the GROMACS 2022.2 simulation package⁷² from the atomic models of the 6mer, 18mer and the 54mer.

Displacement vectors, rotational axes and dihedral angles of atomic models

Symmetry axes were generated with AnAnaS⁷³, and rotation axes and angles were calculated using PyMOL (v.2.5.2) and a compatible script. Displacement vectors were drawn between Cα atoms of the aligned structures using the object argument and cgo-arrow. Dihedral angles between dimers across the fractal interface were calculated in PyMOL (v.2.5.2). The centre of mass of both dimers, as well as of one monomer from each dimer, was first calculated with the com command. The dihedral angle was then calculated using get_dihedral along the axis defined by the two centres of mass of the dimers.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.