For the first time, native proteins self-assemble into the most famous king of fractals, the Sherpinski triangle
Mathematically, fractals are repetitive patterns in which small-scale substructures resemble large-scale structures. They can be described with simple mathematical rules, resulting in a high degree of complexity. Fractals are widely found in nature, such as the branching morphology of plant leaf veins, coastlines, and river systems. Most natural fractals are irregular, and their structures do not match exactly at different scales. Very rare examples of fractal shapes with regular shapes in nature, such as the repetitive structure in Roman cauliflower, have been well studied. These studies have led to a deep understanding of the underlying mechanisms that produce precise self-similarity. All known regular fractals in nature are formed by living organisms and exist on a macroscopic scale. However, despite the great diversity of biomolecular assembly known in science, no such regular fractal structure has been found at the molecular scale to date. This may be due to the fact that fractal structures are difficult to translate into molecular self-assembly.
Figure 1. What is the Sierpiński triangle, the Sierpiński triangle was first proposed by the Polish mathematician Sierpiński in 1915. It is constructed by taking a regular triangle as the initial figure, dividing the regular triangle into 4 small triangles with half the original side length each time, removing the small triangle in the middle, and repeating this process until it can no longer be divided. As shown in Fig. For example, the Sierpiński triangle is one of the most well-known regular fractals, which can be created by trigonometric subdivisions or by random "chaos" that rely on non-local rules, or by all elements with odd binomial coefficients in Pascal's triangle. In contrast, the self-assembly of biomolecules occurs through the orderliness of the arrangement of different functional groups rather than through subdivision, and relies on local contacts between protomers to coordinate the assembly. Synthetic design has overcome these limitations, and one can already use certain organic small molecule constructs to Sierpiński triangles. The key to these designs is the precise control of the structural shape and functional group arrangement of the molecular building blocks, and these designs require special surfaces, precise temperature control during assembly, and fine-tuning of the proportions of different precursors to produce Sierpiński fractals. It is unlikely that such fine assembly requirements will be met in cells, which makes the likelihood of a natural version of these fractals low.
在该工作中，来自德国马普所的Georg K. A. Hochberg教授和Jan M. Schuller教授团队报道了一种天然蛋白质，柠檬酸合成酶的发现，它来自蓝细菌长聚球藻，并发现这种酶可以自我组装成谢尔宾斯基（Sierpiński）三角形。 该工作以题为“Emergence of fractal geometries in the evolution of a metabolic enzyme”发表在《Nature》上。
【Proteins that form the Sierpiński triangle】Bacterial citrate synthase (CS) protein is a homologous oligomerase that can be assembled into dimers and hexamers. The authors found that CS from the cyanobacterial S. elongatus PCC 7942 (SeCS) formed an unusual assembly. Mass photometric (MP) analysis showed that the purified enzyme could form a complex containing 18 CS subunits at nanomolar concentration. The structure of these assemblies was investigated by negative staining electron microscopy (EM), and the assembly of SeCS into regular triangular complexes of different sizes was observed. The triangular complex is 18-mer and contains 9 identifiable density units, each corresponding to a dimer. The three dimers are first arranged into a hexamer ring, and then the three hexamers are joined into a triangle. This 18-mer represents the dominant oligomer species under MP conditions. Larger complexes containing 36 or 54 CS subunits can be observed microscopically, and the protein concentration that forms such a large aggregate is 9-fold higher than that of MP condition (450 nM). The 54 aggregate consists of three 18 aggregates arranged in a larger triangle with a large void in the middle. The 6-, 18-, and 54-mer represent the zero, first, and second order of the Sierpiński triangle, which is a regular fractal geometry. The 36m mer represents another kind of triangle, but they share the building block of the 6m. In addition, the few observed regular assemblies of other shapes also retain triangular edges. To verify that the 18-mer and 54-mer are geometric fractals, the authors estimated their Hausdorff dimension D. For non-fractal shapes, D takes an integer value (2 for a square, 3 for a cube, and so on), while for a fractal, it can be a non-integer, and different fractals have their specific characteristic D values. The estimates were: D(18 mers) = 1.53±0.02, D(54 mers) = 1.67±0.02, and D(Sierpiński triangle) = 1.59. The mathematical fractals are infinitely repeated, and the authors explored whether this protein fractal could be increased to a size of more than 54 subunits. The radius of rotation (Rg) of the solution at a certain concentration was evaluated using small-angle X-ray scattering (SAXS) measurements, and the measured values were compared to the theoretical Rg values calculated by the 6-, 18-, and 54-mer structure models. At concentrations above 100 μM, the measured Rg values exceed the size of 54 mers. While it is not possible to prove that the larger assembly is a Sierpiński triangle and not another type of assembly, these experiments demonstrate that proteins are capable of extended growth.
Figure 2. Protein assembly into morphic structures【Structural basis of Sierpiński assembly】In order to assemble hexamer building blocks into Sierpiński fractals, new interfaces need to be introduced between dimers. There are two conditions that must be met for this interface. First, it must connect the hexahedron along the 120° outer angle, thus forming a triangle. Secondly, only an interface can be formed between the two dimers so that no more subunits are associated with the edge of the triangle. These criteria ensure that the edges of the triangle remain blunted and allow the Sierpiński triangle to be formed. These requirements make it difficult to meet the protein-protein interface typically found in homomers. However, a 120° angle between the hexamers can be achieved by introducing a three-fold symmetry, head-tail C3 interface between the dimers. But such an interface does not satisfy the second requirement, as it allows a third dimer to bind to it, thus forming a triangular lattice instead of a fractal. The second requirement can be met by a bidirectional head-to-head C2 interface between dimers, but this requires that the hexamers no longer interact at the correct angles, resulting in a hexagonal lattice. To understand how the fractal forms are assembled, the authors used cryo-electron microscopy (cryo-EM) to solve for the zero-order (6-mer, 3.1 Å), first-order (18-mer, 3.9 Å), and second-order (54-mer, 5.9 Å) structures of the Sierpiński triangle of the protein. In 18-mers, CS dimers are assembled into hexamers by heterologous interfaces, similar to known hexamer CS proteins. The 18-mer is then formed by additional contact between the two dimers of the adjacent hexamer. The interface has a special geometry so that only one monomer in each dimer participates in the interaction. The interaction between the residue E6 of one monomer and H369 of another monomer establishes an important link, and the formation of fractals can be avoided by mutating E6 or H369 to alanine or by removing amino acids 2-6 (Δ2-6 SeCS). In 18-mers, the inequivalence between the chains allows the interaction between the residues to form at the right angle, so that more dimers cannot bind in. This geometry further proves that the 18-mer lattice is not only a substructure of an ordinary triangular lattice, but actually represents the first order of the Sierpiński triangle. The authors found that the two dimers involved in the fractal connection have a small clockwise rotation relative to their conformation in the free hexamer, which cleverly breaks the D3 symmetry of the hexamer motif in the fractal. Next, the authors further investigated how 18mers are assembled into 54 mers. During structure-function analysis, the authors generated a protein from a point mutant (H369R) that also formed an 18mer at 50 nM that was indistinguishable from wild-type SeCS. In addition, it is able to form larger assemblies and produce less aggregated cryo-EM grids, so the authors can resolve 54-mers with a 5.9 Å structure. For the first time, the structure reveals edge passivation along its outward edge, as well as towards its huge internal voids, which makes it impossible for the hexamer to attach there without introducing a spatial conflict. In addition, the EM structure also revealed a mechanism similar to that of the inclusion of 18mers into 54mers and the interaction of 6mers into 18mers. However, for the two different attachments within the 54-polymer, the dihedral angle of the dimer is different. That is, the interaction between the 6-mers occurs at 60°, which is the same angle that occurs in the free 18-mers. In contrast, in the 54-mer, the angle between the 18-mers is 34°. For larger assemblies, this angle will be further reduced. These observations reveal the basic principle of the generation of Sierpiński triangles. All of the combinations observed seem to minimize the number of undesirable fractal interfaces. In intermediate stoichiometry, proteins are clearly distributed in a collection of non-fractal but still triangular ones. What unites these combinations is their unique triangular shape, which also distinguishes them from several other native proteins that form a two-dimensional lattice.
Figure 3. Assembly Layer
Figure 4. Single-crystal structure analysis [Function of Sierpiński assembly] Next, the authors further investigated whether the assembly of oligomer 18mer, which is dominant at physiological protein concentration, has an effect on the function of the enzyme. CS-catalyzed condensation of acetyl-CoA and oxaloacetate to citrate is the first step in the tricarboxylic acid cycle. The addition of either the substrate or the reaction product results in the structural breakdown into hexamers, which means that the hexamers have stoichiometric catalytic activity. The authors first measured the enzyme kinetics of wild-type and fractal-forming variants (SeCS L18Q). In the conditions of a saturated matrix with complete disruption of the fractal assembly, the kinetic parameters of the two variants are almost identical. Under unsaturated conditions, part of the 18mer remains, and wild-type SeCS is only half as active as the hexamer variant. Second, the authors constructed a mutant (CYS4) that covalently stabilizes the fractal interface via a disulfide bridge. For this variant, a certain percentage of the fractal complex is maintained at the saturated substrate concentration. The catalytic rate constant of cys4 is reduced compared to wild-type SeCS. This decrease in activity is reversible with the addition of a reducing agent, which destroys the disulfide bridge. These results showed that the assembly of constituent complexes significantly reduced the catalytic activity. To explain why 18mer activity is low, the authors unraveled the crystal structure of citric acid-binding hexamer at the 2.7 Å level. Comparing this structure with that of the free hexamer (Δ2-6 SeCS), it was found that the CS dimer in the hexamer underwent counterclockwise rotation after binding to citrate. This rigid body rotation is common in other CS enzymes. It occurs after substrate binding and advances CS from an open conformation to a closed form of catalytic occurrence. The rotation that enters the closed form is in the opposite direction of the clockwise rotation of the dimer into the fractal. This result suggests that the fractal complex must undergo greater conformational movement to induce substrate binding or catalysis, which may have imposed a higher energy barrier, which could explain the decrease in enzyme activity. Next, the authors wondered whether this unusual assembly might have a function in S. elongatus, for example, and whether it could be a means of regulating enzymes. Fractal complexes are pH sensitive. An increase in pH from 7.5 to 9 resulted in a complete decomposition of the structure into hexamers. This behavior is driven by a residue in the fractal interface, H369, which changes to arginine to eliminate pH sensitivity without affecting its assembled constituent form. The diurnal variation of the intracellular pH value of Ginseng longifolia was basically consistent with the pKa of the fractal interface. During the day, its carbon concentration mechanism adjusts the pH to 8.4 by absorbing bicarbonate. In the evening, the pH returns to around 7.3. Changes in diurnal pH can inhibit nocturnal SeCS activity due to reduced activity in fractal assembly. To test this idea, the authors created transgenic strains carrying wild-type CS or mutant CS that could not form fractals at the native site of Staphylococcus longum. The authors quantified their growth under continuous light and a 12-hour circadian cycle, but found no difference between the two conditions. The authors further investigated whether the formation of fractals prevents the depletion of the tricarboxylic acid cycle due to amino acid synthesis during nitrogen starvation. Therefore, the authors tested the ability of two genetically modified Staphylococcus longum to recover from nitrogen starvation. And again, no differences were found between strains carrying wild-type CS or non-fractal CS variants.
Figure 5. Effect of fractal assembly on catalytic function [Evolution of Sierpiński assembly] Based on the above observations and experiments, the authors wondered whether this protein assembly behavior was just a chance event and functionally insignificant. Therefore, the authors did not find 18 mers in its evolutionary history and phylogenetic tree of CS protein, except for phytoplankton thrips of Xylella. This suggests that by expanding the fractal, the 18-mer must be along S. The lineage of elongatus evolved and may have been developed in P. mougeotii and S. Elongatus was the last common ancestor, and then rapidly disappeared along the lineage of cyanobacteria/prochlorococcus. To test this theory, the authors used ancestral sequence reconstructions to resurrect ancestor CS proteins at successive nodes from SeCS to roots, and characterized their assembly by MP. The results show that the relatively weak fractal evolves from the hexamer between ancC and ancB and then becomes stronger between ancB and ancA. The authors then tried to determine which historical amino acid substitutions caused the protein to form a fractal assembly. The side chains E6 and H369 form the key to the fractal interface and confer pH sensitivity to protein assembly, which are already present in ancestors that have not yet formed fractals. When the fractal first evolved, the fractal interface only had one substitution of q18L in the ancC and ancB intervals (uppercase and lowercase letters representing ancestral and derived amino acids, respectively). The introduction of q18L in ancC alone is sufficient to trigger the formation of fractal complexes, including complexes larger than 18 mers. Reversing this substitution (L18q) in SeCS eliminates fractal assembly. In the 18-mer structure, the L18 side chains from the two non-participating monomers are located at the center of the fractal interface, but do not interact at the interface. Thus, q18L may have removed from the interface the repulsive polar interactions that prevent the formation of fractals. However, the introduction of the q18L substitution into an ancestor older than ancC did not produce fractals. As a result, the historical window of opportunity for fractals to evolve by replacing only with q18L is very narrow. Next, the authors searched for factors that strengthened the fractal interface along the interval between ancB and ancA. Only two conservative substitutions occur at the interface of this interval, k8R and y80F. In SeCS, F80 participates in cation-π interactions at the hexamer interface. y80F may have affected the strength of this interaction, potentially making the rotation of the dimer more favorable, which is necessary for binding into 18mer. The results show that only a small number of substitutions are required to establish stable fractals. This result is consistent with previous studies showing that a single substitution can significantly alter the morphology of oligomers or induce supramolecular assembly. This simple origin makes the non-adaptive origin credible. In addition, the pKa of the new interface was ready to match physiological pH fluctuations when assembly appeared: the introduction of q18L into ancC resulted in the interface dissociating within a pH range similar to that of wild-type SeCS. Thus, the apparently physiologically adjusted pKa in the assembly is either a molecular expansion or just a seemingly adaptive coincidence. The large combinations that can be formed by this protein are more obviously evolutionary accidents, and although the higher-order Sierpiński triangles are very regular and complex, they only occur at non-physiological concentrations, and their size is difficult to adapt to S. Cytoplasm of elongatus. Even if 18-mer fulfills a useful function, the protein's ability to make larger combinations is almost certainly a casual byproduct of the unusual symmetry it happened to evolve.
Figure 6. Evolutionary genealogy analysis concludes that this work is based on the accidental discovery of protein self-assembly into an extremely rare fractal structure in nature, and reveals the necessary conditions for the formation of fractals of the protein and the influence of fractal structure on protein function by combining a variety of analytical characterization techniques. This work provides a broader perspective for a deeper understanding of the structure and function of protein assemblies in the future. Source: Frontiers of Polymer Science