Quantitative de-idealization of genericity theorems for binding-induced conformational change and allosteric regulation in proteins

A geometric theory of protein allostery (Zadorin, arXiv:1905.02815, building on Rivoire's evolutionary scenario) proves, using transversality theory on smooth manifolds, that generic solutions of the 'exquisite discrimination' problem — a flexible protein selected to bind a correct ligand more strongly than solvent while binding a very similar incorrect ligand more weakly — must concentrate near a codimension-1 subset of trait space, must undergo a conformational change upon binding, and are generically susceptible to allosteric regulation of their discrimination by binding at a distant site. These theorems, however, are proved only in the zero-temperature limit, in the indistinguishable-ligand idealization, and under the assumption that the discrete sequence space of real proteins is well approximated by a smooth manifold; they are topological and say nothing about the size of the guaranteed conformational change. The open problem is to de-idealize this theory: establish quantitative versions of the three genericity conclusions at nonzero temperature and finite ligand separation, translate them to discrete combinatorial genotype spaces, and formulate a metric/energetic criterion under which the guaranteed conformational change is large enough to be functionally relevant.

Background

Allostery is the regulation of a protein's primary activity by the binding of another molecule at a distant site. Ever since the classical induced-fit picture (ligand binding provokes a conformational change in a flexible protein) and the conformational-selection picture (binding shifts the occupancy among pre-existing conformations), a recurring question has been why conformational change upon binding and the potential for allostery are so widespread in ligand-binding proteins. Rivoire (Physical Review E 100, 032411, 2019) proposed an evolutionary scenario: if a protein must exquisitely discriminate between two very similar ligands — bind the correct one more strongly than solvent and the incorrect one more weakly — then, in any sufficiently flexible system, this selection forces an evolutionary fine tuning that generically produces a conformational change upon binding, and that same fine tuning creates a latent potential for allosteric regulation. The scenario was demonstrated only on two extreme model classes: elastic-network models with one or two degrees of freedom, and a two-dimensional spin-glass model (a disordered model of binary-state 'amino acids') whose intrinsic discontinuity made it unclear whether the effects were general properties of protein-like systems or artifacts. Zadorin (arXiv:1905.02815, 2019, unpublished preprint) closed this gap for smooth models of arbitrary dimension. The setup posits three smooth compact manifolds: a conformation space $X$ , a ligand (environment) space $L$ , and a trait (sequence) space $A$ , linked by a smooth energy function $U(x,\ell,a)$ . In the zero-temperature limit the protein occupies the global minimizer of $U(\cdot,\ell,a)$ , and a phenotype $a$ solves the exquisite-discrimination problem when the ground-state energy with the correct ligand $\ell_r$ is below that with solvent $\ell_\emptyset$ , which is below that with the incorrect ligand $\ell_w$ . Idealizing $\ell_w=\ell_r$ (the 'reduced' problem), and then replacing their difference by an infinitesimal direction in $L$ , the paper applies Thom's transversality theorem and its multijet extension (tools of singularity theory stating that maps are 'generically' — for a residual set in the Whitney smooth topology — transverse to any given submanifold, which forces intersections to be as clean and abundant as dimension counting allows; a $k$ -jet records a map's value and derivatives to order $k$ ) to prove three theorems: (1) generic solutions of the reduced problem form a subset of codimension 1 in $A$ , i.e. evolutionary solutions are finely tuned and rare in a dimension-counting sense; (2) a generic such solution has distinct minimizing conformations with and without the ligand — a conformational change upon binding is generic, and a discriminating protein that does not change conformation is special; (3) the discrimination is generically regulable by binding at a second site, i.e. latent allostery is generic. The author explicitly flags what the theorems do not deliver. They hold at zero temperature and for exactly coincident ligands; the discussion gives only a qualitative, still-zero-temperature sketch in which finite ligand separation 'thickens' the codimension-1 solution set into a slab between two border surfaces. They are topological, not metric: no criterion distinguishes a functionally relevant conformational change from a negligible one, although empirical surveys of apo (unbound) and ligand-bound crystal structures (Gutteridge and Thornton, Journal of Molecular Biology 346, 21–28, 2005, on 60 enzymes) find that induced-fit motions in most enzymes are very small — usually about 1 Å root-mean-square displacement (RMSD, the typical coordinate shift of corresponding atoms between the two structures) — so the size question is physically consequential. And the trait space $A$ is acknowledged to be 'not a smooth manifold in nature' but 'a nondirected graph with high symmetry' (e.g. binary sequences of length $n$ form a hypercube graph), which is assumed, without justification, to be well approximated by a smooth manifold.

Problem Statement

For smooth compact manifolds $X$ (conformations), $L$ (ligands) and $A$ (traits), a smooth energy $U:X\times L\times A\to\mathbb{R}$ , ground-state (zero-temperature) energetics, and the indistinguishable-ligand limit $\ell_w=\ell_r$ , Zadorin's theorems establish, generically in the Whitney smooth topology: solutions of the exquisite-discrimination problem concentrate near a codimension-1 submanifold of $A$ ; such solutions have distinct minimizing conformations with and without the ligand; and the discrimination is regulable by binding at a second site. Establish quantitative counterparts of these three conclusions outside the idealization in which they are proved. The common objective is a single de-idealized theory, physically interpretable and testable, and its components are: (1) Finite temperature and finite ligand separation: replace ground-state energies by free energies $F(\ell,a)=-\beta^{-1}\ln\int_X e^{-\beta U(x,\ell,a)}\,dx$ , where $\beta$ is the inverse temperature, and determine, with quantitative bounds or explicit model computations, how each genericity conclusion — the concentration of discrimination solutions (of which the source's zero-temperature 'thick slab' picture is only a qualitative sketch), the conformational change upon binding, and the allosteric regulability — degrades as $\beta$ becomes finite and as the ligand separation $\|\ell_r-\ell_w\|$ between the correct and incorrect ligands grows from zero, identifying the regimes in which each conclusion survives. (2) Discrete genotype spaces: justify, or replace, the assumption that a combinatorial genotype space such as the $n$ -dimensional hypercube of binary sequences is well approximated by a smooth manifold for the purposes of these genericity statements, and translate the codimension-1 concentration into a quantitative statement about counts, measures, or neighborhoods over discrete genotypes. (3) Metric size: formulate a criterion, in terms of $U$ and physical scales such as bond stiffness and typical mutational energy effects, under which the guaranteed conformational change upon binding is large enough to be functionally relevant — for instance, to permit regulation of the discrimination by a ligand at a distant site — as opposed to negligible, and determine generically, within such a criterion, how large the change is. The results must jointly address this de-idealization of the stated theorems rather than an unrelated restatement of allostery folklore; since those theorems are genericity claims over all smooth systems, the target is a general quantitative statement, for which computations on individual models serve as support.

References

Anton S. Zadorin, 'Allostery and conformational changes upon binding as generic features of proteins: a high-dimension geometrical approach', arXiv:1905.02815 (2019), DOI: 10.48550/arXiv.1905.02815, https://arxiv.org/abs/1905.02815 (unpublished preprint, v1)
Olivier Rivoire, 'Parsimonious evolutionary scenario for the origin of allostery and coevolution patterns in proteins', Physical Review E 100, 032411 (2019), DOI: 10.1103/PhysRevE.100.032411, arXiv:1812.01524, https://arxiv.org/abs/1812.01524
Alex Gutteridge, Janet Thornton, 'Conformational changes observed in enzyme crystal structures upon substrate binding', Journal of Molecular Biology 346, 21–28 (2005), DOI: 10.1016/j.jmb.2004.11.013, https://doi.org/10.1016/j.jmb.2004.11.013 (PMID 15663924)
J. M. McBride, J.-P. Eckmann, T. Tlusty, 'General Theory of Specific Binding: Insights from a Genetic-Mechano-Chemical Protein Model', Molecular Biology and Evolution 39(11), msac217 (2022), DOI: 10.1093/molbev/msac217, https://doi.org/10.1093/molbev/msac217 (PMID 36208205)
Eric Rouviere, Rama Ranganathan, Olivier Rivoire, 'Emergence of Single- versus Multi-State Allostery' (preprint title: 'On the emergence of single versus multi-state allostery'), PRX Life 1, 023004 (2023), DOI: 10.1103/PRXLife.1.023004, arXiv:2111.09377, https://arxiv.org/abs/2111.09377
Olivier Rivoire, 'How flexibility can enhance catalysis', Physical Review Letters 131, 088401 (2023), DOI: 10.1103/PhysRevLett.131.088401, arXiv:2212.08352, https://arxiv.org/abs/2212.08352
Ayaka Sakata, Kunihiko Kaneko, 'Evolutionary Shaping of Low-Dimensional Path Facilitates Robust and Plastic Switching Between Phenotypes', Physical Review Research 5, 043296 (2023), DOI: 10.1103/PhysRevResearch.5.043296, arXiv:2304.11437, https://arxiv.org/abs/2304.11437
A. Sakata, K. Kaneko, 'Dimensional Reduction in Evolving Spin-Glass Model: Correlation of Phenotypic Responses to Environmental and Mutational Changes', Physical Review Letters 124, 218101 (2020), DOI: 10.1103/PhysRevLett.124.218101, https://doi.org/10.1103/PhysRevLett.124.218101