How Flexibility Can Enhance Catalysis

Olivier Rivoire

Phys. Rev. Lett. 131, 088401 – Published 23 August 2023

Abstract

Conformational changes are observed in many enzymes, but their role in catalysis is highly controversial. Here we present a theoretical model that illustrates how rigid catalysts can be fundamentally limited and how a conformational change induced by substrate binding can overcome this limitation, ultimately enabling barrier-free catalysis. The model is deliberately minimal, but the principle it illustrates is general and consistent with unique features of proteins as well as with previous informal proposals to explain the superiority of enzymes over other classes of catalysts. Implementing the discriminative switch suggested by the model could help overcome limitations currently encountered in the design of artificial catalysts.

Received 16 December 2022
Accepted 28 July 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.088401

Physics Subject Headings (PhySH)

Chemical kinetics, dynamics & catalysis Conformation changes Escape problems Nonequilibrium & irreversible thermodynamics

Nonequilibrium lattice models

First passage problems Markovian processes

Statistical Physics & ThermodynamicsInterdisciplinary PhysicsPhysics of Living Systems

Authors & Affiliations

Olivier Rivoire

Center for Interdisciplinary Research in Biology (CIRB), Collège de France, CNRS, INSERM, and Gulliver, CNRS, ESPCI, Université Paris Sciences et Lettres, 75005 Paris, France

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Issue

Vol. 131, Iss. 8 — 25 August 2023

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Images

Figure 1
Model for the spontaneous reaction. (a) Potential of interaction between two particles as a function of their distance $d_{s}$ , parametrized by the barriers $h_{s}^{\pm}$ . No two particles can occupy the same site and therefore $E_{s} (d_{s} = 0) = \infty$ . (b) Two particles occupy distinct sites on a triangular lattice with 12 nodes and periodic boundary conditions along one direction (repeated purple nodes). The particles initially form a dimer ( $d_{s} = 1$ , with energy $h_{s}^{-} - h_{s}^{+}$ ; e.g., top configuration). Each particle can diffuse to a neighboring site with rates given by Metropolis rule. The reaction is completed when the particles are free ( $d_{s} \geq 3$ , energy 0; e.g., bottom configuration). This requires crossing a transition state ( $d_{s} = 2$ , energy $h_{s}^{-}$ ) and takes a mean time $T_{S \to 2 P}$ that scales as $T_{S \to 2 P} \sim e^{h_{s}^{+}}$ for large $h_{s}^{+}$ .
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Figure 2
Model for rigid catalysis. (a) A catalyst consists in two binding sites at fixed locations on the lattice (orange squares). (b) When a particle occupies one of these sites, the energy is decreased by $ε_{c s}$ . (c) Catalytic efficiency $η = T_{S \to 2 P} / T_{C + S \to C + 2 P}$ comparing the mean reaction time in this model, $T_{C + S \to C + 2 P}$ , with the mean time $T_{S \to 2 P}$ for the spontaneous reaction (Fig. 1) as a function of the binding strength $ε_{c s}$ ( $h_{s}^{+} = 6$ and $h_{s}^{-} = 12$ ). This graph is comparable to so-called volcano plots in heterogeneous catalysis [30]. (d) Energy landscape illustrating how a rigid catalyst replaces the single barrier $h_{s}^{+}$ [Fig. 1] by two smaller barriers $h_{s}^{+} - ε_{c s}$ and $ε_{c s}$ . The occupation of the binding sites in each configuration is represented at the bottom. (e) Catalytic efficiency of optimal rigid catalysts as a function of the forward barrier $h_{s}^{+}$ for different reverse barrier $h_{s}^{-}$ , showing that catalysis involves a threshold beyond which the efficiency scales exponentially.
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Figure 3
Model for catalysis with a discriminative switch. (a) We add a degree of freedom in the form of a red particle that can take two positions, down ( $C_{0}$ ) or up ( $C_{1}$ ). (b) Switching from $C_{0}$ to $C_{1}$ involves an energy cost $ε_{c}$ , and the interaction energy at each binding site is $ε_{c s}$ in $C_{0}$ but $ε_{c s} + δ_{c s}$ in $C_{1}$ . (c) Mean time of completion of the reaction in the absence of catalyst (green line), in the presence of an optimal rigid catalyst (blue line), a flexible symmetric catalyst with $ε_{c s} = 0$ and $ε_{c} = δ_{c s} = (h_{s}^{+} + h_{s}^{-}) / 2$ (full red line), or a flexible asymmetric catalyst with $ε_{c s}^{1} = ε_{c s}^{2} = 0$ , $ε_{c} = δ_{c s}^{1} = 4 h_{s}^{+}$ and $δ_{c s}^{2} = 1.5 h_{s}^{+}$ (dotted red line). For large $h_{s}^{+}$ , $T_{C + S \to C + 2 P}$ scales as $e^{a h_{s}^{+}}$ , where $a = 1$ without catalyst, $a \geq 1 / 2$ with rigid catalysts, but $a = 0$ with catalysts having a discriminative switch (with $h_{s}^{-}$ taken to be $h_{s}^{-} = 2 h_{s}^{+}$ ). Standard deviations can also be computed to show that the distributions of first-passage times are increasingly distinct as $h_{s}^{+}$ increases (Fig. S5 [27]). (d) When the conditions given in Eq. (4) are satisfied, the energy along the path from $C_{0} + S$ to $C_{0} + 2 P$ does not increase at any step. Catalysis is then barrierless. Examples of configurations along this path are illustrated at the bottom.
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