Design of a minimal, allosteric, and ATPase-like machine using mechanical linkages

TL;DR

The paper presents a mechanical-linkage model that mimics ATPase allostery using two binding sites arranged in a chain of rigid bars. It demonstrates a cyclic, autonomous ATPase-like machine in which an ATP (S) substrate displaces a non-consumed effector (L), substrate hydrolysis yields Pi and ADP analogs (P1, P2), and the effector displaces P2 to reset the cycle; both forward catalysis and ligation are slower than binding to ensure proper turnover, while two futile pathways (idling and steric displacement) can arise. The authors construct a comprehensive 449-state reaction network and perform stochastic simulations to quantify turnover, ligand activation, and the conditions that maximize the target cycle. They show that a simple, topology-based design can reproduce autonomous, oscillatory behavior and provide design principles for synthetic ATPase-like monomers; the approach suggests paths toward more complex allosteric machines by extending the network topology or using origami-like structures.

Abstract

ATPases cyclically convert chemical energy in the form of ATP gradients into directed motion inside cells. To function, ATPases rely on allosteric communication between at least two binding sites, an internal signaling mechanism that is not well understood. Here, we model an ATPase-like machine by using a system of mechanical linkages to recreate negative allosteric coupling between two binding sites and generate cycles in which the sites alternate occupancy. The ATPase analog has two mechanical degrees of freedom and two discretized binding sites: one for the ATP, Pi and ADP analogs, and one for an allosteric effector analog. The geometry of the ATPase analog allows stepwise binding reactions at each site to capture the two degrees of freedom in a mutually exclusive way. Consequently, the enzyme interconverts between multiple rigid and partially rigid forms, such that neither site can be fully bound when both sites are occupied. Two mechanisms work together to generate an enzymatic cycle: one, in which the tighter-binding ATP analog can bind and displace the effector from the enzyme; and a second, in which flexibility introduced by splitting the ATP analog into two pieces (catalysis) allows the effector to rebind and displace the products (ADP analog). We show that cleavage (forward catalysis) and ligation (reverse catalysis) alter the rigidity of the enzyme complex equivalently to binding and dissociation, respectively, but must do so more slowly for effective cycling to take place. Simple designs for synthetic systems that mimic ATPase monomers can be derived from this work.

Figures (8)

• Figure 1: Three-state switch.A, A single unit. The single unit consists of two rigid squares connected at a flexible node (white). Three conformations are possible: a flexible one (2), in which no additional bars are connected; a rigid one (1), in which a bar (blue) connects the two bottom nodes to form a rigid triangle; and another rigid one (3), in which a bar (red) connects the two top nodes to form a rigid triangle. B, A double unit. A double unit is formed by merging one square (marked with 'x') of two single units. The double unit has four completely rigid conformational states, and four partially rigid conformational states, as described later in more detail.
• Figure 1: Myosin chemomechanical cyle vs simplified chemical cycle.A. Five reactants of a myosin monomer: myosin with a lever arm; polymeric actin; ATP; Pi; and ADP. B. Chemomechanical cycle of a myosin monomer. Myosin goes through six chemical steps and two major mechanical changes of the lever arm, which are the recovery stroke, and the power stroke, where the power stroke is often depicted taking place in two stages. C. Five reactants of the generic ATPase-like machine: the enzyme; the ligand; substrate; P1; and P2. D. Simplified chemical cycle of the generic ATPase-like machine. In this cycle, P1 (the Pi analog) dissociates before ligand (the actin analog) binds.
• Figure 2: The linkage system.A, Five molecules of the linkage system. The enzyme consists of three rigid squares of edge length $\ell$, connected at two points of rotation. Nodes a, b and c form the binding site for the substrate, P1 and P2. Nodes d, e and f from the binding site for the ligand. Complimentary nodes on S, P1, P2 and L, are denoted by the same letter with the superscipt "*". S and L consist of three nodes connected by two bars of length $\ell$, where the center nodes on each are points of flexibility. S is split into P1 and P2 at the special node $\delta$. B, Left, L blocks S. When L is bound at all three nodes, it bends the enzyme down, placing the two outer nodes out of reach for S. The increased distance between S's nodes is $x =\sqrt{3}\ell$. Right, symmetric blocking of L by S. C, Left, L blocks P2. Right, symmetric blocking of L by P2. D, Target cycle and two futile pathways. The target cycle (cycle i; outside black path going clockwise) takes place in thirteen reversible reactions. Each state name (e.g. {S/L}:{10,5}) indicates its composition ({S/L}) and unique linkage state ({10,5})(see SI \ref{['si_crns']}). Forward and reverse rates governing each reaction are labeled on each edge (e.g. between {S/L}:{10,5} and {S/L}:{11,5}, the forward rate is $\text{k}_{\text{off-d}}$, and the reverse rate is $\text{k}_{\text{uni}}$). Reactions numbered 1-6 mark the same changes in composition that take place Eq. 1. Starting at {L}, the allosteric displacement of L by S (light blue bubble) takes place in four intramolecular steps between 1 and 2, and completes with the enzyme bound to only S ({S}). Subsequently (orange bubble), S is cleaved into P1 and P2 (3, ✂). Directly following cleavage, or rounds of ligation (--${}$--) and cleavage, P1 spontaneously dissociates (4), which rectifies catalysis and leaves the enzyme bound to only P2 ({P2}). After L binds (5), the allosteric displacement of P2 by L (green bubble) takes place in three intramolecular steps between 5 and 6, and returns the enzyme to the start of the cycle and bound to only L {L}. The two futile pathways are idling (blue bubble and dashed line) and steric displacement (orange bubble and dashed line). Both pathways are abbreviated, and shown without figures. The change in the bound state along each path is stated: {P1, P2/L} in idling; and {S, P2} in steric displacement. Combined with states along the target path, idling is cycle ii, and steric displacement is cycle iii. E, Simplified network version of the system. Each node here represents a set of states with different intramolecular conformations. F, Idling cycle. G, Steric displacement cycle.
• Figure 2: Basis states. Subset of seventeen states that are used to generate and name the complete set of 449 states. In each of these states only one molecule of S, P1, P2 or L is bound, and the subset of states enumerates all the different ways these four molecules can bind to the enzyme, barring the two states eliminated by the adjacency rule.
• Figure 3: Plots showing behavior of the system.A, Time traces of two stochastic simulations (done with StochPy) comparing the number of P2's released when ligand and substrate are present (blue) versus substrate only (red), where [S] = [L] = 100 $\mu$M. B, Ligand activation ($A$) plotted over a range of substrate concentrations. Because of no, or low activity for simulations done without ligand ($v_{\text{noL}}$) at low [S] ($< 0.1$$\mu$M), the error in $A$ is too large, and these data are left out. C, Turnover rates ($v$) for the most relevant cycles plotted over a range of substrate concentrations. The total rate (black; $v_{\text{total}}$) is the sum of the separated pathways (in color). The target cycle (i, blue; $v_{\text{target}}$) peaks around () and then is overtaken by steric displacements (ii, orange) until the system saturates at high [S]. D, Efficiency ($E$) plotted over a range of substrate concentrations. Efficiency is highest where incidence of the target cycle is highest (compare with C, above).