Figures and data
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.
The linkage system.
A, Five molecules of the linkage system. The enzyme consists of three rigid squares of edge length ℓ, 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 ℓ, where the center nodes on each are points of flexibility. S is split into P1 and P2 at the special node ℓ. 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 
Rates used for the simulations.
These rates are represented in Fig. 2. kreactant-on is either kSon , kLon , kP1on , or kP2on.
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 μM. B, Ligand activation (A) plotted over a range of substrate concentrations. Because of no, or low activity for simulations done without ligand (υnoL) at low [S] (< 0.1 μM), the error in A is too large, and these data are left out. C, Turnover rates (υ) for the most relevant cycles plotted over a range of substrate concentrations. The total rate (black; υtotal) is the sum of the separated pathways (in color). The target cycle (i, blue; υtarget) 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).
Idling and catalysis siloing.
A, Plot of idling silo. While keeping the ligation rate, klig, constant at 10 s−1 , idling becomes the dominant pathway taken by the system as the cleavage rate, kclv, is increased. The crossover point lies where kclv surpasses the ligand’s off rates (between 102 and 103 s−1; vertical blue band). Before this point, in particular where kclv equals klig (at 10s−1), the behavior is optimal, and the target cycle dominates. B, Plot of catalysis silo. While keeping the cleavage rate constant at 10s−1, the turnover rate decreases and approaches zero as the ligation rate, klig, approaches the intramolecular binding rate, kuni. C, Idling silo pathway. Two time points along the target trajectory (top) versus idling silo trajectory (bottom) are shown. Starting at {S/L}:{0, 15} at ti (in blue circle), siloing dominates when kclv is much greater than the ligand off rates (here, koff-e), which biases cleavage to take place before L dissociates, and a transition to {P1, P2/L}: {6, 9, 15} at ti+1 (see SI Fig. 7N for full sequence of idling). By contrast, when koff-e is greater than kclv, L is biased to dissociate before cleavage (top path), and the system transition to {S}:{0} instead, and stays on the target path. The rate conditions governing the top vs bottom paths are shown below. D, Left, Idling silo on the mini-network. Right, Catalysis silo on the mini-network. The starting state for each silo is circled: {S/L} for idling; and {P1, P2} for catalysis. Unlike the idling silo, the catalysis silo is not a cycle, but rather interconversion between {S} and {P1, P2} without turnover. E, Catalysis silo pathway. Two time points along the target trajectory (top) versus catalysis silo (bottom) are shown. Starting at {P1, P2}: {6, 9}, siloing dominates when klig is much greater koff-a, allowing ligation (bottom path) to take place before P1 dissociates (top path). The rate conditions governing the top vs bottom paths are shown below.
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.
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.
Adjacency rule.
Only adjacent bonds are allowed to form or dissociate between two interacting linkages. Left, disallowed substrate state in which the two outer nodes are bound, and the central node is dissociated. Right, disallowed ligand state in which the two outer nodes are bound, and the central node is dissociated.
Rules matrix.
This binary matrix graphically displays the geometric restrictions that exist between the basis states, and which each represent an instance of negative allosteric coupling. Basis states that cannot coexist, and generate a new state, are represented with ∄, for “does not exist”. There are nine conflicts denoted with ∄. Note that P1 and the ligand-bound-states ({L}) do have any conflicts, because P1 is bound monovalently to the enzyme.
Number of states in each set.
Categories of reaction rates
Dissociation rates
Stochastic conversions of rates
Energies of the unbound and bound enzyme.
Exclusive behavior blocks from non-exclusive blocks.
This table shows the sets of nonexclusive behavior blocks (top row, ‘nx’ subscript) that compose each exclusive behavior blocks (left column, no subscript). A ‘1’ indicates that the non-exclusive block is an allowed element, and a ‘0’ indicates that it is not an allowed element. The starred exclusive blocks, lfsd*, lf* and lf-frc*, do not have non-exclusive counterparts, as do the other blocks, and are only defined as compositions of the other non-exclusive blocks.
Behavior block data for ‘basic’ simulations.
Reaction sequences.
Reaction sequences.
Reaction sequences.
Behavior blocks.
Behavior blocks continued.
Movie 1 dwell legend.
This is a labeled version of the legend that appears in Movie 1. This dwell time plot shows a roughly a half second of the simulation, and the species that bound to the enzyme during this time. On the left, the row for each species is labeled (‘species bound’), where S is shown in red, P1 in orange, P2 in green and L in blue. When multiple species are bound the bars of color overlap. For example, when S binds, close to 18.70 seconds, the red and blue bars overlap for a short section, during which the displacement of ligand by substrate takes place (see section labeled {S, L}). There are three more overlapping sections: {P1, P2}, where P1 and P2 are created just after cleavage; {P2, L}*, where short-lived spurious binding of L takes place; and {P2, L}, during which L displaces P2. From the beginning to roughly 19.05 seconds, one cycle (cycle 1) is completed. The beginning of cycle 1, where L first binds, takes place earlier in the simulation and is not shown. The end of cycle 1 overlaps with the beginning of cycle 2, where the overlap is the section where L displaces P2. The two cycles (cycle 1, grey; cycle 2, black) are demarcated at the bottom with horizontal lines.
Movie 2 dwell legend.
A. This is the legend that appears in Movie 2. At the bottom, the seven cycles that take place (though cycles 1 and 7 are not complete) are demarcated. The six reactions are labeled by number for cycle 3. At the top the six cleavage events that take place are demarcated. B. This is the full 100 second simulation from which Movie 2 and Movie 1 were taken.
kcat variation simulations.
klig variation simulations.
Saturation.
A, Pathway to saturation. Beginning with the P2-bound state ({P2}), substrate, at high concentration, binds at node a, followed by ligand. Trivalent ligand binding (abbreviated by ‘…’) displaces P2, which exposes nodes a and b for substrate binding, and finally, substrate saturation. B, Persistence of the saturation state. The saturation state persists because any substrate that dissociates is likely replaced (left branch) before ligand dissociates from one of its nodes and opens up a pathway for the displacement of ligand by substrate (right branch). Substrate replacement beats ligand dissociation because at high [S], the binding rate surpasses the ligand dissociation by orders of magnitude. For example, at 5 mM [S], kon-S = 450 × 102 s−1, whereas koff-d = 680 s−1. The figure shows the saturation state decaying by dissociation of substrate at node a (the second state from top; state {4, 5, 10} in basis form, see SI Fig. 2), but it can also decay by dissociation of substrate from node c (state {3, 5, 10} in basis form).
