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Conceived and designed the experiments: HGZ. Performed the experiments: HGZ. Analyzed the data: HGZ JEH NVM. Contributed reagents/materials/analysis tools: HGZ JEH NVM. Wrote the paper: HGZ.

The authors have declared that no competing interests exist.

The reversal of flagellar motion (switching) results from the interaction between a switch complex of the flagellar rotor and a torque-generating stationary unit, or stator (motor unit). To explain the steeply cooperative ligand-induced switching, present models propose allosteric interactions between subunits of the rotor, but do not address the possibility of a reaction that stimulates a bidirectional motor unit to reverse direction of torque. During flagellar motion, the binding of a ligand-bound switch complex at the dwell site could excite a motor unit. The probability that another switch complex of the rotor, moving according to steady-state rotation, will reach the same dwell site before that motor unit returns to ground state will be determined by the independent decay rate of the excited-state motor unit. Here, we derive an analytical expression for the energy coupling between a switch complex and a motor unit of the stator complex of a flagellum, and demonstrate that this model accounts for the cooperative switching response without the need for allosteric interactions. The analytical result can be reproduced by simulation when (1) the motion of the rotor delivers a subsequent ligand-bound switch to the excited motor unit, thereby providing the excited motor unit with a second chance to remain excited, and (2) the outputs from multiple independent motor units are constrained to a single all-or-none event. In this proposed model, a motor unit and switch complex represent the components of a mathematically defined signal transduction mechanism in which energy coupling is driven by steady-state and is regulated by stochastic ligand binding. Mathematical derivation of the model shows the analytical function to be a general form of the Hill equation (Hill AV (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv–vii).

Bacterial flagellar switching is a process by which the rotation direction of a flagellum reverses. The switching process involves the interaction between a switch complex of the flagellar rotor and a torque-generating stationary unit, or stator. By regulating this interaction, the concentration of an intracellular signaling ligand, CheYP, determines flagellar switching behavior, including the frequency of switching, intervals of counterclockwise (CCW) and clockwise (CW) rotation, and average rotating time in the CW direction, referred to as CW bias

Ligand CheYP binding to the flagellar rotor

In the process of switching, structural changes in the rotor could be transmitted to the stator. The stator comprises approximately 10 motor units, which are connected by the torque transmitted through the rotor

The binding of CheYP shifts the probability of the rotation direction in favor of CW rotation, i.e., CW bias

CheYP has been shown to bind non-cooperatively to the switch complexes of a rotor

Consistent with the binding data, we suggest in this paper that flagellar switching is regulated by stochastic ligand binding and that a cooperative response is generated without an allosteric mechanism. To develop a mathematical model of the flagellar switching mechanism, we start with a model based on the regulation of vertebrate striated muscle, which has been shown to generate cooperative responses from non-cooperative ligand binding _{0} is an apparent constant; and

To model the flagellar switching system, we propose that a motor unit can exist in either a ground state (C) or an excited state (M) and that a switch complex can exist in either a ligand-free (u) or ligand-bound state (U) (_{j}*; j = 1, 2) to represent the intermediates of alternate reactions between the ground (C) and excited (M) states of the motor unit (Reactions 2 and 3, _{1}* and C_{2}* are transition states of an equilibrium pathway (Reactions 1, 2, and 4;

The diagram shows the pathways by which traveling switch components of the rotor (circles) stimulate a motor unit (square sides). At time zero, the motor in the ground state (C) has the null probability of being excited (blue). Formation of a collision complex (C_{j}*; j = 1,2) stimulates a motor unit to an excited state (M) with probability (P_{M}) of unity (red). While P_{M} decays (color key in inset), the rotor traveling at a constant rate, _{r}, breaks contact with one switch complex (U_{1}) at rate _{r−} = _{r} and delivers another switch complex (U_{i}, i>1) to the motor unit at rate _{r+} = _{r}. The ligand concentration, [L], determines whether a switch can form a collision complex (filled circle; U) or not (open circle; u). If by chance, θ = _{L}[L]/(1+_{L}[L]), the switch complex at the dwell site is occupied by ligand, and the formation of C_{j}* restores P_{M} to unity. An equilibrium pathway is required to initiate motor unit excitation (Eq. 1, 2, and 4; _{1}* and C_{2}* are included in the symbols for intermediate products of the equilibrium and second-chance pathways (Reactions 2 and 3), respectively. It should be noted that C_{1}*, C_{2}*, and C_{j}* represent the same change in physical structure of the motor unit as is represented by the transition between diamonds and squares in the schematic. To recapitulate, temporal changes in P_{M} may be traced for the scenario shown (inset). Decay of the excited state is by single exponential (inset). In a different scenario, had the motor unit returned to the ground state before being stimulated again, the equilibrium pathway would have been required to initiate a new excited state.

For modeling an arbitrary system, we use mole fraction quantities for u, U, C, and M, which are dimensionless quantities normalized to the total number of motor units in a given functional ensemble, _{tot}. Thus, given that _{tot} is the sum of the numbers of ground and excited motor units (_{C} plus _{M}), the ratios _{C}/_{tot} and _{M}/_{tot} are equal to the fractions of motors in the ground and excited states, C and M, respectively. Similarly, assuming a one-to-one interaction between a switch complex and a motor unit, the sum of ligand-free and ligand-bound switch complexes, _{u} and _{U}, is _{tot}, and the ratios _{u}/_{tot} and _{U}/_{tot} are equal to u and U, respectively. The distribution of the switch complexes between the u and U states is determined by the ligand concentration, [L], according to mass action: U = _{L}[L]u, where _{L} is the equilibrium constant for the reaction. As defined, u, U, C, and M have values between 0 and 1, and hence behave as probabilities for all _{tot}≥1. L has units of concentration.

At the moment of motor unit stimulation (time, _{M}) will occur is unity. After stimulation (_{M} can return to unity by either the equilibrium pathway or the second-chance pathways (inset;

The essential aspects of second-chance signal transduction can be seen by following the chance events of a single motor unit over time (_{j}* decays to the excited state M (Reaction 4; _{M}) at the moment of M formation (_{j}* decay can return P_{M} to unity (diagram and inset; _{j}* events. A sustaining C_{j}* event could be generated by the equilibrium pathway (scenario not shown;

To derive the rate of change in M by the equilibrium pathway, we define an arbitrary rotor subunit as u_{1}. From Reactions 1, 2, and 4 (_{1} = _{L}[L]u_{1}, C_{1}* = _{1}U_{1} C, and M = _{2} C_{1}* which, by substitution, yields an expression, _{2}_{1} C_{L}[L]u_{1} for the forward rate of M. The reverse rate of M is given by the expression _{−2}_{−1}M. Hence, the rate of change in the equilibrium pathway, _{1}/_{1} and C, the following conservation expressions must hold_{r}). The rotation disrupts C_{2}* at rate _{r−} = _{r}. C_{2}* forms with another switch, U_{i} (i>1), at rate _{r+}, which is limited by either the intrinsic reaction mechanism or the rate of the rotor. Hence, _{r+}≤_{r}.

From the rotation-dependent dissociation of C_{2}* (Reaction 3; _{−2}_{r−}_{i} = _{L}[L]u_{i} (Reactions 1, _{2}* = _{r+}U_{i}C (Reaction 3, _{2} C_{2}* (Reaction 4, _{2}_{r+}C_{L}[L]u_{i} for the forward rate. Hence, the rate of change in the second-chance pathway, _{2}/_{1} is canceled by the delivery of U_{i}, and thus conservation is satisfied overall, and_{r−} = _{r+} = 0,

We devised a computer program (details in Methods S1) to simulate the stimulation of a motor unit by switches being moved by the rotor. In this program, chances for collision complex formation are a sequence of pulses generated at the rate of rotor rotation (_{M} is set to unity; if the switch is ligand-free, P_{M} = _{−2} is the rate of excited state decay and _{M}, the program simulates a stochastic event (

Details of the program are contained in the supplement (Methods S1). Preliminary simulations established the decay rates that correspond best to

We propose that _{0} must be adjusted upward as the value of

_{L} = 3.7 µM _{0}, and _{0}. Shown are the fits for _{0} given by 1 and 2 (purple), 1.5 and 0.7 (blue), and 2 and 0.3 (rose), respectively. _{0} (2 and 0.45, respectively) and varied _{0} = 1, _{L} = 1, and either _{−2}) are 0.8 (squares) or 10 (circles), which were determined in preliminary measurements (details in Methods S1).

Increasing the value of _{0} constant produces a more cooperative response (

The analytical function (Eq. 2.14) is a macroscopic expression of underlying switching events taking place between individual molecules (

To test these novel predictions, we devised a computer program that simulates stochastic binding events at the rate of the rotor using a pulse generator. We constrained the simulation by holding the outcomes of the internal events constant during the period between pulses, which reduces the model to the special condition, _{−1} = 0 (

We also constrained the program to a single pathway. Combining the two pathways should not introduce systematic error when _{M} in this special case of

We use high and low rates of excited state decay to simulate

Given independent motor units, the excited states of an ensemble must be coordinated to achieve a uniform output from a given stator. To achieve an all-or-none output without altering the internal states of each motor unit of an ensemble, the simulated output reverses only when all motor units of the ensemble are in the same state (

This figure demonstrates the criteria used by a computer program to simulate a switch in flagellar direction when multiple motor units act in a concerted manner on the same rotor. The record is a portion of that used in

Based on the flagellar system, we derived a relationship between ligand binding and CW bias (Eq. 2.14) that has the same form as a relationship obtained earlier

The flagellar switching system may prove to be the clearest example of a generalizable biochemical mechanism. The motor unit and switch complex represent fundamental elements of the mechanism, namely, the reader and switch, respectively. The reader and switch form a collision complex that stimulates the reader to an excited state, which corresponds to CW torque of the flagellar motor unit. At the single molecule level, collision complex formation is an event that sets the probability of the excited state to unity at time zero, and as time elapses, the excited state decays with a single exponential rate. At equilibrium, the excited state forms and is sustained by repetitive rounds of collision complex formation and decay. A second-chance pathway for collision complex formation occurs when a steady-state process delivers additional switches to the site of the reader before the excited state has decayed. The steady-state process is mechanical in the flagellar system modeled here, but the model is not specific for the form of work that sustains the steady-state.

In the comparable skeletal muscle model, the second-chance pathway is not driven by mechanical energy. The ATP hydrolysis cycle of myosin provides the steady-state required for the second-chance pathway. In the absence of ATP, the equilibrium pathway of muscle produces M by rigor binding of myosin. Although stable, the rigor bond cycles too slowly to allow movement. In a second-chance pathway, the breaking of the rigor bond by ATP releases one myosin from tropomyosin, only to be replaced by a second myosin that has hydrolyzed ATP before M has decayed

The model described here requires two states for the rotor subunits and two states for the motor. The simplest flagella model would comprise only the CW and CCW states

Cooperative flagellar switching without ligand binding has been observed by over-expressing flagellar subunits locked in either the ligand-bound or ligand-free functional state

Although at least 10 motor units form a stator complex, we chose to fit cooperative CW bias data assuming an ensemble of

Based on _{r+}; _{r+} could be less than the rotor rate, _{r+}≤_{r}.. In contrast, a collision complex must decay back to ground state by the rate of the rotor, _{r−} = _{r}. Under circumstances, where the ratio _{r+}/_{r−} is decreasing owing to constant _{r+} and increasing _{r}, CW bias should asymptotically approach a minimum, which is what was observed

Although the proposed model has two explicit pathways, we constrained the simulation to a single combination pathway for simplicity. The observed fit for

Although it is artificial, constraining the simulation to prevent a ground state event between pulses may be justified by the characteristics of the rotor and stator. A high duty ratio by a motor unit _{−1} (

How does cooperativity improve performance? Considering only the average output, the damped output of ensemble motor units relative to the input of a single component unit (

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^{−1}) reduces the opportunity for a ligand binding event to stimulate the motor to the excited state, which is required for CW output. The CW bias, calculated for one motor (n = 1) using the M function (see below), is shown for three values of α. Conditions: The dwell time interval and ligand binding probability are set in the simulation to unity and 0.5 respectively. Each point represents the average output of 10,000 pulses (Fig. 1S). For simplicity, the coupling and ligand binding constants, _{0} and _{L}, are set to unity. Given these conditions the M function for one motor unit simplifies to M = (1−M)(1+(α−1)M).

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^{13DK106YW}.