Bacterial Flagellar Motor Switch in Response to CheY-P Regulation and Motor Structural Alterations

Abstract

The bacterial flagellar motor (BFM) is a molecular machine that rotates the helical filaments and propels the bacteria swimming toward favorable conditions. In our previous works, we built a stochastic conformational spread model to explain the dynamic and cooperative behavior of BFM switching. Here, we extended this model to test whether it can explain the latest experimental observations regarding CheY-P regulation and motor structural adaptivity. We show that our model predicts a strong correlation between rotational direction and the number of CheY-Ps bound to the switch complex, in agreement with the latest finding from Fukuoka et al. It also predicts that the switching sensitivity of the BFM can be fine-tuned by incorporating additional units into the switch complex, as recently demonstrated by Yuan et al., who showed that stoichiometry of FliM undergoes dynamic change to maintain ultrasensitivity in the motor switching response. In addition, by locking some rotor switching units on the switch complex into the stable clockwise-only conformation, our model has accurately simulated recent experiments expressing clockwise-locked FliG(ΔPAA) into the switch complex and reproduced the increased switching rate of the motor.

Introduction

The rotation of bacterial flagella is powered by a nanoscale rotary molecular machine, the bacterial flagellar motor (BFM). This motor is embedded in the cell envelope and utilizes the electrochemical energy from ion transit across the cell membrane as its power source (1, 2). A single BFM in Escherichia coli can output a power of ∼1.5 × 10⁵ pN·nm·s⁻¹ (3) and rotate the flagellum at ∼300 Hz (4), and the BFM in Vibrio alginolyticus can rotate even faster, reaching ∼700 Hz (5).

The BFM not only propels bacteria, it also plays a crucial role in steering its swimming direction. Each BFM can rotate in two modes, counterclockwise (CCW) and clockwise (CW). When all the motors on an E. coli cell spin CCW, the flagellar filaments coordinate their shape into a corkscrew-shaped bundle that drives the cell steadily forward (run state); when one or more of the motors switch to CW, internal collision between flagellar filaments causes the bundle to break apart and the cell tumbles (tumble state), randomly selecting a new direction in which to swim. In bacterial chemotaxis, the BFM switches stochastically between CCW and CW, and the bacteria repeat a run- tumble-run pattern, which resembles a biased random walk, to enable spatial navigation (6).

Information reflecting changes in the environment, translated through a series of biochemical reactions, can adjust the rotation bias of the BFM. This is coordinated through changes in the phosphorylation level of a key signaling molecule, CheY (7, 8). CheY-Ps are free diffusing molecules in the cytoplasm of the cell, which can bind to the bottom of the switch complex and trigger conformational changes in FliM proteins. Conformational changes in FliM are further coupled to different structural orientations of the FliG protein (7). When interacting with the stator units, different structural orientation of the FliG protein can generate two opposite directions of rotation (CCW or CW) of the BFM (9).

Previous experiments have established that the motor-switching response to changes in CheY-P concentration is ultrasensitive (10), meaning that a small change in CheY-P concentration can lead to a large shift in motor bias. To explain this ultrasensitivity, a conformational spread model has been proposed (11) in which the BFM utilizes the cooperativity of neighbor-neighbor interactions to increase its switching sensitivity, creating a molecular gearbox that is capable of changing direction in milliseconds. Several key predictions of the conformational spread model have been validated by recent experiments (12, 13), implying that the conformational spread model accurately captures the molecular mechanism of BFM switching.

With advances in experimental techniques, more detail surrounding BFM switching has been revealed. Recently, simultaneous recording of motor switching and the number of fluorescent CheY-Ps localized to the switch complex was performed, revealing that the rotational direction of the BFM is directly regulated by the number of CheY-Ps bound to the switch complex (14). A direct counting of the CheY-Ps bound to the switch complex in its CCW or CW rotation state poses additional restrictions to the modeling. More remarkably, a recent experiment by Yuan et al. (15, 16) demonstrated that BFM switching maintains high sensitivity across a broad operating range by dynamically changing the number of FliM units on the switch complex in response to equilibrium levels of CheY-P. Also, recent mutagenesis studies have introduced mutated proteins onto the switch complex, which result in motors that remain stable in the CW state and do not respond to CheY-P regulation (17). The induction of these mutated proteins onto the switch complex led to a series of interesting observations.

In this article, we have modified our original conformational spread model to examine these new features of BFM switching. After thorough search of the parameter space, we saw that these new experimental findings could be quantitatively reproduced by an updated model for conformational spread. This provides further evidence that conformational spread accurately depicts the underlying mechanism controlling BFM switching.

Materials and Methods

In our model, we simplify the switching of the BFM by focusing on the interaction of CheY-P molecules with a ring-shaped assembly of 34 rotor switching units (RSUs), each of which is formed by one FliM protein, four FliN proteins, and approximately one FliG protein. In this ring, each RSU is identical and can exist in either a CCW (Fig. 1, open symbols) or a CW state (Fig. 1, solid symbols), leading to CCW or CW rotation, respectively, when torque is delivered from the stator units. One RSU can be bound (here denoted 1) or not bound (here denoted 0) to one CheY-P molecule. Therefore, the rotational state of each RSU can remain in and transition between the four states: CCW⁰, CW⁰, CW¹, and CCW¹. When the conformational transition from CCW to CW happens in the unbound state, the free-energy change is set to be $E_{\text{A}}^0$ (CCW⁰→ CW⁰), and since CCW⁰ is energetically more favorable, we have $E_{\text{A}}^0>0$. In the bound state, because CW¹ is energetically more stable, we have $E_{\text{A}}^1$ (CW¹ → CCW¹) > 0. The relative positions of the four possible states of each RSU, depicted in a free-energy diagram, are summarized in Fig. 1 B. The free-energy change associated with CheY-P binding can be calculated as

$$\Delta G{\left(\text{CCW}\right)}_{0\to 1}=E_{\text{C}}+E_{\text{A}}^1;$$ (1)

$$\Delta G{\left(\text{CW}\right)}_{0\to 1}=E_{\text{C}}-E_{\text{A}}^0;$$ (2)

$$E_{\text{C}}=-\text{ln}\left(\frac{c}{c_{0.5}}\right),$$ (3)

where c_0.5 is the CheY-P concentration required for neutral bias. Here, neutral bias is the situation where the motor spends the same amount of time spinning in the CCW direction as in the CW direction. Our model further introduces a coupling energy, $E_{\text{J}}$, where the free energy of an RSU would be lowered by $E_{\text{J}}$ if one of its neighboring RSUs is in the same state (Fig. 1 C). The existence of this coupling energy allows conformational spread from one RSU to the rest.

Figure 1.

The conformational spread model. (A) Schematic illustration of the ring complex that contains 34 RSUs. (B) When an RSU has no CheY-P bound, the free energy of conformational change is ±E_A⁰; and when it has CheY-P bound, the free energy of conformational change becomes ±E_A¹. (C) RSUs interact through a coupling energy, E_J, which ensures that neighboring RSUs prefer the same conformation.

To simulate a switch event, we begin by simulating each RSU individually and the possible configurations available to it. The transition rate constants that describe conformational changes between these possible states can be specified as

$$k_{\text{CCW}\to \text{CW}}={\omega }_{\text{a}}\text{exp}\left(-\frac{0.5}{k_{\text{B}}T}\Delta G_{\text{CCW}\to \text{CW}}\right);$$ (4)

$$k_{\text{CW}\to \text{CCW}}={\omega }_{\text{a}}\text{exp}\left(\frac{0.5}{k_{\text{B}}T}\Delta G_{\text{CCW}\to \text{CW}}\right),$$ (5)

where ΔG_CCW→CW represents the free-energy changes associated with this transition and holds one of the following six values: $E_{\text{A}}^0$, ($E_{\text{A}}^0$ − $E_{\text{J}}$), ($E_{\text{A}}^0$ − $2E_{\text{J}}$), − $E_{\text{A}}^1$, (− $E_{\text{A}}^1$ − $E_{\text{J}}$), (− $E_{\text{A}}^1$ − $2E_{\text{J}}$); $k_{\text{B}}$ is Boltzmann’s constant and T is temperature. We set ${\omega }_{\text{a}}$, the fundamental flipping frequency, to be 10⁴ s⁻¹, which is a typical protein conformational changing rate (11).

Next, we calculate the rate constants for CheY-P binding to and dissociating from each RSU. The free-energy change associated with CheY-P binding to and dissociating from an RSU is independent of the states of adjacent RSUs, and is only determined by the conformation of the RSU involved. Therefore, the rate constants can be written as

$$k_{0\to 1}=\frac{c}{c_{0.5}}{\omega }_{\text{b}}\text{exp}\left(-\frac{0.5}{k_{\text{B}}T}\Delta G_{0\to 1}\right);$$ (6)

$$k_{1\to 0}={\omega }_{\text{b}}\text{exp}\left(\frac{0.5}{k_{\text{B}}T}\Delta G_{0\to 1}\right),$$ (7)

where c stands for CheY-P concentration and c_0.5 is the CheY-P concentration required for neutral bias. $\Delta G_{0\to 1}$ is the free energy associated with binding of the ligand, and ${\omega }_{\text{b}}$ stands for the binding rate and was set as 10 s⁻¹ based on the experimental CheY-P binding-rate data (18). We assume that the binding rate is independent of the conformation of the RSU. Under our asymmetric assumption, the value of c_0.5 can only be solved numerically.

In implementing Monte Carlo simulations, a computer program was developed to simulate the dynamic switching of the BFM switch complex. Each RSU(n) was assigned two values of transition time, TA(n), the transition time of the conformational change associated with a change between the CCW and CW states, and TB(n), the transition time of the conformational changes associated with binding or unbinding of the CheY-P molecule. Implementing the Gillespie algorithm iteratively (19), the program locates the single RSU conformational change, either CheY-P binding or mechanical switching, with the earliest execution time in all transition times TA(n) and TB(n). The new transition time is then updated for TA(n) and TB(n) of RSU(n) with

$$t-t_0=-\frac{\text{ln}\left(\text{X}\right)}{k},$$ (8)

where t₀ is the time at which the calculation occurred in the simulation, X is a random variable uniformly distributed between 0 and 1 (X ∼ U(0,1)), and k is the rate constant for the next transition. If the event was the conformational change between the CCW and CW states, the transition time of two neighboring RSUs should also be recalculated and updated. The model records the activity and other details with a time interval of 0.1 ms; thus, we can analyze every switching event and the dynamic behavior of the ring complex.

In this conformational spread model, a motor switching event typically starts with a switch event occurring in a single RSU, and this newly created domain may subsequently either grow to encompass the entire ring or shrink and disappear (Fig. 1 A) as the ring returns to its previous coherent state. The motor spins full speed in the CCW or CW direction while all RSUs are in a coherent state (20). To relate the activity of the ring to output rotational speed, we assume that the speed of the motor can be written as

$$v^{\ast }=\frac{v}{v_{\text{max}}}=1-2\times \frac{N^{\text{CW}}}{N^{\ast }},$$ (9)

where v is the rotational speed and v_max is the maximum speed of the motor. We assume that the speed is positive when the motor spins CCW, that the speed is negative when the motor spins CW, and that the maximum CCW speed is equal to the maximum CW speed. N^∗ is the number of RSUs, whereas N^CW is the number of CW-state RSUs. In this work, we used v^∗ to describe the rotation of the motor; v^∗ = 1 stands for the maximum CCW speed, and v^∗ = −1 stands for the maximum CW speed.

Below a critical coupling energy, the ring exhibits a random pattern of states as the RSUs flip independently of each other. Above the critical coupling energy, switch-like behavior ensues: the ring spends the majority of time in a coherent configuration, stochastically switching between the two extreme states where RSUs are either all CW or all CCW. The proportion of time spent in each of the bistable configurations is governed by the concentration of CheY-P, with the sensitivity of the system to variations in this concentration determined by $E_{\text{A}}^0$ and $E_{\text{A}}^1$.

Results

Direct regulation of BFM switching by CheY-P binding on/off the switch complex

The ultrasensitivity in BFM switching to changes in CheY-P concentration can be explained by several models, including the Monod-Wyman-Changeux (MWC) model (21), the Koshland-Nemethy-Filmer (KNF) model (22), and the conformational spread model. Additional details of the switching dynamics and kinetics provide fine criteria to discriminate different mechanisms. Previous high-resolution measurement of BFM switching revealed that switch events are noninstantaneous, with broadly distributed durations. This result, together with the observation of frequent transient speed fluctuations (slowdowns, pauses, and incomplete switches) strongly favors the conformational spread model or the KNF model over the MWC model. Meanwhile, an in vivo study using FRET to investigate binding between CheY and FliM (18) suggested a weak binding cooperativity between CheY-P and the switch complex. This result implied that the KNF model might not be suitable to explain the BFM switching mechanism either.

However, to discriminate between the KNF and the conformational spread model requires direct quantification of CheY-P bound to the switch complex during the coherent CCW and CW states. Only recently, by simultaneously visualizing fluorescent labeled CheY-P localized to the switch complex and the rotation of a BFM, Fukuoka et al. (14) demonstrated that switching of the BFM was regulated by the binding and dissociation of CheY-P. Using single-molecule counting, they revealed that during CCW rotation, approximately two (average value) CheY-Ps were bound to the switch complex, whereas during CW rotation, ∼13 (average value) CheY-Ps were bound. This result clearly suggested that it is not necessary for CheY-P molecules to be bound to all RSUs to induce a CW rotation, strongly precluding the KNF model. Indeed, their observation provided direct evidence supporting the conformational spread model. In Fig. 2 A, we plotted a sample trace generated from the conformational spread model using the parameter set we confirmed in our previous works ($E_{\text{A}}^0$ = $E_{\text{A}}^1$ = 0.65 $k_{\text{B}}T$, $E_{\text{J}}$ = 4.15 $k_{\text{B}}T$) (20). We saw that motor rotation direction is regulated directly by the number of CheY-Ps bound to the switch complex, and due to the coupling energy between neighboring RSUs, in CCW rotation, ∼12 (average value) CheY-Ps were bound to the switch complex, whereas in CW rotation, ∼22 (average value) CheY-Ps were bound.

Figure 2.

Simultaneous visualization of the flagellar motor rotation and the number of bound CheY-Ps. (A) (Top) The output motor rotation versus time. (Bottom) The number of CheY-Ps bound to the motor versus time predicted by our model. The parameter set we used in this simulation is E_A⁰ = E_A¹ = 0.65 k_BT and E_J = 4.15 k_BT. (B) Cross-correlation profiles between rotational direction and CheY-P number. Gray lines indicate one correlation profile for each five repeats, whereas the red line is the average trace. To see this figure in color, go online.

In Fig. 2 B, we calculated the correlation from several simulation repeats using the formula proposed by Fukuoka et al.,

$$Z\left(\tau \right)=\frac{\frac{1}{N}{\sum }_{t=1}^N[x\left(t\right)\times y\left(t+\tau \right)-\overline{x\left(t\right)}\times \overline{y\left(t\right)}]}{\sqrt{\frac{1}{N}{\sum }_{t=1}^N{[x\left(t\right)-\overline{x\left(t\right)}]}^2}\times \sqrt{\frac{1}{N}{\sum }_{t=1}^N{[y\left(t\right)-\overline{y\left(t\right)}]}^2}}.$$ (10)

In this equation, Z is the function of correlation, τ represents the time lag, t is time, N is the number of sampling points, and x(t) and y(t) are motor rotation and bound CheY-P number, respectively. The correlation calculated from our model simulations shows good qualitative agreement with the result of Fukuoka et al. (14).

With the parameter set determined from fitting Bai et al.’s experimental results ($E_{\text{A}}^0$ = $E_{\text{A}}^1$ = 0.65 $k_{\text{B}}T$, $E_{\text{J}}$ = 4.15 $k_{\text{B}}T$) (12), the conformational spread model predicted ∼12 CheY-Ps bound to the switch complex during CCW rotation and ∼22 CheY-Ps bound during CW rotation. We modified our model and re-searched the parameter space to see if the model could accommodate the discrepancy with experimental data reported by Fukuoka et al. (14) First, we found that our symmetry assumption ($E_{\text{A}}^0$ = $E_{\text{A}}^1$) leads to a conclusion that the sum of the average number of CheY-Ps bound during CCW rotation and the average number of CheY-Ps bound during CW rotation is ∼34, equal to the number of FliM subunits. Thus, to fit the results observed by Fukuoka et al. (14), we must consider asymmetric cases.

To test the interplay between the asymmetry of free-energy changes ($E_{\text{A}}^0$ and $E_{\text{A}}^1$) and the model outputs, we simulated across the parameter ranges 0.2 $k_{\text{B}}T$ ≤ $E_{\text{A}}^0$ ≤ 1.1 $k_{\text{B}}T$ and 0.2 $k_{\text{B}}T$ ≤ $E_{\text{A}}^1$ ≤ 1.1 $k_{\text{B}}T$ when $E_{\text{J}}$ = 4.15 $k_{\text{B}}T$. We introduced the Hill coefficient as another important output of the model. The Hill coefficient describes the sensitivity of BFM switching to changes in CheY-P concentration. The experimentally determined Hill coefficient of ∼10.3 can be chosen as a restriction to parameterize our model (10). The outputs of the model, including the numbers of bound CheY-Ps in the CCW and CW states and the Hill coefficient under different parameter sets are shown in Fig. 3 A. Here, we see clearly that to reproduce Fukuoka’s result, it was required to set $E_{\text{A}}^0$ < $E_{\text{A}}^1$.

Figure 3.

Exploring the parameter space. (A) Two-dimensional contour plot showing the average number of CheY-Ps bound to the motor in CW rotation, the average number of CheY-Ps bound to the motor in CCW rotation, and the Hill coefficient as a function of E_A⁰ and E_A¹. The simulation result shown is with 0.2 k_BT ≤ E_A⁰ ≤ 1.1 k_BT and 0.2 k_BT ≤ E_A¹ ≤ 1.1 k_BT when E_J = 4.15 k_BT. (B) The number of CheY-Ps bound to the motor using the best-fit parameter set (E_A⁰ = 0.45 k_BT, E_A¹ = 1.85 k_BT, and E_J = 4.05 k_BT). Schematic illustrations of the ring states are shown at the bottom of the figure (from left to right, the CW state, the switching state, and the CCW state). There is a lower barrier to CheY-P binding to the ring when each subunit is in the CW state than when it is in the CCW state, due to the free-energy landscape shown in Fig. 1. To see this figure in color, go online.

We then searched a larger parameter space across the ranges 0.15 $k_{\text{B}}T$ ≤ $E_{\text{A}}^0$ ≤ 1.95 $k_{\text{B}}T$, 0.15 $k_{\text{B}}T$ ≤ $E_{\text{A}}^1$ ≤ 1.95 $k_{\text{B}}T$, and 3.75 $k_{\text{B}}T$ ≤ $E_{\text{J}}$ ≤ 4.25 $k_{\text{B}}T$, with a simulation step size of 0.1 $k_{\text{B}}T$, where $E_{\text{A}}^0<E_{\text{A}}^1$. The outputs of our model, including the numbers of bound CheY-Ps during CCW and CW rotation and the Hill coefficient, are listed in Table S1 in the Supporting Material. We compared our model outputs with experimental data, considering a fit good if the mean values of both outputs showed <5% deviation. We found two groups of values that fit the experimental results well: 1) $E_{\text{A}}^0$ = 0.45 $k_{\text{B}}T$, $E_{\text{A}}^1$ = 1.85 $k_{\text{B}}T$, $E_{\text{J}}$ = 4.05 $k_{\text{B}}T$; and 2) $E_{\text{A}}^0$ = 0.45 $k_{\text{B}}T$, $E_{\text{A}}^1$ = 1.95 $k_{\text{B}}T$, $E_{\text{J}}$ = 4.15 $k_{\text{B}}T$.

We then analyzed the additional model outputs of these best-fitting parameter sets. To quantify the behavior of the motor, we defined the locked-state interval (the time the motor rotates in the CCW or CW state) and the switching time (the time the motor takes to make a switch from the CCW to the CW state, or vice versa) (12). We found that the average locked-state interval and switching time were not in agreement with previous experimental measurements (12). To locate the parameter set that reproduced all the experimental results, we further allowed ${\omega }_{\text{a}}$ to float as a free parameter. As discussed in our previous work (20), the flipping frequency, ${\omega }_{\text{a}}$, is a scaling factor of the locked-state interval and switching time, but does not affect other model outputs. We therefore changed the value of ${\omega }_{\text{a}}$ with other parameters defined from the two best-fitting parameter sets (Tables S2 and S3). According to Bai et al. (12), the average locked-state interval was ∼0.75 s and the average switching time was ∼18 ms. Therefore, for our full parameter set, ${\omega }_{\text{a}}$ = 2.6 × 10⁴ s⁻¹, $E_{\text{A}}^0$ = 0.45 $k_{\text{B}}T$, $E_{\text{A}}^1$ = 1.85 $k_{\text{B}}T$, and $E_{\text{J}}$ = 4.05 $k_{\text{B}}T$, our model fit the experimental results perfectly. For the remainder of this work, we use this parameter set unless otherwise specifically mentioned.

We have to point out that although the conformational spread model was able to reproduce these new experimental findings, the variation between individual experimental observations can be explained by the use of different strains and experimental conditions in different experimental studies.

Adaptation through dynamic assembly of RSUs on the BFM switch complex to ensure ultrasensitivity

Adaptation is another important property of the bacterial chemotactic regulation network, because it allows the bacteria to adapt to environmental stimuli and retain ultrasensitivity across a new operating range and thus enables it to respond to further changes. In the chemotactic signal transduction pathway used by E. coli, ligand binding activates the methyl-accepting chemotaxis proteins (MCPs). For instance, when the number of repellents/attractants bound to an MCP increases or falls, CheA autophosphorylates and transfers the phosphorylation group to CheY, increasing the cytoplasmic concentration of CheY-P. For adaptation, the phosphorylation level of CheY-P is gradually removed by the phosphatase CheZ. Meanwhile, the receptor signaling state must also be reset. Adaptation to attractant is mediated by the constitutively active methyltransferase CheR, which adds methyl groups to receptors to increase CheA activity and therefore accelerates CheY phosphorylation. Adaptation to repellent is mediated by the methylesterase CheB through a negative-feedback mechanism; CheB is also phosphorylated by CheA, resulting in elevated levels of CheB-P, which demethylates MCPs to decrease CheA activity and therefore decelerates CheY phosphorylation (23, 24).

Interestingly, recent work by Yuan et al. (15) reported new observations that showed the motor also adapts structurally at the rotor to maintain ultrasensitivity. In a strain lacking CheB and CheR, after adding attractant, the CheY-P concentration was expected to decrease and the BFM to rotate stably with a low CW bias without adaptation. Surprisingly, Yuan et al. (15) observed a slow recovery in the CW bias of BFM switching. This adaptation was subsequently proven to be caused by the incorporation of extra FliM proteins onto the switch complex, instead of by recovery of CheY-P concentration. The authors subsequently found that there was a one-to-one relationship between the number of FliM proteins and the adapted CW bias, and that motors with different numbers of FliM proteins presented different switching response curves. They measured the actual response of motors and pointed out that the Hill coefficient was ∼16–20 (16). This valuable study showed the interplay between CheY-P regulation and structural adaption in BFM switching, and it revealed that the classic switching response curve reported by Cruzel et al. (10) reflected primarily the switching sensitivity of adapted motors.

Our previous works (12, 20) showed that the conformational spread model can perfectly reproduce the switching behaviors of adapted motors with 34 RSUs, but to study the actual response of BFM switching under a varying number of FliM proteins, we have modified our model. Equation 6, describing the rate of CheY-P binding, has been changed to

$$k_{0\to 1}=\frac{c}{c_{0.5}}{\omega }_{\text{b}}\text{exp}\left(-\frac{0.5}{k_{\text{B}}T}\Delta G_{0\to 1}\right);$$ (11)

$$c_{0.5}=\frac{N^{\#}}{N}{c}_{0.5}^{\#},$$ (12)

where N^# is the standard number of RSUs we used in our model and was set to be 34; N is the actual number of RSUs in each simulation; c_0.5 is the modified CheY-P concentration required for neutral bias; and $c_{0.5}^{\#}$ is the CheY-P concentration that results in neutral bias when the number of RSUs is 34. As reported by Yuan et al. (15, 16), as the number of FliM proteins increases, the motor switching sensitivity increases and the response curve is shifted to the left (c_0.5 decreases). Therefore, we assumed that the relationship between c_0.5 and the number of RSUs follows Eq. 12. As a result, increasing the number of RSUs increases the binding rate of CheY-P.

In addition to modifying our model framework, a new parameter set was also needed to perform subsequent simulations. We re-searched the parameter space and found a group of values: $E_{\text{A}}^0$ = 0.85 $k_{\text{B}}T$, $E_{\text{A}}^1$ = 1.95 $k_{\text{B}}T$, $E_{\text{J}}$ = 4.25 $k_{\text{B}}T$, which gave a Hill coefficient around 16.5 when the number of RSUs was set to be 34. Under this parameter set, the model outputs were also in agreement with the observation of Fukuoka et al. (14), who found that during CCW rotation, 2 ± 4 CheY-Ps were bound to the switch complex, whereas during CW rotation, 13 ± 7 CheY-Ps were bound. We varied ${\omega }_{\text{a}}$ (Table S4), and found that when ${\omega }_{\text{a}}$ = 10⁶ s⁻¹, we could fit the experimental data by Bai et al. (12). We also tested the Hill coefficient of different ring sizes under this parameter set (Table S5). Consistent with our previous prediction (20), the Hill coefficient became larger with an increasing number of RSUs, and the model outputs were still in agreement with the data of Yuan et al. (16). Therefore, the new parameter set, ${\omega }_{\text{a}}$ = 10⁶ s⁻¹, $E_{\text{A}}^0$ = 0.85 $k_{\text{B}}T$, $E_{\text{A}}^1$ = 1.95 $k_{\text{B}}T$, and $E_{\text{J}}$ = 4.25 $k_{\text{B}}T$ was chosen to study the actual response with a fixed number of RSUs.

Previous studies have indicated that the motor is at neutral bias when the cytoplasmic CheY-P concentration is ∼3.12 μM (10, 16). Here, we assumed that under this condition, the number of RSUs is 38. Therefore, the value of $c_{0.5}^{\#}$ could be calculated. When the output CW bias of an adapted motor was ∼0.8, the actual number of RSUs was 36, and the c_0.5 of the actual response curve was ∼3.29 μM; when the output CW bias of the adapted motor was ∼0.5, the actual number of RSUs was 38, and the c_0.5 of the actual response curve was ∼3.11 μM, in excellent agreement with the experimental observations of Yuan et al. (16).

As shown in Fig. 4 A, using the new CheY-P binding equation (Eq. 11) and the new parameter set, we tested how our model behaved with varying numbers of RSUs. We plotted response curves for motor switching bias to CheY-P concentration predicted by our model for motors with different numbers of RSUs and saw that the response curve was shifted to the left as the ring size increased. When attractant was added, the cellular CheY-P concentration was lowered below c_0.5 in the response curve of a motor containing 36 RSUs, resulting in low CW-biased rotation. However, after the number of RSUs on the motor was increased to 38, the same CheY-P concentration from the left-shifted curve gave an increased CW bias for motor rotation, an effect equivalent to a recovery in CheY-P concentration. In Fig. 4, B and C, we simulated experimental conditions as follows: attractant was added at time 75 s, the CheY-P concentration was lowered to 3.1 μM, and the CW bias of the BFM switching was lowered to ∼0.25. Then we allowed a gradual incorporation of RSUs into the switch complex at a rate of 0.05 s⁻¹. The data of Yuan et al. only indicated that FliM was incorporated into the switch complex. For our model, we have assumed that the RSUs are replaced as a total unit and have reproduced the slow recovery of CW bias observed in the original data of Yuan et al.

Figure 4.

The actual switching response of motors with a varying number of RSUs and the adaptation behavior of the BFM. (A) The Hill curve for adapted motors measured by Cruzel et al. and the output of our model under different ring sizes. The Hill curve for adapted motors can be viewed as an assembly of the actual response curves of unadapted motors with a mixed number of RSUs. (B) After adding attractant, the motor adapts to the lower CheY-P concentration by gradually increasing the number of RSUs from 36 to 38. (C) The CW bias of motor rotation in response to increasing ring size. To see this figure in color, go online.

What is the biological use for this remodeling capability? The motor expands its operating range, but more importantly, in the absence of adaptation through CheY-P regulation, the motor can dynamically adjust its switching response curve to place the current CheY-P concentration into the most sensitive region, maintaining ultra-sensitivity to subsequent changes in CheY-P concentration. An important prediction arising from our model is that the BFM reduces the number of RSUs when adapting to an increase in CheY-P concentration, an effect after addition of repellent molecules. This prediction awaits experimental validation.

Induction of CW-locked RSUs enhances BFM switching frequency

Apart from altering the number of RSUs on the switch complex, the other form of structural change is to incorporate RSUs with different switching properties. Recent work by Minamino et al. (17) showed that induction of the mutated rotor protein FliG(ΔPAA) affected both the rotational bias and the speed of the flagellar motor. They reported that in a chemotaxis wild-type strain, the motor spun CCW without FliG(ΔPAA) expression; however, when FliG(ΔPAA) was highly expressed, the motor rotated exclusively CW. Finally, when FliG(ΔPAA) was induced at a very low level, the motor switched frequently at neutral bias. Most surprisingly, in a strain completely lacking CheY-P regulation, the motor switched more frequently when FliG(ΔPAA) was expressed into the switch complex. These observations have been confirmed by Lele et al. (25) in a more recent study, where the authors showed that motor switching was triggered by inducing mutated FliG proteins in the absence of CheY-P.

Using our model framework, we simulated the effect of the incorporation of mutated FliG into the switch complex. It is believed that FliG(ΔPAA) forms an RSU that remains stably in the CW conformation and cannot change its conformation upon CheY-P binding. Therefore, the switching of a motor expressing FliG(ΔPAA) can be understood as a ring complex containing several RSUs locked in the CW state. At high induction level, when each RSU on the ring complex is locked in the CW state, it is obvious that the model outputs continuous CW rotation. At low levels of induction, we consider the extreme cases where only one or two CW-locked RSUs are present on the ring complex.

In reproducing the results of Minamino et al. (17), we first noticed that in the wild-type bacterial strain they used, the BFM rotated with very low CW bias (0–0.1), from which we chose [CheY-P] = 1.8 μM (c = 0.6 × c_0.5) for the subsequent simulations. We further assumed that the induction of FliG(ΔPAA) would not change cellular CheY-P concentration. In the conformational spread model, the CW-locked RSU continuously affects the conformation of its neighbors, favoring CW states. In Fig. 5, we showed the switching behavior of the BFM at c = 0.6 × c_0.5 with an increasing number of CW-locked RSUs. When only one CW-locked RSU was present, interestingly, the switching frequency increased and CW bias rose to ∼0.5 (effective as c = c_0.5) due to the existence of this single CW-locked RSU. We further simulated the switching behavior when two CW-locked RSUs were present and we saw the switching increase to high CW bias. When half of the ring was occupied by CW-locked RSUs, the motor rotated stably in the CW state. These simulation results were consistent with the recent experimental observations of Minamino et al. (17).

Figure 5.

Switching behavior of the wild-type motor with increasing expression of CW-locked RSUs at low CheY-P concentration. Shown are the output motor-rotation traces predicted by our model that contains 0, 1, 2, 3, and 17 CW-locked RSUs. For simplification, we assume that CW-locked RSUs are evenly distributed on the rotor ring. To see this figure in color, go online.

We then simulated BFM switching in the absence of CheY-P regulation. In this case, the conformational spread on the switch complex was driven by stochastic flipping of each individual RSU and stabilized only by the coupling energy between neighboring RSUs. Fig. 6 showed the switching behavior of the BFM as the number of CW-locked subunits was increased. With no CW-locked subunits, that is, with all RSUs as wild-type, the motor rotated stably in the CCW state. When one CW-locked RSU was induced, the CCW rotation of the BFM became noisier, generating a high frequency of fluctuations in the number of CCW RSUs. When two CW-locked RSUs were present, these fluctuations increased, and when three CW-locked RSUs were present, the BFM switched erratically. If half of the ring was occupied by CW-locked RSUs, the motor rotated stably in the CW direction. These simulation results were consistent with recent experimental observations of Minamino et al. (17).

Figure 6.

Switching behavior of the wild-type motor with increasing expression of CW-locked RSUs in the absence of CheY-P. Motor-rotation traces predicted by our model that contains 0, 1, 2, 3, and 17 CW-locked RSUs. For simplification, we assume that CW-locked RSUs are evenly distributed on the rotor ring. To see this figure in color, go online.

To facilitate understanding of the switching behavior of the mutant BFM, we built a schematic illustration in Fig. 7, A and B. At a low CheY-P concentration, where a wild-type motor rotated stably in the CCW direction, a mutant motor with one CW-locked RSU resulted in ∼0.5 CW-biased switching. The physical mechanism behind this increased switching rate can be explained as follows: when one RSU on the switching complex is locked in the CW state, this RSU continuously seeds the spread of the CW conformation to its neighbors (Fig. 7 A). However, when the CheY-P concentration is low, the spread of the CW conformation cannot stabilize itself by binding CheY-P to the switch complex, but it is also difficult for the ring to stay in a coherent CCW state owing to the existence of a symmetry-breaking CW-locked RSU. Therefore, we found that the ring oscillates rapidly between coherent CCW and CW states, reflecting the competition between insufficient CheY-P binding and the conformational spread driven by the seed-like CW-locked FliG(ΔPAA) RSU. We must emphasize that mutant RSU switching is distinct from the switching of the wild-type BFM. Fig. 7 C shows a typical switching trace of the wild-type BFM at neutral bias and the distribution of motor rotation speed. For the wild-type motor, we saw that the motor spent most of its time in a stable CCW or CW state, whereas the switching time between states was comparatively much shorter (Fig. 7 C, left column). This is also observed experimentally (12). In comparison, for the motor with one CW-locked RSU (Fig. 7 D, left column), although its switching resembled that of the wild-type motor, we noted a clear difference in the distribution of motor rotation speed (Fig. 7, C and D, right column). In this case, due to the existence of CW-locked RSUs, the coherent CCW state became unstable and disappeared, leaving the coherent CW state as the only stable state and with the motor spending most of its time transitioning between states.

Figure 7.

Effect of FliG(ΔPAA) on motor switching. (A) When CheY-P concentration is low, the induction of a CW-locked RSU makes it easier for the motor to switch from the CCW state to the CW state, but the whole ring cannot stay stably in the CW state due to insufficient binding of CheY-P. Therefore, the motor switches frequently. (B) In the absence of CheY-P, a CW-locked RSU continues to spread its conformation to its neighboring RSUs, but this conformational spread cannot occupy the whole ring. Therefore, the motor shows frequent speed fluctuations. (C) The rotation trace of a wild-type motor at neutral bias (left column, c = c_0.5) and the histogram of its speed (right column) predicted by our model. (D) The rotation trace of a motor with one CW-locked RSU at neutral bias (left column, c = 0.6 × c_0.5) and the histogram of its speed (right column) predicted by our model. (E) The rotation trace of a motor with two CW-locked RSUs in the absence of CheY-P (left column) and the histogram of its speed (right column) predicted by our model. To see this figure in color, go online.

When CheY-P is absent, the CW-locked RSUs continuously seed the spread of the CW conformation to their neighbors. However, there is no CheY-P available to stabilize this conformational spread. The ring relies completely on the neighbor-neighbor coupling energy RSUs to drive the conformational spread, which struggles to propagate over large distances. With one CW-locked RSU, the coherent CCW state was unstable and we saw frequent deviations from the coherent CCW state, reflecting the conformational spread from the CW-locked RSU. When there were two CW-locked RSUs (Fig. 7 E) on the ring, the conformational spread from the CW-locked RSUs lasted longer and sometimes could generate CW rotation of the BFM. Fig. 7 E showed a typical switching trace of the BFM with two CW-locked RSUs (left column) and the distribution of motor rotation speed (right column). With two CW-locked RSUs, there existed no stable rotational states and the motor spent all its time in the transition process.

Discussion

In this article, we tested the applicability of the conformational spread model to explain new features in BFM switching. Our conformational spread model was able to reproduce the latest experimental discoveries and also make to our knowledge new predictions for future experiments.

To build a model where CheY-P binding can influence switching dynamics, it is necessary to assume that CheY-P binds with different affinity to a CCW RSU than to a CW RSU. This enables the binding and dissociation of CheY-P molecules to directly modulate the switching state of each RSU. After incorporating the coupling energy between adjacent RSUs, a direct consequence of the conformational spread model is that the switching of the BFM responds sensitively to changes in CheY-P concentration, but CheY-P molecules are not required to bind/dissociate to/from every RSU to generate CW/CCW rotation of the BFM. Recently, direct imaging of CheY-P in a live E. coli BFM performed by Fukuoka et al. (14) revealed that the number of CheY-P molecules bound to a CW-rotating motor plateaus at ∼13 molecules instead of saturating the motor with 34 molecules. This result supports the conformational spread model, since it indicates that the signal from CheY binding to an RSU is amplified in the motor output. Due to the experimental uncertainties in single-molecule counting, we speculate that their observation (14) that approximately two CheY-Ps were bound in CCW rotation, whereas ∼13 CheY-Ps were bound in CW rotation, will be subject to variation in different bacterial cells and different bacterial strains. Nonetheless, as we have shown in this work, by searching the parameter space of the conformational spread model, we have been able to reproduce their measurement (14), as well as previous works measuring other dynamics of BFM switching (10, 12), demonstrating the flexibility of the conformational spread model.

In this work, we have shown that when our model allows for a dynamic change in the total number of RSUs, the motor is able to adjust the switching response to adapt to the current cellular CheY-P concentration, maintaining ultrasensitivity in subsequent responses. This reproduces the recent experimental observations of Yuan et al. regarding adaptive ultrasensitivity (15, 16). However, our conformational spread model does not offer any explanation for why the RSUs undergo dynamic change and how the motor complex senses the current CheY-P concentration and recruits additional RSUs. Current theories propose that the switch complex senses the direction of BFM rotation rather than the CheY-P concentration. A model from Lele et al. (25, 26) suggested that when the motor spins CCW, FliM binding strengthens and the fraction of FliM molecules that exchange decreases, leading to an increase in the number of RSUs. Here, we ask whether the reverse scenario is also possible, that the motor can reduce the number of RSUs when it needs to adapt to a high CheY-P concentration. Following the theory of Lele et al., we can assume that when the motor spins CW, FliM binding is weakened and the fraction of FliM molecules that exchange increases, resulting in a decrease in the number of RSUs. This decrease, as demonstrated by our simulation, would adjust the switching response curve toward a high CheY-P concentration and therefore cause adaptation in the BFM. This requires experimental testing.

We analyzed BFM switching with some of the RSUs locked in the CW state. Consistent with recent experimental findings, the conformational spread model predicts an enhanced CW bias and switching frequency when FliG(ΔPAA) is induced at a low expression level on the switch complex. However, we also observed that the switching behavior triggered by expression of mutant RSUs was different from the switching behavior in wild-type motors. In agreement with earlier high-resolution measurements of BFM switching (12), we noticed that in wild-type motors, the ratio between the average locked-state interval and switching time at neutral bias was ∼42, indicating that the BFM spends most of its time rotating in a coherent CCW or CW state, with switch events occurring as fast transitions between these states. In a motor with one CW-locked RSU at a low CheY-P concentration, the switching bias and frequency are both increased, but the ratio between the average locked-state interval and switching time is ∼10, indicating that the BFM spends comparatively more time transitioning than in a stable state. With two or more CW-locked RSUs, this phenomenon becomes more prominent, to the point where the motor fluctuates so rapidly that it never dwells in a stable state. Instead of transitioning completely between CW and CCW states, the motor instead undergoes many high-frequency speed fluctuations, some of which cross through zero-speed. This naturally results in a change in rotational direction, but it is distinct from a complete switch event as considered for the wild-type BFM and as such has a markedly different signature in terms of rotation versus time (Fig. 7 E).

In summary, the bacterial flagellar motor continues to impress with its precise and adaptive control. Here, we have updated our model for conformational spread to explain which simple underlying processes can result in complex switching behavior in structurally adaptive systems. To unravel the operational and design principles at this functional end of the chemotaxis pathway requires further collaboration between theoretical modeling and experimental testing.

Author Contributions

Y.S., M.A.B.B., and F.B. designed the research; Q.M. performed the research; Q.M., Y.S., and F.B. analyzed the data; and Q.M., Y.S., M.A.B.B., and F.B. wrote the article.

Editor: Hiroyuki Noji.