We also discuss the concept of motor adaptability to changes in external conditions and set up a general framework to assess its desired level depending on how the motor is expected to function. Our fluctuation analysis provides evidence for cooperativity by leading to an estimate of the stator-stator interaction potential between 1 and 2 for the short-range model, a result that is coherent with this system's expected smooth adaptive nature to external stimuli18,19,20,21,22. We contrast this moderate cooperative behavior found for stator unit binding to the BFM with the highly cooperative behavior of rotor-switching units that are believed to be behind the rapid stochastic switching in BFM rotational direction in the context of bacterial chemotaxis17,23.
In this section, in the spirit of Skoge et al. and Tu, who studied the dynamics of cooperativity in chemical sensing to better understand bacterial chemotaxis, we employ a minimal prototypical one dimensional lattice gas model of cooperative particle binding on a substrate of finite size. We then obtain an analytical expression for the fluctuations of the mean occupancy at equilibrium as a function of the relevant model parameters, including the number of binding sites and the interaction potential between bound particles, which acts as a proxy for system cooperativity. Finally, we apply these results to investigate the characteristic signatures of cooperativity by establishing a general criterion for its manifestation and then propose a procedure to estimate the interaction potential from experimental data.
Notably, a fluctuation analysis of the type used here was previously performed by Liu and Dilger, utilizing finite-size one- and two-dimensional Ising models to investigate cooperative phenomena in ligand-gated ion channels. Modeling of this adsorption system is complicated, in part because of geometrical questions, because it is not clear if the ion channel network can be approximated by a regular Ising like lattice (disorder could play an important role), which could explain why their attempt to quantify ion channel cooperativity was inconclusive. Our work complements and extends the work of Liu and Dilger by providing a comprehensive theoretical analysis: we present the theoretical results in a physically transparent form, explore simple limiting cases in greater detail, and examine the full probability distribution functions (PDFs) for occupation.
Most of the applications of the 1D Ising model, or its generalizations, to the study of cooperativity in biological systems are restricted to the simplifying limit of very large (or infinite) system size for which the type of fluctuation and correlation analysis we perform is not pertinent (fluctuations vanish in this limit and correlations cannot extend over the whole system), see e. g. the recent work on the cooperative binding of cofilin to actin filaments. Here, in contrast, as in the work of Liu and Dilger on fluctuations in ligand binding to ion channels and Rice et al. on cooperative correlations in cardiac thin filament activation, we explicitly focus on the effects that the finite size of the system has on both fluctuations and correlations, the amplitudes and shapes of which can provide valuable insights into the internal configuration of the system.
Our intentional use of a minimalist model strips down the complexity to the bare essentials required to isolate and examine the fundamental roles of cooperativity and anti-cooperativity in characterizing fluctuations. Although derived from a minimal model, the core insights of our study serve as universal principles still applicable to refined and more biologically adapted models. The type of model adopted here is equivalent to the Ising-type models successfully employed to understand the role of (strong) cooperativity in bacterial chemotaxis, although our exact detailed analytical treatment of finite size systems goes beyond the usual treatments proposed for the Ising-type models in this context. Despite the simplicity of the model, the small system size leads to a panoply of physical behavior and key insight into the functioning of a complex biological motor.
Consider a periodic 1D lattice with L binding sites in contact with a heat and particle reservoir of constant temperature T and effective chemical potential . We assign to each binding site i () a binary variable : means that a particle (ligand) occupies the i-th site; otherwise, . The array uniquely defines one of the possible microscopic configurations (or microstates). The relative occupancy in a given microstate is . We say that the system is in mesostate N if the number of particles on the lattice, i. e. the occupancy of the system equals . There are microstates consistent with the occupancy N; in this case, we say that the mesostate N has multiplicity .
The energy of the system depends on the specific microscopic configuration according to the grand canonical effective 1D short-range interaction Hamiltonian:
Since we choose periodic boundary conditions, corresponds to . Furthermore, we introduce with the Boltzmann constant , rendering the two adjustable control parameters of our model, J and , dimensionless. The effective chemical potential, , is considered to be made up of two parts, , where is the chemical potential of the external particle reservoir and is the binding energy of particle adsorption onto the substrate. The nearest-neighbor interaction J is attractive for (cooperativity) and repulsive for negative values (anti-cooperativity). For fixed J the number of bound particles can be modulated by varying through and/or . When we recover the usual non-interacting (Hill-Langmuir) model.
In the Methods Section "Infinite range (IR) model", we will compare the predictions of this short-range model with those of the infinite-range model in which all pairs of particles interact with uniform strength, irrespective of their relative distances on the lattice. As detailed below, this model effectively serves as a proxy for the more general long-range lattice gas model.
The Hamiltonian Eq. (1) falls into the universality class of the 1D short-range lattice gas model. Hence, we can map our model to the 1D Ising model (see for comprehensive overviews). It is, therefore, possible to derive analytical expressions for the grand partition function and both the first and second moments of the relative occupancy in thermodynamic equilibrium by employing the transfer matrix formalism. In what follows, we will express our theoretical results in terms of relative quantities, such as relative occupancy, even when absolute values are experimentally available. This standardized approach allows for consistent qualitative and quantitative comparison across systems of different (finite) size to which our general-purpose approach can be applied, simplifies the analysis, and enhances the applicability across different experimental conditions.
By taking the derivative of the grand partition function with respect to , we can calculate the mean relative occupancy at equilibrium, :
where , and is the equilibrium correlation length of the system (given in number of lattice sites), which can be rewritten in terms of J and as follows (we refer to Section "1D short-range lattice gas (SRLG)" for more details):
The mean equilibrium occupancy equals .
We focus on small systems with around ten binding sites. As discussed below, such systems will turn out to possess the important feature that there should be enough theoretical resolution in the standard deviation to discriminate between different values of J for couplings within the bio-physically expected range (). For purposes of illustration in what follows, unless otherwise stated, we shall use . This particular choice will allow us to both illustrate the general behavior of such small systems and compare with greater ease the theoretical results presented here to the experimental data when we later, in Section "Application to the bacterial flagellar motor (BFM)", apply our model to the study of the recruitment of torque-generating units (or stator units) onto the BFM of E. coli, which has recently shown to recruit up to stator units under high-load conditions, updating earlier results from previous studies. It should be kept in mind that our theoretical results are valid for any value of L and that our conclusions for small systems apply to any sufficiently small value of L. In particular, while the present work considers single-molecule bead assay data from E. coli, our analysis can be readily extended to the BFMs of other bacterial species, which are known to have a different, though still limited, maximum number of torque-generating stator units. This generalization is facilitated by the fact that we express our results in terms of relative, rather than absolute, occupancies.
We propose that the operational definition of "small system" be simply those systems for which fluctuations, as measured by the standard deviation of relative occupancy, are within experimental resolution and can be used to extract the sign and amplitude of the interaction strength within the usual biophysical range. A more careful rendering of this definition can be found in Section "System size, fluctuations, and experimental precision". Such a definition is not contingent on the amplitude of (anti)cooperativity and therefore can encompass even the non-interacting (Hill-Langmuir) model.
Figure 1a depicts the mean relative occupancy from Eq. (2) as a function of the two control parameters of the system: J and . The half-filling (HF) contour line cuts the plot diagonally, meaning that the lattice is, on average, half-filled whenever (or ) holds. Below (above) that line, the system's mean occupancy is lower (higher) than 50 %. Contour lines run together close to the plot's bottom right sector diagonal, corresponding to positive J and negative . In this sector defined by , for , the mean occupancy for a fixed positive value of the interaction potential becomes extremely sensitive to slight chemical potential variations. Systems characterized by parameters in that range behave like biological switches, nearly jumping from zero to full occupancy with small changes in . On the contrary, more adaptive systems, that is, systems whose occupancy varies more smoothly from zero to full occupancy as a function of , would be located in the central and upper left sector of Fig. 1a.
For the short-range lattice gas model from Eq. (1), the standard deviation of the occupancy at equilibrium can be expressed in terms of the derivative of the mean relative occupancy with respect to the chemical potential as follows (cf. Section "1D short-range lattice gas (SRLG)"): . Applying this result to Eq. (2), we obtain a compact analytical expression for as a function of , J, and L:
The dependency of on J and for fixed L is depicted in Fig. 1b. In what follows, we will use as a measure of fluctuations, hence the stochastic nature of the system.
In Fig. 2, we compare the average relative occupancy, , and the standard deviation, , as functions of for two vertical cuts in the above contour plots: (Hill-Langmuir) and . We note that with increasing J the curve sharpens and shifts to lower values of and that the maximum in increases in amplitude and also shifts to lower values of . These shifts are due to the displacement with increasing J of the value of at half-filling, , where the slope of the curve and the amplitude of the standard deviation are both largest.
Fluctuations increase as grows. Along the half-filling diagonal () in Fig. 1b, they saturate at a finite global maximum as J goes to infinity.
By evaluating Eqs. (4) and (3) for , one obtains an explicit formula for the standard deviation at half-filling (HF) as a function of L and J:
with
which represents the maximum equilibrium correlation length at fixed J for any value of mean occupancy.
In the Hill-Langmuir limit (), the correlation length vanishes, and the standard deviation at HF becomes . For low couplings (), the correlation length increases sub-linearly with J and is accurately given by . In this same limit, the standard deviation increases exponentially with J, , an approximation valid for , when (see Fig. 16). This increase can be interpreted as a widening of the probability distribution function of states of a given N around the average value , where the probability is a maximum. In the high cooperativity limit , shows an asymptotic exponential growth, , which is the analog of the well-known zero temperature critical behavior of spin chains; and, consequently, for fixed L, tends to 1/2 when the correlation length surpasses L/2 and the system evolves into a bimodal one (for which only the zero and full occupancy states contribute, see below). We remark that although for fixed J, vanishes as when , as expected, in the limit where L is kept fixed and J tends to , the standard deviation at HF reaches the finite (maximum, or saturation) value of one-half.
Fluctuations decrease as becomes more negative. Along the half-filling diagonal () in Fig. 1b (), they saturate at a finite value as . This is a general result for systems with an odd number of sites. In this strong anti-cooperative limit, for small enough systems, it turns out that fluctuations depend sensitively on whether the number of lattice sites is even or odd. For L even, tends to zero; for odd L, it tends to 1/(2L). This leads to the striking result that in the limit of strong anti-cooperativity, a fluctuation analysis near HF could clearly detect the difference between small systems with even and odd numbers of sites, provided that the experimental precision is high enough.
Often, the experimental parameter one can measure (or control indirectly, as is the case for the BFM by varying the external load, cf. Section "Application to the bacterial flagellar motor (BFM)") is not the chemical potential of the reservoir , but the mean number of bound ligands at equilibrium, . Therefore, it is insightful to plot, in the spirit of, the standard deviation as a function of the mean relative occupancy , which, in general, can be done using a parametric plot.
To unfold this relation, we express and , for different but fixed values of J, as functions of and plot this family of curves in Fig. 3a,b for an odd and even number of lattice sites, respectively (the corresponding contour plots for the case are shown in Figs. 14 and 15, analogously to Fig. 1a,b for ). Each curve for a given J represents the set of points of coordinates with spanning from to 30, which turns out to be a suitable one-dimensional domain to complete the plots. We have chosen the interaction potential J to take values from to 10 in steps of size 1. The standard deviation shows a left-right particle-hole symmetry with respect to the system half-filling: , which is a manifestation of the underlining particle-hole symmetry of the system under study. The system has zero fluctuations when (empty substrate) or (full substrate), which is easy to understand because a distribution of states with non-zero probability around these average values would be incompatible with the average values themselves. For non-negative values of J, the maxima of fluctuations appear at half-filling, taking on the value of for and 1/2 for , and the vs. curves converge asymptotically towards the limiting one. More subtly, results for the anti-cooperative limit depend on whether L is odd or even.
A key result of the above analysis and one of our major conclusions is that for cooperative systems of size and mean relative occupancies not too close to 0 or 1, in the range the curves of vs. are spaced sufficiently far enough apart that the standard deviation is suited for estimating J. The same conclusion can be drawn for anti-cooperative systems of size and mean relative occupancies close to 1/2 in the range . These results are illustrated in Fig. 3a for and Fig. 3b for . Outside these ranges, the standard deviation saturates to limiting values, and therefore experimental data near these limits could only be used to put bounds on the value of J. In the absence of prior knowledge of the system size, the characteristic non-monotonic shape of the parametric plot in Fig. 3b can be exploited to estimate L. For instance, for non-negative values of J, the maximum in occupancy fluctuations corresponds to half-filling.
We will investigate the cooperative and anti-cooperative regimes in more detail in what follows, taking care to study those limits for which we were able to derive simplified exact or approximate analytical expressions for .
In the absence of cooperativity (), which corresponds to the case of a simple adsorption process with on-site volume exclusion interactions only, the free energy calculated from Eq. (1) reduces to an expression determined entirely by the mesostate (the occupancy) N: . The correlation length from Eq. (3) vanishes; hence each adsorbing site acts as an independent particle trap. Thus we recover the Hill-Langmuir adsorption model at equilibrium for which takes the following typical sigmoid form as a function of :
This latter expression exemplifies the general result that, in the thermodynamic limit, , , and hence, as expected, the standard deviation (of the relative occupancy) tends to zero as (self-averaging property at equilibrium in the thermodynamic limit).
In Section "1D short-range lattice gas (SRLG)", we show using a Gaussian approximation for the PDF that for the standard deviation is inversely proportional to the curvature (in absolute value) of the entropy of mixing, which reaches a minimum at HF. Therefore, the standard deviation for reaches a maximum there. More generally, one can show that for not too large, a Gaussian approximation for the PDF is accurate and therefore the standard deviation is inversely proportional to the curvature (in absolute value) of an effective free energy, which reaches a minimum at HF.
The maximal standard deviation at half-filling is due to the large width of the distribution of states with non-zero probability compatible with (see Section "Probability distribution function (PDF) of the occupancy and its dependency on and J"). This limit is easy to understand in terms of a bimodal system: only the empty and full states contribute because the probability of intermediate states is suppressed by the presence of energetically costly domain walls (as will be discussed in detail below). The bimodal system, therefore, behaves as an effective Hill-Langmuir model with a system size equal to 1, as can be seen by taking in Eqs. (9). The purely bimodal model, developed in the context of chemical sensing with applications in bacterial chemotaxis, is commonly referred to as the Monod-Wyman-Changeux (MWC) model.
We have seen in Fig. 3 how, at fixed average occupancy, increasing cooperativity leads to an increase in fluctuations. A simple, insightful, and accurate way of understanding this increase can be formulated by starting from the previously derived result for the Hill-Langmuir model and introducing the concept of block domains. When , all sites are decoupled, and the standard deviation is non-zero for a finite-size system simply because of statistical fluctuations (absence of self-averaging). When , neighboring sites are coupled and can be grouped (approximately) into correlated block domains of size , leading to
where . When , correlations lead to block domains that we take to be of size . The above approximation allows the system to be described by a reduced number, , of independently fluctuating block domains of size b. Since b should also tend to 1, when (zero cooperativity, Hill-Langmuir); and to L, when (strong cooperativity, bimodal behavior), a convenient interpolation formula between the above three cases is
By comparing the predictions of this approximation with the exact results at half-filling, where the standard deviation is a maximum, one can see that this approximation is extremely accurate (see Fig. 16).
Although the choice Eq. (14) for b leads to an accurate approximation for over the whole parameter range, it has the inconvenience for practical applications that, because depends explicitly on , it is not a function of L, , and J alone. To remedy this, a useful and reasonably good global interpolation formula for as an explicit function of L, J, and can be obtained by adopting a choice for [ in Eq. (13)] suggested by an approximate analysis of the Hill coefficient (see Eq. (84), Section "Effective Hill coefficient"). This approximation, which emerges from a simple physically clear approach, is exact at half-filling for all couplings and provides an upper bound for all occupancies that becomes exact at weak and strong coupling. Starting from a weak coupling cumulant expansion, we can find another useful approximate interpolation formula for as an explicit function of L, J, and :
(see Section "Infinite range (IR) model"). For , the window of maximum accuracy for this approximation for is . As we will see below, it encompasses the experimental data window for the system to which we apply our method (the BFM).
The two approximations are complementary (see Figs. 13 and 20) since Eq. (15) provides a lower-bound that is most accurate at low coupling and Eq. (84) provides an upper bound that is a better approximation at strong coupling.
Exact results for the standard deviation can be obtained by writing the mean square relative occupancy at equilibrium,
in terms of the 2-point correlation function, (with ).
Equation (16) can be used as a starting point to gain deeper physical insight into the behavior of the standard deviation than the previous result obtained directly from the grand partition function because it becomes possible to express as an explicit function of and the correlation length , see Eq. (36).
For , the standard deviation is bounded by the zero and infinite cooperativity values, which can be calculated from the limiting forms of . In the absence of cooperativity (), factorizes for : . In this limit, can then be obtained directly from Eq. (16) and simplifies to [Eq. (9)]. In the other limit of infinite cooperativity () and strong correlations, as a function of diverges at HF and away from HF for not too large L remains larger than L/2 (see Fig. 17). From Eq. (30) we see that for L finite in this limit , and saturates at [Eq. (12)].
By comparing the and 14 cases (Fig. 3a,b), we observe that for small enough anti-cooperative systems, there is a clear difference between even and odd system sizes. This difference clearly manifests itself near HF, and arises because of frustration for odd L.
For negative values of J (anti-cooperativity) and L even the curves inflect and tend for towards a characteristic limiting shape with global maxima near and 0.8 and a zero at half-filling (see Fig. 10). A restricted grand partition function approach (presented in Section "Probability distribution function of the occupancy"), which retains only non-overlapping particle-hole pairs, leads to a simple but accurate approximation in the limit for L even (see Fig. 10):
The corresponding standard deviation for can be obtained by exploiting particle-hole symmetry: .
In this limit of strong anti-cooperativity (), the correlation length becomes complex, . Hence, at HF, , leading directly to an oscillating two-point correlation function that describes a sequence of non-overlapping particle-hole pairs (anti-ferromagnetic order in Ising spin language):
where we have taken the real part of and assumed that . alternates between 0 and 1/2, which reflects alternating perfect anti-correlations (nearest neighbors) and correlations (next nearest neighbors) at HF in this limit of strong anti-cooperativity.
We can extract the equilibrium probability distribution function (PDF) of the occupancy [corresponding to the effective Hamiltonian given in Eq. (1)] for fixed J and , through exact enumeration and study how it depends on the interaction potential J. To study for fixed J and , the experimentally accessible quantity, we varied J but kept the mean occupancy fixed (by adjusting to any given choice of J).
Taking , Fig. 4 shows how the PDF behaves for three different positive values of the interaction potential and three characteristic values of the mean relative occupancy: and 2/3, respectively. The correlation length increases with J, and so does the standard deviation; consequently, the PDF profile broadens and flattens. On the one hand, finite-size effects set in when the tails of the PDF touch the system boundaries. On the other hand, when the typical size of a highly correlated particle domain, , becomes comparable to the substrate size, L, finite-size effects dominate, and the PDF saturates to its strong interaction bimodal form.
In the absence of cooperativity, we expect N to be normally distributed with standard deviation , provided the system is large enough, and the position of the mean is far enough from the system boundaries: for (strictly speaking, the variable N being discrete, the probability follows a binomial distribution. But the latter is well approximated by the normal distribution in the large L limit). The half-filling case with in Fig. 4b approximates this Gaussian behavior well because the average value, , is far enough from the boundaries that the tails of the distribution do not reach the boundary values, 0 and L (the distribution decays fast enough to the left and right of the average value). More quantitatively, for any value of J, finite-size effects set in when or (i.e., when the tails of the PDF touch the system boundaries).
The PDF broadens and flattens for intermediate positive values of the interaction potential. For higher values, e. g. , the PDF saturates by accumulating at the boundaries at
(here, is the Kronecker delta) and the system becomes well described by an effective two-state (bimodal) system (this is the limit where there exists only one effective block domain). This result is intuitively plausible and can be obtained by retaining in the full partition sum only the empty and full states (see Section "Probability distribution function of the occupancy" for more details). In this limit, finite-size effects are always dominant, and the standard deviation for the occupancy N can easily be calculated directly from the PDF. As expected, it saturates at ( at half-filling). Owing to the simple form of this limiting PDF, it is easy to calculate all moments of N, leading to (or ) and therefore explicit expressions for low order standardized moments, such as the skewness and kurtosis, see Section "Probability distribution function of the occupancy". This simplification occurs because, with the correlation length, the energy cost needed to create domain walls also increases with J. Hence, a strongly correlated system favors microscopic configurations that minimize the number of domain walls, selecting the mesostates corresponding to either an empty or a fully occupied substrate. This tendency to select extremal occupancies is a characteristic signature of the presence of strong cooperativity effects in small-size systems. Furthermore, for a parameter choice corresponding to an occupancy expectation value different from half-filling, the distribution is skewed, as in Fig. 4a,c, with an asymmetry that increases with the value of the interaction potential.
The situation is more complicated for strong anti-cooperativity () because the system cannot, in general, be reduced to a simple one or two-state system, except near zero and half-filling. Furthermore, near HF, the reduction depends on whether L is even or odd (see Fig. 5). In the following, we will focus on the left half of the plots in Fig. 5, i. e. on occupancies from to L/2, the right half being a reflection of the left through HF owing to the particle-hole symmetry of the system.
For L even, the PDF has a single peak at at zero filling and at at half-filling, leading to vanishing standard deviations in these two limiting cases (see Fig. 3b). At HF, there is only one allowed state, consisting of L/2 non-overlapping particle-hole pairs [] (with a two-fold degeneracy, because the particles can all be on either the even or odd sites).
For L odd, the PDF still has a single peak at at zero filling, but since no single microstate corresponds to HF, there are in this case two peaks with equal (50%) weight at and , leading to a non-vanishing standard deviation, , (see PDF for in Fig. 5). For L odd, particle-hole pairs cannot cover the whole system, and defects must appear to fulfill the HF constraint, either an extra hole () or an extra particle ().
We can summarize the situation as follows. At (or close to) HF, the four different cases studied give rise to very different results for the fluctuations described by the standard deviation and can be considered as clearly distinguishable signatures of different types of particle-particle correlations in small-size systems. At half-filling, the standard deviation takes on the following values in decreasing order: (i) for strong cooperativity (), ; (ii) for no cooperativity (), ; (iii) for strong anti-cooperativity () and L odd, ; and (iv) for strong anti-cooperativity () and L even, .
In contradistinction to the strong cooperativity limit where the two-state approximation is valid over the whole range of relative occupancy (from zero to full-filling) for both L even and odd, the strong anti-cooperativity limit is more complicated because the one or two-state approximation is only valid for zero and half-filling. Although more than two occupancy states are involved between these limits, it is clear that strong anti-cooperativity favors the formation of particle-hole pairs. Therefore for even L, the system can be approximated by a system of non-overlapping particle-hole pairs. For L odd, two extra defect states must be included (an extra particle or extra hole). This restricted grand partition function approach, where only the relevant states participating in the strong cooperativity and strong anti-cooperativity limits are retained, is developed in Section "Probability distribution function of the occupancy" of the Methods.
The theoretical framework outlined in preceding sections is well adapted for examining (anti)co-operativity-mediated adsorption on substrates with a limited number of binding sites -- a fundamental mechanism underlying various critical biological processes. This includes enzyme regulation, chemical sensing, and signal transduction, as well as molecular binding dynamics. A prime example where our methodology finds application is in (homo)oligomeric ring structures. These small-sized, ligand-activated protein structures are widespread across biological systems, and their regulatory mechanisms often involve allosteric interactions between binding sites. Our work complements the study presented in, where it was possible to achieve nearly perfect experimental control and thereby span the whole spectrum of mean occupancies. In contrast, by putting the main emphasis on the analysis of fluctuations, we designed our theoretical approach to work under less controlled experimental conditions, where experimental variables, such as bead sizes in single-molecule experiments, are limited. This is a common scenario in biological research, where perfect control over all variables is often not feasible. Nonetheless, among the diverse suitable applications, we specifically target the stator unit assembly dynamics within the BFM as a primary illustrative application of our theoretical framework.
In what follows, we aim to determine whether cooperativity between BFM stator units, mediated by a (dimensionless) interaction potential J, plays a role in their dynamical assembly at the periphery of the rotor.
Like most microorganisms, bacteria live in fluid environments with low Reynolds numbers, making them experience a viscous force much larger than the inertial ones. They have subsequently evolved a variety of compatible motility mechanisms that have been widely studied. One type of motility involves rotating one or several flagella that propel bacteria through aqueous media. The BFM, whose schematic structure is shown in Fig. 6a, is a transmembrane macromolecular complex that consumes the electrochemical potential across the inner bacterial cell membrane to generate torque and set the flagellum in rotary movement. One of the most compelling properties of the BFM is its ability to change both its conformation and stoichiometry depending on the external medium, allowing it to change the direction of rotation and the magnitude of torque produced. Its mechanisms of adaptation to external stimuli have been widely studied, becoming a model molecular machine to investigate properties such as mechano-sensitivity, chemotaxis and dynamic subunit exchange.
Torque is generated by inner membrane ion channel complexes called stator units (depicted in blue in Fig. 6a) which dynamically bind and unbind to the peptidoglycan (cell wall) at the periphery of the rotor. When unbound, they are inactive and passively diffuse in the inner membrane. In their bound state, anchored to the peptidoglycan, the ion channels are activated, and, through a mechanism not yet fully understood, apply torque to the rotor. The inset of Fig. 6a shows a top-down view of the BFM, illustrating the dynamic recruitment of stator units in analogy with the minimal SRLG model described in Section "Lattice gas model". In this representation, the C ring is modeled as a lattice with a fixed number L of binding sites in contact with a reservoir of stator units, treated as an ideal gas characterized by a chemical potential . Precise measurements of the temporal evolution of the angular velocity of the motor of E. coli, a direct proxy for the number of bound stator units, have shown that the system can recruit up to stator units and that the system is mechanosensitive in that stator unit stoichiometry scales with the external torque induced by the viscous drag acting on the flagellum turning inside an aqueous medium.
In recent years, observations have indirectly suggested that the stator unit recruitment dynamics of the BFM may be consistent with cooperativity among stator units. The first piece of evidence comes from the occupancy dwell time analysis of the BFM remodeling process in tethered cell experiments. In this study, the dwell time in the zero occupancy state is considerably longer than all other non-zero occupancy states, which suggests that once a stator unit is bound, the affinity for successive bindings increases. The second piece of evidence comes from experimental observation that the stator unit adsorption rate onto the rotor depends non-monotonically on the number of bound units. The third line of evidence comes from protein structure studies of the stator units employing cryoelectron microscopy, which led to the proposal of a molecular framework for BFM mechanosensitivity. This framework is coherent with the catch-bond model introduced by, and it is also compatible with incorporating cooperativity through an effective long-range interaction mechanically mediated by the rotor. The authors suggest a dual-state mechanism for adsorbed stator units whose switching from an inactive (non-torque-producing) to an active (torque-producing) state is triggered by the forces exerted on the stator units by the movement of the rotor. This framework offers insight into the longer dwell time for the zero occupancy reported by: the delay can be attributed to the initial unit's engagement with the rotor and the cell wall in a still-inactive state, lacking the motor torque necessary to activate the unit and produce motor rotation.
Much can be learned about the dynamics of the BFM using bead assay measurements. In our experiments performed on E. coli, we attach a microparticle (bead) to the 'hook' (the extracellular portion that joins the motor to the flagellum) (cf. Fig. 6b). We use bacterial strains with genetically deleted cheY. This modification inhibits the production of CheY-P and ensures that the BFM rotates only in the CCW direction, effectively decoupling stator unit dynamics from the CW-CCW switching mechanism. By tracking the bead's off-axis rotation, we can calculate the angular velocity, , and the torque produced, (from the relation , where is the drag coefficient that increases with the bead diameter, yet without obeying a simple power law because of the correction due to rotation near the cell wall). Both and are direct proxies for the number of bound stator units. We can (indirectly) control the mean number of bound stator units at steady state, , by varying the bead size and hence the viscous load, because the binding of stators to the BFM is mechanosensitive (see, e.g.,, and references therein). We can thus measure the temporal evolution of the number of bound stator units on individual motors, as well as the fluctuations around mean occupancy (for more information on the experimental setup, see Section "Experimental data" in the Methods and references).
We describe the mechanosensitive binding and unbinding of stator units in the stationary angular velocity regime of the BFM in terms of our adsorption model [see Eq. (1)], the rotor being a small-size substrate with periodic boundary conditions onto which up to stator units can bind at fixed equally spaced positions. Our working hypothesis, in line with previous studies, is that as long as we only focus on the occupancy of the stator units in the (non-equilibrium) motor stationary state this quantity fluctuates around a fixed mean value as if the stator subsystem were effectively at equilibrium. In this picture, we account for the presence of the unbound (inactive) stator units diffusing freely in the inner membrane by imposing an external reservoir chemical potential, , which is taken to be constant, as we expect depletion effects to be negligible. We incorporate the mechanosensitivity into the model in an average way by assuming that the stator unit binding energy, , depends on the viscous load (in our case dependent on the size of the bead and the viscosity of the surrounding medium). Load-induced changes in will naturally lead to load-dependent average occupancies and fluctuations, as observed experimentally. Furthermore, we assume that the interaction parameter J remains fixed (independent of the load) and check a posteriori if this assumption is consistent with the data.
As explained previously, we attempt to use the fluctuations in the average number of bound stator units in the stationary angular velocity regime to determine whether or not cooperativity is at play in the BFM. Figure 7 shows a comparison between the theoretical standard deviation curves, already presented in Fig. 3a, and the experimental standard deviations from five different applied viscous loads, corresponding to three different beads with different diameters that are indicated in the legend (the values of the experimental data are presented in Table 1). One can see that the experimental data fall between the values of and 2, leading to the conclusion that: (i) a constant (load-independent) cooperativity parameter (J) is a reasonable working hypothesis; (ii) a moderate level of cooperativity in the system, , obtained by fitting the model to the experimental data, is consistent with the experimental observations, see Fig. 7. The standard error of the nonlinear fit is 0.22 and the coefficient of determination is = 0.98. Since is close to its maximum value of 1 and the relative error is less than 20%, we can conclude that there is a good fit of the model to the standard deviation data and therefore that the model is "robust" enough to provide a reliable estimate of J; and (iii) the estimated value of J is coherent with what is expected for typical biological systems exhibiting moderate cooperativity.
To delve into further detail, we can also compare the probability distributions of the occupancy shown in Section "Probability distribution function (PDF) of the occupancy and its dependency on and J" with our experimental ones. Figure 8 shows a comparison between the probability distribution from experiments (black dots, see Table 2) and the probability distributions obtained by exact enumeration for different values of J. Even though, in this analysis of our previously published, but limited, experimental data, we don't observe a precise fit with any of the chosen values, it is clear that the PDFs for the range of J estimated from the standard deviation are coherent with our experimental results. Our results suggest that the interaction potential has a non-zero positive value qualitatively in accordance with the value range deduced from the above fluctuation analysis. (We refer to Fig. 18 in the Methods for a direct comparison of the experimental PDFs and the theoretical PDFs calculated using the best-fit value of the dimensionless interaction potential, .) This qualitative agreement between the evolution of the experimental PDFs with increasing average relative occupancy and the predictions of the underlying lattice gas model with moderate cooperativity therefore lends credence to our choice of minimalist adsorption model. Moreover, as further explained in the Methods section, we observe theoretically that the system becomes bimodal (with only the completely full and empty states having significant probabilities) for slightly higher values of the interaction potential, , behavior that is not seen in the experimental PDF data. We see from Section "Probability distribution function (PDF) of the occupancy and its dependency on and J" (Fig. 18c) that for the chosen system size (), is a threshold value marking a transition from low cooperativity PDFs having a maximum centered on the average occupancy to high cooperativity PDFs exhibiting a bimodal form with local maxima at zero and full filling. In the latter case, microstates with fillings close to the average occupancy have low probabilities, and the motor would undergo discontinuous jerky motion. Such a bacteria would fluctuate between an immobile state (zero stator unit occupancy) and maximum speed (full stator unit occupancy). In the absence of cheY the BFM rotates only in the CCW direction and we expect the stator unit dynamics (now decoupled from the CW-CCW switching mechanism) to be appropriate for a highly and smoothly adaptive molecular machine. High values of stator cooperativity would, therefore, not be expected since such a motor would not be able to adapt smoothly to environmental variations.