The Shaft of the Type 1 Fimbriae Regulates an External Force to Match the FimH Catch Bond

Model system: fluid conditions

Based upon the aforementioned in vivo conditions, for example, the flow in the urethra, the fluid was modeled as a pipe flow with a turbulent core and a thin boundary layer close to the surface (Fig. S2 A). The flow in the layer close to the smooth surface, the viscous sublayer, in which the bacteria attach, was assumed to be laminar with a velocity profile increasing linearly from zero at the surface to a maximum value determined by the core flow.

Moreover, in the presence of turbulence, it has been shown that high-velocity flows occasionally sweep down from the core flow into the thin laminar sublayer (Fig. S2 B) (2,29). These sweeps are created by strong vortices outside the viscous sublayer, temporarily increasing the shear rate near the surface (32). The effect of these temporary sweeps was modeled by momentarily changing the shear rate of the velocity profile in the viscous sublayer (Fig. S2, A and B).

A nominal shear rate, S₀, was defined, chosen to correspond to a reference flow velocity of 8 mm/s at a distance of 3 μm above the surface, which represents a shear rate of ∼2700 s⁻¹. This reference velocity was estimated by considering the general flow in a urethra, i.e., a 5-mm-diameter pipe with a center velocity of 1000 mm/s; the laminar sublayer close to the surface was modeled in a standard fashion using a linear velocity profile determined by the pipe-wall shear stress (30).

To mimic flows and their changes in our model, simulations were made at three different shear rates, S₁, S₂, and S₃, representing values of S₀, 2S₀, and 3S₀, respectively, where the latter two rates predominantly represent an increased rate during a momentary sweep. A transient increase of the flow implies a sudden increase of the drag force adding tensile stress to the pilus and thereby to the adhesin.

Model system: tethered bacterium exposed to fluid flow

Fig. 1 displays a schematic illustration of the model presented in this work. A bacterium is attached to a surface by a pilus and exposed to a linear flow profile. The model takes into account four important conditions: 1), a two-dimensional (2D) motion of bacteria relative to the surface; 2), time-dependent fluid velocity shear profiles; 3), drag-force corrections for near-surface motion, both parallel (33) and perpendicular motion (34); and 4), the allosteric catch-bond properties of the FimH adhesin and the mechanical properties of the shaft.

Pili uncoiling under 1D conditions

The model developed here is based in part on our previous 1D model for damping of the adhesion force of a bacterium attached by a single pilus exposed to a flow (24). That work describes a bacterium modeled as a rigid sphere in a uniform flow attached to a surface by a pilus of length L. The sphere was constrained to move in one dimension, parallel to the flow, exposed to a Stokes drag force given by

where $\eta$ is the dynamic viscosity of the fluid, r is the radius of the sphere, and ${\mathit{v}}_{\text{f}}$ and ${\mathit{\nu }}_{\text{b}}$ are the fluid velocity and the velocity of the bacterium with respect to the surface. The sphere was assumed to be neutrally buoyant and to follow the flow when it suddenly was attached by a pilus (alternatively, when it suddenly was exposed to a flow). Since the sphere is anchored by an extendable pilus, its velocity is equal to the elongation velocity of the pilus, i.e., $v_{\text{b}}=\dot{L}$, which in turn is related to the tensile force to which the pilus is exposed (which is equal to that applied to the sphere as well as that experienced by the adhesin), $F_{\text{P}}$, which can be written as (see Zakrisson et al. (24) for a detailed derivation)

where ${\dot{L}}^{*}$ is the corner velocity for the uncoiling of the pilus, $F_{SS}$ is the steady-state uncoiling force, $\Delta x_{\text{AT}}$ is the turn-to-turn bond length from the ground state to the transition state, $\beta =1/kT$, where k is Boltzmann’s constant and T is the temperature, and $\Delta x_{\text{AB}}=\Delta x_{\text{AT}}$+$\Delta x_{\text{TB}}$, where $\Delta x_{\text{TB}}$ is the distance from the transition state to the open state. Due to the low mass of the bacterium, the effect of inertia was neglected, since it gives rise to forces several magnitudes lower than the fluid forces. Moreover, since the model was 1D, $F_{\text{P}}$ was at every instance considered equal to $F_{\text{D}}$. This model demonstrated that the force to which an adhesin is exposed can be significantly reduced due to the elongation properties of the pilus; as long as the pilus is extending, the force acting on the adhesin is lower (often significantly lower) than that of a stiff linker (for which the force is given by Eq. 1, with $v_b$ = 0). Although this 1D model can illustrate the ability of the pilus to dampen the drag force, it cannot provide a full picture of the adhesion phenomena of a bacterium in a flow. In particular, it does not take the shear profile and near-surface effects into account. To address these effects requires a 2D model.

2D movement of a bacterium attached by a pilus under a linear shear profile

In our refined model, presented here, a tethered bacterium (again modeled as a sphere) is allowed to move in two dimensions in a fluid with a linear shear profile, i.e., with ${\mathbf{v}}_{\text{f}}\left(y\right)=Sy\stackrel{\mathbf{ˆ}}{\mathbf{x}}$, where S is the shear rate, y is the distance from the surface (measured in the direction normal to the surface), and $\stackrel{\mathbf{ˆ}}{\mathbf{x}}$ is the unit vector parallel to the surface (see Fig. 1). Since a bacterium attached to the surface by a pilus and exposed to a force will move in the flow, it will in general have a position-dependent velocity that can be expressed as

where $v_{\text{bx}}\left(y\right)$ and $v_{\text{by}}\left(y\right)$ are the components of the velocity in the $\stackrel{\mathbf{ˆ}}{\mathbf{x}}$- and $\stackrel{\mathbf{ˆ}}{\mathbf{y}}$-directions, respectively, and where $\hat{\mathit{y}}$ is the unit vector perpendicular to the surface (see Fig. 1). Note that whereas $v_{\text{bx}}\left(y\right)$ is positive, $v_{\text{by}}\left(y\right)$ takes a negative value as the bacterium approaches the surface. In addition, the model also allows for the influence of sweeps by allowing the shear rate to be time-dependent, i.e., $S\left(t\right)$. By assuming that y describes the position of the sphere, which in turn is time-dependent, the bacterium will experience a flow velocity, ${\mathbf{v}}_{\text{f}}$, that depends on both its position, $y\left(t\right)$, and time, i.e.,

Hence, the fluid model not only takes into account the altered flow velocity experienced by the sphere as it moves in the fluid, it also depicts the influence of a change in flow velocity on the sphere (due to a time-dependent shear rate).

A bacterium exposed to a flow will be affected by several forces, of which the two dominant forces are the pilus force, ${\mathbf{F}}_{\mathbf{P}},$ which has a magnitude of $F_P$ and is directed along the direction of the pilus, here denoted by $\stackrel{\mathbf{ˆ}}{\mathit{r}},$ and the fluid drag force, ${\mathbf{F}}_{\mathbf{D}}$, originating from the relative motion of the bacterium with respect to the fluid (Fig. 1). These two forces can be written, conveniently, in terms of their x- and y-components as

and

Lift forces arising from shear flow (35,36) and from the presence of a wall (37) can be neglected, since these are insignificant compared to the other forces. As for our 1D model (24), the sphere is assumed to be neutrally buoyant over the timescale studied.

As discussed in Zakrisson et al. (24), the drag force experienced by the sphere will depend on the relative motion of the bacterium with respect to the velocity of the fluid. However, since the sphere is close to a wall, proximity effects need to be taken into account. This is normally done by modifying the Stokes drag force by the use of a correction term. However, since the motion of the sphere has components in both the parallel and perpendicular directions with respect to the surface, and the effect of the presence of the wall is dissimilar in the two directions, it is appropriate to write (34,38)

and

where $C_{\text{x}}\left[\text{y}\left(t\right)\right]$ and $C_{\text{y}}\left[\text{y}\left(t\right)\right]$ are corrections to the Stokes drag force experienced by a bacterium close to a wall moving in the x- and y-directions, respectively. For motions parallel to the surface, we use the fifth-order correction factor derived by Faxen, as given, e.g., by Eq. 7-4.28 in Happel and Brenner (34), which reads

where r is the radius of the bacterium and y is the distance from the surface to the center of the bacterium. To correct for the motion perpendicular to the surface, we use the approximated correction factor that is valid for $y/r>1.1$, which can be written as (39)

Since the effect of inertia can be neglected, force balance must prevail at every moment. This implies that

and

where $\theta \left(t\right)$ is the angle of the pilus with respect to the surface, as indicated in Fig. 1.

These equations need to be solved for the position and velocity of the sphere, as well as the drag and pilus forces, under the condition that the movement of the sphere is constrained by a tether behaving as a pilus, according to the model given above (see Eq. 2). However, the coupling of the position and velocity of the sphere to the instantaneous drag and pilus forces (i.e., Eqs. 7–12), as well as coupling of the pilus force to the elongation velocity of the pilus (Eq. 2), makes the system difficult to solve by analytical means. In short, they imply that as the sphere is exposed to a flow and moves toward the surface, it will experience a position-dependent velocity and be exposed to a position-dependent drag force (both of which are also time-dependent in the presence of sweeps), which in turn give rise to a varying force to which the pilus is exposed. Thus, as described in some detail in the Supporting Material, the set of equations given above have been solved by numerical means.

As discussed in some detail below, and as schematically illustrated in Fig. 2, this system of equations indicates that the sphere will move in curved trajectories. For low flows (and thereby low drag forces), the pilus will not uncoil, and thus its length, L, will not change, being equal to its original coiled length, $L_0$. The sphere will then be constrained into a circular motion controlled by a pilus with a fixed length, $L_0$ (Fig. 2, dashed arrow).

For high enough flows, however, the force acting on the pilus will induce uncoiling (and possibly also subsequent recoiling), which gives rise to an extension (and a possible contraction) of the pilus, i.e., a time-dependent length, $L\left(t\right)$, whose elongation velocity at each moment in time is given by Eq. 2. As further discussed in the Supporting Material, the part of the bacterium velocity due to unfolding will be in the direction of the external force, i.e., in the $\stackrel{\mathbf{ˆ}}{\mathbf{x}}$-direction. This not only will affect the motion trajectory of the bacterium (Fig. 2, solid curved arrows) but also will decrease the force the adhesin experiences in a manner similar to that of the 1D model described in Zakrisson et al. (24).

To investigate the influence of the extendability of the pilus on the model system—in particular, the movement of the bacterium, the force acting on the adhesin, and the survival probability of the bacterium—two types of tethers that constrain the motion of the sphere were considered, a stiff linker and an extendable pilus with biomechanical properties of the type 1 pilus. The motion of a bacterium attached by a stiff linker is therefore used as the reference system to which we compare the biomechanical properties of an extendable pilus with dynamic properties.

The FimH-mannose bond

The type 1 pilus adhesin, FimH, is expressed at the tip of the pilus. It anchors the bacteria to receptors (mannose) on the surface of the host and hence mediates the forces that act on the bacterium. The FimH-mannose bond has been modeled successfully by an allosteric catch-bond model that can predict the lifetime of a bond capable of switching between three different states, a weak, a strong, and an unbound state (22). The model describes the motion of E. coli bacteria rolling on mannose-bovine serum albumin and predicts the force spectroscopy data performed by constant-velocity atomic force microscopy (22,31).

The probability as a function of time of the FimH-mannose bond to reside in state i, $B_{\mathit{i}}$, where i = 1 and i = 2 represent the weak and strong bound states, respectively, can be described by two differential equations (22):

and

where k₁₀, k₂₀, k₁₂, and k₂₁ represent state transition rate coefficients, with an exponential dependence on force, described by the Bell equation, $k_{ij}\left[F\left(t\right)\right]=k_{ij}^o\text{exp}\left[F\left(t\right)\times x_{ij}/kT\right],$ where $k_{ij}^o$ is the thermal rate constant, F represents the applied force, $x_{\mathit{ij}}$ is the distance to the transition state, and kT is the thermal energy (40). These two equations determine the survival probability of the FimH-mannose bond, $B_1$ and $B_2$, as a function of time after bond formation for a given force.

In our simulation of the full fluid-bacteria-pilus-FimH system, described by Eqs. 2, 7, 8, 13, and 14, we used transition-rate parameter values originating from Yakovenko et al. (31), listed in Table S1. As described in some detail in the Supporting Material, the force responses of the two tethers, stiff linker and extendable pilus, were determined first, by solving Eqs. 2, 7, and 8. The time-dependent force data from such simulations were then used to assess the survival probability of the FimH-mannose bond as a function of time by solving Eqs. 13 and 14.

Trajectory for an attached bacterium and the force experienced by the adhesin in a steady flow

To investigate the 2D movement trajectory of a bacterial cell during attachment to a surface exposed to different flow conditions, three shear rates, $S_1$–$S_3$, were considered in the simulations. The bacterium was assumed to have an effective radius of 1.0 μm and to be equipped with a single 2-μm-long tether. It was initially positioned (t = 0) directly above the anchoring point (i.e., at x = 0, y = 3 μm). The flow exposed the bacterium to a drag force (given by Eqs. 7 and 8) that sets the bacterium into motion.

The tether was first given properties of a stiff linker, for all shear rates, (i.e., with no elasticity). For this case, the cell moved along a circular trajectory (Fig. 3 A, solid black line). As alluded to in the Supporting Material, the simulations were terminated when the membrane of the bacterium was at a distance 0.2 μm above the surface (the simulation termination distance, STD). This was caused by the fact that the Stokes drag-force correction coefficients are not valid for distances close to the surface.

The stiff linker was then replaced with an extendable pilus (with biomechanical properties given by those of type 1 pili). Since its elongation velocity, modeled by Eq. 2, depends on the drag force, it is directly related to the shear rate and thereby varies with position above the surface. At high shear rates, the drag force applies sufficient tension to the pilus to uncoil it. Based on previous experimental findings, it was assumed that the uncoiling commences at a force of 30 pN (16). For the case with the lowest shear rate, $S_1$ (Fig. 3 A, green dashed line), this force is obtained at a position of ∼2 μm, after which the bacterium takes a slightly shallower path in comparison to the simulation with the stiff linker.

For the two higher shear rates, $S_2$ and $S_3$, the trajectories become significantly elongated, coinciding at first with that of the stiff linker until the tensile force reaches the threshold value for uncoiling. In the case with a shear rate of $S_2$ (Fig. 3 A, blue dash-dotted curve), the bacterium is translated to a position 11 μm from the tether point before it reaches the STD. Movie S1 shows the movement of the bacterium at a shear rate of $S_2$ (for both a stiff linker and a type 1 pilus). For the case with a shear flow of $S_3$ (Fig. 3 A, red short-dashed curve), the bacterium uncoils completely before it reaches the STD, which it does at a position 14 μm from the tether point. From this position, it acts as a stiff linker.

Fig. 3 B shows the corresponding force acting on the pilus (thereby also on the adhesin-receptor bond at the anchor point) as a function of time during the trajectories explored by the bacterium in Fig. 3 A. As above, the solid curves represent the stiff linker and the various dashed curves correspond to the type 1 pilus, both types exposed to the three different shear rates; the green, blue, and red curves correspond to shear rates of $S_1$, $S_2$, and $S_3$, respectively. The data show that even though the trajectories for the bacterium tethered with a stiff linker (solid curves) are identical for all shear rates (Fig. 3 A, black solid curve), the forces on the anchor point differ significantly. The peak forces for the stiff linker exposed to the three shear rates are ∼100, ∼225, and ∼350 pN, respectively.

At the lowest shear force, $S_1$, the type 1 pilus shows a response similar to that of the stiff linker (Fig. 3 B, green curves), experiencing only a low uncoiling velocity. The simulations indicate that for the higher shear rates, the elongation properties of the pilus give rise to a significant reduction of the peak force. For both the $S_2$ and $S_3$ shear rates (Fig. 3 B, blue and red dashed curves, respectively), the force is reduced to ∼120 pN, which is lower by a factor of 2 and 3, respectively, than in the case where the bacterium is attached by a stiff linker. In addition, the red curve shows that for the highest shear rate, $S_3$, the force eventually increases from 120 pN to 350 pN at ∼1.2 ms. This sudden increase in force originates from the pilus having fully uncoiled, at which point it acts as a stiff linker.

Trajectory for an attached bacterium and the force experienced by the adhesin in a transient flow

The fluid dynamics of the urinary tract system is complex, with peristaltic activities as well as high fluid-flow velocities leading to turbulence. Turbulence in turn gives rise to transient sweeps and ejections that instantaneously change the fluid profile in the viscous sublayer. To better understand how turbulence can affect a tethered bacterium, sweeps were simulated by adding a transient increase of the shear flow at given onset times of 0, 0.25, 0.5, 0.75, and 1 ms.

Fig. 4, A and B, shows simulations of sweeps (modeled as step increases in the shear rate from $S_1$ to $S_2$ and from $S_1$ to $S_3$, respectively) that appear at different onset times. As shown by the solid curve in Fig. 4 A, for a bacterium attached with a stiff linker, the trajectories are again identical (they all overlap). For an extendable pilus exposed to a sudden sweep with a shear rate of $S_2$, the total translation length decreases significantly (from 11 to 4.5 μm) as the onset of the sweep is increased (from 0 to 1 ms). This comes from the fact that for the late-onset times, the bacterium is exposed to a force above the uncoiling force for only a fraction of its trajectory. Hence, it reaches the STD mainly coiled. As illustrated in Fig. 4 B, for the largest shear rate, $S_3$, the pilus reaches full elongation (14 μm) for all onset times up to 0.5 ms.

The force acting on the anchor point as a function of time for three of the above-mentioned cases with sudden onset of sweeps with $S_2$ and $S_3$ shear rates is shown in Fig. 5, A and B, respectively. The solid curves show the force acting on the anchor point for a stiff linker, with different onset times. The red solid curve corresponds to an onset time of 0 ms, whereas the green and blue curves represent onset times of 0.5 and 1 ms, respectively. The simulations show that the onset of the sweep instantaneously increases the force. For the lower sweep shear rate, $S_2$, the force increases from 100 to ∼225 pN (Fig. 5 A, green and blue solid curves), whereas for the highest sweep shear rate, $S_3$, it increases to ∼350 pN (Fig. 5 B, green and blue solid curves).

The dashed curves in Fig. 5 represent a bacterium attaching by a type 1 pilus under the same conditions applied for the stiff linker. Fig. 5 A shows data from the simulations with shear rate S₂, where the uncoiling capability of the type 1 pilus reduces the force on the adhesin (relative to that on the stiff linker) from ∼225 to ∼120 pN irrespective of the onset time of the sweep. For the case with the higher sweep shear rate, $S_3$ (Fig. 5 B), the force is reduced to the same level as with S₂, as long as the pilus uncoils. The steep increases in force to ∼350 pN at 1.2 and 2.1 ms for the cases with 0- and 0.5-ms onset time (Fig. 5 B, red and green dashed curves, respectively) originate from the fact that the pilus has been fully uncoiled, in which state it acts as a stiff linker.

The influence of pilus uncoiling and bacterial trajectory on the survival probability of the FimH adhesin

The results presented above describe how the force acting on the anchor point (and thereby the adhesin-receptor bond) is affected by the properties of the tether and the fluid conditions. To get a better understanding of how these forces affect the lifetime of the adhesin-receptor bond, and thereby the bacterial adhesion lifetime, the model was extended to also include the catch-bond properties of the FimH adhesin (9). The adhesin was modeled as an allosteric catch bond described by Eqs. 13 and 14, with the four force-dependent kinetic coefficients used in Yakovenko et al. (31). In that work, the lifetime of the FimH-mannose bond was investigated for a constant loading rate (i.e., a linearly increasing force with time). In our case, we couple the FimH-mannose bond model to the actual time-dependent force to which the adhesion-receptor bond is exposed under the attachment process simulated above. As the flow exposes the bacterium to a drag force, the tension on the tether changes the force on the adhesin, which in turn modulates the probability of staying bound.

To verify our numerical simulations of the FimH-mannose-bond model, we solved Eqs. 13 and 14 for the three loading rates considered by Yakovenko et al. (31). Our findings are in agreement with the results of that study (31), confirming the numerical procedure used for solving these equations.

The probability of a bacterium staying attached is given by the survival-probability distribution of the adhesion-receptor bond. Fig. 6 shows the probability distribution for a stiff linker and a type 1 pilus (solid and dashed curves, respectively) for shear rate $S_2$, which correspond to the blue curves in Fig. 3 B that initially increase linearly and thereafter flatten out. The figure shows that for a bacterium attached by a stiff linker, the survival probability drops drastically (to values close to zero) after a short time (<0.2 ms). For a bacterium adhering with a type 1 pilus, on the other hand, it remains high (>0.99) through the entire simulation (until the bacterium reaches the STD). In addition, early in the simulation (at ∼0.05 ms), the time development of the survival probability for both linkers changes to a plateaulike dependence. This is attributed to the fact that the bond alters conformation state, from the weak to the strong binding configuration, for which the lifetime is long for the pertinent force.

Survival-probability distributions for a variety of transient flows were also simulated. Fig. 6 shows such distributions for the six cases displayed in Fig. 3 B. Although not explicitly shown here, it was found that the situations presented in Fig. 5, representing delayed onsets of sweep (Fig. 5, A and B, green and blue curves, respectively), gave rise to survival-probability distributions that are similar in form to those shown in Fig. 6, mainly shifted in time. The survival-probability distributions presented in Fig. 6 can therefore be seen as typical for a broad variety of situations.

Influence of the parallel and normal drag correction coefficients on a tethered bacterium under flow

In addition to the above results, we also investigated the impact of the normal and parallel drag correction coefficients (Eqs. 9 and 10) in the simulations. Using the $S_2$ shear rate under the same conditions as presented in Fig. 4 B, four situations were compared, with and without each of the two correction coefficients. The results presented in Fig. S5 show the force on the adhesin for the stiff linker and the type 1 pilus (solid and dashed curves, respectively) with and without each drag coefficient. For the stiff linker, using the model described in this work, which incorporates both correction coefficients, the maximum force was found to be ∼225 pN. For the two cases where only one of the two coefficients was included, either the normal or the parallel, the maximum force was found to be ∼190 pN. For the uncorrected case, it became ∼125 pN. Thus, it was concluded that for a stiff tether, if the correction coefficients are not used, the simulations underestimate the force by almost 50%. For the case with the type 1 pilus, the influence of the correction coefficients on the maximum force is significantly smaller; the maximum forces were found to be ∼120 pN for all four cases. The reason for this is the force-damping ability of the pilus, which disguises the effect of the correction coefficients. However, the time evolution of the motion, and thereby also of the force exposure, are affected by these. The simulations predict that when the correction coefficients are included, it takes five times as long to reach STD as when they are not included.