Noise characteristics of the Escherichia colirotary motor

Analytical results for linear response functions

The receptor activity is a high-pass filter: The magnitude of the response function is small for frequencies ω below ω _M = (γ _R + 3γ _B A*²)∂A/∂M, which is the characteristic frequency due to adaptation. For frequencies above ω _M the response function is a constant, given by the number of receptor complexes N _C participating in the response, and their sensitivity ∂A/∂c to ligand, evaluated at steady-state. The sensitivity is proportional to the receptor complex size N, i.e. it describes the amplification of the response of a single receptor.

The motor is a low-pass filter, i.e. its dynamics introduces a frequency-dependent response, which is constant below the characteristic frequency ${\omega }_X=k_{+}^{*}+k_{-}^{*}$ of the motor due to the steady-state switching rates $k_{+}^{*}$ and $k_{-}^{*}$. The parameter ω₂ describes the sensitivity of motor switching with respect to changes in receptor activity (Methods). At frequencies above ω _X the response is reduced. From Eq. 6 it is obvious that receptors and motor are in a cascade: The motor response introduces a new filter proportional to (ω _i - iω)^-1 which simply multiplies the response function of the response of the receptor activity. The response functions of the full pathway including the phosphorylation reactions are shown in Additional File 1.

where $\left|{\widehat{\chi }}_R\right|$ is the magnitude and ϕ _R is the phase of the response function, which characterise the amplitude and lag of the response behind the input signal, respectively.

Model calibration

Figure 3 shows experimental data for the response function, as well as the fits of our full pathway model. Block et al. [26] and Segall et al. [27] measured the response of the motor using impulses of attractant. For our fit we adjusted adaptation and motor switching rates. Compared to the data by Shimizu et al. [28] at the same temperature, adaptation rates are one order of magnitude larger, i.e. adaptation is faster in these experiments. The parameter ω _X of the motor switching is 2.1/s, consistent with switching rates of about 1 Hz [29]. It is not clear from where the difference in adaptation rates between the two sets of experiments originates. However, different strains and media can lead to large variations in receptor expression level [49]. Besides different experimental conditions, Shimizu et al. [28] used populations of cells, whereas measurements by Segall et al. [27] were done on single cells. For the fit of our model to the data by Shimizu et al. [28], we adjusted only the adaptation rates, as measurements were restricted to low frequencies. The fit at 32°C yields the same adaptation parameters as obtained from fitting dose-response curves of adapting cells [47] (Figure 3B, left). The adaptation rates for room temperature are one order of magnitude smaller. Importantly, fitting to the magnitude of the Fourier transformed response yields a good fit for the phase of the response as well (Figure 3B, right). Fitted parameters are given in the Methods.

Signal filtering along the pathway

Figure 4 shows simulated time courses of the chemotactic response to an concentration impulse and the Fourier transforms of corresponding linear response functions, as well as our analytical results. As can be seen in the figure, linear response functions for the numerically solved non-linear model indeed match the analytically calculated functions, confirming that our calculation results are valid for the chosen input signal. We observe how the input signal is transmitted through the pathway, with the effective pulse durations becoming progressively longer along the pathway (Figure 4, left), including total receptor activity in a cell (A _c ), phosphorylated kinase CheA, phosphorylated response regulator CheY, and finally the motor (X). In Figure 4, middle we show the corresponding linear response functions.

The receptor activity acts as a high-pass filter, i.e. it transmits high-frequency signals, but not low-frequency signals. As can be seen from our simple model (cf. Eq. 5), this property is due to adaptation, which introduces the time-derivative of the signal Δc(t) up to the characteristic frequency ω _M , eliminating the response to slowly changing attractant concentrations. The activity of chemoreceptors is the input to further levels in the pathway. The response of CheA-P is fast, and shows no qualitative difference to the response of receptors in the frequency range shown. In contrast, due to the fast but finite rates of phosphorylation and dephosphorylation, preventing the CheY-P concentration to respond to rapidly changing input signals, the response at the level of CheY is reduced at high frequencies. Similarly, the motor introduces another high-frequency filter due to slow switching between its two states. This additional filter can be deduced from Eq. 6, where the motor response function takes the response of chemoreceptors as input, and additionally introduces a characteristic cut-off frequency ω _X due to slow motor switching rates. Hence, the chemotaxis pathway acts as a band-pass filter [26], which only transmits input signals within a selected frequency range, which is of the order of 1 to 10 s. This time scale corresponds to the average time between two tumbles, allowing sensing of concentration changes during periods of running. As shown in Figure 4, middle the phase tends towards π/2, i.e. a quarter period, at low frequencies. This has been analysed only for the receptor complex activity [28]. This phase difference is due to adaptation and represents the fact that the system takes the time derivative of the stimulus below the characteristic frequency ω _M of adaptation. The phase shift of the receptor activity increases to π at high frequencies, indicating that the activity simply follows the output (a negative sign is due to the negative response of the activity to attractant concentration [28]). The phase at high frequencies for the response of CheA follows the phase of the receptor activity, except for a small increase of the phase shift. In contrast, the phase of CheY and the motor increase significantly beyond π indicating that slow rates of modification and motor switching introduce a lag of the response behind the stimulus.

Analytical results for noise spectra

In this equation we considered fluctuations from receptor switching (first term in numerator), ligand diffusion (second term), as well as the receptor methylation dynamics (third term) at each of the N _C receptor complexes per cell. We have assumed that fluctuations at different receptor complexes are independent. Therefore, we obtain the sum of N _C identical spectra for all complexes. The individual terms S _a (ω), S _c (ω) and Q _M are given by Eq. 30, 34 and 36 in Methods. The frequency dependence of the ligand noise, as well as noise from receptor complex switching, indicates filtering of slowly varying fluctuations with frequencies below the characteristic frequency ω _M due to adaptation. In contrast, only high-frequency fluctuations from the receptor methylation dynamics are filtered by the adaptation dynamics. This is due to finite rates of methylation and demethylation fluctuations introducing correlations in the receptor methylation level.

The first term represents transmitted noise from receptor complexes, including the noise power spectrum of the receptor activity and the sensitivity ${\omega }_2^2$ of motor switching rates to changes in activity. The second term is motor switching noise. Both noises are filtered by the motor, as its finite rates of switching introduce correlations with characteristic frequency ω _X . The noise spectra of the full pathway including the phosphorylation reactions are shown in Additional File 1.

Noise filtering along the pathway

In Figure 4, right we show the power spectrum of fluctuations at the various levels of the signalling pathway, i.e. total receptor activity, CheA-P, CheY-P and the motor. We also plot the individual contributions from processes generating noise, namely ligand diffusion, receptor switching, methylation and demethylation of receptors, and phosphorylation and dephosphorylation of proteins, as well as motor switching. This allows us to follow how noise is generated and transmitted at the various levels of the pathway. The noise spectrum of the receptor activity has its largest contribution at low frequencies, which originates in the receptor methylation and phosphorylation dynamics. Most of the fluctuations from phosphorylation stem from CheB (the separate contributions to the phosphorylation noise are not shown in Figure 4, right). At high-frequencies, the activity noise spectrum is at. This is due to ligand and receptor switching noise, which is removed at low frequencies by adaptation, but not at high-frequencies. The general behaviour of the noise spectrum corresponds to the simplified model (cf. Eq. 8).

The noise spectrum of CheA-P has generally the same shape as the activity spectrum with a large low-frequency component, mainly due to receptor methylation and CheB phosphorylation dynamics. This spectrum also has an almost flat high-frequency behaviour in the frequency range shown. Apart from ligand and receptor switching noise, the flat part of the spectrum is largely determined by fluctuations from CheA autophosphorylation, which has roughly the same shape as activity noise at high frequencies because autophosphorylation depends on the receptor activity.

The noise spectrum of CheY-P is also largest at low frequencies. However, at high frequencies the spectrum falls off as noise is filtered due to the finite rates of CheY phosphorylation and dephosphorylation, which introduce correlations in the fluctuations.

The motor introduces another layer of filtering of transmitted noise with the characteristic motor switching frequency ω _X (cf. Eq. 9). Hence, transmitted noise is reduced by two filters in the frequency range shown, namely due to the CheY-P and motor dynamics. However, the main contribution to the spectrum is due to the motor switching itself, which is reduced only by a first-order filter with characteristic frequency ω _X .

Optimal receptor complex size

Receptor complexes amplify small signals proportionally to their size N. However, also concentration fluctuations are expected to be amplified. Hence, we hypothesise that the receptor complex size could be optimised to yield a balance of advantageous amplification of signals and detrimental amplification of input noise.

In Figure 6A we show the integrated motor response ΔX² (see Methods) to a step stimulus for varying background concentration and receptor complex size. We assume that the step stimulus size is a constant fraction of 10 percent of the background concentration. The integrated response has a characteristic variation with background concentration with the maximum in the sensitivity range of Tar receptors (indicated by their dissociation constants). Furthermore, the response increases with receptor complex size N. We calculated the integrated signal response of the receptor activity (see Methods). This quantity scales linearly with receptor complex size, $\Delta A_c^2\propto N$, due to coherent addition of the signalling responses of different receptor complexes, amplification of concentration changes by receptor complexes, as well as filtering by adaptation.

In Figure 6B and 6C, we show the variance (i.e., the integrated noise power spectrum, see Methods) of the transmitted noise of the pathway at the level of the motor. Only the contribution to the variance from ligand diffusion depends on the background concentration. Compared to the signal response, the maximum of the variance is shifted to a slightly lower concentration. The contribution to the variance from switching of receptor complexes is relatively small compared to the other contributions and roughly constant with receptor complex size, whereas those from ligand diffusion, receptor methylation and phosphorylation dynamics increase with receptor complex size.

To understand these behaviours of the variance more intuitively, we analysed the receptor activity analytically (for details of the calculation, see Methods). We find, the contribution to the variance of the receptor activity from receptor switching is indeed constant, independent of N. The contribution from ligand diffusion scales steeply as N², the difference between ligand noise and ligand signal amplification being due to (i) noise from different complexes is added up incoherently, and (ii) the main contribution to the variance coming from high-frequency ligand noise, which is not filtered by adaptation. The contribution from receptor methylation grows approximately linearly with receptor complex size as a result of the incoherent addition of fluctuations at different receptor complexes and the sensitivity of the receptor complex activity with respect to changes in methylation level increasing proportionally with N. The contribution to the variance from phosphorylation processes grows with receptor complex size similar to the contribution from the methylation dynamics. Overall, the total variance of transmitted noise at the level of the motor has contributions from receptor switching, the dynamics of receptor methylation, and phosphorylation. The latter is approximately constant or grows slower than the amplified signal response, whereas the component from ligand diffusion increases steeper than the signal response with growing receptor complex size.

The resulting SNR, i.e. the ratio of integrated signal response and variance of the noise, is shown in Figure 6D and 6E. The SNR is largest at background concentrations in the sensitivity range of the Tar receptor. Furthermore, due to the different dependencies of the signal and the noise on the receptor complex size, the SNR has a maximum at a particular receptor complex size (Figure 6E). The SNR grows below that complex size due to signal amplification, while the amplified ligand noise from ligand diffusion is still below the internal noise level from receptor switching and receptor methylation and phosphorylation dynamics. Above the optimal receptor complex size, the SNR decreases because the ligand noise is amplified more than the signal.

Optimal adaptation rates

As shown above, adaptation filters slow input signals, with the adaptation speed determining what input frequencies are transmitted by the pathway. Furthermore, the adaptation dynamics filters input noise. Hence, adaptation rates may be expected to be optimised for signal and noise propagation.

Figure 7A shows the integrated signal response at the level of the motor for varying rates of receptor methylation (γ _R ) and demethylation (γ _B ). Varying these parameters describes changing the concentrations of receptor modification enzymes CheR and CheB. The integrated signal response is found to be maximal when both rates of methylation and demethylation become small simultaneously. Then, the adapted receptor activity is in the steep region of dose-response curves, and the initial response to attractant is maximal. At the same time adaptation becomes slow and, therefore, the response lasts long. Interestingly, varying the two parameters independently has different effects on the signalling response: the integrated signal response increases for vanishing γ _R , whereas it decreases for vanishing γ _B . There are two effects that contribute to this behaviour, illustrated in Figure 7B: Firstly, if the concentration of one of the receptor modification enzymes is reduced, the receptors becomes modified predominantly by the opposing enzyme, hence driving the receptor activity towards saturation (A* = 0 or A* = 1). This effect would tend to quench the response by receptors. Secondly, as the enzyme concentration is reduced, adaptation times increase. Hence, this effect increases the integrated signal response as the time the receptor activity deviates from the adapted state increases. According to calculations shown in Methods for the integrated response of receptors, the first effect dominates in the case of reduced γ _B : Due to the strong activity dependence of the demethylation rate, reducing the demethylation rate constant effects the adapted activity of receptors strongly. Hence, receptors are quickly driven into saturation for vanishing γ _B . In contrast, in the case of reduced γ _R the second effect dominates and the increased adaptation time leads to an increased integrated signal response. At large methylation and demethylation rates, adaptation times are reduced leading to a decreasing integrated signal response.

The variance of fluctuations is shown in Figure 7C and 7D. The individual contributions from transmitted noise at the level of the motor look qualitatively similar. For γ _R and γ _B such that the adapted receptor activity is in the steep region of dose-response curves, the variance of transmitted noise is largest. In contrast, for either vanishing γ _R or γ _B all contributions decrease, consistent with calculations for the variance of the receptor activity in Methods. In these cases, the adapted receptor activity becomes saturated, hence, quenching fluctuations transmitted by receptors.

The resulting SNR is shown in Figure 7E. The SNR increases for vanishing γ _R and decreases for vanishing γ _B . According to Figure 7E, a large SNR is obtained for small γ _R and large γ _B , corresponding to the parameters of our model.

Signal response

Hence, the time course of the response is determined by the convolution of the linear response function and the input signal. The linear response function describes the dynamics of the pathway and the convolution with the input signal represents the fact that the current state of the system is determined by the history of the input signal [66]. Importantly, for a nonlinear system to respond linearly to a stimulus, the input signal has to be small. For the receptor complex activity, a signal is small if the change in ligand concentration is small compared to the apparent dissociation constants for the on and off states. These dissociation constants depend on the background concentration as the system adapts to the background concentration c₀, resulting in a particular free-energy difference F*. Thus, the concentration change has to be small compared to $c_0+K_a^{on}$ and $c_0+K_a^{off}$. In this case, the change in free-energy difference due to the concentration change is much smaller than k _B T. In Figure 4, the background concentration c₀ = 0 and the peak concentration of the impulse is 10^-3 mM, compared to $K_a^{on}=0.5\text{mM}$ and $K_a^{off}=0.02\text{mM}$. Hence, the input signal is indeed small.

${\widehat{\chi }}_R\left(\omega \right)$ is also called the frequency-dependent gain [12]. The modulus $\left|{\widehat{\chi }}_R\left(\omega \right)\right|$ describes what frequencies of the input signal are transmitted well, and which ones are attenuated.

Typically, finite activation rates of the system limit the response to rapidly changing input signals, i.e. high-frequency signals. In this case, the Fourier transformed linear response functions falls off at high-frequencies, and the system is called a low-pass filter. If low-frequency components of the input signal are filtered out rather than high frequencies, the system is called a high-pass filter. The chemotaxis pathway is a band-pass filter (see Figure 2), filtering out low and high-frequency components.

Noise

where the Fourier transform is defined on the finite measurement interval T and the average 〈·〉 is over multiple time series.

Determining the noise intensity

The rates r₁ and r₂ typically depend on the concentrations of proteins in the signalling network. The noise term η(t) is composed of two terms η₁(t) and η₂(t), which are associated with the rates r₁ and r₂, respectively. We assume η₁ and η₂ to be independent, i.e. 〈η₁(t) η₂(t')〉 = 0. In general, this is justified as different reactions are catalysed by different proteins. Using 〈η _j (t)η _j (t')〉 = Q _j δ(t - t'), the noise intensities can be calculated if we make the assumption that fluctuations are due to so-called birth and death processes, i.e. creation and destruction of the molecules with average rates $r_1^{*}$ and $r_2^{*}$. Then the associated noise intensities are $Q_1=r_1^{*}$ and $Q_2=r_2^{*}$[8]. The intensity of the total noise η(t) is the sum Q = Q₁ + Q₂ due to the independence of the two noises. As forward and backward rate are equal at steady state, Q is twice the reaction rate in one direction at steady-state.

Switching noise

This result is due to the Fourier transform of the rate of receptor complex switching da/dt yielding $i\omega \widehat{a}$. Hence, the power spectra of a and da/dt differ by a factor ω² as the power spectrum is proportional to the magnitude squared of the Fourier transform according to Eq. 26.

Ligand noise

where k _D L is the rate of molecules moving out of the volume by diffusion, and k _D times the mean concentration c₀ in solution serves as a proxy of the rate of ligand molecules moving into the volume. The noise term η _L (t) is assumed to be Gaussian and white, with zero mean and noise intensity Q _L = Dsc₀.

where s⁶ is the squared volume given by the dimension of the receptor complex. The zero-frequency limit of the power spectrum of the ligand concentration S _c (0) = c₀/(Ds), which corresponds to calculations by Berg and Purcell [71] and Bialek and Setayeshgar [57] for the uncertainty in sensing ligand concentration.

Methylation noise

Motor switching noise

Optimal receptor complex size

where we inserted Eq. 5. Hence, the activity response scales as $\Delta A_c^2\propto {\left(N_{tot}∕N\right)}^2{\left(N^2\right)}^2∕N\propto N$, where we used that N _C = N _tot /N with N _tot the total number of receptors in a cell.

where we used Q _a and inserted Eq. 30 for the power spectrum of receptor switching noise and used that it is almost constant and equal to its zero-frequency value over the integration range. Furthermore, the factor ${\omega }^2∕\left({\omega }^2+{\omega }_M^2\right)\approx 1$ and $A_r^{*}=A^{*}∕N$ is the adapted activity of an individual receptor. Hence, according to this simple calculation the contribution to the variance from receptor switching is roughly constant with receptor complex size.

where 〈δc²〉 = c₀/(Dsτ) is the variance of the ligand concentration measured during the time interval τ. We used Eq. 34 and the same argument as for the switching noise to calculate the integral. Hence, the contribution to the variance from the ligand diffusion grows as ${⟨\delta A_c^2⟩}_c\propto N^{5∕2}$ as a result of incoherent addition of noise from different receptor complexes, the sensitivity ∂A/∂c increasing as N², and the size dependence of receptor complexes s ∝ N^1/2.

where we defined ${\omega }_1={\gamma }_R+3{\gamma }_B{N}^2{\left(A_r^{*}\right)}^2$, inserted $Q_M=2{\gamma }_{R}N\left(1-A_r^{*}\right)$ and ω _M = ω₁(∂A/∂M). Hence, ${⟨\delta A_c^2⟩}_M$ grows approximately linearly with receptor complex size.

The SNR grows linearly with N for small complex sizes, and decreases as N^-2 for larger complex sizes, resulting in an optimal medium receptor complex size, in qualitative agreement with Figure 6.

Optimal adaptation rates

and is only a function of the ratio β = γ _R /γ _B . Expanding the adapted activity around $A_r^{*}=0$ (for γ _R → 0) yields $A_r^{*}\propto {\gamma }_R^{1∕3}$, and around $A_r^{*}=1$ (for γ _B → 0) yields $A_r^{*}\propto {\gamma }_B$. Similarly, ${\omega }_1\propto {\gamma }_R^{2∕3}\left(const.+{\gamma }_B^{4∕3}\right)$. Hence, $\partial A∕\partial c\propto {\gamma }_R^{1∕3}\left({\gamma }_B\right)$ and ${\omega }_1\partial A∕\partial M\propto {\gamma }_R^{4∕3}\left({\gamma }_B\right)$.

The initial response to concentration changes decreases slower than adaptation times, resulting in an increased signal response for vanishing γ _R . For vanishing γ _B , the initial response to concentration changes decreases faster than adaptation speed, hence yielding a vanishing signal response. The overall dependence of the integrated signal response is $\Delta A_c^2\propto {\gamma }_R^{-1∕3}\left({\gamma }_B\right)$ for γ _R → 0(γ _B → 0). For the contributions to the variance of the receptor activity from receptor switching, ligand diffusion and receptor methylation dynamics we obtain ${⟨\delta A_c^2⟩}_a\propto {\gamma }_R^{1∕3}\left({\gamma }_B\right)$, ${⟨\delta A_c^2⟩}_c\propto {\gamma }_R^{2∕3}\left({\gamma }_B^2\right)$and ${⟨\delta A_c^2⟩}_M\propto {\gamma }_R\left({\gamma }_B^{2∕3}\right)$, respectively. Hence, according to our simplified model the SNR of the receptor activity goes as $SNR\propto {\gamma }_R^{-2∕3}\left({\gamma }_B^{4∕3}\right)$, in qualitative agreement with Figure 7.