Experimental measurements of response and noise spectrum
The signal propagation in the chemotaxis pathway has been characterised by the response to small concentration signals (linear response function; see Methods). Specifically, the response has been measured at the level of CheY-P using fluorescence resonance energy transfer (FRET) by Shimizu et al. [28]. In that study the system was stimulated by a periodic variation of the concentration of attractant α-methyl-DL-aspartate (MeAsp). Using a series of frequencies of the stimulation, the magnitude (modulus) and phase, i.e. the lag between signal and response, of the response was determined. In cell-tether experiments of motor rotation, the response to short impulses of attractants was measured at the level of the motor by Block et al. [26] and Segall et al. [27]. Such data determines the linear response function up to a constant factor. Experimental results are shown in Figure 3. Noise propagation in wild-type cells has only been studied at the level of the motor by Korobkova et al. [31]. We use the experimental response functions to calibrate our model, and subsequently study the noise power spectrum and signal-to-noise ratio.
Simplified model for the pathway
Here we consider a simplified pathway to gain intuition of the key processes involved. The simplified pathway consists of chemoreceptor signalling in response to ligand binding and receptor methylation, as well as the rotary motor. Specifically, we use stochastic differential equations in a Langevin approach [41] to describe the dynamics of each type of signalling protein. We assume throughout that fluctuations in concentration are small, allowing us to describe the average behaviour of a signalling molecule by a deterministic dynamics and the fluctuations around the mean by additive noise.
We assume N receptors form cooperative signalling complexes, which can switch between an active (on) and an inactive (off) state. Their activity A is described by the Monod-Wyman-Changeux (MWC) model [42–47]. The activity depends on the external ligand concentration c at the receptor complex, as well as the methylation level M of the complex as detailed in Methods.
We consider N
C
receptor complexes in a cell, and assume that each complex signals independently of the others. The total activity A
c
of all receptors in a cell is determined by the sum over all signalling complexes j. The dynamics of the total activity is
(1)
that is the dynamics of the complex activity is affected by changes in the receptor complex methylation level (first term), changes in ligand concentration (second term), as well as fluctuations due to the switching of the complex between its states (last term). All noise terms η(t) introduced in this section are discussed in Methods.
Changes in the concentration originate from time-varying input signals [c(t)], as well as fluctuations due to ligand diffusion. The dynamics of the concentration at the jth receptor complex is given by
(2)
where the first term captures average concentration changes (indicated by angular brackets 〈...〉), affecting all receptors, and the second term describes concentration fluctuations at each receptor complex, assumed to be uncorrelated between different receptor complexes.
Adaptation is provided by reversible receptor methylation and demethylation, whose dynamics is described by the following equation [47]:
(3)
The total methylation level M
j
of a receptor complex j is changed by methylation of receptors in the inactive state by CheR (first term) and demethylation (second term). This latter rate is assumed to be strongly dependent on the receptor complex activity as only active receptors are demethylated by phosphorylated demethylation enzymes CheB-P. These may act cooperatively, in agreement with time course data of wild-type and a CheB adaptation mutant [47]. The last term represents fluctuations due to the noisy processivity of the methylation and demethylation enzymes.
The motor is described as a two-state system with CW and CCW rotating states, corresponding to running and tumbling modes, respectively. The dynamics of the motor bias X is derived from the Master equation of an ensemble of two-state motors, and is given by
(4)
with X the fraction of motors in the CW rotational state and applying the single-motor limit. In Eq. 4, the first term represents the switching from CCW to CW with the transition rate k+, the second term represents switching from CW to CCW with transition rate k-, and the third term describes temporal fluctuations in switching rates due to the stochastic nature of motor switching. Note that the Langevin equation for the two-state motor is exact [48], see also Additional File 1.
Here, transition rates are modulated by the receptor signalling activity A
c
, whereas in the full pathway model CheY-P modulates motor switching. These rates have been experimentally measured using signalling mutants expressing varying amounts of constitutively active signalling molecule CheY [39]. The switching rates, including a fit of the model we used [39] (cf. Methods) to the data, are shown in Figure 3C.
Signal propagation
We consider the response to input concentration signals at various levels in the signalling pathway to study how signals are transmitted to the rotary motor (see Methods). Briefly, an input signal Δc(t) is a concentration change relative to a constant background concentration c0, affecting all receptors equally and representing a "meaningful" input to the chemotaxis signalling pathway. Hence, the concentration is given by 〈c(t)〉 = c0 + Δc(t). Furthermore, cells are assumed to be adapted to the pre-stimulus concentration c0 with the various levels R of the signalling pathway adapted to their steady-state values R*.
Analytical results for linear response functions
We can analytically calculate the Fourier transformed linear response function from the dynamical equations Eq. 1-4 without noise (see Methods). We can analyse the filtering of the signal at each level of the pathway. The Fourier transformed linear response function for the total activity of all receptors in a cell is
(5)
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.
Similarly, the Fourier transformed response of the motor is given by
(6)
The motor is a low-pass filter, i.e. its dynamics introduces a frequency-dependent response, which is constant below the characteristic frequency of the motor due to the steady-state switching rates and . 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.
For further analysis, we can write the Fourier transformed linear response function as
(7)
where 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.
Noise propagation
To understand the noise characteristics of the motor, we consider the noise sources and their transmission in the pathway. Each step in the signalling pathway is essentially probabilistic, hence, noisy: ligand diffusion and binding, receptor switching between its functional on and off states, as well as receptor methylation and demethylation, phosphorylation and dephosphorylation of signalling proteins CheA, CheY and CheB, and switching of the rotary motor between its two states, CW and CCW rotation. To characterise fluctuations of the phosphorylated signalling protein δR(t) around its mean value 〈R(t)〉, we use the power spectrum S
R
(ω) and the variance 〈δR2〉 = 〈R2(t)〉 - 〈R(t)〉2 (cf. Methods).
Analytical results for noise spectra
Considering Eq. 1-4 with noise, we can analytically calculate power spectra (see Methods). The power spectrum of activity fluctuations is given by
(8)
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 power spectrum of fluctuations in the motor bias is obtained as
(9)
The first term represents transmitted noise from receptor complexes, including the noise power spectrum of the receptor activity and the sensitivity 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
.
Cell-to-cell variation of motor behaviour
How are the signal response, fluctuations and the signal-to-noise ratio (SNR) affected by changing parameters of the pathway such as size of receptor complexes, protein concentrations and reaction rate constants? In this section, we discuss the effect of cell-to-cell variation on the power spectrum of the motor. In the next section, we discuss the SNR and its contributions, and how they depend on receptor complex size and adaptation rates.
According to our model parameters obtained from fitting the Fourier transformed linear response to data, the main contribution to the power spectrum comes from the steady-state switching of the motor between CCW and CW state. However, cell-to-cell variation in protein content and motor switching rates can lead to modifications of the largely Lorentzian-shaped spectrum. These modifications are caused by the transmitted noise from receptor methylation and phosphorylation dynamics (green and blue lines in Figure 4, right). Specifically, Figure 5A shows the motor power spectrum for increased motor switching rates as well as reduced adaptation rates and number of chemoreceptors in a cell. In all cases the low-frequency component of the transmitted noise becomes more prominent.
An increased low-frequency component has been observed in the motor power spectrum for cells with low motor bias [31]. Both, wild-type cells and mutants lacking the signalling pathway were measured. Hence, the mutant's spectrum represents the component to the power spectrum from steady-state motor switching only. Wild-type cells showed a large low-frequency component compared to the mutants. Figure 5B shows that our model can reproduce these experimental data (shown in the Inset), provided we assume a low CCW to CW switching rate leading to small motor bias (see Methods for the details of parameters). Specifically, the low-frequency component of our spectrum originates from noise in the methylation and phosphorylation dynamics in cells with low motor bias. Furthermore, experiments show that the low-frequency component in the motor power spectrum is reduced by increasing the expression level of CheR [31]. In Figure 5C we show that our model reproduces this experimental finding (shown in the Inset). The low-frequency component due to noise from receptor methylation and demethylation is effectively reduced by increasing the methylation rate constant.
Signal-to-noise ratio at the motor
To characterise how signals are transmitted in the presence of noise, we define the SNR at the level of the motor as
(10)
with ΔX2 and 〈δX2〉 defined in Methods. For optimal signalling this ratio should be maximised. For simplicity, we only discuss the receptor activity in the text, while in the figures we additionally show the contribution from phosphorylation processes as transmitted to the motor in the full pathway model.
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, , 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.
Fluctuation-response relationships
Park et al. [50] presented the idea that the signalling response to concentration signals and fluctuations in the chemotaxis pathway are not independent of each other, because they are produced by the same molecular interactions. Specifically, based on measurements at the level of the motor these authors proposed a fluctuation-response theorem, namely an approximate linear relationship between the adaptation time to step stimuli and the variance of fluctuations in CheY-P concentration.
Using our model, we tested this hypothesis and varied the adaptation rates, as well as the total CheY concentration in a cell, resulting in a shifted adapted CheY-P concentration at steady state. We find that the variance of CheY-P (normalised by the squared adapted value) decreases as the adapted CheY-P value increases except for very small adapted CheY-P concentrations (Figure 8A), indicating that the relative strength of fluctuations decreases as expected. In Figure 8B we show the adaptation time, approximated by the inverse of the characteristic frequency due to adaptation, plotted against the variance of CheY-P. We find that at low adaptation times (thick line styles), the adaptation time increases with the variance of CheY-P, indicating that cells with large fluctuations also respond longer to concentration signals. In contrast at long adaptation times, the adaptation time decreases with increasing variance of the pathway (grey parts of the curves). This behaviour can be directly traced back to the non-monotonic variance shown in Figure 7. It is maximal when the adapted CheY-P concentration is about 5 μM, i.e. when typically half of CheY is phosphorylated. The exact relationship depends on what parameters varied, exemplified by the different curves in Figure 8B. For each parameter and small adaptation times, we find an approximate linear relationship in line with Park et al. [50], see Inset.