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Copyright © 2007 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 93, Issue 1, 11-20, 1 July 2007

doi:10.1529/biophysj.106.100263

Biophysical Theory and Modeling

Stochastic Signal Processing and Transduction in Chemotactic Response of Eukaryotic Cells

Masahiro Ueda^*, ,   and Tatsuo Shibata^†

^* Laboratories for Nanobiology, Graduate School of Frontier Biosciences, Osaka University, Suita, Osaka, Japan
^† Department of Mathematical and Life Sciences, University of Hiroshima, Higashi-Hiroshima, Hiroshima, Japan

Address reprint requests to Masahiro Ueda, Laboratories for Nanobiology, Graduate School of Frontier Biosciences, Osaka University, Yamadaoka 1-3, Suita, Osaka 565-0871, Japan. Tel.: 81-6-6879-4632; Fax: 81-6-6879-4634.

Living cells can sense and respond to environmental signals through dynamic signaling processes in the reaction networks of biomolecules. Because biomolecules operate stochastically under the strong influence of thermal fluctuations, living cells can be referred to as stochastically operating biomolecular computation systems. Recent progress in the area of single-molecule detection techniques has identified the stochastic nature of biomolecules in vitro and in living cells 1,2. For example, single ion channels have been observed to exhibit a random transition between open (“on”) and closed (“off”) states in an alternating manner 3. Such stochastic behavior has also been observed in catalytic reactions by single enzyme molecules and in steplike movements by single molecular motors 4,5,6,7. On-off fluctuations in individual molecules inevitably cause number fluctuations in the ensemble of the molecules, thus making intracellular signaling processes inherently noisy. This leads to a fundamental question about intracellular signaling processes in general: how does the signaling system operate reliably under thermal and stochastic fluctuations? To gain insight into how signals are received, processed, and transduced by stochastically operating molecules, we study the chemotactic signaling system of eukaryotic cells as a typical example of a stochastic computation system.

Chemotaxis is a fascinating phenomenon in which cells sense chemical gradients and move with directional preference toward or away from the source of the chemical cues. Eukaryotic cells can sense the differences in chemoattractant concentration across the cell body and respond by extending pseudopods directed up the chemical gradient 8,9,10,11,12. In Dictyostelium cells, extracellular cyclic adenosine 3′, 5′-monophosphate (cAMP) functions as a chemoattractant. Only 2% gradients can induce a biased movement of the cells toward the source of cAMP in a wide range from 10 pM to ∼10μM 13,14,15. Because Dictyostelium cells are 10–20μm in size and contain ∼80,000 receptors on the surface evenly, with an average K_d of ∼100nM 16,17, receptor occupancy is estimated to be ∼16,000 molecules at the maximum efficiency of chemotaxis (25nM), whereas the differences in receptor occupancy between the anterior and posterior halves are ∼130 with 2% gradients. Recently, the lowest gradient value where directed motion is observed was determined by using microfluidic devices 18. The difference in receptor occupancy was estimated to be only on the order of 10 molecules for gradients close to the lower threshold (∼10⁻³nM/μm). Ligand binding to the receptors is a stochastic process, so receptor occupancy should fluctuate with time and space. Assuming a Poisson process, fluctuations in receptor occupancy are the square root of the averaged occupancy, and therefore ∼130, which is comparable to the spatial differences. Around the threshold stimulation (100 pM), occupancy and its fluctuations are ∼80±9 molecules, whereas the differences are ∼1 to ∼8 molecules with 2% to ∼20% gradients. Although this estimation includes many uncertainties, it implies that the input signals for chemotaxis become noisy due to the fluctuations in ligand binding to the receptors. Such fluctuations in signal input have been observed directly by single-molecule imaging of the attractant bound to living Dictyostelium cells 19. Chemotactic signaling systems should amplify small changes in input signals. However, by the same system, small random changes (noise) in the input signal would be amplified also, resulting in the propagation of noise as well as signal. Thus, how chemotactic cells reliably obtain information regarding the gradient from such noisy input is a critical question for directional sensing in chemotaxis.

Stochastic signaling processes in living cells have been studied theoretically. Oosawa constructed a theory of spontaneous signal generation in living cells based on thermal fluctuations of biomolecules 20,21,22. Berg and Purcell have shown that chemoreception by receptors is limited by molecular counting noise 23. For the chemotaxis of amoeboid cells, Tranquillo and colleagues constructed a stochastic model in which the ligand-receptor binding reaction generates stochastically the intracellular messenger that is the critical regulator of the motile system to modulate turning frequency of cells. Based on kinetic fluctuations in ligand-receptor binding, the model explains well the characteristic features of leukocyte random motility and chemotaxis 24,25. Recently, generation and propagation of noise in intracellular processes have been studied in engineered transcriptional regulatory networks 26,27,28,29. Elowitz and colleagues have clearly shown experimentally that noise propagates along a cascade of gene expression. Paulsson unified the gene network experiments by analyzing the propagation of noise in gene networks from a theoretical point of view 30. Shibata and Fujimoto addressed how noise relates to the amplification of signals in intracellular signaling processes, which is summarized as the gain-fluctuation relation 31. The relation tells us that signal and noise propagation along the signaling cascade can be characterized by the gain and characteristic time of the signaling reactions, which can be applied generally to intracellular signaling reactions including Michaelis-Menten, allosteric, and push-pull reactions. Recent progress in imaging techniques to monitor directly intracellular signaling reactions makes it possible to determine stochastic properties of signaling molecules, and therefore a theoretical framework is required to evaluate quantitatively how the properties of signaling molecules affect cellular response.

Here we consider a simple but general model in which receptors receive ligands stochastically and the resulting active receptors generate second messengers stochastically. We applied the gain-fluctuation relation to this model, by which the signal/noise ratio (SNR) of chemotactic signals can be calculated based on the properties of the signaling molecules obtained experimentally. Analysis of the SNR reveals that directional sensing in eukaryotic chemotaxis is limited by receptor-generated stochastic noise, and also reveals how the stochastic nature of the receptor and second messengers affects the SNR of chemotactic signals, which suggests regulatory mechanisms for the noisy signal transduction in chemotactic cells. Our model provides a theoretical framework with experimental approaches to the chemotactic signaling system and can further be applied to stochastic signaling systems in general.

Single-molecule imaging of ligand binding to chemoattractant receptors in living Dictyostelium cells demonstrates that signal inputs fluctuate with time and space (Figure 1A)19,32. The lifetime of ligand binding shows an exponential distribution, with time constants ranging between ∼1 and ∼3s (Figure 1B). The time series of receptor occupancy exhibits fluctuations (Figure 1C) with exponential time correlations. These results demonstrate that ligand binding can be described basically as a Poisson process. This means that the chemotactic ligand binds to the receptor randomly and, hence, that input signals are noisy. Note that such fluctuations in input signal are not derived from an error of experimental measurements. The fluctuations are due to the stochastic nature of the ligand-binding process, which is accompanied inherently by the ligand-binding reaction.

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Figure 1

Fluctuations in signal inputs for chemotactic response. (A) Single-molecule imaging of a fluorescent-labeled cAMP (Cy3-cAMP) bound to the receptor in living Dictyostelium cells. Cy3-cAMP was added uniformly to Dictyostelium cells at 10 nM. The basal surface of the cells was observed by using total internal reflection fluorescence microscopy, as described previously 19,32. Individual white spots are single molecules of Cy3-cAMP bound to the receptors in living cells. Time, h:min:s. Scale bar, 5μm. (B) Cumulative frequency histogram of lifetime of Cy3-cAMP spots. The lifetimes of individual Cy3-cAMP molecules were obtained by counting the time duration between the appearance and disappearance of the fluorescent spots. The line represents the fitting of data to a sum of two exponential functions, where a₁, a₂, k₁, and k₂ are fitting parameters. k₁ and k₂ are dissociation rates, the inverse of the average lifetime. The number of Cy3-cAMP spots analyzed was 1024. k₁=1.0 and k₂=0.13s⁻¹. a₁=74.3% and a₂=31.4%. (C) Time course of the number of Cy3-cAMP spots bound to the basal surface of the cells, showing the number fluctuations of signal inputs for chemotactic response.

We have developed a stochastic model that describes the signal and noise propagation along the transmembrane signaling pathway by receptors. As shown in Figure 2A, we assume that receptors receive ligands randomly as signal input, leading to the stochastic generation of intracellular messengers as output. The second output messengers then degrade with time. This scheme is representative of many signaling pathways. In the chemotactic signaling system of Dictyostelium cells, the first and subsequent reactions correspond to cAMP binding to the receptor and G-protein activation, respectively 10,11,12. The noise of the active receptors is the deviation from the average amount of active receptors, which can be quantified by where R^* is the molecular number of active receptors per cell and is its average. Assuming that the receptors distribute uniformly on the surface of the cells, the noise, is given by the gain-fluctuation relation 31, as follows,

(1)

(2)

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Figure 2

Stochastic model of chemotactic signaling. (A) Signal transduction reactions by chemoattractant receptors. The ligand (L) binds to the inactive receptor (R), leading to the formation of an active receptor (R^*), which produces the active second messenger (X^*) from the inactive precursor (X). The active X^* is switched off to the inactive state, X, in due time. These reactions can be described by Michaelis-Menten kinetics. (B) The cell is placed under a ligand-concentraion gradient. L, average concentration; ΔL, difference in ligand concentration between the anterior and posterior ends of the cell. The anterior and posterior halves sense and on average, respectively. The difference in receptor occupancy ΔR^* is produced from ligand-concentration differences, which lead to the difference in second messenger, ΔX^*, between the anterior and posterior halves. The differences ΔR^* and ΔX^* should include the noise and around average values and respectively.

From Eq. (2), it is clear that a reaction with higher gain is more sensitive to small changes in input signal, resulting in higher amplification of the signals. However, Eq. (1) tells us that the reaction with higher gain also generates larger noise, because noise is proportional to g_R. That is, a higher gain is required for higher amplification of input signals, but it also inevitably and simultaneously increases noise. When the ligand-binding reaction is described by the gain, g_R, decreases as the increase of ligand concentration, L (Figure 3A; see Eq. (9)), and, hence, the relative noise, decreases (Figure 3C). Thus, chemotactic cells receive noisier signals at lower ligand concentrations.

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Figure 3

Relationship between gain and noise. (A) Active receptor concentration (R^*; solid line) and gain (g_R; dashed line) plotted as functions of ligand concentration. (B) Dependence of the gain g_X on receptor occupancy. (C) Dependence of relative noise in R^* on ligand concentration. (D) Dependence of relative noise in X^* on ligand concentration. Extrinsic noise and intrinsic noise are represented by black dashed and solid lines, respectively. The total noise strength is represented by the red solid line. The parameter values used for the calculation are summarized in Table 1. (Inset) Log-log plot. Extrinsic noise contributes dominantly to the total noise in the lower ligand-concentration range, whereas intrinsic noise contributes dominantly in the higher ligand-concentration ranges.

The active receptor, R^*, leads to the stochastic activation of intracellular messenger X to the active form X^* as output. The noise of the active second-messenger concentration is described by 30,31

(3)

The second term on the righthand side of Eq. (3) is the extrinsic noise 26, which describes how the noise of active receptor R^* propagates into the noise of second-messenger concentration. When the time constant of the second-messenger production is faster than that of the active receptor (), the noise of the active receptor is propagated more efficiently into the noise of the second messenger, with a decrease in τ_X, because the term increases gradually and reaches unity as increases. In this case, the second-messenger-production reaction can follow rapid temporal changes of the active receptor. On the other hand, in the case of the second-messenger reaction cannot follow the noise of the active receptor. Instead, the noise of the active receptor is averaged temporally, and the extrinsic noise decreases. In the extreme case, the extrinsic noise is eliminated from the total noise by time-averaging effects. Thus, the relatively slower reaction is required in the second-messenger production to reduce the extrinsic noise generated by the ligand-binding reaction, whereas the relatively faster reaction causes the noise propagation.

Even if the amount of active receptor is constant without noise (σ_R=0) or the extrinsic noise is almost neglected by the effect of temporal averaging in the second-messenger reaction, the second messenger should be accompanied by noise, because the active receptors activate stochastically the second messenger. Such intrinsically generated noise by the second-messenger-activation reaction itself is called intrinsic noise 26, which is given by the first term on the righthand side of Eq. (3). The intrinsic noise is included inevitably in the total noise of second messengers. Figure 3D shows the relative contributions of intrinsic and extrinsic noise to the total noise. The total noise, increases with decreasing ligand concentration. Because extrinsic noise is proportional to the square of the gain, whereas intrinsic noise is proportional to the gain (Eq. (3)), extrinsic noise contributes dominantly to the total noise in the lower ligand-concentration ranges, where the gain, g_X, becomes relatively higher (Figure 3B). On the other hand, intrinsic noise contributes dominantly in the higher ligand-concentration ranges (Figure 3D, inset). Thus, the receptors generate noisier signals in the lower ligand-concentration range, which would take into account the inefficient chemotaxis in the corresponding ranges, as described in the next section.

To explain the mechanisms whereby cells sense chemical gradients, two representative mechanisms have been proposed: temporal sensing and spatial sensing mechanisms 8,9,23,33. In the temporal sensing mechanism, the movements of cells or their parts, such as pseudopods, are essential for gradient sensing, in which the spatial differences in chemoattractant concentration are converted into temporal changes through the movements. In the spatial sensing mechanism, cells detect the signals simultaneously at different points over their surfaces. As a result of comparison of the detected signals, the cells sense the direction of the chemical gradient. Dictyostelium cells can form positive or negative gradients of some signaling molecules, such as PI3-kinase and PTEN (tensin homology protein), inside the cell along the gradient of cAMP without cell movements and pseudopod extensions, indicating that the cells can sense the higher-concentration side of cAMP across the cell body without motion, which provides strong evidence that the origin of chemotactic signals is spatial differences in receptor occupancy 10,11,12. Thus, the cells do not necessarily require temporal sensing mechanisms for gradient sensing.

Devreotes and colleagues propose an alternative mechanism, the so-called local excitation global inhibition (LEGI) mechanism, in which temporal and spatial mechanisms are integrated to take into account the behavior of chemotactic cells 10,34,35. In the LEGI mechanism, receptor occupancy in a local area determines the local level of excitation, whereas the average level of receptor occupancy over the entire surface of the cell determines the level of inhibition in all regions of the cell. Although this mechanism does not assume direct comparison of the ligand concentration between different points over the cell surface, spatial differences of the ligand concentration are sensed through a comparison between the excitatory signals and the inhibitory signals at each of the local areas. Thus, chemotactic signals in the LEGI mechanism are derived from differences in receptor occupancy between the local region and the total surface of the cells. With regard to the origin of chemotactic signals, the LEGI mechanism can be thought of as an extension of the spatial sensing mechanism, which provides the molecular basis for the comparison of the spatial differences in receptor occupancy across the cell body. Although the temporal sensing mechanism may have a role for gradient sensing of chemotactic cells, the spatial sensing mechanism is essential, as described above. Here, we discuss the signal and noise propagation based on the spatial sensing mechanism, in which the chemotactic signals are the spatial differences in receptor occupancy across the cell body. We did not consider sensory adaptation in our model, because G-protein activation does not exhibit adaptation in Dictyostelium cells when ligand stimulation is applied continuously to cells 36.

We consider the differences in second-messenger concentration, ΔX^*, between the higher- (anterior) and lower (posterior)-ligand-concentration regions of chemotactic cells placed under a chemical gradient. As shown in Figure 2B, the concentration difference in the ligand concentration, ΔL, may produce the difference in receptor occupancy, ΔR^*, which may then lead to the difference in second-messenger concentration, ΔX^*, between the anterior and posterior regions of chemotactic cells. The ΔR^* and ΔX^* should include the noise, and around the average values, and respectively.

To evaluate the effects of the noise on gradient sensing, we studied the SNR, defined as From Eq. (3), we obtain the following relation between and (see Appendix for derivation).

(4)

Figure 4A shows dependence of the SNR, on the average concentration of ligand. The parameter values to calculate the SNR for Dictyostelium cells are summarized in Table 1 (see Appendix and Eq. (11)). We also performed stochastic numerical simulation showing agreement with our theory (Figure 4A). The SNR of chemotactic signals attains a maximum at the ligand concentration between the affinity of the receptor, K_d, and the EC₅₀ concentration, where the G-protein activation reaches half-maximum. This optimal concentration value is dependent mainly on the receptor affinity K_d, and is relatively unaffected by the EC₅₀ variation of G-protein activation (data not shown). In the lower-ligand-concentration range, the SNR is determined mainly by the contribution of the extrinsic noise, meaning that the fluctuations in active receptor dominantly affect the quality of the chemotactic signals. In the higher-ligand-concentration range, the SNR deteriorates with an increase in ligand concentration, because receptors are gradually saturated, making them unable to produce the large differences in second-messenger concentration between the anterior and posterior halves of cells, leading to an increase in intrinsic noise.

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Figure 4

SNR of chemotactic signals. (A) Dependence of the SNR on ligand concentation obtained theoretically by Eq. (4) (red line) and numerically (green diamonds). The cell is located without locomotion under a linear chemoattractant gradient of 2% along the anterior-posterior axis with the midpoint concentration L. The parameter values used for the calculation are summarized in Table 1. For the simulation, the spatial coordinate of the cell is discretized into small boxes appropriately, and the reactions take place in each box according to the Gillespie’s algorithm 44. (B) Relative contributions of extrinsic (blue) and intrinsic (green) noises on the total SNR of chemotactic signals (red). (C) Numerical calculation of chemotactic signals. Time course of second-messenger concentration difference (ΔX^*) between anterior and posterior halves of single cells. L=0.01μM. (D) A proportional relation between the SNR of chemotactic signals and γ, which represents the ratio between the total time durations with ΔX^*>0 and ΔX^*<0. Dashed line, where Erf(x) is the error function. (E) Comparison of the SNR with Fisher’s experimental data for chemotactic accuracy of Dictyostelium cells 14. The SNR obtained theoretically (red line) was overlaid on the experimental data (red circles). (Adapted from Fig. 5 of Fisher et al. 14 with permission). The SNR was plotted on the same scale as in A.

We next examined the relationship between the SNR of chemotactic signals and the signaling accuracy. As shown in Figure 4C, the time series of ΔX^* obtained by numerical calculation indicates that ΔX^* can sometimes be negative. This means that the concentration gradients of second messengers can be reversed against ligand-concentration gradients by fluctuations in ligand binding and second-messenger production reactions. Because chemotaxis is expected to be more accurate when ΔX^*>0 is produced more frequently, the ratio between the total time durations with ΔX^*>0 and ΔX^*<0, γ, can be used as an index of chemotactic signaling accuracy. As shown in Figure 4D, the ratio γ increases in proportion to the SNR. Thus, when chemotactic signals have a higher SNR, ligand gradients are represented on the second-messenger gradient for a longer time, which would lead cells to exhibit chemotaxis more accurately.

The chemotactic accuracy of Dictyostelium cells has been measured experimentally by Fisher et al. 14. The dependence of chemotactic accuracy on ligand concentration exhibits a profile similar to our calculated SNR (Figure 4AE). In the experiment, the cell’s movements were biased toward the higher concentration of cAMP over a range of 10 pM to 10μM, and chemotactic accuracy attained a maximum at 25 nM of cAMP concentration. This optimal value is almost the same as the concentration at which the SNR reaches the maximum (Figure 4E). The agreement between the SNR and chemotactic accuracy indicates that the ability of directional sensing is limited by the inherently generated stochastic noise during the transmembrane signaling of receptors. Note that Eq. (4) does not depend on a particular detail of the spatial sensing mechanism, and can be applied to other systems. In fact, similar dependence of chemotactic accuracy has been observed in mammalian leukocytes and neurons, although these cells exhibit chemotaxis at different ranges of ligand concentration 37,38. Such differences in the dependence of chemotactic accuracy on ligand concentration can be explained by cell-type-specific parameters, such as the ligand-binding affinity of receptor, K_d, and the EC₅₀ concentration for second-messenger activation.

When the ligand concentration L is sufficiently small (L ≪ K_R, K_XR in Eq. (11)), the SNR of the chemotactic signals changes in a manner of If the cell requires a signal exceeding a threshold SNR to detect chemical gradients, the cell will exhibit a threshold, ΔL_threshold, for each ligand concentration L for chemotaxis. Then, supposing that such threshold SNR is independent of ligand concentration L, we obtain the relation ΔL_threshold ∝ L^0.5. The threshold gradient in a given concentration of ligands can be measured experimentally. In fact, Van Haastert 39 reported the relation between the average concentration of ligand and the corresponding threshold gradient at which 50% of the cells can respond in the chemotactic assay. He found that α for ΔL_threshold ∝ L^α was estimated to be 0.35, which largely agrees with our estimation. In the experiments, the relatively high background cAMP concentrations were used to reveal sensory adaptation processes. Then, the threshold relation is not simply applicable at experimental conditions. Sensory adaptation, which was not considered in our model, may contribute to the α value being lower than theoretical estimation. To further evaluate our model, similar experiments would be required at the lower background concentrations in shallow gradients.

According to Eq. (11), the SNR changes in proportion to ΔL at a given concentration of ligands, L. Fisher et al. also studied the dependence of chemotactic accuracy on ΔL at 25nM cAMP 14. The accuracy was reduced almost linearly with a decrease of ΔL, vanishing at 10pM/μm, which was a 0.3% to ∼0.6% gradient. From our formula, the SNR for the 0.3% gradient around 25nM was estimated to be ∼0.07. Supposing that such a minimum SNR is a threshold for chemotaxis at any chemoattractant concentration, the ligand concentration required for chemotaxis at 2% gradient ranges from ∼200pM to 1μM (Figure 4AE), which is narrower than the observed range of chemotaxis in experiments. Those mechanisms not considered in our model, such as the temporal sensing mechanism or sensory adaptation, may contribute to chemotaxis at the lowest and highest concentration ranges. The role of adaptation in the SNR of chemotactic signals will be discussed elsewhere.

Despite the qualitative agreement between the SNR of chemotactic signals at the receptor level and chemotactic accuracy, the quantitative relationship between the two remains to be clarified. The chemotactic signaling system of Dictyostelium cells and other cell types has many components between the receptors and motile apparatus to convert the signals from receptors into unidirectional cell movement. Devreotes and colleagues have revealed that one of the key reactions in the chemotactic signaling system is a distinctive localization of phosphatidylinositol 3,4,5-trisphosphates (PI(3,4,5)P₃) on the membrane facing a higher concentration of cAMP 10,11,12. The PI(3,4,5)P₃ localization takes place in an all-or-none manner, meaning that noisy input is processed and transduced to generate a clear signal reflecting the gradient direction of chemoattractants through the cascades upstream of PI(3,4,5)P₃. It would be valuable to examine how the SNR of chemotactic signals at the receptor level is reflected in the dynamics of PI(3,4,5)P₃ localization.

Our results suggest how the SNR of chemotactic signals is improved by the properties of the receptors and the downstream second messenger. First, the SNR can be improved with a decrease of τ_R, meaning that faster transitions between the ligand-binding (on) state and ligand-unbinding (off) state of the receptors can produce chemotactic signals with higher SNR (Figure 5A). When ligand concentration is increased, the ligand association rate (k_onL) to the receptor is accelerated, resulting in a decrease in the time constant, τ_R (Eq. (10)). That is, an increase in ligand concentration results in better efficiency of chemotactic signals not only by increasing the average concentration of the active receptor but also by decreasing the characteristic time of fluctuations of the active receptor. Moreover, signal improvements are possible by increasing the on-rate (k_on) and/or off-rate (k_off). For example, when the potential barrier between the on-state and off-state of the receptor becomes lower, the cycling between the two states is accelerated by the acceleration of the on-rate (k_on) and the off-rate (k_off) resulting in a decrease in τ_R and, hence, improvement of the SNR of chemotactic signals. Ueda et al. 19 reported polarity in receptor kinetic states along the length of chemotactic cells, which suggests that the SNR is higher at the pseudopod region than at the tail region. Such polarity in the SNR of chemotactic signals may provide a basis for the polarity observed in the response of Dictyostelium cells 40.

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Figure 5

Signal improvements. (A) Receptor fluctuation-dependent signal improvements. The dissociation rates of a ligand ( k_off) were changed: (blue line) 0.1s⁻¹; (green line) 1s⁻¹ (standard condition); (red line) 3s⁻¹. (B) Time-averaging effects. The SNR was improved by increasing the time constants of the second messenger. The degradation rates (k_d) were changed and the corresponding SNR was calculated: (blue line) 10s⁻¹; (green line) 1s⁻¹; (red line) 0.1s⁻¹. (C) Dependency of the SNR on the expression levels of receptors. The receptor numbers per single cell are 16,000 (blue), 80,000 (green), and 400,000 (red) molecules/cell. (D) Effects of affinity modulation on the SNR. Ligand-binding affinity: 18nM (red), 180nM (green), and 1800nM (blue).

Second, the SNR can be improved by increasing τ_X. Longer lifetime of the second messenger, which corresponds to slower degradation, causes noise reduction more effectively through time-averaging of the extrinsic noise (Figure 5B). This means that the regulatory mechanism for the degradation or the inactivation of second messengers has a pivotal role on the signal improvements for chemotaxis. In the case of G-protein, the hydrolysis rates of the bound GTP on the α-subunit and the reassociation rates with the βγ-subunit mostly determine the lifetimes of active G-proteins, and thus affect the SNR of chemotactic signals. This suggests that the GTPase-activating protein, such as regulators of G-protein signaling (RGS), can regulate the quality of the signal by modulating the inactivation rates of G-protein.

Third, an increase in gain g_X can contribute to improved SNR. A high gain that is larger than unity can be obtained for reactions with some cooperativity or ultrasensitivity 31. Eq. (4) can be applied for such reactions. When the cells use cooperative or ultrasensitive reactions for second-messenger activation, chemotactic signals can be improved through reduction of the intrinsic noise.

Fourth, the SNR depends on the total amount of receptor expressed in cells (Figure 5C). The SNR is improved in the lower and higher concentration ranges of chemoattractant by increasing and decreasing receptor number, respectively. Receptor internalization can contribute to SNR improvements in the higher concentration ranges by decreasing membrane-bound receptors. Also, receptor affinity for the chemoattractant is an important factor in adjusting the concentration ranges in chemotaxis (Figure 5D). Modification in the affinity causes a shift in the dependence of SNR on chemoattractant concentration, which would be a basis for a wider-range response. It is well known that the cAMP receptors in Dictyostelium cells are phosphorylated with cAMP stimulation, leading to a three- to approximately sixfold decrease in ligand-binding affinity 41. According to our formula, such an affinity shift of receptors contributes to an SNR increase in the higher-ligand-concentration range, and thus extends the response range to higher ligand concentrations.

Our discussion on the minimum model of chemotactic signaling cascade can be generalized for a longer cascade including multistep reactions. In such a case, extrinsic noise would be amplified by the gain or reduced by time-averaging effects at each step. The gain depends on the type of reaction 31 and also on the concentration ranges of the reaction (e.g., Figure 3AB). Time-averaging of the extrinsic noise depends on the time constants of the reaction at each step, which are usually determined by both the production and the degradation rates of the messenger molecules. Intrinsic noise would be added inevitably to the extrinsic noise at each step. When the time constants of the reactions become longer along the signaling cascade, the SNR of chemotactic signals would have a more improved effect at the lower reactions through time-averaging effects. In such a signaling system, the shallow gradient can be detected at downstream reactions of the cascade even if it does not generate effectively a clear signal at upstream reactions, suggesting that the downstream molecules have a pivotal role in the detection of a faint signal. Dictyostelium cells treated with a PI3-kinase inhibitor can exhibit chemotaxis, but it is restricted to the higher concentration range 42, suggesting that PI3-kinase and the PTEN system are required for detection of a faint signal in a noisy environment. Similar reasoning can be applied to parallel cascades with different time constants. Thus, our model can evaluate the quality of signals in the chemotactic signaling system, which can be further applied to stochastically operating signaling systems in general. To reveal how signal and noise are propagated in a stochastic signaling system, it is important to determine experimentally the gains and time constants of reactions along the signaling cascade. Noise propagation along longer signaling cascades will be discussed elsewhere.

Appendix

Derivation of snr of chemotactic signals

We define chemotactic signals as the difference in the concentration of X^*, ΔX^*, between the anterior and posterior halves of chemotactic cells (Figure 2B). Since we are interested in chemotaxis in a shallow gradient, the difference ΔL is small enough so that we only consider linear terms with respect to the differences. The average differences, and between the anterior and posterior halves are defined as

(5)

where and are the average numbers of active receptor and the active second messenger, respectively, in the cell. The noise of the differences in active receptor, is approximately equal to the summation of the noise in the anterior and posterior regions, given by where the subscripts “anterior” and “posterior” indicate the regions of the cell. From Eq. (1), we find

(6)

Similarly, the noise of chemotactic signals, is approximated by Since the concentration gradient of ligand is so small, we may expect that the gain, g_X, and the formed gradient of the active receptor are almost constant

(7)

From Eqs. (5) and (7), the relative noise strength of chemotactic signals, is obtained in Eq. (4). The SNR of chemotactic signals is the square root of the inverse of the relative noise strength.

To calculate the parameters in Eq. (4) for Dictyostelium cells, we consider the simplest reaction scheme:

According to this scheme, the average number of active receptor, and second messenger, are given by Michaelis-Menten kinetics,

(8)

where R_total is the total molecular number of receptors per single cell, the affinity for the ligand with association and dissociation rate constants k_on and k_off, X_total the total molecular number of second messenger per cell, the concentration of active receptor where the activation of the second messenger reaches half-maximum with production and degradation rates k_p and k_d of the second messenger. The gains of active receptor to the ligand concentration and the second messenger to the active receptor number are given by

(9)

The time constants of the reactions are calculated by

(10)

When Eqs. (5), (6), (8), and (9) are substituted into Eq. (4), the SNR is obtained as a function of L and ΔL by

(11)

Here, is the ligand concentration with which the activation of X reaches the half-maximum value, and According to Eq. (11), the SNR is proportional to when ligand concentration is much smaller than K_R and K_XR. Supposing that cells can sense the gradient if the SNR of chemotactic signals is larger than the threshold SNR, with constant C, and hence we have (SNR_threshold/C) for chemotaxis. Therefore, the minimum differences of ligand concentration for chemotaxis are proportional to the square root of L, ΔL_threshold ∝

We should note that Eq. (1) may have an additional noise derived from the fluctuation of the ligand concentration in extracellular solution, which can be given by

(12)

where D is the diffusion constant of the ligand and Φ is the cell size 23,43. Using D ≈ 10³μm²/s for cAMP and the other parameter values shown in Table 1 we find indicating that the second term in Eq. (12), which is derived from the ligand concentration fluctuation in extracellular solution, is negligible.