A simple two state model of adaptive response

To study the essential properties of fluctuation in an adaptive system and its underlying mechanism, we here study a simple two state model that responds and adapts to a change in the environmental conditions. In the present simplified two-state model, each molecule is in one of the two states, active and inactive, between which transition reactions take place. The rates of the transition reactions are dependent on the environmental ligand concentration. When the environmental ligand concentration is changed, the equilibrium between the two states is broken, leading to a transient increase or decrease in the number of active molecules, . Such cases are often observed in sensory systems, such as receptors, which are activated upon binding or unbinding of ligands. For instance, in the case of bacterial chemotaxis, the activation probability of the chemoreceptor decreases as the increase of chemoattractant concentration. In the case of chemotactic cells Dictyostelium, the activation probability of the G-protein coupled receptor cAR1 (cAMP receptor) increases with the cAMP concentration.

Adaptation occurs when modification of molecules can also affect the equilibrium between the active and inactive states. After a transient response to a change in stimulus, the modification or demodification reactions occur, which shift the equilibrium between two states to compensate for the transient response. As a result, the number of active molecules, , exhibits an adaptation. The adaptation can be perfect, when the rates of modification and demodification reactions are determined by alone. Here, for simplicity, we consider a single modification step. Thus, each molecule is either modified or unmodified. The equilibrium constant between active and inactive states is dependent on these modification states. In the extreme case, the unmodified state, denoted by , is always inactive. The modified state consists of an active state A and inactive state , between which transition reactions can take place. There are many possible kinetic schemes for the adaptive response. Here, we consider the following simple kinetic scheme (Fig. 1):(1)where , , and are the number of molecules of A, and , respectively, is the ligand concentration, and are the rate constants of the activation and deactivation reactions between A and in the modified state, which are dependent on the ligand concentration , and and are the rate constants of the modification and demodification reactions, respectively. Here, for the modification and demodification reactions, we consider the enzymatic reaction described by the Michaelis-Menten equation, given by and , respectively. We note that this mechanism requires that the system works far from thermodynamic equilibrium.

At steady state, since the modification and demodification reactions are balanced, we obtain from scheme 1. To achieve a perfect adaptation of activity , the modification reaction should perform at a saturating level, , giving [2]. As a result, the rate of modification and its reverse reactions are dependent solely on the activity . Under such a condition, we obtain the equation at steady state. Notice that no parameter dependent on is included, indicating that the steady state level of obtained as a solution of this equation is independent of and the model exhibits a perfect adaptation against changes in .

Since the total concentration is conserved and the modification reactions are working under the saturation condition , the reaction scheme can be reduced to(2)We consider the stochastic kinetics described by the chemical Langevin equation [21], given by(3)The last three terms in each equation are noise terms because of the stochastic occurrence of the reactions. Here, is white Gaussian noise with zero mean and .

Stochastic fluctuations of activity in adaptive systems is not adaptive

To study whether the property of stochastic fluctuation in the activity is adaptive or not, we calculate the variance of by solving the chemical Langevin equation shown in Eq. (3), adopting the linear noise approximation. In a steady state, the fluctuation intensity is given approximately by(4)with , , and (see Materials and Methods for details). Here, is a constant, and is regarded as a measure of the non-first order (nonlinear) degree of the demodification reaction, given by, . Thus, from Eq.4 the fluctuation intensity can be dependent on the absolute concentration through , and it is a nonadaptive property of adaptive sensory systems. Since reactions specific to a particular system are restricted to the form of and , this nonadaptive property in the stochastic fluctuation is considered as a property common to the class of models considered here.

What property of our model makes the fluctuation of activity dependent on the ligand concentration ? When the demodification reaction performs as a first order reaction, is unity. In such a case, according to Eq. (4), the fluctuation becomes and is independent of . When the inactivation reaction is not present, vanishes and the fluctuation becomes , which is also insensitive to . Therefore, to have nonadaptive fluctuation these two depletion pathways for the active form A, both the inactivation and demodification reactions, are necessary, where at least one of them should be a non-first order reaction. As we shall see later, the major reaction to deplete A changes from the demodification reaction to the inactivation one with the increase of .

Nonadaptive fluctuations in reduced activity dependent kinetics

When the transition between active and inactive states is faster than the modification and demodification reactions as is often supposed, the present model can be further simplified. In the case of a bacterial chemoreceptor, the methylation and demethylation reactions are usually supposed to be much slower than the activation and inactivation reactions. In the case of ion channels, the transition between conducting and non-conducting states involves a conformational change, which is expected to be much slower than the transition between open and closed states. The internalization of receptors is also a slow process compared with the activation and inactivation reactions. In the present model, the modified state M, which consists of active state A and inactive modified state , is produced and depleted according to the following reaction scheme,(5)When the state transition between A and is much faster than the modification and demodification reactions, it can be regarded as in equilibrium. Then the number of A is determined by the reaction,(6)Thus, the number of is expected to follow the binomial distribution conditional on a given number of the molecules in modified state M, . Thus, the present two state model can be described by the chemical Langevin equation for modified level , given by(7)with(8)where is the Gaussian white noise with and , and is a random number of the normal distribution with zero mean and the variance given by . We notice that Eqs.(7) and (8) are considered an extension of the system level approach to an adaptive system proposed in Ref. [22] to include the effect of stochastic fluctuations.

By solving the chemical Langevin equation (7) with the linear noise approximation, we obtain the variance of as (see Materials and Methods for details). Thus, by adding the variance of the binomial distribution, the variance of the activity , , is given by(9)Since and are dependent on the ligand concentration , the relative noise intensity is dependent on the absolute ligand concentration , indicating again that it is a nonadaptive property of adaptive sensory systems. We note that the expression in Eq.(9) can be obtained from Eq. (4) by assuming that the activation reaction is much faster than the demodification reaction, i.e., .

Stochastic fluctuation is not an adaptive property of bacterial chemoreceptor system

In the case of bacterial chemoreceptors, the modification and demodification reactions are performed by methyltransferases CheR and CheB, respectively. The unmethylated state is usually assumed to be inactive [23]. For simplicity, we consider first the case with a single methylation step. For the methylated receptors, the forms of activation and inactivation reaction rates and in scheme 2 are chosen so as to satisfy bacterial chemotaxis where the tumbling rate decreases (increases) when the ligand concentration increases (decreases). Thus, and are respectively decreasing and increasing functions of . We adopt simplest forms given by , and , where, and are the maximum velocities, is the dissociation constant of the ligand, and and are constants with and . Even under the no chemoattractant condition, the activation and inactivation rates are respectively given by and , which makes and able to reach equilibrium even without the methylation and demethylation reactions. The steady state level of is obtained as , which is independent of showing a perfect adaptation.

We performed a stochastic simulation of scheme 2 for the case of bacterial chemoreceptor as shown in Fig. 3A (see Materials and Methods for the detail of the simulation method). The time course shows that the increase (decrease) of results in a transient decrease (increase) of activity and thus the tumbling frequency. After the transient response, the stochastic time course of exhibits a perfect adaptation (Fig. 3A) over a range of more than six orders of ligand concentration (Fig. 3B). The most probable value of perfectly adapts to the background ligand concentration. The ensemble average of , , deviates slightly from the stationary value under low background, because the distribution of is skewed to the right (Fig. 3C inset). The modification level increases with the increase of , which is consistently observed in the experiments of bacterial chemoreceptor reported previously [23].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. Stochastic property of the two-state bacterial sensory model.

A. Stochastic response and adaptation obtained by scheme 2. The activity is plotted as a function of time. For step increase and decrease in ligand concentration (blue line), the activity shows the response and adaptation. The time series obtained by the corresponding kinetic equation without the noise term is shown by the green line. B. Adaptation of activity . The ensemble average of , , and the most probable value of , obtained by the stochastic simulation are plotted as functions of the ligand concentration. The activity obtained by the kinetic equation without the noise term (green line) is also depicted. The deviation between and is due to the nonlinearity of the demodification reaction. The theoretical result including the nonlinear effect is also shown (red line, see Materials and Methods). C. The dependence of activity fluctuation on the ligand concentration. The relative fluctuations of activity , , obtained by stochastic simulation are plotted as functions of chemoattractant concentration for scheme 2(), without the inactivation reaction with , and without the nonlinear effect in which . The respective theoretical lines are also plotted. Inset: The distributions of activity for several ligand concentrations. D. The effect of the activation and inactivation reaction rates, and , on the nonadaptive fluctuation. The fluctuation strength is calculated according to Eq. (4) for each and . The other parameter values are the same as indicated in Materials and Methods. E. The effect of the ligand independent inactivation reaction, , on the nonadaptive fluctuation. For each value of as indicated in the figure, the fluctuation strength is calculated according to Eq. (4).

https://doi.org/10.1371/journal.pone.0011224.g003

Fig. 3C shows the relative fluctuation , which is not a constant but a decreasing function of . As decreases to zero, approaches a saturation level, while is decreasing to a lower bound level with increasing . Thus, while the mean level of activity is an adaptive property, the stochastic fluctuation is a nonadaptive property sensitive to the ligand concentration. Therefore, the stochastic activity can still bear information of the chemoattractant ligand concentration.

In Fig. 3C, the theoretical result of stochastic fluctuation in the two state model given by Eq.(4) is applied to the case of bacteria, exhibiting a good agreement with the numerical result. For this case, is an increasing function of . According to Eq. (4), when the ligand concentration is sufficiently small, is given by . As increases, decreases approximately in proportion to . Then, as increases further, approaches .

Such a decrease of the fluctuation with the increase of is due to the shift of effective depletion pathway of A from the demodification reaction to the inactivation reaction. As the ligand concentration decreases to , the rate of the inactivation reaction is reduced. Since in such a case the time constant of , given by , is much smaller than that of , given by , the fluctuation in is effectively averaged out and has no significant effect on the fluctuation of . Thus, can be replaced by its average. Therefore, the reaction of the active state A is effectively reduced to be,(10)which consists of the production reaction with the constant reaction rate and the depletion reaction with rate . According to Ref. [16], the fluctuation strength of A at is given by , where is the gain of for the increase of the rate constant , i.e., . For scheme 10, the gain is given by the nonlinear degree , i.e. . Thus, the fluctuation strength can be rewritten as , showing that the large fluctuation at is due to the large gain , which is a result of the demethylation reaction with nonlinear rate . This scheme indicates the effective pathways with the strongest flux under the wild type condition. We should note that this reduced scheme does not necessarily mean that the chemoreceptor does not undergo reversible transitions between active and inactive states and not obey the detailed balance in the CheB and CheR deleted mutant cells without methylation and demethylation reactions.

For sufficiently large , both the activation and inactivation reaction rates, and , are much larger than the demethylation reaction rate , i.e., . Thus, in this range of , the demethylation reaction can be neglected. The number increases as the increase of , since the modification level increases as mentioned before. This indicates that for sufficiently large the fluctuation of relative to its mean can be neglected due to its large concentration. It follows that the activation reaction rate is effectively constant. Therefore, the reaction of active state A is reduced to be:(11)where the distribution of follows a Poisson distribution, which gives . We also note that since the gain is unity for this scheme, the fluctuation intensity is given by . Thus, the decrease of the nonlinear demodification reaction rate relative to the activation and inactivation reaction rates is essential for the decrease of the fluctuation intensity of as the ligand concentration increases.

We should notice that the present result of nonadaptive fluctuation does not depend strongly on the several parameter values. As shown in Fig. 3D, even when the maximum rates and of activation and inativation reactions were increased or decreased ten times, the dependence of fluctuation on the ligand concentration was almost unchanged. Therefore, our result is applicable to the case when the transitions between active and inactive states are not fast processes and are considered to be away from equilibrium.

Note that the transition rates should be fast enough, otherwise cannot decrease when the ligand concentration is high (Fig. 3D red line). We also studied the dependence of the nonadaptive property in the activity fluctuation on the ligand-independent inactivation reaction, which is characterized by in . Such a ligand-independent inactivation reaction is expected for the case of bacterial chemotaxis. To obtain sufficiently strong response and to satisfy the bacterial chemotaxis (see below), should be much smaller than in . In the present case, we set . Thus, we have . As shown in Fig. 3E, the nonadaptive property in the activity fluctuation is not strongly dependent on the parameter .

In the present case, the variance is a decreasing function of since is an increasing function. This property of is required to satisfy the bacterial chemotaxis where the tumbling rate decreases when the ligand concentration increases. If is a decreasing function of , which is expected for a chemorepellent [8], the variance can be an increasing function. Therefore, the behavioral fluctuation of bacterial chemotaxis is dependent on the properties of the biochemical network.

The same result can be obtained in the detailed bacterial chemoreceptor model, in which multiple methylation sites are considered [24] (Fig. 2A). As shown in Fig. 2B, the stochastic time course of activity exhibits adaptive responses to the steplike changes in the chemoattractant concentration. However, the fluctuation intensity shown in Fig. 2C indicates clearly its ligand dependence, which is a decreasing function of the ligand concentration as is obtained in the simple two state model. Moreover, the parameter dependence of the fluctuation property is essentially the same in both models. From Eq. 4, the increase of results in a decrease of the fluctuation intensity when as shown in Fig. 4A. Since is the maximum reaction rate constant of the demodification reaction, it is equivalent to the increase of CheB concentration. Fig. 4B shows the ligand dependence of the fluctuation intensity in the detailed model. One finding, that an increase of CheB concentration results in a decrease of the ligand dependence and the reduction of the fluctuation intensity, shows good agreement with our theoretical result. Note that a five-fold increase of CheB concentration is sufficient for the fluctuation intensity to approach the Poissonian fluctuation, . Therefore, our analysis extracts an essential feature of adaptive sensory systems, irrespective of the detailed aspects of the chemoreceptor reactions.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 4. The fluctuation strength of the activity in our simple two-state model and the detailed model shown in Fig. 2.

A: In our simple two-state model, the fluctuation strength of the activity and its ligand dependence decrease with the increase of . B: In the detailed model, an increase of CheB concentration results in a decrease of the fluctuation intensity of the activity and its ligand dependence.

https://doi.org/10.1371/journal.pone.0011224.g004

Behavioral variability is dependent on ligand concentration

The above result indicates that bacterial behavior can exhibit a ligand dependence in its behavioral fluctuation, which is compatible with the property of adaptation. We study first the dependence of bacterial behavior on the chemoattractant concentration under spatially homogeneous conditions.

Following the previous studies [25], [26], we extended our model to include motile machinery, the flagellar motor, which stochastically switches between the two states, “run” and “tumble” with transition rates from run to tumble and from tumble to run (see Materials and Methods). Here, denotes the deviation of from the steady state value, and , , and are constant parameters. Thus, the transition rates can be time-dependent. If the fluctuation of activity is small enough to be ignored, the transition rates and are constants and the run and tumble durations follow exponential distributions. This simple model can reproduce the switching behavior of the mutant cell, which expresses the constitutive active form of CheY [18]. For this mutant cell, the switching rates of rotation were decoupled from the receptor activity, showing that CCW duration, which is equivalent to the “run” state, was exponentially distributed. However, large fluctuations generated in the adaptation reaction can propagate to the motile machinery with ultrasensitivity [27], where it is amplified. Such fluctuations may affect the run and tumble duration distributions. In fact, when the chemoattractant is absent, the run duration exhibits a heavy-tailed distribution, where the probability to have a longer run duration is not bounded by an exponential distribution (Fig. 5 ) [18], [26]. Correlated with the decrease in the fluctuation of activity as the ligand concentration increases (Fig. 3C), such behavioral fluctuation is reduced and approaches an exponential distribution at high chemoattractant concentration (Fig. 5 and ). Thus, the run duration is changed from a heavy-tailed distribution to an exponential one as the chemoattractant concentration increases.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 5. Probability distribution of run duration under the uniform background ligand concentrations.

Run durations are measured for . Inset: Logarithmic view of the identical plot.

https://doi.org/10.1371/journal.pone.0011224.g005

The distribution of durations of counterclockwise (CCW) and clockwise (CW) rotations has been measured in the absence of chemoattractant [18]. The CCW and CW rotations correspond to run and tumble, respectively. The duration of CCW rotation was found to obey the heavy-tailed distribution, whereas CW duration was distributed exponentially. This experiment suggested that the temporal fluctuation generated at the chemoreceptor propagates to the motor, leading to the run duration being distorted from an exponential distribution. We also note that the earlier experiment by Block et al. demonstrated that in the presence of chemoattractant, the CCW duration was distributed exponentially [10]. The amount of ligand in their experiment was comparable to the dissociation constant of chemoreceptor. Our result above could consistently explain the apparent discrepancy between the two experiments, by considering the dependences of fluctuation on the ligand concentration.

Chemotactic performance can be enhanced by fluctuations

Such dependence of the run length distribution on the ligand concentration would enable the bacterial motility to depend on the chemoattractant level, even though the sensory system exhibits the property of perfect adaptation.

In particular, the heavy-tailed distribution of run length could give rise to a motility spreading in an area larger than the motion of an ordinary random walk. Thus, higher mobility would be expected when the environmental ligand concentration is low. The mobility of bacteria in a uniform chemoattractant concentration can be characterized by the mean square displacement (MSD) (Fig. 6), calculated as , where is the position at time . The MSD is the variance of the distribution of bacteria at time starting from the same position at , which increases linearly for sufficiently long time scales as , where is the effective diffusion constant. The result shows that is larger in the absence of ligand and decreases as increases. Such a dependence of motility on the ligand concentration is a consequence of the fluctuation in the adaptation reaction that is dependent on the chemoattractant concentration. Considering a “noiseless cell” in which the activity does not contain intrinsic fluctuations, the effective diffusion constant of such a noiseless cell is constant without dependence on . The high motility of the wild-type cells in a low concentration regime is also seen in the directional persistence of cell migration, , defined as the ratio between the net displacement equivalent to and the total length of the motional trajectory in an interval . The persistence is a decreasing function of time . As shown in Fig. 6 inset, the more ligand concentration increases, the faster persistence falls with time. Therefore, for low ligand concentrations, the bacteria can spread into a wider area within a short time scale.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 6. Mean Square Displacements for the bacterial motility.

MSDs are plotted as functions of time for . At the long time scale, , bacterial motion is regarded as a random walk, with linear dependence on time. Inset: Persistence of the bacterial motion plotted as functions of time . The persistence is unity for a short time, indicating that the motion is ballistic, then, it decays in proportion to . The broken line is proportional to .

https://doi.org/10.1371/journal.pone.0011224.g006

Under the chemoattractant gradient a bacterium exhibits a directional motion, which can be quantified by the net velocity for relatively short periods of time [28], [29] (see Materials and Methods). shows a linear dependence on the steepness of chemoattractant gradient (Fig. 7A). For low background concentration, the velocity of the wild-type cell is larger than that of the noiseless cell, , indicating that the fluctuation improves the chemotactic performance. For a high background concentration, is smaller than . The ratio shown in Fig. 7B indicates that the enhancement of chemotactic performance is prominent when the background concentration is low and the gradient is shallow. While the fluctuation in the sensory apparatus may disturb the ability of gradient sensing as noise, our result reveals the opposite role in chemotaxis. Particularly in the low concentration regime, bacteria search for a chemoattractant in a wider area. Once they reach a shallow gradient, they climb up quickly.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 7. Short term velocity along the chemical gradient.

A. Relationship between short term velocity and the gradient. Open circles represent for the wild-type, and closed circles for the noiseless cell, . is measured for . B. , the normalized velocity of by , is plotted against the , the ligand concentration at the starting point.

https://doi.org/10.1371/journal.pone.0011224.g007

To clarify the reason for this increase, we investigated several mutants. We first studied mutant I, which has the modification and demodification rates, and , that are 10 times faster than those rates of the wild-type cell. The activity of this mutant has a correlation time of fluctuation that is faster than that of the wild-type cell. In mutant I, the fluctuation intensities of the switching rates of the motility, and , become smaller, resulting in the disappearance of the tail in the run length distribution despite the large activity fluctuation. To study the effect of the stochastic fluctuation of activity, we also investigated the noiseless cell of the mutant I. In Fig. 8A, we plot the ratio between the velocity of the mutant I and the velocity of its noiseless cell, i.e., . The ratio is less than unity, indicating that the performance is not improved by the stochastic fluctuation of in mutant I. Next we studied another mutant II, in which the inactivation pathway is deleted (see Fig. 3C green). Mutant II shows a large fluctuation and a heavy-tailed distribution of run length irrespective of the environmental ligand concentration. We also investigated the corresponding noiseless mutant. As shown in Fig. 8B, the ratio between the velocities of mutant II, , is larger than that of the wild-type cell for any background concentration of unless the ligand gradient is quite steep . These results indicate that the increase in the velocity is mainly the consequence of the heavy-tailed distribution of run length.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 8. The ratio between the velocities of mutant I (A) and mutant II (B) and their noiseless cells plotted as functions of ligand concentration .

https://doi.org/10.1371/journal.pone.0011224.g008

Bacteria spread with time, even from a source of chemoattractant, because of the biased random walk. To quantify the degree of spreading with time, we measured the temporal change of for the bacterial population put on the tip of the exponential gradient for various value of steepness and concentrations at the tip. To compare of the wild-type, , with that of the noiseless cell, , we introduce their time averaged ratio . As shown in Fig. 9B, is a decreasing function of the background concentration. Therefore, whereas the fluctuation of the adaptation reaction enhances the spread of bacteria from an area of low chemoattractant concentration, in a high concentration area the spreading to weaken aggregation is not increased.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 9. Mobility from the chemoattractant source.

A: The mean square displacement, , in the presence of the chemical gradient, . are shown by solid lines, and by broken lines. The individual bacterium starts to swim in a one dimensional field from the higher end of the gradient, . B: Time averaged ratio against the background concentration for various value of steepness. MSDs are averaged for , which is long enough to capture the bacterial long term behavior.

https://doi.org/10.1371/journal.pone.0011224.g009

The above results indicate that the stochastic fluctuation of the sensory system does not reduce the chemotactic performance in most situations. In particular, under the low background concentration, the large fluctuation of the sensory system leads to increasing the cell motility and chemotactic speed. We should note that to study the increase of the chemotactic performance, the cell with stochastic sensory system (wild type) was compared with the cell with the sensory system without stochasticity (noseless cell). The time constants and the response sensitivity are the same between these two types of cells. Thus, the increase in the chemotactic performance is purely the consequence of the effect of noise, but is not the effect of the increase in time constant as in the case of Emonet and Cluzel [19]. When reaching a high concentration area, bacteria suppress spreading by decreasing the stochastic fluctuations of the chemoreceptor circuit. Consequently, a bacterial population achieves higher aggregation performance toward the chemoattractant by switching its behavior depending on the chemoattractant concentration.

Conclusion

In the present paper, we studied the stochastic nature of the adaptive sensory systems, such as ion channels, and membrane receptors. While the activity shows adaptation on average, its temporal fluctuation is a nonadaptive property, which is sensitive to the environmental ligand concentration. The ligand dependence is revealed when there exist two depletion pathway of the activity, one of which exhibits a non-first order property (Fig. 10). Since our analysis is performed in a simple prototypic model, the nonadaptive fluctuation is a property common to the adaptive systems studied here. In the present paper, we further analyzed the bacterial chemoreceptor, which is the best studied adaptive sensory system. We have shown that the nonadaptive fluctuation influences the motile property through the switching reaction of the flagellar motor, resulting in the behavioral fluctuation being dependent on the background ligand concentration. The ligand dependence of bacterial behavior influences cell motility under uniform environment, which increases the chemotactic performance. Therefore, the nonadaptive fluctuation can carry information of the environmental ligand concentration. By altering the behavior depending on the fluctuation intensity of the sensory system, the cell can adaptively change its behavior to suit the environmental conditions. Our result indicates a possible function of stochastic fluctuation in that it can transmit information downstream, even though this cannot be done by its average.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 10. Schematic diagram of the adaptation motif with nonadaptive fluctuations.

The motif consists of a set of chemical species. In the figure, only molecule A is shown. All the reaction paths are drawn in thick line. The output is the number of molecules of A, which is activity of the motif. The input signal modulates the activity through the activation and inactivation reaction rates. The influx is the supply reaction for the motif, whereas the efflux is the depletion reactions from it. The efflux reaction is typically the enzymatic reaction, which generate a larger fluctuation. When the efflux is dependent only on the activity, and both the influx and efflux are independent of other molecules, the activity exhibits a perfect adaptation at steady state. For the nonadaptive fluctuation, at least two depletion pathways are necessary for molecule A. One pathway in the motif should be dependent on the input, and one must be a nonlinear reaction.

https://doi.org/10.1371/journal.pone.0011224.g010

Derivation of fluctuation strength in the two state model

To derive Eq. (4), we consider the chemical Langevin equation for scheme 2, given by Eq. (3). The concentrations and can be written as(12)(13)where and are the steady state solution without noise, and and are the time dependent deviations about and . To obtain the variance of , we performed the linear noise approximation, which gives the equation,(14)where , is the linear regression matrix given by(15)and is the white Gaussian noise with zero mean and with(16)Using the relation with the covariance matrix , we obtain(17)Introducing new variables, and , we obtain Eq. (4).

Fluctuation strength in the reduced activity dependent kinetic model

The linearized equation of Eq. (7) with linear noise approximation is(18)where, . Under the equilibrium condition between and , the fluctuation strength of , , is given,(19)With Eq. (8), we have Eq. (9).

Simulation of the receptor sensory reactions

The stochastic simulation of scheme 2 is performed by τ-leap algorithm [30], which approximately simulates the stochastic dynamics of the chemical reactions in discrete time steps. For the numerical simulation of noiseless cell, we calculated the ordinary differential equations, where the noise terms are omitted in Eq. (3), with the Euler method.

The response and adaptation times of the sensory system of bacterial chemotaxis are respectively and for a small step increase of chemoattractant [8]. We determined the reaction parameters to reproduce these time constants. In the model given by scheme 2, the response time and adaptation time are approximately given by(20)(21)In the case of , and , we obtained and for the low background , which is consistent with the above time constants.

The effect of nonlinearity on average activity

We notice that the statistical average is larger than the solution of the kinetic equation without the noise term, , particularly for a low background ligand concentration, as shown in Fig. 3B. The deviation is also found in the fluctuation intensity as shown in Fig. 3C, where the estimated variance given in Eq. (4) is smaller than the variance obtained by numerical simulations. These deviations are due to the nonlinearity of the demodification reaction. From Eq. (12), the statistical average of at steady state is given by with . The average can be obtained by solving the equation,(22)By performing a Taylor expansion at up to the second order of , we have(23)Note that increases in proportion to the increase of . Thus, we expand and as a series of as and . By equating the terms in each power of in Eq. (23), we have , and . Thus, the steady state value up to the zeroth order of is given by,(24)which shows good agreement with the numerical result shown in Fig. 3B.

Model of bacterial motility

The motion of a bacterium consists of “run” and “tumble”. Between these two states, we consider stochastic transitions as(25)where and are the activity-dependent switching rates as shown in the text [25], [26]. The parameter values that we used in the simulation were , , , and . The combination of and determines the dependence of the fraction of CW state in time on the activity . The parameter values of and were chosen to reproduce the reported experimental result [27]. The value of was chosen to have an exponential distribution for the tumbling duration even in the absence of ligand, as was reported in Ref. [18]. For each state of “run” and “tumble” motions, bacteria are considered to show a rotational Brownian motion with respective constant speeds, and . The direction of motion in a two dimensional space, , follows a stochastic differential equation given by , where is a white Gaussian noise with and , and is the strength of noise with indicating the state of motion, i.e., “r” and “t” for “run” and “tumble” motions, respectively. In the present paper, we use , , , and . For the stochastic simulation of bacterial motility, we used the τ-leap algorithm for the receptor reactions and motor switching, and the Euler-Maruyama method for the bacterial movement in the same discrete time step.

Measurement of chemotactic performance

The short term velocity, , under the chemoattractant gradient [28] is given by(26)where represents the mean duration of successive run and tumble intervals for upward motion, starting at a given position and for downward motion. Multiplying by the swimming velocity , represents the averaged upward velocity along the chemical gradient between two tumbling motions. is measured in the presence of the linear gradient, , where , and denote the chemical gradient, the position of a bacterium, and the ligand concentration at starting point, . At , cells are adapted to the ligand concentration, , and start to move along or against the gradient.

To obtain in the presence of the chemical gradient, the bacterial population is placed at the top of the gradient. As the initial condition, the receptor activity reaches the steady state at the concentration of the top of the gradient. is calculated from the bacterial population, which consists of cells. There exists a reflective wall at , prohibiting the bacterium from going across the boundary . This boundary condition is equivalent to the ligand distribution spreading exponentially on both positive and negative sides of .