Relevance of intracellular polarity to accuracy of eukaryotic chemotaxis

We consider a spontaneously maintained internal polarity in the intracellular process (figure 1(b)). The cells migrate in the direction of the internal polarity, which is subject to change by intra- and extra-cellular perturbations (figure 1(c)). The intra-cellular perturbation originates from stochastic variation of signal reactions and other intracellular processes, which introduces randomness to the polarity direction, leading to random cell migration. For the extra-cellular perturbation, we consider the bias of chemoattractant gradients, and the random variation due to the stochastic binding between receptors and chemoattractant ligands. Cells sense the gradient through the inference process of external gradients that exerts the effective bias on the direction of motion. Here, we denote the bias by a non-dimensional parameter A (figure 1(c)). For simplicity, we assume that the magnitude of the internal polarity and the migration speed are constant.

The time-evolution of the polarity direction is given by [20, 23, 24, 30]

$$\begin{eqnarray}&&\frac{{\rm d}}{{\rm d}t}{{\theta }_{q}}(t)=-S{\rm sin} {{\theta }_{q}}(t)+\xi (t)\ ,\end{eqnarray} \tag{ 1 }$$

where ${{\theta }_{q}}(t)$ is the angle between the directions of the internal polarity and chemical gradient at the time t. The first term on the right hand side of equation (1) describes the deterministic driving force which orients the internal polarity into the gradient direction. The coefficient S indicates the strength of the driving force, and ${\rm sin} {{\theta }_{q}}(t)$ is the first term in the Fourier series that satisfies the symmetry with respect to ${{\theta }_{q}}(t)\to {{\theta }_{q}}(t)+2\pi$ and ${{\theta }_{q}}(t)\to -{{\theta }_{q}}(t)$. The second term $\xi (t)$ describes the stochastic variation in the intra- and extra-cellular perturbations; $\xi (t)={{\xi }^{\operatorname{int}.}}(t)+{{\xi }^{{\rm ext}.}}(t)$. The terms ${{\xi }^{\operatorname{int}.}}(t)$ and ${{\xi }^{{\rm ext}.}}(t)$ are given as the white Gaussian noises satisfying $\langle {{\xi }^{*}}(t)\rangle =0$ and $\langle {{\xi }^{*}}(t^{\prime} ){{\xi }^{*}}(t)\rangle =2{{D}^{*}}\delta (t-t^{\prime} )$ with the magnitudes D^* (rad² s⁻¹) of the dispersion. Here, the superscript $^{*}$ is $^{\operatorname{int}.}$ or $^{{\rm ext}.}$. The extra-cellular noise term ${{\xi }^{{\rm ext}.}}$ describes the stochastic variation in the number of ligand–receptor complex. Thus, its dispersion ${{D}^{{\rm ext}.}}$ depends on the chemoattractant concentration. The intra-cellular noise term ${{\xi }^{\operatorname{int}.}}$ describes the fluctuation in the chemotactic signal reactions. For simplicity we consider that the dispersion ${{D}^{\operatorname{int}.}}$ is independent of the chemoattractant concentration. The total dispersion is given by the sum of these dispersions as $D={{D}^{\operatorname{int}.}}+{{D}^{{\rm ext}.}}$. The intrinsic property of the cell, ${{D}^{\operatorname{int}.}}$, is independent from the extracellular perturbation, S and ${{D}^{{\rm ext}.}}$. Dynamics and statistics of the direction θ_q of the internal polarity and migration are determined by all these properties as a consequence of equation (1). In particular, the persistence of the direction θ_q depends on both ${{D}^{\operatorname{int}.}}$ and ${{D}^{{\rm ext}.}}$ as briefly explained in section 1. More rigorous formulation of equation (1) and the relation between the coefficients and physical parameters are shown in appendix A.1.

In [20, 23], equation (1) is derived by the symmetry argument as in the present paper. In [21, 22], the dependence of the driving force S in equation (1) on the extracellular chemoattractant concentration was taken into account. In these works [20–23], the dispersion of noise term $\xi (t)$ is characterized by a phenomenological parameter independent of the chemoattractant concentration. Here, for the noise term, we explicitly consider the noise contribution in the stochastic process between ligand and receptor. As a result, the dependence on the chemoattractant concentration naturally appears. We introduce this dependence with an intuitive consideration in section 3.1 and with the mathematical derivation in appendix A.1.

D. discoideum AX2 cells (wild type) were starved by suspension in development buffer (DB: 5 mM Na phosphate buffer, 2 mM MgSO₄, 0.2 mM CaCl₂, pH 6.3) for 1 h and were then pulsed with 10 nM cAMP at 6 min intervals for up to 3.5 h at 21 ${}^\circ$C, resulting in elongated cells with chemotactic competency [35]. The prepared cells were settled on a glass dish (IWAKI) at cell density $1.0\times {{10}^{5}}$ cell ml⁻¹ (to preclude explicit cell–cell interactions), containing DB supplemented with a given concentration of cAMP and 2 mM dithiothreitol (DTT, Sigma) was added to inhibit phosphodiesterase in order to keep the cAMP concentration in the medium. Cells were incubated under these conditions for 20 min. Phase-contrast imaging was performed using an inverted microscope (TiE, Nikon) with a 40× phase-contrast objective, equipped with an EMCCD camera (iXon+, Andor). Images of cells were taken every second for 30 min. To obtain the trajectories of cell area centroids, the cell periphery of individual cells was detected by enhancing the image contrast using the function histeq in Matlab 7.6 (mathworks) and then thresholded. All trajectories analyzed were longer than 5 min. The cell velocity ${{{\boldsymbol{v}} }_{n}}$ in frame n was obtained as ${{{\boldsymbol{v}} }_{n}}={{{\boldsymbol{r}} }_{n+1}}-{{{\boldsymbol{r}} }_{n}}$, where ${{{\boldsymbol{r}} }_{n}}$ denotes the centroid trajectory. The migration direction θ_n at frame n was defined as ${{{\boldsymbol{v}} }_{n}}={{v}_{n}}({\rm cos} {{\theta }_{n}},{\rm sin} {{\theta }_{n}})$, where v_n is the speed. The temporal autocorrelation function of the migration direction, C(n), given by, $C(n)=\langle {\rm cos} ({{\theta }_{n}}-{{\theta }_{0}})\rangle$ was calculated for individual cells. Then, the average autocorrelation functions were obtained by taking the averages over ensembles, as shown in figure 2(a) (red points). The autocorrelation functions (red points) exhibits delta-function like behavior at n = 0, indicating the presence of a time scale much shorter than 1 s and/or an uncorrelated noise due to the measurement error including image processing. Because the cell velocity ${{{\boldsymbol{v}} }_{n}}$ was obtained by one-step difference of centroid positions, the velocities in subsequent two steps exhibit anti-correlation. Thus, the sudden drops in the values at n = 1 are artifacts of data analysis. Therefore, we considered the autocorrelation function from n = 2 to 200 s to determine the characteristic time. Then, equation (2) was used for the fitting using the function lsqcurvefit of Matlab 7.6 (mathworks). The fitted line was indicated by black lines in figure 2(a). The numbers of cells for individual conditions were shown in the legend of figure 2.

We obtained the trajectories of cell centroids at 1 s intervals for longer than 5 min up to 30 min. The velocity vector of the area centroid of cells, ${\boldsymbol{v}} (t)=v({\rm cos} \theta (t),{\rm sin} \theta (t))$, was determined, from which we calculated the temporal autocorrelation function C(t) of the migration direction $\theta (t)$, given by $C(t)=\langle {\rm cos} (\theta (t)-\theta (0))\rangle$ as above. figure 2(a) plots C(t) at different cAMP concentrations. The correlation function C(t) can be fitted by a sum of two exponential functions given by

$$\begin{eqnarray}&&C(t)={{C}_{1}}{{{\rm e}}^{-t/{{\tau }_{1}}}}+{{C}_{2}}{{{\rm e}}^{-t/{{\tau }_{2}}}}\ .\end{eqnarray} \tag{ 2 }$$

The time constants τ₁ and τ₂ $({{\tau }_{1}}\lt {{\tau }_{2}})$ obtained by the curve fitting showed a dependence on cAMP concentration. The time constant τ₁ is within ranges of about 10–20 s and τ₂ is in several hundreds of seconds. The shorter time constant τ₁ is considered to characterize rapid deformations in the cell shape, whereas the longer one τ₂ quantifies the persistence time in the migration direction. Since the time scale of the persistence is the focus of this paper, we plot τ₂ as the correlation time in figure 2(b). We note that the correlation times of the centroid velocity ${\boldsymbol{v}} (t)$ and the direction $\theta (t)$ essentially correspond with each other (figure 2(b)). This indicates that the constant speed approximation used in equation (A.3) is valid in this time scale.

We first consider cell migration under uniform conditions without gradients of chemoattractant. In this case, the time-evolution of polarity direction θ_q is given by equation (1) with S = 0. In the absence of extra-cellular perturbation, the total noise strength is $D={{D}^{\operatorname{int}.}}$. When the cell is exposed to a spatially uniform chemoattractant, the stochastic binding of receptors and chemoattractant ligands is another noise contribution, ${{D}^{{\rm ext}.}}$. The information of the binding states at a particular time decays exponentially with time constant ${{\tau }_{R}}$. Here, ${{\tau }_{{\rm R}}}$ is the correlation time of the receptor state, given by [24]

$$\begin{eqnarray}&&{{\tau }_{{\rm R}}}({{C}_{0}})=\frac{1}{{{k}_{{\rm d}}}+{{k}_{{\rm a}}}{{C}_{0}}}=\frac{1}{{{k}_{{\rm d}}}(1+K_{{\rm d}}^{-1}{{C}_{0}})}\ ,\end{eqnarray} \tag{ 3 }$$

where k_d and k_a are the dissociation and association rates, respectively, C₀ is the chemoattractant concentration, and ${{K}_{{\rm d}}}={{k}_{{\rm d}}}/{{k}_{{\rm a}}}$ is the dissociation constant. The strength of noise applied during time interval ${{\tau }_{{\rm R}}}$ is proportional to ${{f}_{q}}{{\tau }_{{\rm R}}}$, where f_q is the responsiveness that characterizes the response to the extra-cellular perturbation. The number of independent such noise per unit time is estimated as $N\sim 1/{{\tau }_{{\rm R}}}$. Thus, the dispersion of the extracellular-noise contribution ${{D}^{{\rm ext}.}}$ through receptors is estimated by taking the average over N independent noises as ${{D}^{{\rm ext}.}}\sim {{({{f}_{q}}{{\tau }_{{\rm R}}}N)}^{2}}/N=f_{q}^{2}{{\tau }_{{\rm R}}}$. By a detailed calculation of ${{D}^{{\rm ext}.}}$ shown in appendix (equations (A.7), (A.9) and (A.10)), we have the similar expression with the numerical factor $1/2$ as

$$\begin{eqnarray}&&{{D}^{{\rm ext}.}}=\frac{f_{q}^{2}{{\tau }_{{\rm R}}}}{2}\ .\end{eqnarray} \tag{ 4 }$$

The direction of cell migration is determined by the direction of internal polarity, which shows stochastic fluctuation as a result of intra- and extra-cellular perturbations, as explained above. With the increase in total dispersion, $D={{D}^{{\rm ext}.}}+{{D}^{\operatorname{int}.}}$, the correlation time τ_c of the migration direction decreases. Considering that the angular diffusion constant D has an inverse time unit, the correlation time τ_c is given by ${{\tau }_{{\rm c}}}={{D}^{-1}}$. (For a detailed derivation of this relation, see section appendix A.3). Thus, from equations (3) and (4), the correlation time τ_c is given by

$$\begin{eqnarray}\begin{array}{rcl} {{\tau }_{{\rm c}}}({{C}_{0}}) & = & {{\left( {{D}^{{\rm ext}.}}+{{D}^{\operatorname{int}.}} \right)}^{-1}} \\ {} & = & {{\left( \frac{f_{q}^{2}}{2{{k}_{{\rm d}}}\left( 1+K_{{\rm d}}^{-1}{{C}_{0}} \right)}+{{D}^{\operatorname{int}.}} \right)}^{-1}}\ . \\ \end{array}\end{eqnarray} \tag{ 5 }$$

This equality indicates that the motile behavior of cells depends upon chemoattractant concentration, even if the chemoattractant is homogeneously distributed in space. When the chemoattractant concentration C₀ is much smaller than the dissociation constant K_d, ${{C}_{0}}\ll {{K}_{{\rm d}}}$, the correlation time ${{\tau }_{{\rm c}}}({{C}_{0}})$ is approximately constant with respect to C₀. When the concentration is comparable to or larger than the dissociation constant, i.e., ${{C}_{0}}\gtrsim {{K}_{{\rm d}}}$, but ${{C}_{0}}\lesssim {{K}_{{\rm d}}}\;f_{0}^{2}{{\left( 2{{k}_{{\rm d}}}{{D}^{\operatorname{int}.}} \right)}^{-1}}$, equation (5) indicates that ${{\tau }_{{\rm c}}}({{C}_{0}})$ increases with C₀ (figure 2(b), blue solid line). When ${{C}_{0}}\gg {{K}_{d}}\;f_{0}^{2}{{(2{{k}_{{\rm d}}}{{D}^{\operatorname{int}.}})}^{-1}}$, ${{\tau }_{{\rm c}}}({{C}_{0}})$ is approximately constant and given by ${{\tau }_{{\rm c}}}={{({{D}^{\operatorname{int}.}})}^{-1}}$. Altogether, our theory indicates that the correlation time τ_c shows an increase with the concentration C₀ at ${{C}_{0}}\sim {{K}_{{\rm d}}}$. By fitting equation (5) to the measured concentration dependence of the correlation time, and knowing the values of the receptor parameters, we can estimate the values of parameters f_q and ${{D}^{\operatorname{int}.}}$. To perform this, since we focus on the time scale of the persistence here, we compare τ₂ with the correlation time τ_c in equation (5).

In the presence of chemoattractant gradients, cells perform gradient sensing, and orient their internal polarity in the direction of the gradient. The quality of the gradient sensing may depend on the stochasticity in the receptor states. When the noise increases or decreases relative to the signal strength, the accuracy of gradient sensing may decrease or increase, respectively [36, 37]. Thus, the effective gradient information, quantified by the bias A, may decrease if the stochastic noise increases. The bias A also depends on the chemoattractant concentration and the steepness of the gradient. The dependence of A on the gradient information and ligand–receptor reaction can be explicitly derived by considering the gradient sensing process (see equation (A.25) for detail).

The response of cells to gradient perturbations should depend on the responsiveness f_q . As f_q increases, the chemoattractant gradient should exert greater effect. Thus, the driving force S exerted on the internal polarity depends on both the bias and the responsiveness, which may be expressed as

$$\begin{eqnarray}&&S=\frac{{{f}_{q}}A}{2}.\end{eqnarray} \tag{ 6 }$$

A more detailed derivation is given in appendix A.1.

Fisher et al [4] specifies the accuracy κ by the sharpness of the distribution $P({{\theta }_{v}})$ of migration directions θ_v , defined by $P({{\theta }_{v}})\propto {\rm exp} (\kappa {\rm cos} {{\theta }_{v}})$. The CI is ${\rm CI}=\langle {\rm cos} {{\theta }_{v}}\rangle$ with the statistical average $\langle \cdot \rangle$ according to the distribution $P({{\theta }_{v}})$. When the chemotactic accuracy κ is small, the CI is roughly $\kappa /2$ [4]. κ is also expected to be proportional to the driving force S. Furthermore, κ should improve as the stochastic perturbations ${{D}^{{\rm ext}.}}+{{D}^{\operatorname{int}.}}$ decrease, or conversely, as the correlation time τ_c increases. These conditions are satisfied if κ takes the form

$$\begin{eqnarray}&&\kappa ={{\tau }_{{\rm c}}}S=\frac{A}{{{f}_{q}}k_{{\rm d}}^{-1}{{(1+K_{{\rm d}}^{-1}{{C}_{0}})}^{-1}}+2f_{q}^{-1}{{D}^{\operatorname{int}.}}}\ .\end{eqnarray} \tag{ 7 }$$

Details are provided in appendix A.4.

To see the significance of the internal polarity on the chemotactic accuracy, we consider the amplification ratio r given by the ratio between the chemotactic accuracy κ with and κ without internal polarity. Here, the accuracy κ without internal polarity is given by the bias A. (We can see this fact by comparing (A.4) and (A.17) in appendix for small A.) This ratio $r=\kappa /A$ quantifies the extent to which the internal polarity enhances the chemotactic accuracy. From equation (7), the amplification ratio r is obtained as

$$\begin{eqnarray}&&r=\frac{1}{{{f}_{q}}{{\tau }_{{\rm R}}}+2f_{q}^{-1}{{D}^{\operatorname{int}.}}}\ ,\end{eqnarray} \tag{ 8 }$$

which depends on both ${{D}^{\operatorname{int}.}}$ and C₀ through ${{\tau }_{{\rm R}}}$. In figure 3(a), the amplification ratio r is plotted as a function of f_q and ${{D}^{\operatorname{int}.}}$ in the low concentration limit ${{C}_{0}}\to 0$, i.e. ${{\tau }_{{\rm R}}}=1$ s. To remove the dependences on ${{D}^{\operatorname{int}.}}$ and C₀, we study the scaled amplification ratio $\hat{r}$ given by

$$\begin{eqnarray}&&\hat{r}\equiv \frac{r}{\sqrt{1/(8{{\tau }_{{\rm R}}}{{D}^{\operatorname{int}.}})}}=\frac{2}{\widehat{{{f}_{q}}}+{{\widehat{{{f}_{q}}}}^{-1}}}\ ,\end{eqnarray} \tag{ 9 }$$

which depends only on the scaled responsiveness $\widehat{{{f}_{q}}}={{f}_{q}}/\sqrt{2{{D}^{\operatorname{int}.}}/{{\tau }_{{\rm R}}}}$. Figure 3(b) plots the rescaled amplification ratio $\hat{r}$ as a function of the rescaled responsiveness $\widehat{{{f}_{q}}}$. From this figure, we observe that the amplification is maximized ($\hat{r}=1$) at $\widehat{{{f}_{q}}}=1$. These results suggest that, for a given value of ${{D}^{\operatorname{int}.}}$, there exists an optimum responsiveness f_q that maximizes the chemotactic accuracy. This maximum accuracy is achieved when the external and intrinsic perturbations are balanced. If the responsiveness f_q is small, the correlation time τ_c in equation (5) is almost independent of f_q , because the dispersion is dominated by the intrinsic stochastic perturbation ${{D}^{\operatorname{int}.}}$. Simultaneously, the external driving force S in equation (6) is strengthened as f_q increases. Therefore, for small responsiveness f_q , the accuracy κ increases proportionally to the responsiveness f_q . Conversely, if the responsiveness f_q is large, the correlation time τ_c is dominated by extrinsic stochastic perturbations as ${{\tau }_{{\rm c}}}\propto f_{q}^{-2}$ in equation (5). In this case, the accuracy κ in equation (7) decreases in proportion to $f_{q}^{-1}$.

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In this subsection, we study the influence of internal polarity on chemotaxis based on the values of the parameters f_q and ${{D}^{\operatorname{int}.}}$ of Dictyostelium cells. To determine these values, we investigated the dependence of the correlation time of the migration direction on the extracellular concentration of the chemoattractant cAMP. We tracked the migration of single Dictyostelium cells in uniform cAMP concentrations (see section 2.2 for detail). In figure 2(b), the relevant time constant τ₂ obtained by the curve fitting were plotted against cAMP concentration. As predicted in our theory given by equation (5), the experimental data showed the increase of ${{\tau }_{{\rm c}}}({{C}_{0}})$ for increasing C₀ at ${{C}_{0}}\sim {{K}_{{\rm d}}}$. Then, to obtain the values of f_q and ${{D}^{\operatorname{int}.}}$, we fitted equation (5) to the time constants τ₂ with the known receptor constants k_d and K_d. As shown by the solid curve in figure 2(b), the dependence of τ₂ on the cAMP concentration were well fitted by equation (5), with fitting parameters

$$\begin{eqnarray}&&{{f}_{q}}\sim 0.0539\ {\rm rad}\;{{{\rm s}}^{-1}}\end{eqnarray} \tag{ 10 }$$

and

$$\begin{eqnarray}&&{{D}^{\operatorname{int}.}}\sim 0.00163\ {\rm ra}{{{\rm d}}^{2}}\;{{{\rm s}}^{-1}}\ .\end{eqnarray} \tag{ 11 }$$

Note that although the time constant at 10 nM exhibited a deviation from the tendency in those of the other concentrations, we included this point for the data fitting. Whether this deviation at 10 nM is due to some experimental problem or a limitation of our theory remains to be clarified for future problem.

Using the parameter values specified by equations (10) and (11), we first study the stationary distribution of the migration direction. As an example, we consider a particular steepness of chemical gradient that induces the effective bias on the polarity direction with the value A = 0.1. In figure 4(a), the distributions of migration direction are plotted for A = 0.1 in the absence (blue broken line) and presence (red solid line) of internal polarity (equation (A.4) and equation (A.17), respectively). The migration direction is distributed more sharply around the gradient direction when the internal polarity is present.

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We next study the dependence of chemotactic accuracy κ and chemotaxis index ${\rm CI}$ on the chemoattractant concentration C₀. As shown in figure 4(b), both the accuracy and dynamic range of chemotaxis, given by equation (7) (red solid line), are larger in the presence of internal polarity than those in the absence of the polarity (blue broken line). Together, these results indicate that internal polarity contributes not only to chemotactic accuracy but also to the dynamic range of chemotaxis. Figure 4(c) shows the dependence of the ${\rm CI}$ on the concentration C₀. The stochastic signal processing and transduction during chemotactic signaling has been investigated theoretically [36] and experimentally [37]. These previous studies showed that the relationship between chemotactic accuracy and extracellular cAMP concentration mirrors the signal-to-noise ratio at the level of the second messenger [36, 37]. Therefore, here, we also take into account the intracellular noise in the second messenger reaction to the bias A (see equations (A.25) – (A.27) in appendix for details), and investigate its effect. The resultant chemotactic accuracy κ and ${\rm CI}$ as functions of cAMP concentration are shown in figure 4(d) and 4(e) (red solid lines), respectively. Those peak concentrations are lower than the case without the second messenger reaction shown in figure 4(b), (c). Again, both the accuracy and dynamic range of chemotaxis are improved by internal polarity. Comparing figure 4(d) with figure 4(b), we find that the accuracy κ in the high concentration region ${{C}_{0}}\gtrsim {{K}_{{\rm d}}}$ is reduced by stochasticity in the second messenger reaction. The slope at the higher concentration ${{C}_{0}}\gt {{K}_{{\rm d}}}$ becomes steeper than that at ${{C}_{0}}\lt {{K}_{{\rm d}}}$ as shown in figure 4(d)(red solid lines), which agrees with the experimental result by Fisher et al [4]. Thus, the present theoretical result reproduces the ligand concentration of the highest chemotactic accuracy and the dynamic range of chemotaxis as shown in the Fisherʼs result [4] by taking into account the second messenger reaction and adjusting the parameter values.

The effect of internal polarity on chemotaxis can be characterized by the amplification ratio r introduced in equation (8). The ratio r depends on the values of f_q and ${{D}^{\operatorname{int}.}}$ as shown in figure 3(a). For the values of f_q and ${{D}^{\operatorname{int}.}}$ of Dictyostelium cells, indicated by the red circle in figure 3(a), the amplification factor is $r\sim 8.74$. The scaled amplification ratio $\hat{r}$ given by equation (9) depends only on the scaled responsiveness $\widehat{{{f}_{q}}}$, as explained above. The scaled responsiveness of Dictyostelium cells is estimated as $\widehat{{{f}_{q}}}\sim 0.997$, 0.705, and 0.301 for ${{C}_{0}}\to 0$, ${{C}_{0}}={{K}_{{\rm d}}}=180\;{\rm nM}$, and ${{C}_{0}}=10\times {{K}_{{\rm d}}}=1800\;{\rm nM}$, respectively (see figure 3(b), broken lines). For ${{C}_{0}}\lesssim {{K}_{{\rm d}}}$, the responsiveness is distributed around the maximum amplification ratio, indicating that the responsiveness parameter f_q of Dictyostelium cells is almost optimal in this concentration range.

3.4.1. Chemotaxis ability of mutant cells

3.4.2. Other contributions relevant to chemotactic accuracy

Several factors that may reduce chemotactic accuracy have been excluded from the model. The experimental accuracy of chemotaxis obtained by Fisher et al [4] is smaller than that obtained in our theory (see figure 4(d)), indicating that some of these factors are significant.

To obtain the maximum feasible accuracy, we assumed that all $80\;000$ receptors are located at the periphery of the cell. However, in reality, the receptors in the vicinity of the vertical point cannot contribute to determining the migration direction. This effect reduces the effective number of receptors and hence decreases the chemotactic accuracy. In the same way, when the number X of second messengers is reduced, the accuracy also decreases. As an extreme example, we consider the cases when N = 250 for the receptors (green dotted lines) and X = 1250 for the second messengers (purple dashed–dotted lines) as shown in figures 4(d), (e). In both cases, the accuracy is predicted to be decreased. We showed more detailed dependence of the accuracy κ and index ${\rm CI}$ on N and X in figure A.2 in Appendix.

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Spatial and temporal stochastic variations in the distribution of receptors and intracellular signaling molecules may also affect the dispersion in the chemotactic accuracy. For example, we have neglected the correlations in the spatial density fluctuations of receptors and the activated second messengers at different times and positions along the cell membrane (see appendix A.5 for details). However, these spatiotemporal stochastic fluctuations and correlations are expected because the receptors and activated second messengers can diffuse. In addition, the activation time is also subject to stochastic delays. An intriguing future problem is to theoretically investigate the consequences of spatial and temporal correlations on the density fluctuations of activated second messengers. Such studies might elucidate how cells overcome these noises.

We also postulated that the migration direction θ_v adiabatically follows the polarity direction θ_q on the time scale of interest. However, θ_v may deviate from θ_q over time scales exceeding the correlation time of polarity direction. These deviations, which would further contribute to fluctuation in the motile direction, are observed in cell deformation processes. Previously, we have reported that cell deformation can alter the gradient sensing ability, with subsequent effect on the probability distribution of chemotactic migration directions [29, 30]. Therefore, the temporal fluctuations observed in Dictyostelium cells may also affect the dispersion of motile directions.

3.4.3. Cell polarity and gradient-sensing response

In this paper, we developed a theory of chemotactic cell migration that demonstrates the connection between cell intrinsic polarity and chemotactic accuracy. The time-evolution of the direction of the polarity is described by a stochastic differential equation (1), which consists of the deterministic driving force due to the guidance cue and stochastic noise due to both intra- and extra-cellular perturbations. We connected their coefficients with the physical quantities on chemical environment and signal processing by examining the statistics of chemo-ligand and receptors as a source of external perturbation. Based on this theory, firstly, we demonstrated that the direction of polarity persists over a correlation time that is predicted to increase as the chemoattractant concentration is increased (equation (5)). Next, we theoretically derived the chemotactic accuracy as a function of both the bias due to the guidance cue and the correlation time of polarity direction (equation (7)). The results indicated that the accuracy can be improved by the presence of intracellular polarity. Furthermore, the analysis of chemotactic accuracy suggests that accuracy is maximized at some optimal responsiveness to extracellular perturbations (equation (9)). To obtain the model parameters, we studied the correlation time of random cell migration in cell tracking analysis of Dictyostelium cells. The persistence time depended on the chemoattractant concentration as predicted. From the fitted parameters (10) and (11), we inferred that polarized Dictyostelium cells can respond optimally to a chemical gradient as shown in figure 3(b). Chemotactic accuracy was almost nine times larger than can be achieved by non-polarized gradient sensing. Using the obtained parameter values, we show that polarity also improves the dynamic range of chemotaxis. We conclude that cell polarity established spontaneously in the absence of external guidance cues plays a pivotal role in efficient chemotaxis. This conclusion is consistent with the recent numerical study based on a detailed mathematical model [45].

The present model, which incorporates internal polarity and gradient sensing, is applicable to several different cases, including chemotaxis with cell shape deformation. In particular, slight deviations of the motile direction distribution from the normal circular distribution can be explained by the present model, provided that the process of cell deformation and the influence of cell shape on the gradient sensing are included [30]. The current model can be used to study chemotaxis towards time-varying gradients, such as chemoattractant waves. By introducing cell–cell interactions into the model, the collective chemotaxis of cell populations could also be investigated. The connection between microscopic models based on signal reaction kinetics [13, 14, 32, 33] and macroscopic models, such as our present theory, is an intriguing future topic.

We consider the spontaneous formation of the internal polarity ${\boldsymbol{q}} =({{q}_{x}},{{q}_{y}})$, incorporating gradient sensing and internal and external perturbations. The simplest evolution equation that describes such internal polarity ${\boldsymbol{q}} =({{q}_{x}},{{q}_{y}})$ is given by [30]

$$\begin{eqnarray}&&\frac{{\rm d}}{{\rm d}t}{\boldsymbol{q}} ={{I}_{q}}(1-|{\boldsymbol{q}} {{|}^{2}}){\boldsymbol{q}} +{{{\boldsymbol{ \Xi }} }^{\operatorname{int}.}}(t)+{{{\boldsymbol{ \Xi }} }^{{\rm ext}.}}(t)\ .\end{eqnarray} \tag{ A.1 }$$

The first term on the right hand side describes the formation of polarity. In particular, by the first part ${{I}_{q}}{\boldsymbol{q}}$ of this term, the isotropy of ${\boldsymbol{q}}$ is spontaneously broken. The same formulation has been proposed for an active Brownian particle with energy supply [47, 48]. The other part $-{{I}_{q}}|{\boldsymbol{q}} {{|}^{2}}{\boldsymbol{q}}$ is the lowest order term for preventing the divergence of ${\boldsymbol{q}}$ [46]. In this article, we assume ${{I}_{q}}\to \infty$ for further simplicity, so that the internal polarity ${\boldsymbol{q}}$ is reduced to ${\boldsymbol{q}} (t)=({\rm cos} {{\theta }_{q}}(t),{\rm sin} {{\theta }_{q}}(t))$ with $|{\boldsymbol{q}} |=1$. The second term ${{{\boldsymbol{ \Xi }} }^{\operatorname{int}.}}(t)$ describes the internal noise, assumed as Gaussian white noise with $\langle {{{\boldsymbol{ \Xi }} }^{\operatorname{int}.}}\rangle =0$, and

$$\begin{eqnarray}&&\langle \Xi _{i}^{\operatorname{int}.}(t)\Xi _{j}^{\operatorname{int}.}(t^{\prime} )\rangle =2{{D}^{\operatorname{int}.}}{{\delta }_{ij}}\delta (t-t^{\prime} ).\end{eqnarray} \tag{ A.2 }$$

The last term ${{{\boldsymbol{ \Xi }} }^{{\rm ext}.}}(t)\equiv {{f}_{q}}{\boldsymbol{e}} (t)$ describes the gradient sensing with responsiveness f_q . The unit vector ${\boldsymbol{e}} ({\boldsymbol{t}} )$ specifies the inferred direction of the external gradient, $\phi (t)$, i.e. ${\boldsymbol{e}} (t)=({\rm sin} \phi (t),{\rm cos} \phi (t))$. Here, we consider that the cell estimates the extracellular gradient direction from the distribution of its chemoattractant-occupied receptors or activated second messengers (see appendix A.5). In this paper, we consider cell motions in the time scale longer than the persistence time of cell migration. On these time scales, the velocity ${\boldsymbol{v}}$ of the cell immediately follows the internal polarity as

$$\begin{eqnarray}&&{\boldsymbol{v}} ={{v}_{0}}{\boldsymbol{q}} \ ,\end{eqnarray} \tag{ A.3 }$$

where the migration speed v₀ is assumed constant as previously reported [24].

For a given spatial distribution of chemoattractant-occupied receptors, we consider the most probable estimate of the gradient direction, ϕ. The simplest probability distribution $P(\phi )$ of the estimated direction for cells with circular shape is given by [29]

$$\begin{eqnarray}&&P(\phi )=\frac{1}{2\pi }+\frac{A}{2\pi }{\rm cos} \phi \ ,\end{eqnarray} \tag{ A.4 }$$

where the true gradient direction is $\tilde{\phi }=0$. Here, A is the bias strength, which can be derived by considering the gradient sensing process, as shown in equation (A.25) of appendix A.5. Equation (A.4) is the first order approximation of the circular-normal distribution with respect to the bias strength A. In the absence of internal polarity, the CI gained from directional inference alone is given as

$$\begin{eqnarray}&&{\rm CI}=\int _{0}^{2\pi }{\rm d}\phi {\rm cos} \phi P(\phi )=A/2.\end{eqnarray} \tag{ A.5 }$$

The characteristic time ${{\tau }_{{\rm R}}}$ in equation (3) is much smaller than the persistence time of 300 s [8] in the migrating direction. Thus, we assume that the characteristic time τ_c , during which the polarity direction persists, is also much longer than the correlation time ${{\tau }_{{\rm R}}}$ of the estimated direction. We consider the integral of the driving force over time interval $\Delta t\gg {{\tau }_{{\rm R}}}$, ${\boldsymbol{ \Delta }} {{W}^{{\rm ext}.}}=\int _{t}^{t+\Delta t}{{{\boldsymbol{ \Xi }} }^{{\rm ext}.}}(s){\rm d}s$. Then, by the central limit theorem, ${\boldsymbol{ \Delta }} {{W}^{{\rm ext}.}}$ follows the Gaussian distribution. The average and mean square displacement of ${\boldsymbol{ \Delta }} {{W}^{{\rm ext}.}}$ are given by

$$\begin{eqnarray}&&\langle {\boldsymbol{ \Delta }} {{W}^{{\rm ext}.}}\rangle =\left( 0,\frac{{{f}_{q}}A}{2}\Delta t \right)\end{eqnarray} \tag{ A.6 }$$

and

$$\begin{eqnarray}&&\left\langle {{\left| {\boldsymbol{ \Delta }} {{W}^{{\rm ext}.}} \right|}^{2}} \right\rangle =2f_{q}^{2}{{\tau }_{{\rm R}}}\Delta t.\end{eqnarray} \tag{ A.7 }$$

Equations (A.6) and (A.7) are derived from the second term on the right hand side of equation (A.4) and the first term up to O(A), respectively. The driving force S in equation (6) is given by the y-component of $\langle {\boldsymbol{ \Delta }} {{W}^{{\rm ext}.}}\rangle$ in equation (A.6) divided by $\Delta t$. Next, consider the change in the direction of internal polarity within $\Delta t$, $\Delta {{\theta }_{q}}={{\theta }_{q}}(t+\Delta t)-{{\theta }_{q}}(t)$. From equation (A.1) with ${{I}_{q}}\to \infty$, equations (A.2), (A.6), and (A.7), the moments of $\Delta {{\theta }_{q}}$ up to $O(\Delta t)$ are given by

$$\begin{eqnarray}&&\langle \Delta {{\theta }_{q}}\rangle =-\frac{1}{2}{{f}_{q}}A{\rm sin} {{\theta }_{q}}(t)\Delta t\end{eqnarray} \tag{ A.8 }$$

$$\begin{eqnarray}&&\langle {{(\Delta {{\theta }_{q}})}^{2}}\rangle =(f_{q}^{2}{{\tau }_{R}}+2{{D}^{\operatorname{int}.}})\Delta t\ .\end{eqnarray} \tag{ A.9 }$$

Here, since $\Delta {{\theta }_{q}}$ is the projection of the 2D vector $\Delta {\boldsymbol{q}}$ into 1D, the numerical factor 2 in equation (A.7) disappears in equation (A.9). In this way we obtain equation (1) and, using the Kramers–Moyal expansion with equations (A.8) and (A.9) the Fokker–Planck equation

$$\begin{eqnarray}&&\frac{\partial }{\partial t}P({{\theta }_{q}},t)=\frac{\partial }{\partial {{\theta }_{q}}}\left[ {{c}_{q}}({{\theta }_{q}})P({{\theta }_{q}},t) \right]+D\frac{{{\partial }^{2}}}{\partial \theta _{q}^{2}}P({{\theta }_{q}},t)\ ,\end{eqnarray} \tag{ A.10 }$$

where ${{c}_{q}}({{\theta }_{q}})=({{f}_{q}}A{\rm sin} {{\theta }_{q}})/2$, and D is the diffusion constant $D={{D}^{{\rm ext}.}}+{{D}^{\operatorname{int}.}}$, which consists of two independent dispersions ${{D}^{{\rm ext}.}}=f_{q}^{2}{{\tau }_{{\rm R}}}/2$ and ${{D}^{\operatorname{int}.}}$, respectively denoting the diffusion strengths introduced by external and internal perturbations.

Note that, in the main text, we began with equation (1), in which the directionality of the deterministic driving term is given by ${\rm sin} {{\theta }_{q}}$ and the noise dispersion D is independent from the polarity direction θ_q . This equation is justified with the central limit theorem when the gradient strength A is sufficiently small, or up to O(A), as shown in equations (A.6) and (A.7). This approximation does not depend on particular gradient sensing mechanisms of cells excepting the case in which the second order term ${\rm cos} (2\phi )$ is included in equation (A.4) up to O(A). In this exception, the noise dispersion ${{D}^{{\rm ext}.}}$ depends on the polarity direction θ_q [30].

Here, we consider the contribution of the receptor noise to the downstream signals. Consider a linear signaling cascade, in which the activity of each reaction step is modulated by the one-upstream step. Then, the stochastic temporal evolutions of the small deviations η_i from their stationary average at each step are described by the following linearized Langevin equations [49]

$$\begin{eqnarray}&&{{\tau }_{i}}\frac{{\rm d}}{{\rm d}t}{{\eta }_{i}}=-{{\eta }_{i}}+{{\eta }_{i-1}}+{{\xi }_{i}}(t)\ ,\end{eqnarray} \tag{ A.11 }$$

where τ_i is the time constant (given by the inverse depletion rate) for each reaction step, and the dimensionless quantity η_i includes signal amplification effects. The noise in the activated receptor is given by η₀, with $\langle {{\eta }_{0}}(t^{\prime} ){{\eta }_{0}}(t)\rangle =\sigma _{0}^{2}{\rm exp} (-|t^{\prime} -t|/{{\tau }_{{\rm R}}})$. The last term ξ_i denotes the noise in the ith signal transduction.

From equation (A.11), the autocorrelation function is given by

$$\begin{eqnarray}&&\langle {{\eta }_{n}}(t^{\prime} ){{\eta }_{n}}(t)\rangle ={{\langle {{\eta }_{n}}(t^{\prime} ){{\eta }_{n}}(t)\rangle }_{0}}+\Delta \ ,\end{eqnarray} \tag{ A.12 }$$

where the first term on the right hand side denotes the contribution of the noise in the receptor signal, η₀, to the noise at the nth step of the signal cascade, given by

$$\begin{eqnarray}&&{{\left\langle {{\eta }_{n}}(t^{\prime} ){{\eta }_{n}}(t) \right\rangle }_{0}}=\sigma _{0}^{2}{{\Lambda }_{n}}(t^{\prime} -t)\ .\end{eqnarray} \tag{ A.13 }$$

The function ${{\Lambda }_{n}}(\Delta t)$ describes the time-dependence of the receptor noise in ${{\eta }_{n}}(t)$

$$\begin{eqnarray}&&{{\Lambda }_{n}}(\Delta t)=\mathop{\sum }\limits_{i=0}^{n}\frac{{{\tau }_{{\rm R}}}}{{{\tau }_{i}}}\left( \mathop{\prod }\limits_{j=0,j\ne i}^{n}\frac{\tau _{i}^{2}}{\tau _{i}^{2}-\tau _{j}^{2}} \right){{{\rm e}}^{-\left| \Delta t \right|/{{\tau }_{i}}}}\end{eqnarray} \tag{ A.14 }$$

with ${{\tau }_{0}}\equiv {{\tau }_{{\rm R}}}$. The second noise term Δ on the right hand side of equation (A.12) is the contribution of the various noises in the signal cascade downstream of the receptor, ξ_i . Since Δ is included in ${{D}^{\operatorname{int}.}}$ but not in ${{D}^{{\rm ext}.}}$, we focus on the receptor noise given by equation (A.13).

In Dictyostelium cells, the receptor time constant ${{\tau }_{{\rm R}}}$ is about 1 s, while the time constants τ_i of cascade reactions $(i=1,2,\ldots ,n)$ are comparable to or less than several seconds [50, 51]. On the other hand, the persistence time of the migration direction is about ${{\tau }_{c}}\sim 300$ s. Because the time constants ${{\tau }_{i}}\quad (i=0,1,2,\ldots n)$ are much smaller than τ_c, we can replace ${{{\rm e}}^{-|\Delta t|/{{\tau }_{i}}}}$ by $2{{\tau }_{i}}\delta (\Delta t)$. Thus, we obtain⁸

$$\begin{eqnarray}&&{{\Lambda }_{n}}(\Delta t)=2{{\tau }_{{\rm R}}}\delta (\Delta t)\sim {{{\rm e}}^{-|\Delta t|/{{\tau }_{{\rm R}}}}}\ ,\end{eqnarray} \tag{ A.15 }$$

and hence ${{\langle {{\eta }_{n}}(t^{\prime} ){{\eta }_{n}}(t)\rangle }_{0}}=\langle {{\eta }_{0}}(t^{\prime} ){{\eta }_{0}}(t)\rangle$. Therefore, the time constant ${{\tau }_{{\rm R}}}(={{\tau }_{0}})$ can be used in equations (4) and (A.7). Intuitively, this argument implies that the time constant ${{\tau }_{{\rm R}}}$, considered as the interval of signaling events at the receptor, is not affected by the time delay between a particular signaling event and the resulting directional changes along the signaling cascades (figure A.1).

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The persistence of migration direction is characterized by the autocorrelation function of the migration direction. In the presence of a uniform chemoattractant with ${{C}_{0}}\gt 0$ and A = 0, the autocorrelation function of ${\boldsymbol{q}} (t)$, $C(t)=\langle {\boldsymbol{q}} (t)\cdot {\boldsymbol{q}} (0)\rangle =\langle {\rm cos} ({{\theta }_{q}}(t)-{{\theta }_{q}}(0))\rangle$ is obtained from equation (A.10) with A = 0 as

$$\begin{eqnarray}\begin{array}{rcl} C(t) & = & \int _{0}^{2\pi }{\rm cos} {{\theta }_{q}}P({{\theta }_{q}},t|{{\theta }_{q}}=0,t=0){\rm d}{{\theta }_{q}} \\ {} & = & {\rm exp} \left( -\frac{|t|}{{{D}^{-1}}} \right)\ , \\ \end{array}\end{eqnarray} \tag{ A.16 }$$

where the diffusion constant $D={{D}^{{\rm ext}.}}+{{D}^{\operatorname{int}.}}$. In this manner, we obtain the correlation time τ_c in equation (5).

Solving equation (A.10), the stationary probability distribution ${{P}_{s}}({{\theta }_{q}})$ of the polarity and migration direction ${{\theta }_{v}}={{\theta }_{q}}$ is given by the circular normal distribution as

$$\begin{eqnarray}&&{{P}_{s}}({{\theta }_{q}})=\frac{1}{{{I}_{0}}(\kappa )}{\rm exp} (\kappa {\rm cos} {{\theta }_{q}})\ ,\end{eqnarray} \tag{ A.17 }$$

where the accuracy κ is obtained from equation (7). Here, ${{I}_{0}}(\kappa )$ is the modified Bessel function, defined as ${{I}_{0}}(\kappa )=\int {\rm d}{{\theta }_{q}}{\rm exp} (\kappa {\rm cos} {{\theta }_{q}})$. Equation (A.17) is plotted as a function of migration direction in figure 4(a) (red solid line). The parameter values of a Dictyostelium cell are assumed. From equation (A.17), the chemotaxis index is obtained as

$$\begin{eqnarray}&&{\rm CI}\equiv \int {\rm d}{{\theta }_{q}}{\rm cos} {{\theta }_{q}}{{P}_{s}}({{\theta }_{q}})={{\left. \frac{\partial }{\partial \kappa }{\rm log} {{I}_{0}}(\kappa ) \right|}_{\kappa =({{f}_{q}}A)/(2D)}}\ .\end{eqnarray} \tag{ A.18 }$$

The ${\rm CI}$, given by equation (A.18), saturates at 1; that is, ${\rm CI}\sim 1-({{f}_{q}}{{\tau }_{{\rm R}}}+2f_{q}^{-1}{{D}^{\operatorname{int}.}})/(2A)\to 1$ as $A\to \infty$, whereas ${\rm CI}\sim A/(2{{f}_{q}}{{\tau }_{{\rm R}}}+4f_{q}^{-1}{{D}^{\operatorname{int}.}})=\kappa /2$ as $A\to 0$. Figure 4(c) plots the CI (A.18) with equations (A.25) and (A.27) (red solid line) as functions of chemoattractant concentration assuming the experimental parameters ${{k}_{{\rm d}}}=1\ {{{\rm s}}^{-1}}$ and ${{K}_{{\rm d}}}=180\;{\rm nM}$ and the parameter values (10) and (11).

Here, we extend the maximum likelihood estimates of gradient direction reported in [11, 29] to include the effect of stochasticity at the level of the second messenger [36, 37]. We denote the concentrations of activated receptor and the activated second messenger at θ by ${{r}^{*}}(\theta )$ and ${{x}^{*}}(\theta )$, respectively. Under an exponential chemoattractant gradient of steepness p, the chemoattractant concentration along the periphery of the cell is given by $C(\theta )={{C}_{0}}{\rm exp} ((p/2){\rm cos} (\theta -\phi ))$. When cells estimate the gradient direction from the distribution of the activated second messenger, the most probable direction ϕ is given by $s{\rm cos} \phi ={{Z}_{1}}$ and $s{\rm sin} \phi ={{Z}_{2}}$, where ${{s}^{2}}=Z_{1}^{2}+Z_{2}^{2}$ and

$$\begin{eqnarray}&&{{Z}_{1}}=\int _{0}^{2\pi }{{x}^{*}}(\theta ){\rm cos} \theta {\rm d}\theta ,\end{eqnarray} \tag{ A.19 }$$

$$\begin{eqnarray}&&{{Z}_{2}}=\int _{0}^{2\pi }{{x}^{*}}(\theta ){\rm sin} \theta {\rm d}\theta .\end{eqnarray} \tag{ A.20 }$$

The distribution of the estimated ϕ can be derived from the distributions of Z₁ and Z₂. Here, we consider the distribution of ϕ up to the first order of p, from which we calculate the average and the dispersion of ${{x}^{*}}(\theta )$. The averages of r^* and x^* are obtained as

$$\begin{eqnarray}&&\langle {{r}^{*}}(\theta )\rangle =\frac{C(\theta )}{C(\theta )+{{K}_{d}}}{{r}_{0}},\end{eqnarray} \tag{ A.21 }$$

$$\begin{eqnarray}\begin{array}{rcl} \langle {{x}^{*}}(\theta )\rangle & = & \frac{\langle {{r}^{*}}(\theta )\rangle }{\langle {{r}^{*}}(\theta )\rangle +{{K}_{X}}}{{x}_{0}} \\ {} & = & \frac{C(\theta ){{r}_{0}}}{C(\theta ){{r}_{0}}+{{K}_{X}}(C(\theta )+{{K}_{d}})}{{x}_{0}}\ , \\ \end{array}\end{eqnarray} \tag{ A.22 }$$

where ${{r}_{0}}=N/2\pi$, ${{x}_{0}}={{X}_{t}}/2\pi$ and ${{K}_{X}}={{k}_{xd}}/{{k}_{xp}}$, and N and X_t are the total concentrations of receptor and second messenger, respectively. The production and degradation rates of the second messenger are ${{k}_{xp}}$ and ${{k}_{xd}}$, respectively. The fluctuation densities of ${{r}^{*}}(\theta )$ and ${{x}^{*}}(\theta )$, defined by $\langle {{r}^{*}}(\theta ){{r}^{*}}(\theta ^{\prime} )\rangle =\sigma _{r}^{2}(\theta )\delta (\theta -\theta ^{\prime} )$ and $\langle {{x}^{*}}(\theta ){{x}^{*}}(\theta ^{\prime} )\rangle$ $=\;\sigma _{x}^{2}(\theta )\delta (\theta -\theta ^{\prime} )$, are given by

$$\begin{eqnarray}&&\sigma _{r}^{2}(\theta )=\frac{C(\theta ){{K}_{{\rm d}}}}{{{\left( C(\theta )+{{K}_{{\rm d}}} \right)}^{2}}}{{r}_{0}}\ ,\end{eqnarray} \tag{ A.23 }$$

$$\begin{eqnarray}\begin{array}{rcl} \sigma _{x}^{2}(\theta ) & = & {{g}_{{\rm X}}}(\theta )\langle {{x}^{*}}(\theta )\rangle \\ {} & {} & +{{g}_{{\rm X}}}{{(\theta )}^{2}}\frac{{{\tau }_{{\rm R}}}(\theta )}{{{\tau }_{{\rm R}}}(\theta )+{{\tau }_{X}}(\theta )}\frac{\sigma _{r}^{2}}{{{\langle {{r}^{*}}(\theta )\rangle }^{2}}}{{\langle {{x}^{*}}(\theta )\rangle }^{2}}, \\ \end{array}\end{eqnarray} \tag{ A.24 }$$

where ${{g}_{x}}(\theta )={{K}_{X}}/({{K}_{X}}+{{r}^{*}})$, ${{\tau }_{{\rm R}}}(\theta )={{({{k}_{{\rm on}}}C(\theta )+{{k}_{{\rm off}}})}^{-1}}$ and ${{\tau }_{x}}(\theta )={{({{k}_{xp}}{{r}^{*}}(\theta )+{{k}_{xd}})}^{-1}}$. Using the average and dispersion of ${{x}^{*}}(\theta )$, the distribution of ϕ was evaluated up to the first order of p (see equation (A.4)) with

$$\begin{eqnarray}&&A=\frac{1}{2}p{{\pi }^{1/2}}{{\mu }^{1/2}},\end{eqnarray} \tag{ A.25 }$$

and

$$\begin{eqnarray}\begin{array}{rcl} {{\mu }^{-1}} & = & \frac{2{{({{C}_{0}}/{{K}_{{\rm d}}}+1)}^{2}}}{\pi {{C}_{0}}{{K}_{X}}{{r}_{0}}{{x}_{0}}}\left\{ \frac{{{[{{K}_{{\rm d}}}{{K}_{X}}+{{C}_{0}}({{K}_{X}}+{{r}_{0}})]}^{2}}}{{{C}_{0}}+{{K}_{{\rm d}}}} \right. \\ {} & {} & +\left. \frac{{{K}_{{\rm d}}}{{K}_{X}}[{{K}_{X}}({{C}_{0}}+{{K}_{{\rm d}}})+{{C}_{0}}{{r}_{0}}]{{x}_{0}}}{({{C}_{0}}+{{K}_{{\rm d}}})[{{K}_{X}}+({{K}_{{\rm d}}}+{{C}_{0}}){{k}_{{\rm on}}}/{{k}_{{\rm xp}}}]+{{C}_{0}}{{r}_{0}}} \right\}. \\ \end{array}\end{eqnarray} \tag{ A.26 }$$

Taking ${{x}_{0}}\to \infty$, followed by the limit ${{k}_{xp}}\to \infty$ with fixed K_X , we have $\mu ={{\mu }_{{\rm mee}.}}$ where

$$\begin{eqnarray}&&{{\mu }_{{\rm mee}.}}\equiv \frac{{{C}_{0}}{{K}_{d}}N}{4{{({{C}_{0}}+{{K}_{d}})}^{2}}}\ .\end{eqnarray} \tag{ A.27 }$$

This form of ${{\mu }_{{\rm mee}.}}$ is consistent with the result based on the most efficient estimation (maximum likelihood estimate without stochasticity in the second messenger reaction) [29]. We also note that, as ${{C}_{0}}\to \infty$, we have $\mu \sim (\pi /2){{K}_{X}}K_{{\rm d}}^{2}{{r}_{0}}{{x}_{0}}C_{0}^{-2}/{{({{K}_{X}}+{{r}_{0}})}^{2}}\propto C_{0}^{-2}$. Thus, this function decays much faster than ${{\mu }_{{\rm mee}.}}$ derived from the most efficient estimate, which decreases as $\propto \ C_{0}^{-1}$ as ${{C}_{0}}\to \infty$. Moreover, the coefficient is proportional to the second messenger concentration x₀. Therefore, consistent with a previous report [36], the activation of the second messenger becomes the bottleneck of the chemotactic signal processing at high chemoattractant concentrations.

Figure A.2 shows the chemotactic accuracy in equation (7) and chemotaxis index given by equation (A.18) with equations (A.25) and (A.26) for several different numbers of receptors N and second messengers X, with the experimental parameter values ${{k}_{{\rm d}}}=1\ {{{\rm s}}^{-1}}$ and ${{K}_{{\rm d}}}=180\ {\rm nM}$ and the values of f_q and ${{D}^{\operatorname{int}.}}$ given by (10) and (11). As shown in figures A.2(a), the peak concentration of the accuracy increases and get close to the dissociation constant K_d when only the numbers N of receptors are reduced. If we also reduce the numbers X of second messengers, the peak concentration is decreased as shown in figures A.2(c). In each case of figures A.2(c), the peak concentration is lower than that of the red solid line in figure 4(d) and the purple dashed-dotted line in figure A.2(c).