Extracellular and intracellular factors regulating the migration direction of a chemotactic cell in traveling-wave chemotaxis

We consider the motion of a single D. discoideum cell in one dimension exposed to a periodic traveling wave of cAMP concentration (figure 1(a)). The model equation in this study is such that the position of a single cell, x(t), is governed by the following equation of motion,

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\rm d}}{{\rm d}t}x(t) & = & \chi (S(x(t-{{\tau }_{{\rm mot}}}),t-{{\tau }_{{\rm mot}}})) \\ {} & {} & \times \nabla S(x(t-{{\tau }_{{\rm pol}}}),t-{{\tau }_{{\rm pol}}}), \\ \end{array}\end{eqnarray} \tag{ 1 }$$

where $S(x,t)$ is a periodic function with respect to both x and t. This quantity indicates the strength of the stimulation, which is positively correlated with the concentration of cAMP. Therefore, S is assumed to be an increasing function of the concentration of cAMP. The function, χ, referred to as the chemotactic coefficient in what follows, is also a periodic function that increases when $S(x(t-{{\tau }_{{\rm mot}}}),t-{{\tau }_{{\rm mot}}})$ increases. The functional forms of S and χ are determined later. The biological meanings of the two parameters, ${{\tau }_{{\rm mot}}}$ and ${{\tau }_{{\rm pol}}}$, are explained in following paragraphs.

This model equation is constructed as follows (further mathematical derivation based on some experiments [16, 25–27] is shown in appendix A). To describe the chemotaxis, we use the widely accepted assumption that the velocity of the cell is proportional to the spatial gradient of S around the cell [14, 28, 29]. We also assume that stimulation by cAMP induces two responses in the cell (figure 1(b)): changes in the cell polarity and motility. Recent studies on intracellular signal transduction pathways have indicated that these responses should be separately recognized [16, 30–33].

The polarity is a directional pseudopod extension toward the gradient of the cAMP concentration. This phenomenon is induced by the localization of proteins including phosphatidylinositol 3, 4, 5-trisphosphate along the plasma membrane resulting from sensing of the gradient of the cAMP concentration [32]. In the present model, we attribute the magnitude of this response to $\nabla S$. The parameter, ${{\tau }_{{\rm pol}}}$, is introduced to incorporate the response time from stimulation to the change in the polarity.

The motility is the magnitude and the frequency of the pseudopod extension. Mathematically, the chemotactic coefficient, χ, describes the motility. We assume that the motility depends on the magnitude of the stimulation, S. Based on this assumption, the chemotactic coefficient, χ, oscillates with the same period as the periodic stimulation by cAMP, S. The parameter, ${{\tau }_{{\rm mot}}}$, is introduced to incorporate the response time from stimulation to the change in the motility.

The assumption of oscillatory behavior for the motility is supported by some previous studies. Soll et al measured the time series of the instantaneous velocity of a single D. discoideum cell in detail when the cell was exposed to temporally periodic changes in cAMP concentration in the absence of a spatial gradient [16, 25–27]. The magnitude of the instantaneous velocity oscillated with the same period as the extracellular cAMP concentration. This result implies that periodic stimulation by cAMP induces periodic changes in motility. In the situation we consider here, the cell is periodically stimulated by the traveling wave. Therefore, we assume that the chemotactic coefficient oscillates with the same period as the periodic stimulation. Because a significant constant phase difference between the oscillation of the extracellular cAMP concentration and that of the instantaneous velocity was observed in those experiments, the response time, ${{\tau }_{{\rm mot}}},$ corresponding to the phase difference is introduced in the model.

First, we consider a case in which the waveform of the stimulation is a sinusoid, namely

$$\begin{eqnarray}&&S(x,t)={{S}_{{\rm ave}}}+{{S}_{{\rm osc}}}{\rm sin} \left( \frac{2\pi }{\lambda }x+\frac{2\pi }{T}t \right),\end{eqnarray} \tag{ 2 }$$

where λ and T denote the wavelength and the period, respectively. S_ave represents the average of the stimulation, and S_osc represents the amplitude of the oscillation, satisfying ${{S}_{{\rm osc}}}\leqslant {{S}_{{\rm ave}}}$. We determine the waveform (2) from previous experimental results [9, 16, 34, 35]. The concentration of the cAMP wave changes exponentially between 10⁻⁸ and 10⁻⁶ M, and the waveform is sharply peaked, which is well approximated by ${\rm exp} [{\rm sin} \left( 2\pi x/\lambda +2\pi t/T \right)]$ [9]. Additionally, the velocity of the cell varies roughly with the spatial gradient of the logarithm of the concentration [35]. Thus, we use a sinusoid for S. Note that, as we discuss later, the choice of the waveform is not essential to the result.

The chemotactic coefficient is also assumed to be a sinusoid, namely

$$\begin{eqnarray}\begin{array}{rcl} \chi (t) & = & {{\chi }_{{\rm ave}}}+{{\chi }_{{\rm osc}}}{\rm sin} \left[ \frac{2\pi }{\lambda }x(t-{{\tau }_{{\rm mot}}}) \right. \\ {} & {} & \left. +\frac{2\pi }{T}(t-{{\tau }_{{\rm mot}}}) \right], \\ \end{array}\end{eqnarray} \tag{ 3 }$$

where ${{\chi }_{{\rm ave}}}$ and ${{\chi }_{{\rm osc}}}$ are constant parameters satisfying the condition, ${{\chi }_{{\rm osc}}}\leqslant {{\chi }_{{\rm ave}}}$. This assumption is derived from the observation that the amplitude of the fluctuations of the velocity of the cell is proportional to the logarithm of the wave amplitude of the cAMP concentration when there is no spatial gradient [16, 34].

We show that the propagation of the traveling wave induces a migration in one direction. We can generate the approximations $x(t-{{\tau }_{{\rm pol}}})\approx x(t)$ and $x(t-{{\tau }_{{\rm mot}}})\approx x(t)$ because the velocity of the traveling wave, $\lambda /T\sim 300$ μm min⁻¹ [9], is much greater than the speed of the cell, which varies between 0 and 7 μm min⁻¹ [16]. By substituting the equations (2) and (3) into the equation of motion (1), the velocity of a single cell is reduced to (further detailed derivation of equation (4) is shown in appendix B)

$$\begin{eqnarray}\begin{array}{rcl} \frac{{\rm d}}{{\rm d}t}x(t) & = & \;\frac{2\pi {{\chi }_{{\rm ave}}}\;{{S}_{{\rm osc}}}}{\lambda }{\rm cos} \left[ \frac{2\pi }{\lambda }x(t) \right. \\ {} & {} & +\left. \frac{2\pi }{T}(t-{{\tau }_{{\rm pol}}}) \right] \\ {} & {} & +\frac{\pi {{\chi }_{{\rm osc}}}{{S}_{{\rm osc}}}}{\lambda }{\rm sin} \left[ \frac{4\pi }{\lambda }x(t) \right. \\ {} & {} & +\frac{2\pi }{T}\left. (2t-{{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}) \right] \\ {} & {} & -\frac{\pi {{\chi }_{{\rm osc}}}{{S}_{{\rm osc}}}}{\lambda }{\rm sin} \left[ \frac{2\pi }{T}({{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}) \right]. \\ \end{array}\end{eqnarray} \tag{ 4 }$$

The third term on the right hand side of the equation (4) is a constant, whereas the first and second terms are periodic with respect to time as long as x(t) is treated as a constant during one wave period. Therefore, on the average over one wave period, the third term contributes to the translocation of the cell much more than the first and second terms. By eliminating those periodic terms, the averaged velocity, $\bar{v}$, is obtained as follows

$$\begin{eqnarray}&&\bar{v}=-\frac{\pi {{\chi }_{{\rm osc}}}{{S}_{{\rm osc}}}}{\lambda }{\rm sin} \left[ \frac{2\pi }{T}({{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}) \right].\end{eqnarray} \tag{ 5 }$$

Figure 2 illustrates the numerical solution of the differential equation (1) for parameters typical in the aggregation state of a cell (see table 1). It is also apparent that the averaged velocity, $\bar{v}$, well approximates the net migration of a cell.

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Table 1.  Typical values for D. discoideum cells in the aggregation process. They are used in a numerical calculation for producing figure 2. Although $2\pi {{\chi }_{{\rm ave}}}{{S}_{{\rm osc}}}/\lambda$ and $2\pi {{\chi }_{{\rm osc}}}{{S}_{{\rm osc}}}/\lambda$ have never been directly measured, we set these values to the half of maximum velocity [16]. Although ${{\tau }_{{\rm pol}}}$ has not been directly measured, it is inferred from the [36]. Note that ${{\tau }_{{\rm mot}}}$, and ${{\tau }_{{\rm pol}}}$ would depend on T (see the main text for details).

Parameter	Definition	Value	Source
λ	Wavelength of cAMP wave	2000 μm	[37]
T	Period of cAMP wave	7 min	[16, 27, 37]
$2\pi {{\chi }_{{\rm ave}}}{{S}_{{\rm osc}}}/\lambda$	Coefficient in equation (4)	3.5 μm min−1	[16]
$2\pi {{\chi }_{{\rm osc}}}{{S}_{{\rm osc}}}/\lambda$	Coefficient in equation (4)	3.5 μm min−1	[16]
${{\tau }_{{\rm mot}}}$	Response time to change in motility	6 min	[16, 27]
${{\tau }_{{\rm pol}}}$	Response time to change in polarity	1 min	[36]

According to equation (5), the wave period, T, and the difference between the response times, ${{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}$, are important parameters that determine the direction and the magnitude of the net migration. The cell migrates toward or away from the wave source on the average when ${\rm sin} [2\pi ({{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}})/T]\lt 0$ or ${\rm sin} [2\pi ({{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}})/T]\gt 0$, respectively. Figure 3(a) shows the phase diagram of the direction of net migration on the $\left( {{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}},T \right)$-plane.

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A recent study, in which motions of a cell under waves with several periods were investigated, allows us to estimate the dependence of the difference between the response times, ${{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}$, on a wave period. We assume a linear dependence for simplicity and set it to ${{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}=5+\alpha (T-7)$ in order to satisfy ${{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}=5$ when T = 7. Under this assumption, averaged velocities (5) for $\alpha =0.40$, 0.45, and 0.50 are obtained as figure 3(b). By comparing the averaged velocities with those measured by Skoge et al [24], we estimate α at 0.45. Using the linear dependence, ${{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}=5+0.45(T-7)$, we obtain instantaneous velocities of a cell by numerically solving equation (1). Figure 4 shows the instantaneous velocities under the waves with the periods of 6, 10, and 16 min.

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To see that the essence of the result does not depend on the waveform, consider another situation in which the stimulation by the traveling wave of cAMP is sharply peaked and represented by

$$\begin{eqnarray}&&S(x,t)={{S}_{0}}{\rm exp} \left[ {{S}_{1}}{\rm sin} \left( \frac{2\pi }{\lambda }x+\frac{2\pi }{T}t \right) \right],\end{eqnarray} \tag{ 6 }$$

where S₀ and S₁ are positive constant parameters, and the chemotactic coefficient is given by equation (3). In the same manner as for the sinusoidal traveling wave, we ignore the periodic function and obtain

$$\begin{eqnarray}&&\bar{v}=-\frac{\pi {{\chi }_{{\rm osc}}}{{S}_{0}}{{S}_{1}}I}{\lambda }{\rm sin} \left[ \frac{2\pi }{T}\left( {{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}} \right) \right],\end{eqnarray} \tag{ 7 }$$

where $I\int _{-1}^{1}{\rm d}t\;\sqrt{1-{{t}^{2}}}{\rm exp} ({{S}_{1}}t)\gt 0$. We see that the direction of the net migration of the cell is determined by the sign of ${\rm sin} [2\pi ({{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}})/T]$ again.

Even when the waveform of the cAMP is not as simple as in the above examples, as long as the waveform is periodic and mono-phasic, the wave period, T, and the difference between the two response times, ${{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}$, are still key parameters governing the direction of the net migration. This is explained by a qualitative discussion in the following way.

Figures 5(a) and (b) illustrate two situations in which the wave periods, T, are different. Because the wave, $S(t)\equiv S(x(t),t)$, is mono-phasic and periodic, the spatial gradient, $\nabla S(t)$, becomes a biphasic wave whose time average is 0. When the wave period is such that the peak of the motility, $\chi (t)$, appears while $\nabla S(t-{{\tau }_{{\rm pol}}})$ is positive in each cycle of the cAMP wave, the time average of the product of $\chi (t)$ and $\nabla S(t-{{\tau }_{{\rm pol}}})$ becomes positive (figure 5(a)). This result implies that the averaged velocity, $\bar{v}$, is positive, and the cell migrates toward the wave source (figure 5(a)). Conversely, when awave period is such that the peak of the motility $\chi (t)$ appears when $\nabla S(t-{{\tau }_{{\rm pol}}})$ is negative in a cycle, the time average of the product of $\chi (t)$ and $\nabla S(t-{{\tau }_{{\rm pol}}})$ becomes negative (figure 5(b)). This result implies that the averaged velocity, $\bar{v}$, is negative, and the cell migrates away from the wave source (figure 5(b)). Thus, the wave period, T, determines the direction of cell migration.

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Figures 6(a) and (b) illustrate two situations in which the differences between the two response times, ${{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}$, are different. When two response times are such that the peak of the motility, $\chi (t)$, appears while $\nabla S(t-{{\tau }_{{\rm pol}}})$ is positive or negative in each cycle of the cAMP wave, the time average of the product of $\chi (t)$ and $\nabla S(t-{{\tau }_{{\rm pol}}})$ becomes positive or negative, respectively (figures 6(a) and (b)). Thus, the direction of the cell migration also depends on the difference between the two response times, ${{\tau }_{{\rm mot}}}-{{\tau }_{{\rm pol}}}$.

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The directional net migration of a cell has been reproduced in several models in which the time dependence of the chemotactic coefficient is taken into account [14–23]. We conclude that all of these models have been constructed in such a way that the appropriate phase difference between the stimulation and the response is generated.