Figure 2 shows typical behavior of a specimen in a dead-ended capillary [(KCl) = 4 mM]. Figure 2A shows a time-course of the distance from the capillary end. When the forward swimming Paramecium bumped against the capillary end, the specimen began to swim backward before reversing direction and swimming forward again. This behavior was repeated many times. Shortly after the first collision with the capillary end (in the region labeled T1 in Figure 2), the backward swimming distance was 0.3–0.5 mm, which corresponds to one to two times the length of the body. The period of backward swimming was 2–5 s. The backward swimming distance gradually became longer (region T2 in Figure 2A), and the period increased to 5–10 s. Finally, in region T3 in Figure 2A, the distance reached 3–4 mm and the period increased to 5–15 s. Figure 2B shows the distribution of backward swimming distances in Figure 2A. Two peaks are present at 0.3–0.5 mm and 3–4 mm. This behavior was observed in 13 of 15 tested individuals under the same experimental conditions.

These results imply that Paramecium exhibits two types of backward swimming, which are distinguishable by their distances and periods. The first type can be referred to as short-term backward swimming (SBS) with distances of 0.3–1 mm and periods of 2–10 s. The second type is long-term backward swimming (LBS) with distances of 3–4 mm and periods of 10–15 s.

Figure 3 shows effect of potassium ion concentration (2, 4, 8, and 20 mM) on backward swimming of Paramecium in a dead-ended capillary. Figure 3A shows the trajectory of backward and forward swimming motions in the time and one-dimensional space plot, and the distance of backward swimming was measured from the swimming trajectory. The statistical occurrence of this distance was shown in Figure 3B. They had peaks at 0.3–1 mm (open triangles in Figure 3B) and 3–4 mm (closed triangles in Figure 3B) at 2, 4, and 8 mM, and there was a single peak at 2 mm (closed triangles in Figure 3B) at 20 mM. In Figure 3C, the distribution of backward swimming periods is shown for each K⁺ concentration; at 2, 4, and 8 mM there were peaks at 3–9 s (open triangles in Figure 3C), and 10–15 s (closed triangles in Figure 3C), and at 20 mM there was a single peak at 17 s (closed triangles in Figure 3C). Figure 3D shows the dependency of LBS distance on potassium ion concentration. The LBS distance decreased with increasing concentration.

In summary, Paramecium in a dead-ended capillary exhibited both SBS and LBS at K⁺ concentrations of 2–8 mM, and only LBS at 20 mM.

The experimental results described above provide clear evidence that repeated mechanical stimulus in a dead-ended capillary can increase the distance of backward swimming in Paramecium. The modification of behavior in Paramecium is a novel behavior, which develops from the restriction of the moving region. This implies that there is an adaptive capacity for spatial navigation in Paramecium. The important thing for understanding such behaviors or capacities in organisms is an understanding of the physiological mechanism.

The movement of cilia occurs due to the ciliary motion. The ciliary motion is regulated by the membrane potential change or the biochemical reactions in the cell and the cilium. In Paramecium the relationship between the behaviors and the membrane potential changes has been well studied (Eckert, 1972; Naitoh, 1974; Naitoh and Sugino, 1984). Therefore, we now attempt to understand the underlying mechanisms of these behaviors in Paramecium through mathematical modeling of membrane potential change.

Based on the Hodgkin–Huxley-type model for the excitation dynamics of membrane potentials (Hodgkin and Huxley, 1952; Naitoh and Sugino, 1984), we propose a simplified model to explain the behavior observed in our biological experiment. In the mathematical modeling that follows we consider only the case of LBS, even though the experiments also revealed SBS. Before constructing the model, we explain the mechanism of the membrane potential response in Paramecium induced by mechanical stimulus.

The swimming behavior of Paramecium depends on its ciliary motion, which is in turn controlled by a membrane potential caused mainly by the difference in concentration of Ca²⁺ and K⁺ between the interior and exterior of the cell. The mechanism by which the membrane potential changes is essentially the same as that in nerve cells and muscle cells, although the ionic species involved are different.

When the forward swimming Paramecium collides with a solid object, extracellular Ca²⁺ ions flow into the cell, mediated by opening of the mechano-sensitive Ca²⁺ channels that are distributed in the anterior region of the cell (Ogura and Machemer, 1980; Satow et al., 1983; Machemer and Machemer-Röhnisch, 1984; Tominaga and Naitoh, 1994). This depolarizes the membrane potential. The depolarized membrane is more permeable to Ca²⁺ ions due to opening of the voltage-sensitive Ca²⁺ channels, which are localized in cilia. This results in a large regenerative depolarization (Dunlap, 1977; Machemer and Ogura, 1979). The increased concentration of Ca²⁺ in cilia leads to the reversal of ciliary beating and hence to backward swimming. The increased membrane potential also opens the voltage-sensitive K⁺ channels, allowing intracellular K⁺ ions to flow out of the cell. This outflow leads to repolarization of the membrane. The cilia gradually resume their original direction of beating as the membrane potential decreases, and the Paramecium begins to swim forward again. The duration time of backward swimming corresponds to the duration time of Ca²⁺ current flow.

Naitoh and Sugino (1984) have reported that the avoidance response of Paramecium, which corresponds to SBS in our report, can be represented by a Hodgkin–Huxley-type equation on the basis of the above scenario for the physiological mechanism of backward swimming. However, the mechanism of the membrane response to repeated current stimuli is unclear. Naitoh (1990) has suggested that Ca²⁺ channels with slow time constants (which deactivate slowly) might carry Ca²⁺ current after the action potential has gone, even though Ca²⁺ channels with fast and slow time constants are the same physically. Hinrichsen et al. (1984) reported that the backward swimming of a mutant specimen of Paramecium, whose Ca²⁺ current deactivated poorly, persisted for longer under high concentrations of K⁺ ions than that of wild-type Paramecium.

According to this line of understanding, we can assume that the time constants of the Ca²⁺ channels in Paramecium can be changed by the stimulus. LBS is induced by the Ca²⁺ current, which is deactivated slowly due to Ca²⁺ channels with slow time constants that remain activated after the action potential is gone. We also assume that the membrane potential in the cell corresponds to that in the cilia without regard to the localization of Ca²⁺ channels for simplicity. We formulated an equation for the membrane potential relevant to LBS in Paramecium (regions T2 and T3 in Figure 2A). The time-course of the membrane potential can be expressed as: (1)CmV˙(t)=δ(t)Iapp(t)−ICa(t, V)               −IK(t, V)−Ileak(t, V).

Here t is the time, C_m is the membrane capacitance, δ is a function that switches the outward current, and I_app(t) is the outward electric current, which corresponds to the mechanical stimulus applied by bumping against the end of the capillary in our experiment. I_Ca is the Ca²⁺ current, I_K is the K⁺ current, and I_leak is the leak current. These three quantities can be expressed as: (2)ICa(t,V)=gCa(V(t)−ECa), (3)IK(t,V)=gK(V(t)−EK), (4)Ileak(t,V)=g¯leak(V(t)−Eleak), where g_Ca and g_K are the ionic conductance (the rate of passage of ions), the definition of which will be given later. g_leak is the maximum ionic conductance of the leak ion channel. E_Ca and E_K are the equilibrium potentials of Ca²⁺ and K⁺, respectively. These two quantities can be described by the Nernst equation (Equation 5) for the diffusion of electric charge. E_leak is the equilibrium potential of the leak ions, and is given approximately by Equation (6).

Here, R is a gas constant, T is the absolute temperature, z is the ionic valence, and F is the Faraday constant. The parameters [X^z+]_o and [X^z+]_i are the extracellular and intracellular concentrations of X^z+ ions, respectively. g_LCa and g_LK are the ionic conductance of the Ca²⁺ and K⁺ channels, which are insensitive to changes in the membrane potential.

The Ca²⁺ and K⁺ currents are controlled by the conductance of the Ca²⁺ and K⁺ channels, respectively. The Ca²⁺ conductance (g_Ca) is described using the activation gate factor m and the deactivation gate factor h. The K⁺ conductance (g_K) is described by the activation gate factor n. These factors are the variables that represent the gating of the voltage-sensitive ionic channels.

The occurrence of LBS requires long-term activation of the Ca²⁺ channel, because the duration time of backward swimming in Paramecium depends on the time for which Ca²⁺ current flows. We assume that the Ca²⁺ channels are in one of two modes, with a fast or slow time constant with respect to the deactivation process. We now introduce two more parameters, g_CaS and P(t), to represent the conductance and occurrence rate of Ca²⁺ channels with the slow time constant, respectively. We include these parameters in the original equation for Ca²⁺ conductance in the Hodgkin–Huxley-type model for Paramecium reported by Naitoh and Sugino (1984). The definition of P(t) will be given later. The total Ca²⁺ conductance is given as the sum of the products of the occurrence rate with the conductance of the Ca²⁺ channels with fast and slow time constants. The resulting equations for ionic conductance are as follows: (7)gCa(t, V)=(1−P(t))gCaF(t, V)+P(t)gCaS(t, V), (8)gCaF(t, V)=g¯CamF(V)5{1−(1−hF(V))5}, (9)gCaS(t, V)=g¯CamS(V)5{1−(1−hS(V))5}, (10)gK(t,V)=g¯Kn(V), where g_CaF is the Ca²⁺ conductance with the fast time constant. g_Ca and g_K are the maximum Ca²⁺ and K⁺ conductance of the membrane when all the Ca²⁺ and K⁺ channels are activated, respectively. j_F and j_S (j ∈ {m, h}) are the gate variables for the channels with fast and slow time constants. The equations for the gate variables are as follows: (11)x˙=αx(V)(1−x)−βx(V)x,      x∈{mF, mS, hF, hS, n}, where x is the opening rate of the gates. α_x and β_x are functions of the membrane potential, which are determined by fitting the rate constants using analytical expressions as follows: (12)αmF(V)=−0.0224(V+2.5809)               /(exp​(−(V+2.5809)/0.7331)−1), (13)βmF(V)=0.1426exp​(−V/39.3464), (14)αhF(V)=0.1exp​(−(V+30)/5), (15)βhF(V)=1/(exp​((38.2866−V)/30.9397)+1), (16)αn(V)=0.0375(58.5845−V)             /(exp​((58.5845−V)/8.16699)−1), (17)βn(V)=0.1015exp​(−V/68.0968), (18)αjS(V)=αjF(V)γ, βjS(V) = βjF(V)γ,                j∈{m,h}, (19)γ={γ¯δ=0,1δ=1.

Here, γ (0 ≤ γ ≤ 1) is a constant. For each fixed V in Equation (11), the time constants (τ_x(x ∈ {m_F, m_S, h_F, h_S, n})) and equilibrium values (x_∞(x ∈ {m_F, m_S, h_F, h_S, n})) of the gate variables are given as follows: (20)τx=1/(αx+βx), x∞=αx/(αx+βx).

P (0 ≤ P ≤ 1) is the rate of the Ca²⁺ channel with the slow time constant in the deactivation process. The Ca²⁺ channels with the fast and slow time constants are the same physically, and it is assumed that the mode of the Ca²⁺ channel is changed by the stimulus-induced membrane potential response. The relevant equation is as follows: (21)P(t)=(P(ti)+ΔP)exp​(−(t−ti)/τP), where t_i is the finishing time of the i^th stimulation, ΔP is the maximum rate of the Ca²⁺ channel changing the mode, and τ_P is the decay time constant of P.

The numerical parameters and initial values for our proposed model are based on the experimental conditions and the measurements of membrane potential reported by Naitoh et al and are as follows: C_m = 2 [μF/cm²], E_Ca = 125 [mV] ([Ca²⁺]_i = 6.06 × 10⁻⁸ [M], [Ca²⁺]_o = 10⁻³ [M]), E_K = −57 ~ 0 [mV] ([K⁺]_i = 20 [mM], [K⁺]_o = 2–20 [mM]), g_LCa = 0.1 [mS/cm²], g_LK = 0.8 [mS/cm²], T = 298 [K], R = 2.0 (cal/K·mole), F = 23,000 (cal/V·mole), g_Ca = 0.6667 [mS/cm²], g_K = 1.3333 [mS/cm²], g_leak = 0.42 [mS/cm²], γ = 0.005–0.05, C_P = 0.2, τ_P = 700.0, V(0) = −30 [mV], m_F(V(0)) = 0.2911, m_S(V(0)) = 0.01, h_F(V(0)) = 0.162, h_S(V(0)) = 0.1, n(V(0)) = 0.0163, P(0) = 0, I_app = 100.

We simulated the membrane potential responses when the outward electric current is injected periodically, because LBS occurred by repeating the colliding with the end of capillary. Figure 4A shows time-courses of the membrane potential, the Ca²⁺ current, the gate parameter m_S, and the rate of the Ca²⁺ channel with the slow time constant when the outward electric current is injected periodically. Here it was assumed that [K⁺] = 4 mM (E_K = −40 mV). When the outward electric current was injected before all the Ca²⁺ channels with the slow time constant had returned to the mode with the fast time constant (P > 0), the rate of the Ca²⁺ channel with the slow time constant increased gradually. This resulted in an increase in magnitude of the Ca²⁺ current after the action potential had passed.

Figure 4B shows the dependency of the duration for which Ca²⁺ current flows (τ_b) on the number of trials (the number of collisions with the capillary end). The duration time is defined as the time for which the Ca²⁺ current is smaller than a threshold value. The duration time increased with the number of trials and eventually became saturated. The time for which Ca²⁺ current flows corresponds to the duration time of backward swimming in Paramecium, and the results of this simulation represent the behavior displayed during regions T2 and T3 in Figure 2A.

Figure 4C shows the dependency of the duration time of Ca²⁺ current flow on the equilibrium potential of the K⁺ ion. The threshold for this duration time was set to −0.1 (nA) based on the experimental data reported by Naitoh and Kaneko (1972). The duration time is maximum at E_K = −40 mV, which corresponds to [K⁺] = 4 mM.

The maximum value of the duration time of Ca²⁺ current flow depends on the time constants of the gate parameters and on the threshold value of the duration time (data not shown). As shown in Figure 6B, the time constant of h, which influences the duration time of Ca²⁺ current flow, reached its maximum value when the membrane potential was −20 mV, which is the same value of the resting potential as E_K and E_Ca were −40 and −125 mV, respectively. Therefore, when the threshold value for the duration time is sufficiently low, the Ca²⁺ current-flow duration time is maximum at E_K = −40 mV as shown in Figure 4C. When the threshold value for the duration time is sufficiently high, the Ca²⁺ current-flow duration time decreases with increasing E_K. The latter case was experimentally observed, as illustrated by the result in Figure 3D.

To understand the mechanism of the long-term Ca²⁺ current, which is invoked by the Ca²⁺ channel with slow time constant, we analyzed the relationship between the change of membrane potential and the move of nullcline.

Figures 5A,B show the stimulus-induced membrane potential responses in the absence (P = 0) and presence (P = 1) of Ca²⁺ channels that have slow time constants with respect to the deactivation process. Figure 6 shows the dependence of the gate parameters on the membrane potential.

The injection of an outward electric current into the cell produced a passive exponential membrane depolarization, and a V nullcline (V˙ = 0) that was shifted from (0) to (1) in Figure 5A. As the time constant of m_F was smaller than those of h_F and n (Figure 6B), both m_F and n increased as the membrane potential became positive, whereas h_F decreased (Figure 6A). m_F increased before changes occurred in h_F and n. Here Ca²⁺ ions flow into the cell when the Ca²⁺ channels are opened. The magnitude of the membrane potential increased with the magnitude of the Ca²⁺ current.

When the membrane potential became larger than 20 mV, n began to increase as shown in Figure 6A. Now K⁺ ions flow out of the cell due to the opening of the K⁺ channels. The magnitude of the membrane potential was controlled by increasing the magnitude of the K⁺ current. When the membrane potential approached E_Ca, it began to decrease because the magnitude of the Ca²⁺ current was small and the magnitude of the K⁺ current was large. Although the V nullcline then shifted from (1) to (2), there was little change in the equilibrium values of the gate parameters (m_F, h_F, n) when the magnitude of the outward electric current was large enough. Therefore, the Ca²⁺ channels remained in the activated state.

The membrane potential decreased sharply after the outward electric current was gone, and then the V nullcline shifted from (2) to (3). As the time constant of m_F was smaller than those of h_F and n, the decrease of m_F occurred before the decrease of n and the increase of h_F. The K⁺ current was still carried after the Ca²⁺ current had stopped, and the membrane potential became hyperpolarized. The V nullcline then shifted from (3) to (4), and the membrane potential neared the resting state. Paramecium swims backward while Ca²⁺ current flows due to activation of the Ca²⁺ channels.

As shown in Figure 5B, the V nullclines shifted successively from (5) to (9) in the presence of the Ca²⁺ channel with the slow time constant (P = 1). In Figure 5B2 it is apparent that the orbit of the solution was deformed due to the slow time constants of m_S and h_S in the deactivation process of the Ca²⁺ channel. As the time constants of m_S and h_S were larger than that of n, the Ca²⁺ current still flowed due to the slow convergence of m and h after the K⁺ current had stopped. Although the Ca²⁺ current flowed while the Ca²⁺ channels were activated, because the Ca²⁺ current was small [(7–8) in Figure 5B], the change in membrane potential was also small. Therefore, given the presence of Ca²⁺ channels with the slow time constant, our proposed model reproduces the Ca²⁺ current that invokes backward swimming in Paramecium after the action potential is gone.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.