Cellular Ethological Dynamics in Diorama Environments

Amoeboid crawling, which involves a sol–gel transformation of the cell, can also be viewed as a coupled system of local motion that is often bistable, excitable, or oscillatory in terms of dynamics. It is known that these three dynamic behaviors arise from the same equations of motion (with different parameters).⁴³⁾ In fact, these dynamic behaviors observed in Physarum and Dictyostelium amoebas were modeled by differential equations.⁴⁴⁾

In the giant amoeba Physarum plasmodium, local rhythmic contraction propagates throughout the body, thereby receiving a force (reaction force) from the substrate surface to propel it forward. Such movement is called peristaltic crawling (the word peristalsis means propagation of contraction waves) and can be considered in the same way as the movement of earthworms and gastropods because all these organisms can migrate with periodic propagation of contraction waves along the body. It is known that these organisms can switch the moving directions depending on the situation, but the dynamic mechanism has not been obvious.⁴⁵⁾

Various mechanical models of peristaltic crawling have been proposed.⁴⁶⁾ Here, we will introduce a simple model.^47–49) For the sake of simplicity, the body of the organism is assumed to be a one-dimensional body consisting of alternating mass points and springs (Fig. 1). The point mass has weight and only has kinetic friction (the static friction ignored) with the substrate surface. The equation of motion was \begin{equation} \xi \dot{X}_{n} = K \{(X_{n+1}-X_{n}-l_{n+\frac{1}{2}})-(X_{n}-X_{n-1}-l_{n-\frac{1}{2}})\}, \end{equation} (1) where \(X_{n}\) was the position of mass point n, ξ was the viscous friction coefficient, \(l_{n+1/2}\) was the natural length of the Hookean spring between the mass points n and \(n+1\), K was the spring constant, and \(\dot{X}\) was the time derivative of X. In this equation, an overdamping regime was assumed.

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Figure 1. A model for peristaltic crawling in one-dimensional space.⁴⁷⁾ (a) Schematic diagram for multi-block model. Each block had the mass and a dynamic friction coefficient with the substrate surface and was connected to each other by Hookean springs. However, the natural length of this spring changed depending on time and space. This represented wave propagation of peristaltic motion. (b) In the non-dimensionalized model, the peristaltic wave e (wavelength λ) propagated at velocity v, shown at the bottom of the figure. The periodic rise (δ-function) of friction coefficient η was given at a certain phase of the wave and moved with the wave at velocity v, as shown in the upper panel. (c) The dependency of migration velocity on the phase of friction rise in the non-dimensionalized model. The migration velocity was averaged over one period (λ) of the peristaltic wave. The solution \(u^{(0)}\) was obtained in the unperturbed condition (the friction coefficient was constant), and the friction rise was given at any phase β. Depending on β, the migration velocity V (dotted line) changed. When \(V<0\) (the left half of the figure), \(u^{(0)}_{t}>0\) (this part of the body was moving to the right), and \(u^{(0)}_{s}<0\) (this part was contracting and denser). Conversely, when \(V>0\) (the left half of the figure), that part was moving to the left while relaxing and elongating. Please note that the part of the body that was moving to the right was always contracted while the part moving to the left was always elongated. When the brakes (the friction rise) were applied in the contracted part (moving to the right), the body migrated to the left (\(V<0\)). Contrary to this, when the body migrated to the right, the brakes were applied in the elongated part (moving to the left).

To represent active peristaltic motion, the natural length of a spring (Hookean spring) was varied periodically, and its phase propagated along the body as \begin{equation} l_{n+\frac{1}{2}} = l_{0} +a \sin (\omega t-2 \pi n/N), \end{equation} (2) where ω was angular velocity, N was the spatial period, and a and \(l_{0}\) were the constants.

These equations were rewritten as \begin{align} \xi_{n} \dot{U}_{n} &= K[U_{n+1}+U_{n-1}-2U_{n}\notag\\ &\quad-a \{\sin (\omega t-2\pi n/N)-\sin (\omega t-2\pi (n-1)/N)\}], \end{align} (3) by introducing the new variable \(U_{n}=X_{n}-n l_{0}\), where \(U_{n}\) was the displacement of mass point n from the standard position.

By introducing the reference coordinate \(s=n l_{0}\) and the dimensionless variables and parameters, \(\eta_{n}(s, t)=\xi_{n}(t)/l_{0}\) (rescaled friction constant), \(\kappa =l_{0} K\) (rescaled spring constant), \(e_{0}=a/l_{0}\), \(k=2\pi/(Nl_{0})\), and by taking the limit \(l_{0}\rightarrow 0\), the continuum equations were obtained \begin{align} \eta(s, t) \frac{\partial u}{\partial t} = \kappa \left(\frac{\partial^{2} u}{\partial s^{2}}-\frac{\partial e}{\partial s}\right), & \end{align} (4) \begin{align} e(s, t) = e_{0} \sin (\omega t-ks). & \end{align} (5)

When the friction coefficient η was constant in time and space, the net displacement in one cycle of contraction motion was zero, although local cyclic displacement could be observed. If the friction coefficient was constant and anisotropic (different in anterior-posterior directions), it moved in the direction of low friction, but in this case, it was not able to move in the opposite direction.

What mechanism enables the variability of forward and backward movement? This issue was considered next. So, it was assumed that the friction constant changed with periodic motion as the body of the organism expanded and contracted due to rhythmic contraction. For example, in Physarum plasmodium, it might change due to some biochemical and/or mechanical conditions on the substrate surface. In worms and gastropods, it was expected that the body was pressed against the substrate surface or lifted. Although the details were still unknown, it was plausible that the friction constant changed at a certain phase of rhythmic motion.

For simplicity and convenience of formulation, a rise in friction constant was introduced with δ-function at the certain phase (δ-function pulse of anchoring) as \(\eta\rightarrow \eta\{1+\epsilon p(s, t)\}\), where \(p(s, t)=\sum_{n=-\infty}^{\infty}2\pi\delta \{\omega t -k(s-B-n\lambda)-\alpha\}=\sum_{n=-\infty}^{\infty}2\pi\delta \{\omega t -(ks-\beta-2n\pi)-\alpha\}\) (where \(\beta=kB\)), ϵ was the small parameter for perturbation expansions, β was the phase distance of the δ-function pulse measured from a peak of peristaltic wave \(e(s, t)\), λ was the wave length, and α was constant. The governing equation of motion to be solved was \begin{equation} \{1+\epsilon\,p(s, t)\} \frac{\partial u}{\partial t} = D_{0}\left\{\frac{\partial^{2} u}{\partial s^{2}}+v_{0} \cos (\omega t-k s)\right\}, \end{equation} (6) where \(D_{0}=\kappa/n_{0}\) and \(v_{0}=ke_{0}\).

By assuming the perturbation expansion of solution \(u=u^{(0)}+\epsilon\,u^{(0)}+\cdots\), the governing equations with the 0-th and 1st order of ϵ were obtained as \begin{align} \frac{\partial u^{(0)}}{\partial t} = D_{0} \left\{\frac{\partial^{2} u^{(0)}}{\partial s^{2}}+v_{0}\cos(\omega t-ks)\right\}, & \end{align} (7) \begin{align} \frac{\partial u^{(1)}}{\partial t} = D_{0} \frac{\partial^{2} u^{(1)}}{\partial s^{2}}-p(s, t)\frac{\partial u^{(0)}}{\partial t}, & \end{align} (8) and solved as \begin{equation} \left\langle \frac{\partial u^{(1)}}{\partial t} \right\rangle_{\lambda} {}= - \omega A \sin \beta, \end{equation} (9) where \(\langle\bullet \rangle_{\lambda}=\frac{1}{\lambda}\int^{\lambda}_{0}(\bullet)\,ds\). The migration distance averaged over the wavelength λ is \(V=\epsilon\langle u_{t}\rangle^{(1)}_{\lambda}=-\epsilon\omega A\sin \beta =-\epsilon e_{0}\frac{\omega}{k}\cos \alpha\sin \beta\).

When the frictional anchoring took place in a compressed part of the body, the direction of migration was opposite to the direction of the peristaltic wave. Conversely, when the anchoring was in a stretched part of the body, the migration direction was the same as the peristaltic wave. The conclusion was that (1) even if the peristaltic wave propagated in the same direction, the phase difference of anchoring switched the migration direction.

From the symmetry of the equation of motion, (2) even if the phase of anchoring was the same, the migration direction was reversed when the propagation direction of the peristaltic wave was reversed. Furthermore, (3) even if both the propagation direction of the peristaltic wave and the phase of anchoring were the same, the migration direction reversed when the frictional pulse switched from anchoring (increase of friction constant) to releasing (decrease of frictional constant).

As described above, three factors contribute to the reversal of the direction of movement. Both directions of movement are inherent in the mechanism of peristaltic crawling. This physical nature gives the potential variability of behavior to the crawling cells and organisms. This insight is to be stressed, when thinking about the evolutionary development of behavioral flexibility and diversity.

Therefore, it is important to efficiently control the friction constant accompanied by the local periodic motion. If there were some mobile appendages on the body, efficient control would be possible, and the legs of animals like lugworms, centipedes, millipedes, caterpillars, and insects might be considered as the appendages that acquired such functionality.

In the future, if we can formulate the meta-clonal waves of ciliary beating in a similar framework of mechanical modeling, we will be able to describe the crawling motion and the swimming motion in a unified manner.

A microalga Chlamydomonas shows phototaxis. Its sign depends on the light intensity: positive (attractive) to weak light and negative (repulsive) to overpowering light. When it receives light, photosynthesis takes place and the redox potential within the cell changes. This redox potential controls the sign of phototaxis. There is an intracellular mechanism to switch the sign of phototaxis according to the environment.

Conversely, even under the same experimental conditions, there are cell-to-cell differences. Even under conditions where most cells show positive phototaxis, a few cells show negative or neutral motility. Individual cells that eventually exhibit positive phototaxis may initially be temporarily away from light. Although most individual cells show positive taxis, each cell takes a different option of behavioral responses.

The mechanism of such behavioral variability and diversity has been investigated with a mechanical model. Chlamydomonas has a spheroid-like body, with a single eye spot in the middle of the body that senses light from the surroundings (Fig. 2). At the tip of the body (one end of the long axis of the spheroid), two cilia move like the arms in human breaststroke, so the body moves forward. At this time, the cell body rotates around a certain axis (an axis with some angular difference from the long axis of the spheroid). This allows Chlamydomonas to trace a spiral trajectory of swimming. Inevitably, the eye spot senses light while rotating 360 degrees.

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Figure 2. Model of spiral swimming in Chlamydomonas.²⁵⁾ The coordinate system \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) was attached to the body of the organism and represented the direction of the body orientation. The axis \(\vec{c}\) was the anterior-posterior axis of the body, and it was assumed that the organism moved at a constant speed \(v_{0}\) along the \(\vec{c}\) direction. Note that the eye spot was directed toward the direction \((\vec{a}+\vec{b})/2\), and the cilium closer to this eye spot was called cis-cilium (or cis-flagellum), and the other one was called trans-cilium (or trans-flagellum). Chlamydomonas was assumed to be a rigid body and to rotate around the \(\vec{\omega}\)-axis. The body orientation of Chlamydomonas was described in the coordinate system \((x, y, z)\) fixed in the laboratory.

When the eye spot receives light (turns in the direction from which the light comes), the rhodopsin photoreceptor mechanism in the eye spot is activated, and with a short delay time, the beating patterns of cilia are changed temporarily. Then the body axis of Chlamydomonas tilts and the swimming trajectory changes. The intensity of the light received affects the temporal modulation of ciliary beating. In addition, even under the same experimental conditions, the temporal modulation of ciliary beating and the lag time from photoreception varies considerably among individuals. As described above, some variability of phototactic response is observed in Chlamydomonas.

To consider the phototactic variability, various simple models of spiral swimming have been proposed.^25,26) The Chlamydomonas body was replaced by a rigid spheroid. A Cartesian coordinate system with the long axis of the spheroid as the c-axis (with the other two axes as the a- and b-axes) was introduced to always determine the anterior-posterior, left-right, and dorsoventral axes of the body. The spheroidal Chlamydomonas was assumed to move always forward at a constant speed \(v_{0}\) in the positive direction of the c-axis while rotating around the ω-axis, which has some angular differences with the a-, b-, c-axes. The eye spot was assumed to be oriented to the direction of \((\vec{e}_{a}+\vec{e}_{b})/2\). However, here, \(\vec{e}_{a}\) and \(\vec{e}_{b}\) were unit vector in the positive direction of the a- and b-axes, respectively.

Assuming that the ω-axis tilted temporarily when the eye spot received light. In this case of photoresponse, there were three parameters: the tilting direction and magnitude of ω-axis, and the lag time between the photo-reception and the tilting. The photo-reception \(I(t)\) (photo-reception function) was given by the inner product for the direction from which the light came (parallel rays) and the direction to which the eye spot faced, \(I(t)=I_{0}(-\vec{e}_{\textit{light}}\cdot\vec{e}_{\textit{eyespot}}+1)/2\).

The rotation vector \(\vec{\omega}\) (rotation around the ω-axis) was in general decomposed into three rotation components around three axes, \(\vec{\omega}=\omega_{a} (t)\vec{a} +\omega_{b} (t)\vec{b} +\omega_{c} (t)\vec{c}\), where \(\omega_{a} (t) =\omega_{a}^{(0)}-p(t)\), \(\omega_{b} (t) =0\), and \(\omega_{c} (t) =-\omega_{c}^{(0)}\) [1/s], in a specific model.²⁵⁾ Here \(p(t)\) was photo-induced tilting of \(\vec{\omega}\), given by \(p(t)=\gamma_{0}\frac{d I(t-\tau_{0})}{dt}\), where \(\tau_{0}\) was the lag time, and \(\gamma_{0}\) was −1 or +1 for the two opposite directions of tilting (this came from the experimental results). At last, all three parameters were of the tilting magnitude \(I_{0}\), the tilting direction \(\gamma_{0}\), and the lag time of tilting \(\tau_{0}\).

The translational and rotational motion of spheroid was described in 3-d space by the space-fixed Cartesian coordinate \((\vec{x},\vec{y},\vec{z})\) as \begin{align} \frac{d\theta_{1}}{dt} = - \frac{\cos \theta_{3}}{\sin \theta_{2}}(\omega_{a}^{(0)}-p(t)), \end{align} (10) \begin{align} \frac{d\theta_{2}}{dt} = \cos \theta_{3} (\omega_{a}^{(0)}-p(t)), \end{align} (11) \begin{align} \frac{d\theta_{3}}{dt} = - \omega_{c}^{(0)}+\cos \theta_{3} \frac{\cos \theta_{2}}{\sin \theta_{2}} (\omega_{a}^{(0)}-p(t)), \end{align} (12) where \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) were Euler angles. These were the governing equations for the model.

After the study on the dynamic behavior of the model, the following was found. Depending on the parameters of \(\tau_{0}\) and \(\gamma_{0}\), these were the finding. Sign of the chemotaxis was switched. In addition, the movement of temporarily moving away from the light and finally approaching to the light source was reproduced. The swimming trajectory did not always approach towards (recede from) light monotonically. Between the photo-induced transient modulation of ciliary beating and the final expression (phenotype) of chemotaxis, there was an inherent gap that gave rise to potential behavioral diversity, and the final destination changed depending on external and internal conditions.

What was interesting from a mathematical point of view was that there existed a swimming trajectory in which the eye point always faced at the constant angle with respect to the direction in which the light ray came (steady-state solutions). There were two steady-state solutions for each positive and negative phototaxis. In a set of parameters of the model, the linear stability of these steady-state solutions was determined. For the steady-state solutions of positive or negative phototaxis, one was stable and the other was unstable. No bistable regions were present. Both of positive and negative phototaxis lay in the dynamical mechanism for spiral swimming. Such latent behavioral variability can be a source of diverse behaviors even in very similar conditions.

Rotating spiral motion observed in Dictyostelium was observed in other multicellular organisms.⁵⁰⁾ It may be seen in the contraction of the heart.^51–59) When the heart was in a disordered state of tachycardia, a rotating spiral wave of contraction was observed, and later contraction pattern of the heart could be a critical stage of fibrillation. In this case, the rotating spiral wave is a nuisance to be removed, and its effective removal method has been physically studied. Additionally, in connection with the fertilization of Xenopus oocytes, a rotating spiral wave of calcium ion was observed on the cell surface.⁶⁰⁾ The rotating spiral wave is common in both protists and vertebrates as a dynamical behavior of excitable cells.

In the formation of the transport network observed in the Physarum plasmodium, the network shape was similar to that of public transportation in the Tokyo region in the sense of multi-objective optimization: cost of total length, fault-tolerance of global connectivity against accidental disconnection of local edge, and efficiency of transportation between two sites. This similarity could be based on the similar mechanism of current reinforcement. Each edge was variable adaptively to the protoplasmic and traffic flow in Physarum and public transportation network, respectively.¹⁶⁾ It is interesting that a similar mechanism worked in these two substantially different systems although it might be coincidental. A similar mechanism to the current reinforcement is found in all of three groups of multicellular eukaryotes: fungi,⁶¹⁾ plants,^62–64) and animals.⁶⁵⁾

In addition to amoeboid tactic movement, ciliary tactic movement was observed in Chlamydomonas and was found in many other species of protist. This ciliary motion was characterized commonly by the spiral trajectory of swimming. Sperms in multicellular organisms swam using flagella (its molecular structure is the same as cilium) and showed a similar spiral trajectory to an ovum or multiple ova by sensing chemo-attractant. In the sense of spiral swimming with sensing, methods of tactic behavior by cilia could be compared in protist and multicellular organisms.

In fertilization, so many sperms swim together in a semi-closed internal space of a female body or an open space of natural habitat with an inevitable disturbance.^66–70) At the time, they not only competed but also cooperated with each other. This interaction among ciliary spiral swimmers is interesting from a hydrodynamic point of view. Collective motions of red tide algae and Euglena could be a research subject to be studied by both hydrodynamics and cell biology.^71–73) In a colony of microalgae like Gonium and Volvox, phototactic behavior was established through hydrodynamic interaction between many cilia, while similar hydrodynamic interaction could be expected in 2d sheet of ciliary cells on the surface of the mammalian airway and the surface of the internal cavity in sponges. Further, comparative studies on ciliary interaction in protists and multi-cellular organisms are needed.

The outdoor environment inhabited by protists is never uniform and flat and changes over time, considering their body size and range of migration. In this sense, they live in a sufficiently complex environment. Simultaneously, there is also the pressure of predator-prey relationships. As in some examples already mentioned in this review, protist behavior is smarter and more variable than one might expect, even though they are single-celled organisms. Consequently, their behavior shows diversity and advanced ability in problem-solving. These abilities have commonalities with cell-level behaviors in multi-cellular organisms and appear to be inherited from protists to multi-cellular organisms. Therefore, the elucidation of the diversity of cell behavior and the mechanism of problem-solving ability is useful for a basic understanding of the ability of information processing in multi-cellular organisms.

In terms of basic science, this research squarely tackles the classic question of how an organism's ability for information processing arises from the physical motion of matter. A cell's response to a single stimulus is generally simple and transient. However, due to the complexity of the environment and the active migration into new environments, cells continue to receive various stimuli in a complicated temporal sequence. Such sequential stimuli have complex effects on multiple time-scale dynamics within the cell body, resulting in diverse and smart behavioral outputs. Therefore, smart behavior can only be demonstrated in the corresponding complex environment.

To summarize, we shall conclude this review by quoting a short excerpt from the project website³⁵⁾ that includes the three keywords.

Intelligence broadly describes an ability to adapt to the environment. In this sense, single-celled organisms like protists (eukaryotic unicellular organisms) have a prototype of intelligence, or rather they can demonstrate skillful behavior in complex field environments due to their sophisticated evolution over hundreds of millions of years. This behavioral ability seems to be inherited as “single-cellular” behavior in multicellular organisms (sperm motility during fertilization, cell motility in the internal environment, etc.).

In this Research Area, we define “proto-intelligence” as the fundamental adaptability to the environment that single-celled organisms potentially possess. We name such artificial conditions as “diorama environments” where organisms can show their potential proto-intelligence. Diorama environments may mimic the complexity of a habitat but in a setup designed for testing proto-intelligence. For example, one such instance is that of an amoeboid organism of slime mold, which displays the ability to find the shortest path in a maze of diorama environments.

Because the mechanisms of proto-intelligence can often be formulated using coupled kinetic equations of cell motion and the environment, such environment-coupled mechanics will be thoroughly applied. We will challenge and advance the algorithms (heuristics) of proto-intelligence.