Control of interaction strength in a network of the true slime mold by a microfabricated structure

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1 BioSystems 55 (2000) Control of interaction strength in a network of the true slime mold by a microfabricated structure Atsuko Takamatsu a, *, Teruo Fujii a b, Isao Endo a a Biochemical Systems Laboratory, RIKEN (The Institute of Physical and Chemical Research), 2-1, Hirosawa, Wako-shi, Saitama, , Japan b Institute of Industrial Science, Uni ersity of Tokyo, , Roppongi, Minato-ku, Tokyo, , Japan Abstract The plasmodium of the true slime mold, Physarum polycephalum, which shows various nonlinear oscillatory phenomena, for example, in its thickness, protoplasmic streaming and concentration of intracellular chemicals, can be regarded as a collective of nonlinear oscillators. The plasmodial oscillators are interconnected by microscale tubes whose dimensions can be closely related to the strength of interaction between the oscillators. Investigation of the collective behavior of the oscillators under the conditions in which the interaction strength can be systematically controlled gives significant information on the characteristics of the system. In this study, we proposed a living model system of a coupled oscillator system in the Physarum plasmodium. We patterned the geometry and dimensions of the microscale tube structure in the plasmodium by a microfabricated structure (microstructure). As the first step, we constructed a two-oscillator system for the plasmodium that has two wells (oscillator part) and a channel (coupling part). We investigated the oscillation behavior by monitoring the thickness oscillation of the plasmodium in the microstructure with various channel widths. It was found that the oscillation behavior of two oscillators dynamically changed depending on the channel width. Based on the results of measurements of the tube dimensions and the velocity of the protoplasmic streaming in the tube, we discuss how the channel width relates to the interaction strength of the coupled oscillator system Elsevier Science Ireland Ltd. All rights reserved. Keywords: Physarum polycephalum; Coupled oscillators; Microfabrication 1. Introduction Nonlinear dynamics becomes important for the studies of collective cells, for example, cardiac cells, neurons and other oscillatory or excitable cells (Winfree, 1980). In these biological systems, * Corresponding author. Tel: ; fax: address: takamatu@cel.riken.go.jp (A. Takamatsu) the connections between the elements are not highly designed individually. Nevertheless, the collectives of such elements self-organize to show various biological functions, such as information processing in nervous systems and cooperative beating in cardiac systems. In order to understand the mechanism of biological self-organization, it is important to investigate the behavior of the systems under conditions in which the connections of the elements can be systematically controlled. In /00/$ - see front matter 2000 Elsevier Science Ireland Ltd. All rights reserved. PII: S (99)

2 34 A. Takamatsu et al. / BioSystems 55 (2000) Fig. 1. Microstructure for coupled plasmodial oscillators. (a) Two-oscillator system. (b) Cross section of A A in (a). (c) Cross section of B B in (a). this study, we adopted the true slime mold, Physarum polycephalum, as a living nonlinear element. The plasmodium of P. polycephalum is a huge amoeboid multinucleated unicellular organism. The plasmodium is an aggregate of endoplasm without any highly differentiated structure like a nervous system. Multiple plasmodia can be fused into a single one and a single plasmodium can be easily divided into multiple ones. Despite its simple structure, the plasmodium demonstrates sophisticated biological functions. Namely, a large cluster of the plasmodium is able to behave as a single entity based on the information from its surroundings. The information acquired by a local partial body drives the cooperative behavior by transmitting the information throughout the whole cluster. The information is considered to be processed through the interaction among the partial bodies. This leads to, for example, the attracting/escaping behavior of the plasmodium to/from attractants/repellents (Knowles and Carlile, 1978). These functions, information transmission and processing, are said to be realized through the intrinsic oscillatory phenomena in the plasmodium, such as thickness oscillation (Baranowski, 1978) which synchronizes with the shuttle streaming of the protoplasm (Kamiya, 1950), and the oscillation in the concentrations of ATP (Yoshimoto et al., 1981) and Ca 2+ (Natsume et al., 1992). These phenomena are supposed to be generated by mechanochemical reactions caused by complicated interactions among intracellular chemicals, proteins, organelles, etc. (Ueda et al., 1986). From the viewpoint of nonlinear dynamics, the plasmodium can therefore be modeled as a collective system that consists of nonlinear oscillators interconnected with certain interaction strength. In this model, an oscillator can be defined for each partial body in the plasmodium, and the interaction among the oscillators would be related to the protoplasmic streaming observed inside the tube structures which interconnect the partial bodies. It is therefore expected that, by controlling the growth pattern and the dimensions of the tube structure, we could observe various types of collective behavior which could help us to understand the characteristics of the plasmodium as a nonlinear dynamic system. In this paper, we describe the advanced patterning method of the plasmodium by a microfabricated structure (microstructure) and the results of observation of the plasmodium in the microstructure for a two-oscillator system, as the first step. 2. Materials and methods (Takamatsu et al., 1998, 2000) 2.1. Geometry control of the plasmodium The plasmodium prefers a wet medium such as an agar plate (1.5%), on which the plasmodium is usually cultured, to a dry one. It is possible to control the geometry of the plasmodium by using the agar plate covered with a dry medium having variously designed openings. As the dry medium, we adopted an ultra-thick negative photoresist resin (NANO SU-8 50, Microlithography Chemical) which enabled us to fabricate microstructures with about 10 m precision that is the same order as the tube diameter of the plasmodium in the natural situation. The microstructures for the two-oscillator system is a thick sheet ( 100 m) having an opening consisting of two circular wells (oscillator part) connected by a channel (coupling part) as shown in Fig. 1. The diameters of the wells are determined to be 2 mm

3 A. Takamatsu et al. / BioSystems 55 (2000) In order to enhance the change in thickness in the images obtained by the above method, we calculated differential images between the two images at time t and t+ t ( t=4 s), where the black and white images mean an increase and decrease in the thickness of the plasmodium, respectively (Fig. 3(a)) Measurement of the tube diameter and elocity of the protoplasmic streaming Fig. 2. Growing process of the plasmodium on the microstructure. (Fig. 1(b)), so that we can consider the plasmodium in a well as a single oscillator. The size is comparable to half of the spatial wavelength of oscillation in the natural plasmodium and, within this area, the plasmodium shows almost synchronized oscillation. The plasmodium was cultured on 1.5% agar medium with oatmeal as the food. The tip portions of the plasmodium were placed at the both wells in the microstructure on the agar plate. The plasmodia in the wells grew along the channel pattern during the culture for 5 15 h under the conditions of 25 C and RH 85%, as shown in Fig. 2. Finally, the plasmodia from both sides of the wells contacted at around the center of the channel to be physically connected through the tube structure (Fig. 1(a)), where the endoplasmic streaming was observed Obser ation of the thickness oscillation The plasmodia grown in the microstructure were placed under a stereoscopic microscope (SMZ-2T, Nikon). The transmitted light through the plasmodium was detected by a CCD camera (C2400, Hamamatsu). The image data converted by the flame grabber (LG-3, Scion) were sequentially stored in the PC (Power Macintosh G3, Apple) every 4 s. We have confirmed that the change in the transmitted light intensity is inversely proportional to the change in thickness within the range of thickness oscillation. The experiments were performed in a thermostat and humidistat chamber (PR-2K, ESPEC) under the condition of C and RH %. The plasmodia with the microstructure were placed under a inverted microscope (TMD series, Nikon, objective lens 10 or 20) and the protoplasmic streaming in the tube structure was observed. The images were recorded by the VCR (CCD-TRV92, Sony). The tube diameter was defined as the width of the area where the protoplasmic streaming was observed. The velocity of the protoplasmic streaming was determined by measuring the time that organelles observed at around the center of the tube structure passed a certain distance on the video monitor. 3. Results and discussion The thickness oscillation of the plasmodium on the microstructure for the two-oscillator system was observed over various channel width (W) ranges from 50 to 800 ( m) where the channel length was fixed at L=4 (mm). Fig. 3(a) shows the spatio-temporal patterns of the change in the thickness oscillation when W=400 ( m). Alternating wave propagation from one plasmodial oscillator to another was observed. The upper graph of Fig. 3(b) shows the time course of the thickness of each oscillator that was obtained by averaging the images over each well. The lower graph of Fig. 3(b) shows the time course of the phase difference between the two oscillators that was determined by a conventional method (for example, Nakata et al., 1998). These graphs demonstrate that the two oscillators are synchronized and the phases are locked in anti-phase. When W is smaller (W= m), the phase difference sometimes comes off the anti-phase line (180 ) as shown in the lower graph of Fig. 3(c),

4 36 A. Takamatsu et al. / BioSystems 55 (2000) which can be categorized as quasi-periodic oscillation. When W is much smaller (W= m), the oscillators were not synchronized and oscillated in their own periods (Fig. 3(d)). In the extremely narrow channel below 50 m in width, the tube structure was never formed. On the contrary, in a wide channel of more than 400 m in width, the oscillations became very complicated. Here, the wave propagation was observed in directions other than the major axis of the channel. This oscillation behavior depending on the channel width suggests that the interaction strength between the oscillators is controlled by the channel dimension of the microstructure in a systematic manner. In order to confirm how the interaction strength is controlled by the channel width, we investigated the relation of the channel width to the diameter of the tube structure in the channel. The plasmodium in the channel formed a single tube structure up to W=400 m and the diameter increased depending on W as shown in Fig. 4. Beyond this critical width, the increase in the tube diameter became saturated and the plasmodium in the channel formed a dendritic tube structure (Takamatsu et al., 2000). The complicated oscillation behavior in the wide channel would be caused by the dendritic tube structure. Hereafter, we considered only the case under the critical channel width (W=400 m). Fig. 5(a d) show the time course of the velocity of the protoplasmic streaming with various channel widths, where periodic changes in the velocity can be observed. In addition, the amplitude of the velocity is dependent on the channel width. Fig. 5(e) shows the averaged absolute velocity of the Fig. 3. Thickness oscillations of the plasmodium on the two-oscillator system. (a) Spatio-temporal pattern of the oscillation. (b), (c) and (d) Thickness oscillation in each oscillator (upper graphs) and phase difference between the two oscillators (lower graphs). (a) and (b) W=400 m, L=4 mm. (c) W=200 m, L=4 mm, (d) W=100 m, L=4 mm.

5 A. Takamatsu et al. / BioSystems 55 (2000) Fig. 4. Relation of channel width W to tube diameter. The data were obtained by averaging the original data over ten points. This result indicates that the amount of protoplasm transported by the streaming is controlled by the channel width. The amount transported by the protoplasmic streaming is estimated to be about 0.1 l per half of the period of oscillation, when W=400 m (the tube diameter is about 100 m and the velocity is about 0.25 mm/s). On the other hand, the amount of the protoplasm changed according to the thickness oscillation in the oscillator part, whose change was about 100 m inour observation, is estimated to be about 0.3 l. This is the same order as the estimated amount calculated from the velocity of the protoplasmic streaming. It can therefore be considered that the protoplasmic streaming directly affects on the change in the thickness of the oscillator parts. The anti-phase oscillation which appears in the system with the relatively large channel width (W=400 m) can be explained by the fact that the effects of the transportation of the protoplasm on the thickness in the oscillator parts are reciprocal. Consequently, the thickness in one oscillator part increases, and that in another decreases. It is concluded that the interaction strength between the oscillators is closely related to the amount of the protoplasm transported by the streaming and, accordingly, is also related to the channel width. 4. Conclusion Fig. 5. Relation of channel width W to protoplasmic streaming inside the tube. (a) W=400 m, (b) W=300 m, (c) W=200 m, (d) W=100 m. (e) Averaged velocity of the absolute value over 10 periods of oscillation for each channel width. protoplasmic streaming. As the channel width becomes larger, the averaged velocity increases. In conclusion, we succeeded in controlling the growth pattern of the Physarum plasmodium by the microfabricated structure. Using this microstructure for a two-oscillator system, we elucidated that the strength of interaction among the plasmodia depends on the tube dimensions which can be systematically controlled by the channel width. With this system, it is also possible to integrate a stimulator and observe the response to stimuli in a systematic way. Furthermore, rich oscillation behavior will be observed using the 2-D microstructures as shown in Fig. 6, which gives us further information about the system as a collective nonlinear oscillator system.

6 38 A. Takamatsu et al. / BioSystems 55 (2000) Fig. 6. Multiple-oscillator system. (a) Triple oscillators. (b) Multiple oscillators. Acknowledgements The authors thank Dr Hosokawa for helping in the microfabrication process. This study was partly supported by Special Coordination Funds for promoting Science and Technology of the Science and Technology Agency of the Japanese Government. References Baranowski, Z., The contraction-relaxation waves in Physarum polycephalum plasmodia. Acta Protozool. 17, Kamiya, N., The rate of the protoplasmic flow in the myxomycete plasmodium. II. Cytologia 15, Knowles, D.J.C., Carlile, M.J., The chemotactic response of plasmodia of the myxomycete Physarum polycephalum to sugars and related compounds. J. Gen. Microbiol. 108, Nakata, S., Miyata, T., Ojima, N., Yoshikawa, K., Self-synchronization in coupled salt-water oscillators. Physica D 115, Natsume, K., Miyake, Y., Yano, M., Shimizu, H., Development of spatio-temporal pattern of Ca 2+ on the chemotactic behavior of Physarum plasmodium. Protoplasma 166, Takamatsu, A., Fujii, T., Hosokawa, K., Endo, I., Microfabricated structure for nonlinear dynamics observation of cell network, Proceedings for the 20th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp Takamatsu, A., Fujii, T., Yokota, H., Hosokawa, K., Higuchi, T., Endo, I., Controlling the geometry and the coupling strength of the oscillators system in Physarum plasmodium by microfabricated structure. Protoplasma, in press. Ueda, T., Matsumoto, K., Akitaya, T., Kobatake, Y., Spatial and temporal organization of intracellular adenine nucleotides and cyclic nucleotides in relation to rhythmic motility in Physarum plasmodium. Exp. Cell. Res. 162, Winfree, A.T., The Geometry of Biological Time. Springer-Verlag, NY, Heiderberg, Berlin. Yoshimoto, Y., Sakai, T., Kamiya, N., ATP oscillation in Physarum plasmodium. Protoplasma 109,