PROTOPLASMA
Protoplasma (1996) 194:243-249
9 Springer-Verlag 1996 Printed in Austria
Dynamic complexity in Physarum polycephalum shuttle streaming Steven J. Coggin 1"* and James L. Pazun 2 Department of Biology, Catawba College, Salisbury, North Carolina and 2 Department of Chemistry, Pfeiffer University, Misenheimer, North Carolina Received July 11, 1995 Accepted July 1, 1996
Summary. The nature of the oscillator controlling shuttle streaming in Physarumpolycephalum is not well understood. To examine the possibility of complex behavior in shuttle streaming, the time between reversal of streaming direction was measured over several hours in an intact plasmodium to produce a time series. Time series data were then used to analyze shuttle streaming dynamics. Complexity in shuttle streaming is revealed by an inverse frequency (I/y') power spectrum where the amplitude of reversals is plotted against their frequency. The complex dynamics of shuttle streaming is also shown by a trajectory in phase space typical of a strange attractor. Finally, shuttle streaming time series data have a dominant Lyapunov exponent of approximately zero. Dynamic systems with a Lyapunov exponent of zero exist in a state at the edge of chaos. Systems at the edge exhibit self-organized criticality, which produces complex behavior in many physical and biological systems. We propose that complex dynamics in Physarum shuttle streaming is an example of self-organized criticality in the cytoplasm. The complex behavior of Physarum is an emergent phenomenon that probably results from the interaction of actin filaments, myosin, ATP, and other components involved in cell motility.
Keywords: Physarumpolycephalum; Shuttle streaming; Complexity; Self-organized criticality; Emergent behavior. Introduction
The myxomycete Physarumpolycephalum has been an important model in the study of cytoplasmic streaming for over 50 years (Kamiya 1940). The plasmodial stage of Physarumis a giant multinucleate cell that may cover an area of several square centimeters. Within the anastomosing strands of the plasmodium are nuclei, vacuoles, and other organelles that can * Correspondence and reprints: Department of Biology, Catawba College, 2300 West Innes Street, Salisbury, NC 28144, U.S.A. E-mail:
[email protected]
move at rates of more than 1 mm/s. The mechanism of cytoplasmic streaming in Physarum is well characterized. Organelle movement within the plasmodium is caused by pressure-induced flow of the endoplasm sol resulting from contraction of the ectoplasm gel. Ectoplasmic contraction is caused by actomyosin sliding regulated by cytoplasmic Ca ++ concentration (Kessler 1982). A unique aspect of Physarum cytoplasmic movement is the rhythmic reversal of direction, called shuttle streaming (Tyson 1982). Cytoplasm and organelIes flow through the strands of the plasmodium toward the terminal fan, then the movement slows, stops, and reverses direction. Shuttle streaming is coordinated in Physarum with synchronous reversal throughout the plasmodium. The interval between streaming reversals ranges from several seconds to several minutes, with an average reversal time of about 1 min (Kamiya 1959). Shuttle streaming has been described as periodic or regular but the nature of the oscillator is not well understood (Smith 1994a, Wohlfarth-Bottermann 1986). The oscillator must control the rhythm of streaming over a large area in the plasmodium as well as over time to produce synchronous reversal. Candidates for the oscillator include fluctuations of Ca ++ concentration (Wohlfarth-Bottermann and Gotz von Olenhusen 1977), ATP levels (Yoshimoto et al. 1981), actin filaments (Nakajima and Allen 1965), and surface tension instability (Smith 1994a). We propose that the oscillator for Physarum shuttle streaming is the result of complex interactions in the cytoplasm.
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S.J. Coggin and J. L. Pazun: Dynamic complexity in Physarum shuttle streaming
Complexity theory has had many applications in the biological sciences from evolutionary computer simulations (Langdon 1992), epidemiology (Schaffer and Kot 1985), to ecology (Hastings et al. 1993) and physiology (Goldberger et al. 1990). An excellent use of complexity theory has been made in the analysis of cardiac dynamics in the healthy and diseased heart. Goldberger et al. (1988) have shown that in a healthy heart the interval between beats is highly irregular and adaptable allowing the heart to more easily respond to physiological or environmental changes. A diseased heart hours before a fatal heart attack has a very regular beat with little time variation in heart rate. The regularity in the diseased heart indicates lack of flexibility leading to a heart attack. Dynamic complexity occurs in systems that are far from thermodynamic equilibrium and under certain conditions show self-organization. Self-organization produces spatial and temporal patterns based on local interaction of the components that make up the system (Nicolis and Prigogine 1989). The dynamics of complex systems are neither regular like a true periodic oscillator, nor random like white noise. Complex systems can appear random, but their underlying order can be revealed by several types of analysis. One technique to detect complex behavior is to use time series data to produce a power spectrum. A power spectrum is the plot of the time series amplitudes versus frequencies. Periodic systems show high amplitude peaks at certain frequencies in their power spectrum while complex systems have a wide range of amplitude peaks at many frequencies. A common feature in complex systems is an inverse frequency (l/f) power spectrum where the amplitude of the spectrum is inversely proportional to frequency. At low frequencies a l/f power spectrum has high amplitudes that decrease at higher frequencies (Van Vleit 1987). Many complex systems including earthquake magnitude (Bak and Tang 1989), extinction events (Nicolis and Prigogine 1989), human cognition (Gilden et al. 1995), and normal heart function (Goldberger et al. 1988) exhibit such l/f power spectra. Complex systems are also characterized by local interaction of their components resulting in emergent behavior which cannot be predicted a priori based on an understanding of the system's components. A chemical system exhibiting emergent behavior is the Belouzov-Zhabotinsky (BZ) reaction. In this reaction, cerium ions catalyze the oxidation of an organic fuel by bromate that may produce two distinct emergent outcomes. In a stirred system, the BZ reaction
produces quasi-periodic color changes that are in essence a noisy chemical clock. In an unstirred system, the BZ reaction produces spatial patterns of color that appear as rings, waves, or spirals. The patterns of the BZ reaction, both temporal and spatial, emerge from the complex chemical interaction of the reagents (Winfree 1973). Spiral waves of intracellular Ca ++ have been reported in amphibian oocytes (Lechleiter et al. 1991) and in aggregating amoebae of the cellular slime mold Dictyostelium discoideum (Goldbeter and Martiel 1987). These biological examples of emergence result from interactions at the cellular and intercellular levels to produce complex forms and behavior. The dynamics of a system can be analyzed by the behavior of its attractor in phase space. The phase space attractor of a system is a map of the changing conditions in the system. Each point on the attractor is a summary of all the variables affecting the system at a moment in time. As the system evolves, changes in these variables result in a different location of the point in phase space. The points in phase space trace a trajectory that summarizes the changes of the system. A system with regular oscillations, like a pendulum, traces a regular, closed trajectory in phase space. Such a trajectory, called a limit cycle attractor, reflects the regular motion of the pendulum as it returns to the same starting point. In contrast, a random system produces a stochastic pattern of points in phase space, tracing out an attractor with no discernable pattern. The trajectory dynamics of a complex system, like a normal heart or the BZ reaction, when mapped in phase space is called a strange attractor. A strange attractor has orbits that lie within a defined region of phase space but the orbits never intersect and never follow the same trajectory twice (Packard et al. 1980). Another tool to describe the dynamics of a system is the Lyapunov exponent. The Lyapunov exponent of a system is a measure of the rate of divergence between adjacent points on an attractor. When applied to a system with regular oscillations like a pendulum, adjacent points on its limit cycle attractor do not diverge with time and yield a negative Lyapunov exponent. In a chaotic system, adjacent points on its attractor diverge exponentially with time producing a positive Lyapunov exponent (Nicholis and Prigogine 1989). Between these two limiting cases, chaotic and periodic, lie complex systems exhibiting self-organized criticality. A self-organized system evolves to the point where it is poised to exhibit a wide range of response
S. J. Coggin and J. L. Pazun: Dynamic complexity in Physarum shuttle streaming
to perturbation and typically has a l/f power spectrum (Bak and Chen 1991). Complex systems exist in a state called the ,edge of chaos that lies between the periodic and the chaotic and have a Lyapunov exponent near zero (Langdon 1992). Many physical and biological systems exhibit self-organized criticality including weather patterns (Tziperman et al. 1994), earthquakes (Bak and Chang 1989), chemical reactions (Scott 1991), organ function (Babloyantz and Destexhe 1986, Goldberger et aI. 1988), and morphogenesis (Goodwin and Kauffman 1990, Goodwin and Briere 1992). Ueda et al. (1990) proposed that shuttle streaming in Physarum represents self-organization in a system far from equilibrium. We present evidence that the temporal pattern of shuttle streaming in P. poIycephalum is a complex system and that shuttle streaming is an emergent behavior resulting from self-organization in the cytoplasm.
Material and methods Organism and culture conditions Physarum polyeephalum was cultured from dried sclerotia (Carolina Biological Supply Company, Burlington, NC) on 2% water agar in 15 • 100 mm petri plates. A piece of sclerotium on filter paper was placed at the center of an agar plate and two rolled oat flakes were added to opposite sides of the petri plate, equidistant from the sclerotium. The cultures were incubated at 22-25 ~ in the dark. Young plasmodia germinated from scterotia in 8 to 24 h began shuttle streaming and actively migrated across the surface of the agar. The migrating plasmodia usually had multiple anterior fans and numerous strands of protoplasm. One or both of the rolled oat flakes were engulfed by the plasmodium at the time of streaming reversal measurements. Sporulation usually occurred between 72 and 96 h after germination.
Microscopy and time series determination Time series measurements of streaming reversal were made using young plasmodia, 8-30 h after germination. No measurements were made after a culture was 48 h old to avoid the effects sporulation might have on streaming dynamics. A major strand of the plasmodiurn, 50-t00 ~tm in diameter, was chosen and the movement of organcUes past a point on the strand was observed with a Reichert-Jung compound light microscope (Leica Inc., Deer field, IL). The time between reversals of cytoplasmic streaming direction was measured to the nearest 0.01 s. The time between reversals was recorded and pIotted against elapsed time to produce the time series. A typical time series data set contained 200 to 350 points and covered 1M h. The power spectrum of streaming reversal was determined from a time series using the Fast Fourier Transform (FFT). The FFT decomposes a complex function into its component frequencies with corresponding amplitudes. The results of the FFT were graphed as a loglog plot of mean-square amplitude versus frequency (Nussbaumer 1982). A linear least squares fit of the power spectrum was made and the slope of the regression line, b, was calculated. A three-dimensional phase space diagram of the attractor describing
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the time series was produced using the time delay method of Packard et al. (1980). The time series gives the interval between streaming reversals (T,, Tn+l, Tn+2, T,~,...) with differences in reversal times reflecting the interaction between cytoplasmic components involved in shuttle streaming. The one-dimensional time series, and thus all the factors affecting it, can be represented by the trajectory of points in three-dimensional phase space. The coordinates of the first point of the attractor in phase space was derived from the first point in the time series (Tt), the second point in the time series (T2), and the third point in the time series (T3). The point with these coordinates (T~ = T,,, T2 = T,,+j, T3 = T,,+2) was plotted on the three-dimensional phase space diagram. This procedure was repeated for each successive point in the time series and the resultant points were connected producing the phase space trajectory. Lyapunov exponents of the attractor were determined using the program "Lyapunov Exponent for Noisy Nonlinear Systems" (LENNS) (Ellner et al. 1992). This program uses a neural network to calculate the dominant Lyapunov exponeut and standard deviation for time series data over a range of embedding dimensions. At embedding dimension d = 1 the divergence of adjacent points on the attractor is measured. At successively higher embedding dimensions (d = 2,3, 4,...) the divergence of a point on the attractor is compared with those 2,3,4,... points away. The Lyapunov exponent should remain constant at and above the embedding dimension large enough to contain the attractor. For example, an attractor with a dimension of 3.5 should have a constant Lyapunov exponent above embedding dimension 4.
Results and discussion
We have found that the oscillator controlling shuttle streaming in Physarum polycephalum exhibits complex dynamics rather than being periodic or random. Coordination of shuttle streaming over long ranges of both time and space is an emergent phenomenon resulting from self-organized criticality in the cytoplasm. The cytoplasm of Physarum, like that of all eukaryotes, is structurally and chemically complex (Alberts et al. 1994). Interaction between membranes, organelle compartments, ions, small organic molecules, proteins, and the genome can produce complex dynamics at the cellular level (Hess and Mikhailov 1994). Evidence for dynamic complexity in Physarum comes from the analysis of shuttle streaming time series data. Three different lines of evidence show complex behavior in shuttle streaming; a 1/f power spectrum, a strange attractor in phase space, and a Lyapunov exponent of zero. These results show streaming exhibits complex dynamics rather than being periodic or random. Sixteen time series data sets of Physarum shuttle streaming were collected and all showed the characteristics of dynamic complexity. The behavior of the system was not affected by whether the plasmodium
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TIME (s) Fig. 1. A time series of streaming reversals in an intact plasmodium of Physarumpolycephalum. The time between reversal of streaming direction is plotted against elapsed time over a 4 h period. The time series shows a wide variation in the interval between reversals that range from less than 1 to over 70 s. No obvious periodic structure is evident in the time series
had engulfed none, one, or both of the rolled oat flakes on the petri dish. A single time series consisting of 350 points (Fig. 1) is analyzed here. The time series in Fig. 1 shows the intervals between streaming reversals in a single strand of cytoplasm over a 4 h period. The time between reversals ranges from less than 1 s to over 70 s with an average of 48 s. In some times series the interval between reversal of streaming direction can range over three orders of magnitude in intact plasmodia. A comparable range of variability in reversal times is found in isolated plasmodial strands measured in a double chamber experiment (Kamiya 1959) and so is not due to plasmodial feeding or migration. If shuttle streaming is controlled by a regular oscillator, the time series would take the form of a regular curve and would not show the wide variations in time between reversals seen in Fig. 1. Since a similar degree of variability in the time between reversals is found in both isolated strands of cytoplasm and intact plasmodia the variability is an intrinsic characteristic of Physarum shuttle streaming.
Power law behavior The time between reversals in shuttle streaming is not random but shows a definite pattern revealed by the power spectrum of the time series. A random system, such as one producing Gaussian white noise, would have a straight line power spectrum where amplitude is independent of frequency and a slope of b = 0 (Schroeder 1991). A system with periodic dynamics
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-1 0.0 0.5 1.0 1.5 2.0 2.5 LOG FREQUENCY Fig. 2. Power law behavior in Physarum shuttle streaming. Fast Fourier Transform of points (n = 350) from the time series in Fig. 1 was carried out. A log-log plot of the transform was produced and a linear least squares fit of the spectrum was made. The slope of the best fit line is b = -0.92 that lies within the 95% confidence interval of a true inverse frequency (l/j) spectrum
would show a peak at certain frequencies but complex systems have a power spectrum that lacks a predominant frequency but rather covers a broad range of frequencies. The power spectrum of shuttle streaming dynamics derived from the FFT of the time series data set (Fig. 2) shows a broad band spectrum with high amplitudes at low frequencies and lower amplitudes at higher frequencies. A linear least squares fit of the power spectrum has a slope b - - - 0 . 9 2 . An ideal 1/f line (b = -1.00) falls within the 95% confidence interval for the best fit line. Shuttle streaming reversal in Physarum is therefore described by a 1/frequency (l/j*) power law, which is a strong indicator of complex dynamics. A 1/f power spectrum is also consistent with a system showing self-organized criticality where slight perturbations cause many events at low frequency and fewer events at higher frequencies (Bak and Chen 1991).
Phase space diagram The phase space portrait of the attractor produced by the time delay method (Fig. 3) is also indicative of complexity. The attractor has an irregular shape with orbits that return to the same area of phase space but not to the same point. Most of the orbits remain in the central area of the attractor but some orbits extend away from the center indicating a noisy system. This behavior in phase space is characteristic of a dynamic system described by a strange attractor (Nicolis and Prigogine 1989). Each point on a phase space diagram gives a summary of the dynamic state of the system. Every cellular component that plays a role in the dynamics of the
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Fig. 3. Phase space portrait of Physarum shuttle streaming dynamics. The time series data in Fig. 1 were used to produce the attractor in three-dimensional phase space by the time delay method. The coordinates for the first point on the attractor were determined using the first point in the time series (T 0 plotted on the T,, axis, the second point in the time series (T2) plotted on the T,,+I axis, and the third point in the time series (T3) plotted on the/',,+2 axis. This procedure was repeated for each successive point in the time series and the points were connected producing the attractor. This procedure converts the one-dimensional information in the time series to a threedimensional attractor reflecting the changing dynamics of shuttle streaming
system affects the position of the point in phase space. As the system changes, the point follows a trajectory in phase space that reflects these changes. Smith (1994b) modeled the dynamics of shuttle streaming measured in a double chamber experiment. Smith's model shows an oscillator with the regular rhythm of a periodic system controlled by cytoplasmic Ca ++ concentration with a phase space diagram typical of a limit cycle attractor. The phase space diagram from our time series data (Fig. 3) has characteristics of a strange attractor, indicating complex rather than periodic dynamics. A mathematical model of shuttle streaming would be useful in testing hypotheses on the mechanism of this process. To successfully simulate shuttle streaming, the dynamic nature of the plasmodium would need to be taken into account. A model of shuttle streaming should include all the cytoplasmic components involved in streaming, their changing concentrations, and their interactions.
Lyapunov exponent The dominant Lyapunov exponent for the time series in Fig. 1 was calculated over a range of embedding dimensions (Fig. 4). At low embedding dimensions the Lyapunov exponent is negative but it asymptotically approaches zero as the embedding dimension increases. By embedding dimension 4, the Lyapunov exponent is statistically indistinguishable from zero.
Fig. 4. Lyapunov exponent of the time series in Fig. 1. The program LENNS was used to calculate the average Lyapunov exponent (LE) plotted against embedding dimension (d). The LE should remain constant at embedding dimensions greater than the dimension of the attractor. A negative LE indicates a periodic system, a positive LE indicates a chaotic system, and LE = 0 indicates a complex system. Error bars are the standard deviation at each embedding dimension
Saturation of the Lyapunov exponent by embedding dimension 4 is significant because it agrees with the fractal dimension of the attractor (D ~ 3.5) calculated from the integral correlation function of the attractor (Coggin and Pazun unpubl. 1995). A Lyapunov exponent of zero shows shuttle streaming is neither regular nor chaotic but complex. Systems with a Lyapunov exponent of zero are associated with a state called the edge of chaos, where complex behavior is the rule. The edge of chaos has been found in computer models of coevolution (Kauffman and Johnsen 1991) and cellular automata (Langdon 1992). These computer simulations exhibit many properties of life including response, adaptation, and emergent behavior.
Conclusion Complex chemical systems, such as the BZ reaction, exhibit emergent oscillations that have both temporal and spatial components (Winfree 1973). Like the BZ reaction, Physarum oscillations are coordinated spatially and temporally producing simultaneous reversal throughout the plasmodium. No single pacemaker is needed to control this oscillation in Physarum since it emerges from the local interactions in the cytoplasm. Complex behavior allows a system to respond quickly to perturbation and return to a normal state (Goldberger et al. 1988). The irregular rhythm of shuttle streaming, with its complex dynamics, can be effectively maintained despite fluctuations in environmental conditions and internal physiological state. Rather than being under the control of a single cytoplasmic entity, shuttle streaming is the result of lineraction between many cellular components. These
S. J. Coggin and J. L. Pazun: Dynamic complexity in Physarum shuttle streaming
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may include actin filaments and their polymerization dynamics, soluble myosin and its state of phosphorylation, intracellular Ca ++ concentration, kinase activity, ATP levels, and actin-binding proteins. Interactions of cytoplasmic components could cause emergent behavior such as synchronous reversal of streaming. In the future we plan to experimentally manipulate these cytoplasmic components and examine the resulting changes in the dynamics of shuttle streaming. It may be possible to shift the dynamics of shuttle streaming to a periodic or chaotic regime by altering the plasmodial motility. Shuttle streaming in Physarurn is probably a particularly easy to observe case of dynamic complexity that is characteristic of all cells,
Acknowledgements We wish to thank Ms. Julie Flatter, Holly Bowers and Jennifer Hale for their excellent technical assistance. We would also like to thank Dr. Randy Wayne for his critical reading and helpful suggestions on the original manuscript. Lyapunov exponents were determined using the facilities of the North Carolina Supercomputing Center, Research Triangle Park, NC and their generous support is gratefully acknowledged.
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