Journal of Theoretical Biology 364 (2015) 260–265
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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Nonlinearity in cytoplasm viscosity can generate an essential symmetry breaking in cellular behaviors Masashi Tachikawa a,n, Atsushi Mochizuki a,b a b
Theoretical Biology Laboratory, RIKEN, Japan Core Research for Evolutional Science and Technology, Japan Science and Technology Agency, Japan
H I G H L I G H T S
Protoplasmic flow in a giant ameboid cell is modeled and the impact of nonlinearity in viscosity on its behaviors is considered. The nonlinearity in the viscosity generates a novel type of symmetry breaking in the protoplasmic flow. The symmetry breaking causes the interaction between different behavioral modes implemented on different time scales. The symmetry breaking makes transportation of molecular signals from the front to the rear of the cell during the locomotion.
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Article history: Received 11 March 2014 Received in revised form 6 September 2014 Accepted 16 September 2014 Available online 27 September 2014
The cytoplasms of ameboid cells are nonlinearly viscous. The cell controls this viscosity by modulating the amount, localization and interactions of bio-polymers. Here we investigated how the nonlinearity infers the cellular behaviors and whether nonlinearity-specific behaviors exist. We modeled the developed plasmodium of the slime mold Physarum polycephalum as a network of branching tubes and examined the linear and nonlinear viscous cytoplasm flows in the tubes. We found that the nonlinearity in the cytoplasm's viscosity induces a novel type of symmetry breaking in the protoplasmic flow. We also show that symmetry breaking can play an important role in adaptive behaviors, namely, connection of behavioral modes implemented on different time scales and transportation of molecular signals from the front to the rear of the cell during cellular locomotion. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Physarum polycephalum Locomotion Cytosol flow Symmetry breaking
1. Introduction Cellular materials frequently show complex responses to external perturbations, which are essential for cell–environment interactions. For example, cytoplasm displays non-Newtonian fluid properties, such as nonlinear viscosity and yield stress (Kamiya, 1950; Sato et al., 1983; Guy et al., 2011; Tsai et al., 1993). Such complex intracellular responses arise from the complex structures and behaviors of the constituent biopolymers (Eisenberg, 1990). By controlling the amount, localization and interactions of biopolymers, cells control the responses of their materials to external perturbations, and thereby design their adaptive behaviors. In this paper, we consider how the nonlinearity in the cytoplasm's viscosity can affect ameboid cell behaviors, focusing on Physarum plasmodium cells. n Corresponding author. Present address: Theoretical Biology Laboratory, RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan. Tel.: þ 81 48 467 9546; fax: þ81 48 462 1709. E-mail address: [email protected] (M. Tachikawa).
http://dx.doi.org/10.1016/j.jtbi.2014.09.023 0022-5193/& 2014 Elsevier Ltd. All rights reserved.
The plasmodium of Physarum polycephalum is a multinucleated cell growing up to several centimeters that self-organizes into complex shapes (Wohlfarth-Bottermann, 1979; Nakagaki and Guy, 2008). Its cytoplasm contains abundant actin and myosin filaments, which form meshes of various rigidities depending on the intra- and extra-cellular environment (Kamiya et al., 1988; Ueda et al., 1990). These meshes are cross-linked into a hard gel (the cytogel) that forms a vein-like structure; a large network of branching tubes through which the cytoplasm (cytosol) flows. The diameter of a developed tube is the order of 0.1 mm (Wohlfarth-Bottermann, 1979). At the periphery of the network, the cytoplasm spreads into a sheet formation (Kamiya et al., 1988). Physarum plasmodium also exhibits self-organized behaviors. The peripheral cytoplasmic sheet undergoes cycles of gelation/ contraction and solation/relaxation. The cycles are generated by a mechano-chemical oscillator involving the extent of cross-linking in the acto-myosin mesh (Wohlfarth-Bottermann, 1979; Yoshimoto et al., 1981a, 1981b; Oster and Odell, 1984; Yoshiyama et al., 2009). The oscillating cross-linking in the mesh is reflected in the cell thickness, which oscillates with a period of approximately 100 s. This
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mechano-chemical oscillator performs as a periodic pump, shuttling the streaming flow of cytosol into the cytogel tube network (Odell, 1984). The speed of the streaming flow can reach up to 1 mm/s (Nakagaki and Guy, 2008). Oscillation frequently induces simultaneous wave patterns across the entire cell (Takamatsu et al., 1997; Takagi and Ueda, 2010) and may be triggered by environmental stimuli (Nakagaki et al., 2000; Takagi and Ueda, 2008). The wave pattern is thought to correlate with the cytosol flow pattern, although simultaneous occurrence of both patterns has yet to be directly observed. On a slower timescale than the oscillatory behavior, the cytoplasm is transported in a direction selected by the net flow, namely, the difference in the flow volumes between opposing shuttle streaming flows. Such directional transportation deforms the cell and alters its migration direction (Satoh et al., 1985; Matsumoto et al., 2008). On yet slower scales, the cell adapts its behaviors in response to various environmental stresses, such as light, attractant/repellent substances and ground condition (Takamatsu et al., 2009; Tero et al., 2010; Ueda et al., 2011). Since Physarum plasmodium lacks a nervous system or similar apparatus, organized movement through the entire cell body is thought to occur by self-organization of spatially distributed mechanochemical dynamics. That is, the cell dynamics are governed by the physical properties of the intracellular chemical substances. Therefore, Physarum polycephalum is a suitable organism for investigating the relationship between the physical properties of cellular materials and their adaptive behavior, which remains largely unknown. Numerous mathematical models have been proposed for understanding the mechanical properties of Physarum plasmodium (Tsai et al., 1993; Oster and Odell, 1984; Teplov et al., 1991; Smith, 1994; Yamada et al., 1999). Most of these models (Oster and Odell, 1984; Teplov et al., 1991; Smith, 1994; Yamada et al., 1999) assume linear viscosity of the cytoplasm. Because previous models qualitatively reconstruct certain cellular behaviors, it is possible that nonlinearity exerts a second-order effect on behavioral properties. However, linear viscosity models cannot reveal nonlinear-specific behaviors. For example, a cytoplasm model with yield stress was recently proposed to realize tube formation in developing plasmodium cell (Guy et al., 2011). Here we question whether the nonlinear viscosity of the cytoplasm affects the relationships among different behavioral modes operating over different time scales. Analogous to the nervous system (Klimesch et al., 2010), the state of the rapid oscillatory mode should govern the slow mode behaviors. Information transport between modes with different time scales will enable Physarum to generate selforganized adaptive behaviors across the entire cell. The present paper investigates this idea in simplified models of plasmodium cells. To this end, we implemented linear and nonlinear viscous cytosol in the constructed models, and tested whether the model behaviors qualitatively differ between the linear and nonlinear models. We found that nonlinearity in the cytosol viscosity causes symmetry breaking in the cytosol streaming flow, followed by accumulation of cytosol at one side of the cell on a slower timescale.
2. Cytosol flow in a dumbbell shaped cell We model the Physarum plasmodium as a dumbbell consisting of three parts; two circular sheets of protoplasm (chambers) connected by one straight cytogel tube (Fig. 1A). Both chambers act as pumps, inducing periodic pressure changes by contraction– relaxation oscillations to stream the cytosol through the connecting tube. This model mimics the simplest Physarum with a developed tube. Physarum can also be experimentally manipulated
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Fig. 1. Time-averaged flow rate of nonlinear viscous cytosol in the tube of a dumbbell- shaped Physarum model. (A) Schematic of the model. (B) Time-averaged flow rate of power law fluid with exponent 3, versus phase difference between the chambers ðΔφ ¼ φ1 φ2 Þ. The inset plots the flow rate as a function of pressure difference between the chambers. (C) Flow volume of a Bingham fluid versus phase difference between the chambers. β was fixed at 1:0 while other parameters ðγ; ϕÞ were varied. The line colors in (B) and (C) correspond to the parameter values shown at the top of the figure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
into a given shape, and dumbbell shaped Physarum has been actually forged and examined by Takamatsu et al. (2000), in which the diameter of circular sheets was set to be 2 mm and the tube length was 4 mm. The oscillations in both chambers are reportedly entrained with each other, and their phase relationship temporally alters. Approximating the connecting tube as a straight, rigid cylinder, and assuming that Stokes' approximation is valid for cytosol (Teplov et al., 1991; Smith, 1994), the volumetric flow rate ðU ðt ÞÞ in the cylinder can be written as a function of the pressure difference between both ends of the tube ΔP ðt Þ, U ðt Þ ¼ νðΔP ðt Þ; L; RÞ;
ð1Þ
where νð U Þ denotes the relationship between pressure difference and flow rate (pressure–flow relation), calculated from the viscosity of the focal material. L and R are the tube length and crosssectional radius, respectively. The pressure–flow relation is linear
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Table 1 Properties of flows through cylinders in three fluid flow models. Model
Gradient of flow velocity ν~ ðλÞ
Volumetric flow rate νðΔP; L; RÞ
Newtonian
∂uðr Þ 1 ¼ λ ∂r ηlin
U¼
Power law
∂uðr Þ ¼ ∂r
λ
!1=n
n þ 1 πR3 þ 1=n 1=n : ¼ αpow ΔP 1=n 1=n ΔP 3n þ 1 2Lη pow 8 0; ΔP o ΔP 0 > > ! < πR4 4 1 ΔP 0 4 U¼ ΔP ΔP 0 þ ; ΔP Z ΔP 0 > 3 > 3 3 ΔP : 8LηBin 2Lλ0 ΔP 0 ¼ R
U¼
ηpow
∂uðr Þ ∂r 8 0; > < 1 ¼ ð λ λ Þ; > 0 : ηBin
Bingham
πR4 ΔP : ¼ αlin ΔP 8Lηlin
λ o λ0 λ Z λ0
Δp ¼pressure difference between the ends of the cylinder. ηlin ; ηpow ; and ηBin denote the viscosity coefficient, consistency constant, and plastic viscosity coefficient, respectively. n is the flow behavior index. λ0 is the yield stress.
(nonlinear) for fluids of linear (nonlinear) viscosity (the calculations are detailed in Appendix A and summarized in Table 1). Let P 1;2 ðt Þ be the pressures generated by the two chambers at time t. The time-averaged flow rate in an interval T is then given by Z 1 T U¼ νðP 1 ðt Þ P 2 ðt Þ; L; RÞdt; ð2Þ T 0
with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β ¼ β21 þ β22 2β1 β2 cos 2π φ2 φ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi γ ¼ γ 21 þ γ 22 2γ 1 γ 2 cos 4π ψ 2 ψ 1
where the flow from chambers 1 to 2 is selected as the positive flow. If the cytosol behaves as a Newtonian fluid with linear viscosity, the pressure–flow relation is U ðt Þ ¼ αlin ΔP ðt Þ (Table 1), and the integral in the flow volume equation is decomposed as follows: Z T Z 1 1 T U lin ¼ αlin P 1 ðt Þdt P 2 ðt Þdt : ð3Þ T 0 T 0
φ ¼ φ1 þ
Therefore, in the Newtonian fluid model, the time-averaged flow rate is proportional to the difference between the time averaged pressures. If both chambers have the same averaged pressure, the time-averaged flow rate is zero. However, if the pressure–flow relation is nonlinear, the time integral in Eq. (2) cannot be decomposed. To compare the flow properties of nonlinear viscous cytosol with those of the linear case, we express the periodic pressure profiles in the two chambers as P i ðt Þ ¼ pi þ βi sin 2π t þ φi þ γ i sin 4π t þ ψ i ; ð4Þ where the subscripts i ¼ 1; 2 are indices of the chambers, pi is the average pressure in chamber i, and φi and ψ i are the phase shifts in the i th chamber in the first and second Fourier modes, respectively. For a Newtonian fluid, the time-averaged flow rate integrated over one period is proportional to the average pressure difference between the chambers ðδp ¼ p1 p2 Þ (Z ) Z 1 1 U lin ¼ αlin P 1 ðt Þdt P 2 ðt Þdt ¼ αlin δp: ð5Þ 0
0
Note that in Newtonian fluids, unidirectional net flow is not caused by differences in the periodic pressure profiles. However, if the cytosol is nonlinearly viscous, unidirectional net flow is expected even when the averaged pressure is identical in both chambers. Here we consider a power law fluid, which models polymer solutions with shear thinning (Sato et al., 1983). For a power law fluid with behavior index n ¼1/3, the pressure–flow relation is U pow ¼ αpow ðΔP Þ3 (Table 1), and the time-averaged flow rate becomes Z 1 U pow ¼ ðP 1 ðt Þ P 2 ðt ÞÞ3 dt ¼ δp3 αpow 0 3 3 2 2 þ δpfβ þ γ 2 g þ β γ sin ð4π ðφ ψ ÞÞ; ð6Þ 2 4
and
! β2 sin 2π φ2 φ1 1 arctan 2π β1 þ β2 cos 2π φ2 φ1 ( 0; if β1 þ β2 cos 2π φ2 φ1 Z0 þ 0:5; if β1 þ β2 cos 2π φ2 φ1 o0 ! γ 2 sin 4π ψ 2 ψ 1 1 ψ ¼ ψ 1 þ arctan 4π γ 1 þ γ 2 cos 4π ψ 2 ψ 1 ( 0; if γ 1 þ γ 2 cos 4π ψ 2 ψ 1 Z 0 : þ 0:5; if γ 1 þ γ 2 cos 4π ψ 2 ψ 1 o 0 The last term of Eq. (6) indicates that, even if δp ¼ 0, non-zero net flow is contributed by the difference in the periodic pressure profiles. The flow profiles for several parameter sets are shown in Fig. 1B. Fig. 1C plots the profile of a Bingham fluid, which models fluids with yield stress (Table 1) (Chotard-Ghodsnia and Verdier, 2007). In these figures, the pressure profiles in both chambers are chosen to satisfy P i ðt Þ ¼ p0 þ β sin 2π t φi þ γ sin 4π t φi þ 2πϕ : ð7Þ
The states of the chambers differ only by a phase difference ðΔφ ¼ φ1 φ2 Þ. Although this function generates more symmetrical pressure profiles than the general profile discussed above, it ensures net flow. Therefore we conclude that nonlinear cytosol viscosity can convert the phase relationship between mechanically equivalent chambers into directional cytosol flow. The phase relationship and net cytoplasmic flow are fast-mode and slow-mode states, respectively. Thus, this result implies that the nonlinear viscosity transports information from the fast to the slow behavioral modes. We now investigate the relationship between the flow rate and the symmetry of the system. Suppose that the pressure profiles in both chambers are identical but symmetrically reversed in time, i.e., P trs t φi ¼ P trs t 1 φi ¼ P trs θ t þφi ; ð8Þ where the period is chosen to be 1. In other words, the pressure profile is an even function with origin at ðθ=2Þ þ φi . Consequently, the pressure difference profile becomes an odd function with origin at t ¼ ðθ þ φ1 þ φ2 Þ=2 ΔP trs ðt Þ P trs t φ1 P trs t φ2 ¼ P trs θ t þ φ1 P trs θ t þ φ2 ¼ P trs θ þφ1 þ φ2 t φ1 þ P trs θ þ φ1 þφ2 t φ2 ¼ ΔP trs θ þφ1 þ φ2 t ð9Þ
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Because of the exchange symmetry of the system, the viscosity function νð U Þ should also be an odd function. Therefore, a time reversal pressure profile implies that the time-averaged flow rate in one period is exactly zero Z τþ1 U¼ νðΔP trs ðt ÞÞdt ¼ 0; ð10Þ τ
where τ is an arbitrary constant that determines the starting point of the integral. The pressure profile (7) possesses time-reversal symmetry when γ ¼ 0 or ϕ ¼ 0:25. As shown in Fig. 1B and C, no net flow occurs under these conditions, regardless of the phase difference between the chambers.
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tube 3. Thus, rotational flows occur in the regions ( þ,þ ,þ ) and ( , , ). Under these conditions, net flow enters and exits all chambers. Fig. 2C shows the accumulated cytosol volume in chamber 1 after one oscillation period. The white lines indicate where no cytosol has accumulated. The rotational flow areas are bounded by the black dashed line. These figures indicate that cytosol can accumulate in a particular chamber while also leaving the chamber (the regions in which relationships occur are indicated by stars in Fig. 2C). Since cytosol accumulation induces migration of Physarum, this result indicates that, if the cytosol is nonlinearly viscous, Physarum can back-transport cytosol against the migration direction.
3. Cytosol flow in cells with general forms 4. Summary Next, we consider the net cytosol flow induced by nonlinear viscosity in more complex Physarum cells. We model the developed cytogel network by branching hard cylindrical tubes and the spread cyotoplasmic sheets at each end of the network by periodic pumps. Stokes' approximation is assumed in the cytosol. Here we only consider branching nodes which connect three tubes, because the nodes connecting larger number of tubes can be represented by the combination of three-tube nodes. The flow volume is conserved at each node,
∑ ð11Þ ν P i ðt Þ P j0 ðt Þ; Lij0 ; Rij0 ¼ 0;
j0 ¼ j;j þ 1;j þ 2
where P i ðt Þ is the pressure at the focal node and P j;j þ 1;j þ 2 ðt Þ are pressures at the opposite ends of the connected tubes. Lij;j þ 1;j þ 2 and Rij;j þ 1;j þ 2 are the tube lengths and cross-sectional radii, respectively. Applying Eq. (11) to all nodes of any given tube network, we uniquely determine the flow rate in every tube and the pressure at every node, regardless of the viscosity function. If the viscosity is linear, the cytosol flow in the network is analogous to electric current with a given static electric potential. The time-averaged flow also becomes a potential flow, whose potential function is the time averaged pressure. Under these circumstances, Kirchhoff's loop law holds for any closed loop in the network n
8Lii þ 1 ηlin
i¼1
πR4ii þ 1
∑
U ii þ 1 ¼ 0;
ð12Þ
where the nodes i ¼ 1; …; n are arranged in a circle, and ηlin is the viscosity coefficient (Table 1). Lii þ 1 , Rii þ 1 and U ii þ 1 are the lengths, cross-sectional radii and the time-averaged flow rate of their connecting tubes, respectively. The positive flow direction is that of the order of chambers. There are two states of flow to satisfy Eq. (12); all U ii þ 1 are zero, or some U ii þ 1 are positive while others are negative. Therefore, rotational flow, in which all U ii þ 1 have the same sign, is impossible. Since Eq. (12) holds in any closed loop of a given network, the flow through the entire network is irrotational. Conversely, a nonlinear viscous cytoplasm admits rotational flow. To see this, we construct a triangular network of chambers and tubes, and assume a power law fluid (Fig. 2). Physarum of this shape have also been forged and examined in previous experimental studies (Takamatsu et al., 2001). The pressure profile in each chamber is given by Eq. (7), where i ¼ 1–3, and we set Δφ12 ¼ φ1 φ2 , Δφ13 ¼ φ1 φ3 . The cytosol flow directions in tubes j ¼ 1–3 are shown in Fig. 2B on the Δφ12 ; Δφ13 plane. Clockwise and counterclockwise flows are indicated by positive and negative signs, respectively. For example, (þ, þ, ) denotes clockwise flow in tubes 1 and 2, and counter-clockwise flow in
In summary, we have considered the impact of nonlinear cytosol viscosity on the behavior of Physarum plasmodium. We modeled the developed Physarum body by a network of hard tubes, fluid cytosol and periodic pumps at the network periphery. We examined both linear and nonlinear viscous cytosols, and revealed qualitative differences in their flow properties. The timeaveraged flow of linearly viscous cytosol is analogous to flow through an electric circuit, whose potential function is the timeaveraged pressure. In this scenario, the time-averaged flows through tubes, and the averaged pressures at network nodes or in the pumps, satisfy Kirchhoff's Laws (see Eq. (12)). However, the flow of nonlinearly viscous cytosol cannot be modeled in this way. We examined how nonlinearity alters the flow in dumbbell plasmodia and in a triangular network of plasmodium compartments. In the dumbbell model, we found that the time-averaged flow can be non-zero even when the oscillation behavior of the two chambers differs by phase alone. In the triangular model, nonlinearity admits rotational flow, and cytosol can both exit and accumulate in a given chamber. Several studies have reported symmetric fluid flows generated by periodic mechanical movements. Examples are peristaltic pumps (Teplov et al., 1991; Hoepffener and Fukagawa, 2009) and flapping filaments (Bagheri et al., 2012; Cartwright et al., 2006). In these studies, inertial effects (Hoepffener and Fukagawa, 2009; Bagheri et al., 2012) or non-reciprocal boundary deformations (Teplov et al., 1991; Cartwright et al., 2006) are important for asymmetric flow. In Physarum plasmodium cells, tubes with insufficiently hard cytogel walls can deform, yielding a similar effect (Teplov et al., 1991). However, the phenomena revealed in our study differ from the asymmetric flow phenomena reported in previous studies. Incorporating both effects would complicate the model results, and is therefore not attempted here. The multiple behavioral modes of Physarum plasmodium operate on different time scales. For instance, the periodic pump oscillation and migration operate on rapid and slow time scales, respectively. In the dumbbell model, we found that the state of the rapid mode (phase relationship between the chambers) inferred the behavior of the slower mode (net flow and migration), provided that the cytosol was nonlinearly viscous. If linear viscosity of the cytosol is assumed, migration and other slower behaviors are independent of the oscillatory mode. However, the oscillatory patterns of real Physarum plasmodia are highly variable and sensitive to environmental conditions (Takamatsu et al., 1997; Takagi and Ueda, 2010), and appear to be correlated with migratory behavior (Takamatsu et al., 1997; Nakagaki et al., 2000; Takagi and Ueda, 2008). Therefore, if Physarum is to generate self-organized behaviors throughout its cell body, its various behavioral modes must influence each other. This influence appears only if nonlinear viscosity of the cytoplasm is assumed.
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Fig. 2. Time-averaged flow rate in a triangular Physarum model. A power law fluid with exponent 3 is assumed, and ðβ; γ; ϕÞ ¼ ð1:0; 0:4; 0:0Þ. (A) Schematic of the model. (B) Signs of the time-averaged flow in the three tubes of the model, shown on the Δφ12 ; Δφ13 plane. The plane is divided into eight regions. The parentheses in each division indicate (sign of the flow in tube 1, sign of the flow in tube 2, sign of the flow in tube 3). Clockwise and counterclockwise flows are denoted byþ and , respectively. (C) Color map of the cytosol volume accumulated in chamber 1 during one period, shown on the Δφ12 ; Δφ13 plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The triangular model indicates that the net flow can oppose the migration direction, and transport chemical signals from the front to the rear of the cell. Such inverse flow will markedly affect the behavioral decision of cells, but is precluded in the linearly viscous model, in which the rear region of the cell cannot know the condition of the front region. Because no detailed constitutive model of Physarum cytosol has been established, we tested two simple nonlinear viscous fluids as toy models of cytosol material; a power-law fluid demonstrating shear thinning and a Bingham fluid. Quantitative discussions were not attempted. In fact, the net flow volume primarily depends on the deviation of the actual viscosity from the linear viscous model, for which reliable values have not been reported. However, the symmetry breaking observed in our results does not require the details of the nonlinearity in viscosity. Our findings are expected to be general and to describe the cytosol dynamics in real organisms. For example, if the cytoplasm shows shear thickening behaviors (which is modeled by power law fluid with behavior index larger than one) as suggested in (Sato et al., 1983), the flow characteristics will differ from the results shown in Figs. 1 and 2, but the shear thickening will nonetheless lead to symmetry breaking. The cytosol viscosity, the relation between the fluidity and the driving force generated by cell itself, depends on the states, amounts and localizations of the constituent biopolymers. Consequently, Physarum cytosol responds in various ways to environmental conditions, leading to qualitatively diverse behaviors. The nonlinearity in cytosol viscosity induces a novel type of behavioral symmetry breaking in Physarum's cytosol flow models. The new motion generated by the symmetry breaking will provide significant insight into adaptive behaviors. In particular, the nonlinear viscosity model will be key for understanding the connection of behavioral modes operating on different time scales and the molecular signal transport from the front to the rear of the cell during plasmodium locomotion.
Acknowledgment The authors are grateful to Dr. T. Nakagaki and Dr. Y. Tanaka for fruitful discussions.
Appendix A The cylindrical Physarum tube is represented in cylindrical coordinates with radius r and length z. The equation of motion of the cytosol is given by ∂P ðzÞ 1 ∂ðλr Þ ¼ ; ∂z r ∂r
ð13Þ
where P ðzÞ is pressure and λ is the shear stress in the z direction. The visco-plastic property of cytosol is given by the functional relationship ν~ ð U Þ between λ and the gradient of the flow velocity (uðr Þ) in the r direction ∂uðr Þ ¼ ν~ ðλÞ; ∂r From Eq. (13), the integral form of the flow velocity is Z R Z R 1 uðr Þ ¼ ΔPr 0 dr 0 ; ν~ ðλÞdr 0 ¼ ν~ 2L r r
ð14Þ
ð15Þ
where R is the radius of the cylinder. The total flow rate becomes Z 2π Z R U¼ uðr Þr dr dθ νðΔP; L; RÞ: ð16Þ 0
0
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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Nonlinearity in cytoplasm viscosity can generate an essential symmetry breaking in cellular behaviors Masashi Tachikawa a,n, Atsushi Mochizuki a,b a b
Theoretical Biology Laboratory, RIKEN, Japan Core Research for Evolutional Science and Technology, Japan Science and Technology Agency, Japan
H I G H L I G H T S
Protoplasmic flow in a giant ameboid cell is modeled and the impact of nonlinearity in viscosity on its behaviors is considered. The nonlinearity in the viscosity generates a novel type of symmetry breaking in the protoplasmic flow. The symmetry breaking causes the interaction between different behavioral modes implemented on different time scales. The symmetry breaking makes transportation of molecular signals from the front to the rear of the cell during the locomotion.
art ic l e i nf o
a b s t r a c t
Article history: Received 11 March 2014 Received in revised form 6 September 2014 Accepted 16 September 2014 Available online 27 September 2014
The cytoplasms of ameboid cells are nonlinearly viscous. The cell controls this viscosity by modulating the amount, localization and interactions of bio-polymers. Here we investigated how the nonlinearity infers the cellular behaviors and whether nonlinearity-specific behaviors exist. We modeled the developed plasmodium of the slime mold Physarum polycephalum as a network of branching tubes and examined the linear and nonlinear viscous cytoplasm flows in the tubes. We found that the nonlinearity in the cytoplasm's viscosity induces a novel type of symmetry breaking in the protoplasmic flow. We also show that symmetry breaking can play an important role in adaptive behaviors, namely, connection of behavioral modes implemented on different time scales and transportation of molecular signals from the front to the rear of the cell during cellular locomotion. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Physarum polycephalum Locomotion Cytosol flow Symmetry breaking
1. Introduction Cellular materials frequently show complex responses to external perturbations, which are essential for cell–environment interactions. For example, cytoplasm displays non-Newtonian fluid properties, such as nonlinear viscosity and yield stress (Kamiya, 1950; Sato et al., 1983; Guy et al., 2011; Tsai et al., 1993). Such complex intracellular responses arise from the complex structures and behaviors of the constituent biopolymers (Eisenberg, 1990). By controlling the amount, localization and interactions of biopolymers, cells control the responses of their materials to external perturbations, and thereby design their adaptive behaviors. In this paper, we consider how the nonlinearity in the cytoplasm's viscosity can affect ameboid cell behaviors, focusing on Physarum plasmodium cells. n Corresponding author. Present address: Theoretical Biology Laboratory, RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan. Tel.: þ 81 48 467 9546; fax: þ81 48 462 1709. E-mail address: [email protected] (M. Tachikawa).
http://dx.doi.org/10.1016/j.jtbi.2014.09.023 0022-5193/& 2014 Elsevier Ltd. All rights reserved.
The plasmodium of Physarum polycephalum is a multinucleated cell growing up to several centimeters that self-organizes into complex shapes (Wohlfarth-Bottermann, 1979; Nakagaki and Guy, 2008). Its cytoplasm contains abundant actin and myosin filaments, which form meshes of various rigidities depending on the intra- and extra-cellular environment (Kamiya et al., 1988; Ueda et al., 1990). These meshes are cross-linked into a hard gel (the cytogel) that forms a vein-like structure; a large network of branching tubes through which the cytoplasm (cytosol) flows. The diameter of a developed tube is the order of 0.1 mm (Wohlfarth-Bottermann, 1979). At the periphery of the network, the cytoplasm spreads into a sheet formation (Kamiya et al., 1988). Physarum plasmodium also exhibits self-organized behaviors. The peripheral cytoplasmic sheet undergoes cycles of gelation/ contraction and solation/relaxation. The cycles are generated by a mechano-chemical oscillator involving the extent of cross-linking in the acto-myosin mesh (Wohlfarth-Bottermann, 1979; Yoshimoto et al., 1981a, 1981b; Oster and Odell, 1984; Yoshiyama et al., 2009). The oscillating cross-linking in the mesh is reflected in the cell thickness, which oscillates with a period of approximately 100 s. This
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mechano-chemical oscillator performs as a periodic pump, shuttling the streaming flow of cytosol into the cytogel tube network (Odell, 1984). The speed of the streaming flow can reach up to 1 mm/s (Nakagaki and Guy, 2008). Oscillation frequently induces simultaneous wave patterns across the entire cell (Takamatsu et al., 1997; Takagi and Ueda, 2010) and may be triggered by environmental stimuli (Nakagaki et al., 2000; Takagi and Ueda, 2008). The wave pattern is thought to correlate with the cytosol flow pattern, although simultaneous occurrence of both patterns has yet to be directly observed. On a slower timescale than the oscillatory behavior, the cytoplasm is transported in a direction selected by the net flow, namely, the difference in the flow volumes between opposing shuttle streaming flows. Such directional transportation deforms the cell and alters its migration direction (Satoh et al., 1985; Matsumoto et al., 2008). On yet slower scales, the cell adapts its behaviors in response to various environmental stresses, such as light, attractant/repellent substances and ground condition (Takamatsu et al., 2009; Tero et al., 2010; Ueda et al., 2011). Since Physarum plasmodium lacks a nervous system or similar apparatus, organized movement through the entire cell body is thought to occur by self-organization of spatially distributed mechanochemical dynamics. That is, the cell dynamics are governed by the physical properties of the intracellular chemical substances. Therefore, Physarum polycephalum is a suitable organism for investigating the relationship between the physical properties of cellular materials and their adaptive behavior, which remains largely unknown. Numerous mathematical models have been proposed for understanding the mechanical properties of Physarum plasmodium (Tsai et al., 1993; Oster and Odell, 1984; Teplov et al., 1991; Smith, 1994; Yamada et al., 1999). Most of these models (Oster and Odell, 1984; Teplov et al., 1991; Smith, 1994; Yamada et al., 1999) assume linear viscosity of the cytoplasm. Because previous models qualitatively reconstruct certain cellular behaviors, it is possible that nonlinearity exerts a second-order effect on behavioral properties. However, linear viscosity models cannot reveal nonlinear-specific behaviors. For example, a cytoplasm model with yield stress was recently proposed to realize tube formation in developing plasmodium cell (Guy et al., 2011). Here we question whether the nonlinear viscosity of the cytoplasm affects the relationships among different behavioral modes operating over different time scales. Analogous to the nervous system (Klimesch et al., 2010), the state of the rapid oscillatory mode should govern the slow mode behaviors. Information transport between modes with different time scales will enable Physarum to generate selforganized adaptive behaviors across the entire cell. The present paper investigates this idea in simplified models of plasmodium cells. To this end, we implemented linear and nonlinear viscous cytosol in the constructed models, and tested whether the model behaviors qualitatively differ between the linear and nonlinear models. We found that nonlinearity in the cytosol viscosity causes symmetry breaking in the cytosol streaming flow, followed by accumulation of cytosol at one side of the cell on a slower timescale.
2. Cytosol flow in a dumbbell shaped cell We model the Physarum plasmodium as a dumbbell consisting of three parts; two circular sheets of protoplasm (chambers) connected by one straight cytogel tube (Fig. 1A). Both chambers act as pumps, inducing periodic pressure changes by contraction– relaxation oscillations to stream the cytosol through the connecting tube. This model mimics the simplest Physarum with a developed tube. Physarum can also be experimentally manipulated
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Fig. 1. Time-averaged flow rate of nonlinear viscous cytosol in the tube of a dumbbell- shaped Physarum model. (A) Schematic of the model. (B) Time-averaged flow rate of power law fluid with exponent 3, versus phase difference between the chambers ðΔφ ¼ φ1 φ2 Þ. The inset plots the flow rate as a function of pressure difference between the chambers. (C) Flow volume of a Bingham fluid versus phase difference between the chambers. β was fixed at 1:0 while other parameters ðγ; ϕÞ were varied. The line colors in (B) and (C) correspond to the parameter values shown at the top of the figure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
into a given shape, and dumbbell shaped Physarum has been actually forged and examined by Takamatsu et al. (2000), in which the diameter of circular sheets was set to be 2 mm and the tube length was 4 mm. The oscillations in both chambers are reportedly entrained with each other, and their phase relationship temporally alters. Approximating the connecting tube as a straight, rigid cylinder, and assuming that Stokes' approximation is valid for cytosol (Teplov et al., 1991; Smith, 1994), the volumetric flow rate ðU ðt ÞÞ in the cylinder can be written as a function of the pressure difference between both ends of the tube ΔP ðt Þ, U ðt Þ ¼ νðΔP ðt Þ; L; RÞ;
ð1Þ
where νð U Þ denotes the relationship between pressure difference and flow rate (pressure–flow relation), calculated from the viscosity of the focal material. L and R are the tube length and crosssectional radius, respectively. The pressure–flow relation is linear
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Table 1 Properties of flows through cylinders in three fluid flow models. Model
Gradient of flow velocity ν~ ðλÞ
Volumetric flow rate νðΔP; L; RÞ
Newtonian
∂uðr Þ 1 ¼ λ ∂r ηlin
U¼
Power law
∂uðr Þ ¼ ∂r
λ
!1=n
n þ 1 πR3 þ 1=n 1=n : ¼ αpow ΔP 1=n 1=n ΔP 3n þ 1 2Lη pow 8 0; ΔP o ΔP 0 > > ! < πR4 4 1 ΔP 0 4 U¼ ΔP ΔP 0 þ ; ΔP Z ΔP 0 > 3 > 3 3 ΔP : 8LηBin 2Lλ0 ΔP 0 ¼ R
U¼
ηpow
∂uðr Þ ∂r 8 0; > < 1 ¼ ð λ λ Þ; > 0 : ηBin
Bingham
πR4 ΔP : ¼ αlin ΔP 8Lηlin
λ o λ0 λ Z λ0
Δp ¼pressure difference between the ends of the cylinder. ηlin ; ηpow ; and ηBin denote the viscosity coefficient, consistency constant, and plastic viscosity coefficient, respectively. n is the flow behavior index. λ0 is the yield stress.
(nonlinear) for fluids of linear (nonlinear) viscosity (the calculations are detailed in Appendix A and summarized in Table 1). Let P 1;2 ðt Þ be the pressures generated by the two chambers at time t. The time-averaged flow rate in an interval T is then given by Z 1 T U¼ νðP 1 ðt Þ P 2 ðt Þ; L; RÞdt; ð2Þ T 0
with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β ¼ β21 þ β22 2β1 β2 cos 2π φ2 φ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi γ ¼ γ 21 þ γ 22 2γ 1 γ 2 cos 4π ψ 2 ψ 1
where the flow from chambers 1 to 2 is selected as the positive flow. If the cytosol behaves as a Newtonian fluid with linear viscosity, the pressure–flow relation is U ðt Þ ¼ αlin ΔP ðt Þ (Table 1), and the integral in the flow volume equation is decomposed as follows: Z T Z 1 1 T U lin ¼ αlin P 1 ðt Þdt P 2 ðt Þdt : ð3Þ T 0 T 0
φ ¼ φ1 þ
Therefore, in the Newtonian fluid model, the time-averaged flow rate is proportional to the difference between the time averaged pressures. If both chambers have the same averaged pressure, the time-averaged flow rate is zero. However, if the pressure–flow relation is nonlinear, the time integral in Eq. (2) cannot be decomposed. To compare the flow properties of nonlinear viscous cytosol with those of the linear case, we express the periodic pressure profiles in the two chambers as P i ðt Þ ¼ pi þ βi sin 2π t þ φi þ γ i sin 4π t þ ψ i ; ð4Þ where the subscripts i ¼ 1; 2 are indices of the chambers, pi is the average pressure in chamber i, and φi and ψ i are the phase shifts in the i th chamber in the first and second Fourier modes, respectively. For a Newtonian fluid, the time-averaged flow rate integrated over one period is proportional to the average pressure difference between the chambers ðδp ¼ p1 p2 Þ (Z ) Z 1 1 U lin ¼ αlin P 1 ðt Þdt P 2 ðt Þdt ¼ αlin δp: ð5Þ 0
0
Note that in Newtonian fluids, unidirectional net flow is not caused by differences in the periodic pressure profiles. However, if the cytosol is nonlinearly viscous, unidirectional net flow is expected even when the averaged pressure is identical in both chambers. Here we consider a power law fluid, which models polymer solutions with shear thinning (Sato et al., 1983). For a power law fluid with behavior index n ¼1/3, the pressure–flow relation is U pow ¼ αpow ðΔP Þ3 (Table 1), and the time-averaged flow rate becomes Z 1 U pow ¼ ðP 1 ðt Þ P 2 ðt ÞÞ3 dt ¼ δp3 αpow 0 3 3 2 2 þ δpfβ þ γ 2 g þ β γ sin ð4π ðφ ψ ÞÞ; ð6Þ 2 4
and
! β2 sin 2π φ2 φ1 1 arctan 2π β1 þ β2 cos 2π φ2 φ1 ( 0; if β1 þ β2 cos 2π φ2 φ1 Z0 þ 0:5; if β1 þ β2 cos 2π φ2 φ1 o0 ! γ 2 sin 4π ψ 2 ψ 1 1 ψ ¼ ψ 1 þ arctan 4π γ 1 þ γ 2 cos 4π ψ 2 ψ 1 ( 0; if γ 1 þ γ 2 cos 4π ψ 2 ψ 1 Z 0 : þ 0:5; if γ 1 þ γ 2 cos 4π ψ 2 ψ 1 o 0 The last term of Eq. (6) indicates that, even if δp ¼ 0, non-zero net flow is contributed by the difference in the periodic pressure profiles. The flow profiles for several parameter sets are shown in Fig. 1B. Fig. 1C plots the profile of a Bingham fluid, which models fluids with yield stress (Table 1) (Chotard-Ghodsnia and Verdier, 2007). In these figures, the pressure profiles in both chambers are chosen to satisfy P i ðt Þ ¼ p0 þ β sin 2π t φi þ γ sin 4π t φi þ 2πϕ : ð7Þ
The states of the chambers differ only by a phase difference ðΔφ ¼ φ1 φ2 Þ. Although this function generates more symmetrical pressure profiles than the general profile discussed above, it ensures net flow. Therefore we conclude that nonlinear cytosol viscosity can convert the phase relationship between mechanically equivalent chambers into directional cytosol flow. The phase relationship and net cytoplasmic flow are fast-mode and slow-mode states, respectively. Thus, this result implies that the nonlinear viscosity transports information from the fast to the slow behavioral modes. We now investigate the relationship between the flow rate and the symmetry of the system. Suppose that the pressure profiles in both chambers are identical but symmetrically reversed in time, i.e., P trs t φi ¼ P trs t 1 φi ¼ P trs θ t þφi ; ð8Þ where the period is chosen to be 1. In other words, the pressure profile is an even function with origin at ðθ=2Þ þ φi . Consequently, the pressure difference profile becomes an odd function with origin at t ¼ ðθ þ φ1 þ φ2 Þ=2 ΔP trs ðt Þ P trs t φ1 P trs t φ2 ¼ P trs θ t þ φ1 P trs θ t þ φ2 ¼ P trs θ þφ1 þ φ2 t φ1 þ P trs θ þ φ1 þφ2 t φ2 ¼ ΔP trs θ þφ1 þ φ2 t ð9Þ
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Because of the exchange symmetry of the system, the viscosity function νð U Þ should also be an odd function. Therefore, a time reversal pressure profile implies that the time-averaged flow rate in one period is exactly zero Z τþ1 U¼ νðΔP trs ðt ÞÞdt ¼ 0; ð10Þ τ
where τ is an arbitrary constant that determines the starting point of the integral. The pressure profile (7) possesses time-reversal symmetry when γ ¼ 0 or ϕ ¼ 0:25. As shown in Fig. 1B and C, no net flow occurs under these conditions, regardless of the phase difference between the chambers.
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tube 3. Thus, rotational flows occur in the regions ( þ,þ ,þ ) and ( , , ). Under these conditions, net flow enters and exits all chambers. Fig. 2C shows the accumulated cytosol volume in chamber 1 after one oscillation period. The white lines indicate where no cytosol has accumulated. The rotational flow areas are bounded by the black dashed line. These figures indicate that cytosol can accumulate in a particular chamber while also leaving the chamber (the regions in which relationships occur are indicated by stars in Fig. 2C). Since cytosol accumulation induces migration of Physarum, this result indicates that, if the cytosol is nonlinearly viscous, Physarum can back-transport cytosol against the migration direction.
3. Cytosol flow in cells with general forms 4. Summary Next, we consider the net cytosol flow induced by nonlinear viscosity in more complex Physarum cells. We model the developed cytogel network by branching hard cylindrical tubes and the spread cyotoplasmic sheets at each end of the network by periodic pumps. Stokes' approximation is assumed in the cytosol. Here we only consider branching nodes which connect three tubes, because the nodes connecting larger number of tubes can be represented by the combination of three-tube nodes. The flow volume is conserved at each node,
∑ ð11Þ ν P i ðt Þ P j0 ðt Þ; Lij0 ; Rij0 ¼ 0;
j0 ¼ j;j þ 1;j þ 2
where P i ðt Þ is the pressure at the focal node and P j;j þ 1;j þ 2 ðt Þ are pressures at the opposite ends of the connected tubes. Lij;j þ 1;j þ 2 and Rij;j þ 1;j þ 2 are the tube lengths and cross-sectional radii, respectively. Applying Eq. (11) to all nodes of any given tube network, we uniquely determine the flow rate in every tube and the pressure at every node, regardless of the viscosity function. If the viscosity is linear, the cytosol flow in the network is analogous to electric current with a given static electric potential. The time-averaged flow also becomes a potential flow, whose potential function is the time averaged pressure. Under these circumstances, Kirchhoff's loop law holds for any closed loop in the network n
8Lii þ 1 ηlin
i¼1
πR4ii þ 1
∑
U ii þ 1 ¼ 0;
ð12Þ
where the nodes i ¼ 1; …; n are arranged in a circle, and ηlin is the viscosity coefficient (Table 1). Lii þ 1 , Rii þ 1 and U ii þ 1 are the lengths, cross-sectional radii and the time-averaged flow rate of their connecting tubes, respectively. The positive flow direction is that of the order of chambers. There are two states of flow to satisfy Eq. (12); all U ii þ 1 are zero, or some U ii þ 1 are positive while others are negative. Therefore, rotational flow, in which all U ii þ 1 have the same sign, is impossible. Since Eq. (12) holds in any closed loop of a given network, the flow through the entire network is irrotational. Conversely, a nonlinear viscous cytoplasm admits rotational flow. To see this, we construct a triangular network of chambers and tubes, and assume a power law fluid (Fig. 2). Physarum of this shape have also been forged and examined in previous experimental studies (Takamatsu et al., 2001). The pressure profile in each chamber is given by Eq. (7), where i ¼ 1–3, and we set Δφ12 ¼ φ1 φ2 , Δφ13 ¼ φ1 φ3 . The cytosol flow directions in tubes j ¼ 1–3 are shown in Fig. 2B on the Δφ12 ; Δφ13 plane. Clockwise and counterclockwise flows are indicated by positive and negative signs, respectively. For example, (þ, þ, ) denotes clockwise flow in tubes 1 and 2, and counter-clockwise flow in
In summary, we have considered the impact of nonlinear cytosol viscosity on the behavior of Physarum plasmodium. We modeled the developed Physarum body by a network of hard tubes, fluid cytosol and periodic pumps at the network periphery. We examined both linear and nonlinear viscous cytosols, and revealed qualitative differences in their flow properties. The timeaveraged flow of linearly viscous cytosol is analogous to flow through an electric circuit, whose potential function is the timeaveraged pressure. In this scenario, the time-averaged flows through tubes, and the averaged pressures at network nodes or in the pumps, satisfy Kirchhoff's Laws (see Eq. (12)). However, the flow of nonlinearly viscous cytosol cannot be modeled in this way. We examined how nonlinearity alters the flow in dumbbell plasmodia and in a triangular network of plasmodium compartments. In the dumbbell model, we found that the time-averaged flow can be non-zero even when the oscillation behavior of the two chambers differs by phase alone. In the triangular model, nonlinearity admits rotational flow, and cytosol can both exit and accumulate in a given chamber. Several studies have reported symmetric fluid flows generated by periodic mechanical movements. Examples are peristaltic pumps (Teplov et al., 1991; Hoepffener and Fukagawa, 2009) and flapping filaments (Bagheri et al., 2012; Cartwright et al., 2006). In these studies, inertial effects (Hoepffener and Fukagawa, 2009; Bagheri et al., 2012) or non-reciprocal boundary deformations (Teplov et al., 1991; Cartwright et al., 2006) are important for asymmetric flow. In Physarum plasmodium cells, tubes with insufficiently hard cytogel walls can deform, yielding a similar effect (Teplov et al., 1991). However, the phenomena revealed in our study differ from the asymmetric flow phenomena reported in previous studies. Incorporating both effects would complicate the model results, and is therefore not attempted here. The multiple behavioral modes of Physarum plasmodium operate on different time scales. For instance, the periodic pump oscillation and migration operate on rapid and slow time scales, respectively. In the dumbbell model, we found that the state of the rapid mode (phase relationship between the chambers) inferred the behavior of the slower mode (net flow and migration), provided that the cytosol was nonlinearly viscous. If linear viscosity of the cytosol is assumed, migration and other slower behaviors are independent of the oscillatory mode. However, the oscillatory patterns of real Physarum plasmodia are highly variable and sensitive to environmental conditions (Takamatsu et al., 1997; Takagi and Ueda, 2010), and appear to be correlated with migratory behavior (Takamatsu et al., 1997; Nakagaki et al., 2000; Takagi and Ueda, 2008). Therefore, if Physarum is to generate self-organized behaviors throughout its cell body, its various behavioral modes must influence each other. This influence appears only if nonlinear viscosity of the cytoplasm is assumed.
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Fig. 2. Time-averaged flow rate in a triangular Physarum model. A power law fluid with exponent 3 is assumed, and ðβ; γ; ϕÞ ¼ ð1:0; 0:4; 0:0Þ. (A) Schematic of the model. (B) Signs of the time-averaged flow in the three tubes of the model, shown on the Δφ12 ; Δφ13 plane. The plane is divided into eight regions. The parentheses in each division indicate (sign of the flow in tube 1, sign of the flow in tube 2, sign of the flow in tube 3). Clockwise and counterclockwise flows are denoted byþ and , respectively. (C) Color map of the cytosol volume accumulated in chamber 1 during one period, shown on the Δφ12 ; Δφ13 plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The triangular model indicates that the net flow can oppose the migration direction, and transport chemical signals from the front to the rear of the cell. Such inverse flow will markedly affect the behavioral decision of cells, but is precluded in the linearly viscous model, in which the rear region of the cell cannot know the condition of the front region. Because no detailed constitutive model of Physarum cytosol has been established, we tested two simple nonlinear viscous fluids as toy models of cytosol material; a power-law fluid demonstrating shear thinning and a Bingham fluid. Quantitative discussions were not attempted. In fact, the net flow volume primarily depends on the deviation of the actual viscosity from the linear viscous model, for which reliable values have not been reported. However, the symmetry breaking observed in our results does not require the details of the nonlinearity in viscosity. Our findings are expected to be general and to describe the cytosol dynamics in real organisms. For example, if the cytoplasm shows shear thickening behaviors (which is modeled by power law fluid with behavior index larger than one) as suggested in (Sato et al., 1983), the flow characteristics will differ from the results shown in Figs. 1 and 2, but the shear thickening will nonetheless lead to symmetry breaking. The cytosol viscosity, the relation between the fluidity and the driving force generated by cell itself, depends on the states, amounts and localizations of the constituent biopolymers. Consequently, Physarum cytosol responds in various ways to environmental conditions, leading to qualitatively diverse behaviors. The nonlinearity in cytosol viscosity induces a novel type of behavioral symmetry breaking in Physarum's cytosol flow models. The new motion generated by the symmetry breaking will provide significant insight into adaptive behaviors. In particular, the nonlinear viscosity model will be key for understanding the connection of behavioral modes operating on different time scales and the molecular signal transport from the front to the rear of the cell during plasmodium locomotion.
Acknowledgment The authors are grateful to Dr. T. Nakagaki and Dr. Y. Tanaka for fruitful discussions.
Appendix A The cylindrical Physarum tube is represented in cylindrical coordinates with radius r and length z. The equation of motion of the cytosol is given by ∂P ðzÞ 1 ∂ðλr Þ ¼ ; ∂z r ∂r
ð13Þ
where P ðzÞ is pressure and λ is the shear stress in the z direction. The visco-plastic property of cytosol is given by the functional relationship ν~ ð U Þ between λ and the gradient of the flow velocity (uðr Þ) in the r direction ∂uðr Þ ¼ ν~ ðλÞ; ∂r From Eq. (13), the integral form of the flow velocity is Z R Z R 1 uðr Þ ¼ ΔPr 0 dr 0 ; ν~ ðλÞdr 0 ¼ ν~ 2L r r
ð14Þ
ð15Þ
where R is the radius of the cylinder. The total flow rate becomes Z 2π Z R U¼ uðr Þr dr dθ νðΔP; L; RÞ: ð16Þ 0
0
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