# Oscillatory fluid flow drives scaling of contraction wave with system size

See allHide authors and affiliations

Edited by Herbert Levine, Rice University, Houston, TX, and approved September 9, 2018 (received for review April 6, 2018)

## Significance

Long-range fluid flows are crucial for the functioning of many organisms, as they provide forcing for migration and development and spread resources and signals. How flows can span vastly different scales is unclear. Here, we develop a minimal, two-component model, coupling the mechanics of a cell’s cortex to a contraction-triggering chemical. The chemical itself is spread with the fluid flows that arise due to the cortex contractions. Through theoretical and numerical analysis, we find that the oscillatory component of the flows can give rise to robust scaling of contraction waves with system size—much beyond predicted length scales. This mechanism is likely to work in a broad class of systems.

## Abstract

Flows over remarkably long distances are crucial to the functioning of many organisms, across all kingdoms of life. Coordinated flows are fundamental to power deformations, required for migration or development, or to spread resources and signals. A ubiquitous mechanism to generate flows, particularly prominent in animals and amoebas, is actomyosin cortex-driven mechanical deformations that pump the fluid enclosed by the cortex. However, it is unclear how cortex dynamics can self-organize to give rise to coordinated flows across the largely varying scales of biological systems. Here, we develop a mechanochemical model of actomyosin cortex mechanics coupled to a contraction-triggering, soluble chemical. The chemical itself is advected with the flows generated by the cortex-driven deformations of the tubular-shaped cell. The theoretical model predicts a dynamic instability giving rise to stable patterns of cortex contraction waves and oscillatory flows. Surprisingly, simulated patterns extend beyond the intrinsic length scale of the dynamic instability—scaling with system size instead. Patterns appear randomly but can be robustly generated in a growing system or by flow-generating boundary conditions. We identify oscillatory flows as the key for the scaling of contraction waves with system size. Our work shows the importance of active flows in biophysical models of patterning, not only as a regulating input or an emergent output, but also as a full part of a self-organized machinery. Contractions and fluid flows are observed in all kinds of organisms, so this concept is likely to be relevant for a broad class of systems.

Fluid flows are fundamental to the functioning of all organisms. They play an important role in homeostasis, by spreading resources and biochemical signals (1⇓–3). They power deformations driving migration of many motile cells (4⇓–6) and can even directly impact organism size (7). Surprisingly, even in the absence of a pacemaker like a heart, flows are coordinated on vastly different scales ranging from the size of a single migrating cell of about 20 μm (8, 9), via the *Caenorhabditis elegans* gonad of about 450 μm (10, 11), to the *Drosophila* embryos of about 500 μm (12), to acellular slime molds of more than 2 cm in size (13). The physical mechanism of how coordinated flows can self-organize particularly in a single cellular envelope remains unknown.

Animal and slime mold cells are lined with an actomyosin cortex situated just below their cellular envelope, enclosing the cells’ fluid cytoplasm. This actomyosin meshwork forms an active viscoelastic material (14, 15). It contracts under myosin motor activity and thereby drives the enclosed cytoplasm to flow into a less contracted part of the cell (1, 16). Long-range flows adapting to system size therefore require a spatial organization of cortex contractility. The mechanism driving the coordination of cortex contraction across a cell or an organism is unclear.

Already in the early 1980s Oster and Odell (17, 18) explored the idea that a contraction-triggering chemical, like calcium, could explain dynamic, oscillatory patterns of actomyosin activity in the cell cortex. However, the dynamic patterns’ spatial component was not investigated. Calcium is necessary to generate actomyosin contractions in many biological systems (19⇓⇓–22). Additionally, calcium is regulated by mechanical stretching, via mechanosensitive channels (23⇓⇓–26). Consequently, cortex expansion triggers the influx of calcium which in turn leads to contraction. Due to the widespread importance of cortex activity in developmental processes, this feedback is of general interest, investigated in mechanochemical models (27). Models describing the cortex as a fluid (28) or as a poroelastic medium (29), where active stress is up-regulated by a chemical immersed in this medium, account for short-range traveling waves of contractions and oscillatory flows. However, the mechanistic insight is missing that can account for coordination of contractions on scales beyond the intrinsic length scale of the dynamic system and thus account for very long-range fluid flows scaling with system size.

Particularly, the slime mold *Physarum polycephalum* is renowned for long-range coordination of cortex contractions. Here, fluid flows scale with organism size from 2 mm to at least 2 cm (13). Again, flows are known to power organism migration (6, 30). Moreover, stimulants that alter cortex contractility have recently been found to be advected with the fluid flows inside the cell (2). This observation suggests that the physical transport by fluid flows is the key to long-range spatial coordination of cortex contractions and fluid flows.

Here, we investigate in a tubular geometry the self-organization of cortex contractions, coupled to a contraction-triggering chemical which is advected with the flows of the fluid cytoplasm. The simple two-component model is unstable toward self-sustained cortex oscillations as cortex stretching triggers the increase of the contraction-triggering chemical concentration. A linear analysis of the model predicts traveling-wave solutions of cortex shape and the intrinsic wavelength of the dynamic instability is derived. In contrast to our analytic prediction, numerical solutions of the model in a tube with periodic boundaries show a probabilistic distribution of five different patterns of traveling waves. Although the tube is twice as long as the expected intrinsic wavelength, in one of these patterns the traveling wave scales with tube length. Further analysis shows that scaling can be robustly generated in growing tubes with periodic boundary conditions or by flow-generating boundary conditions in nongrowing tubes. We identify oscillatory flows as the key to the scaling of contraction waves with system size. The ubiquity of fluid flows in biological and nonliving systems suggests that this nontrivial scaling could be broadly relevant in active matter.

## Results

### Coupling of Tubular-Shaped Cortex with Contraction-Triggering Chemical.

A cell showing coordinated cytoplasmic fluid flows in general has a distinctive viscous fluid phase separated from a surrounding viscoelastic actomyosin cortex. The nature of long-range flows typically entails a tubular cell shape. We here consider as a minimal model an active, viscoelastic tube of length L, filled with a fluid. Tube shape is fully defined by the tube’s radius

In light of the role of calcium in coordinating actomyosin activity, we describe the strength of the cortex contractile stress to be proportional to the concentration of a contraction-triggering chemical c. In addition, contractions may self-amplify as more actin fibers overlap in a contracted cortex following observations for low myosin concentrations typical for nonmuscle cells (31, 32). Inversely, overlap decreases in an expanded cortex, reducing potential contractility. Consequently, the contractile stress is represented by*SI Appendix, Linear Stability Analysis*).

### Stretch-Activated Chemical Inflow Controls Self-Sustained Oscillations.

At zero fluid flow the tube’s radius is uniformly at its rest value of *B*). Thus, the relative stretch parameterized by *SI Appendix, Instability Condition*). The wavelength of the most unstable mode is given by*B*). The oscillation frequency ω can be derived (*SI Appendix*, Eq. **S1**). The result confirms the intuitive idea that oscillations occur if the stretch-activated chemical release, controlled by

### Multiple Patterns of Contractions Arise in a Periodic Tube.

To study the self-organization of contractile waves in organisms of varying sizes, we numerically solve model Eqs. **1** and **5** in tubes of different lengths L and measure the sizes of the contractile wave patterns λ (Fig. 2). As model parameters, we choose physiological values for calcium kinetics and actomyosin cortex mechanics (*Materials and Methods*). Tube radius is chosen to match *P. polycephalum*—most renowned for scaling contraction waves. The model is further verified by comparing the numerically observed phase relationship between fluid flow and contraction-triggering chemical to experimental data of *P. polycephalum* (34). The model predicts the nontrivial change of phase relationship along the tube (*SI Appendix*, Fig. S3), robust against changes in model parameters. Among ambiguous observations on the role of calcium in *P. polycephalum* (35, 36), complemented by theoretical models (37), this experimental verification of our model promotes that calcium activates actin–myosin contractions in *P. polycephalum* as is common in living organisms.

To determine the size of the waves, we computed the power spectral density of the radius *B* and *E*, for examples of patterns with different wavelength and equal wave size). Simulations with closed boundary conditions (Fig. 2*A*) fully match our expectations from linear stability analysis, namely waves increasing with tube length up to an upper bound given by the intrinsic wavelength corresponding to the most unstable mode *B*) we observe waves exceeding

Characterizing more precisely the variety of wave patterns, we screen multiple runs with different initial perturbations, for the intermediate tube length *SI Appendix*, Fig. S2). Observed wave patterns can be divided into five cases, by wave size and period-averaged flow rate along the tube (see *SI Appendix*, Fig. S6 for details on the identification of the patterns). There are one, two, or three waves traveling in the same direction (Fig. 3 *A*, *B*, or *C*, respectively), two antisymmetric waves (Fig. 3*E*), or two asymmetric waves (Fig. 3*D*). The single wave matching tube length (Fig. 3*A*) generates the strongest net fluid flow, exceeding unidirectional multiple wave patterns (Fig. 3 *B* and *C*) and asymmetric waves (Fig. 3*D*). Patterns with antisymmetric waves (Fig. 3*E*) do not create a net flow due to their invariance under space flipping, thus not providing any mass transport or long-range mixing. Patterns occur with very different probabilities with only 12% showing the most efficient pattern regarding mixing and transport, where the wave scales with tube length. In general, net flow and maximum flow increase with wave size (38) (*SI Appendix*, Fig. S5), and thus generating waves scaling with system size is fundamental for organisms whose size varies vastly. What measures can make this most efficient pattern more robust? What mechanism drives the scaling of the contractions with the size of the tube?

### Growth of the Tube Leads to the Robust Scaling of the Wave.

To investigate robustness and the mechanism behind the scaling of contractile waves we performed simulations of growing tubes and measured wave size for periodic or closed boundaries (Fig. 4). Tubes grow linearly, starting from

In agreement with linear stability analysis we find that waves in tubes with closed boundaries grow with tube length only up to the upper bound *SI Appendix*, Fig. S1), suggesting another mechanism beyond linear stability analysis at play. Above this limit to the scaling *SI Appendix*, Fig. S5), in accordance with observations (39). Results are robust against variations in parameters. Particularly, changing the fluid viscosity μ varies the scaling limit *SI Appendix*, Fig. S7 for **7**, we can see that the predicted wavelength scales like **5** leads to a typical scale proportional to *SI Appendix*, Fig. S7). As

### Scaling of the Wave Is Due to Oscillatory Flows.

To distinguish the role of net flow *SI Appendix*, Eq. **S1**). The values of *SI Appendix*, Fig. S8).

Imposed flow boundary conditions result in long-range contraction patterns (Fig. 5). Interestingly, *SI Appendix*, Fig. S8). Thus, we find that oscillating flow at the boundary, rather than net flow, is the key to scaling with system size much beyond the intrinsic length scale of the instability.

## Discussion

We have studied the self-organization of long-range fluid flows in tubular-shaped cells, due to the coupling of cortex contractions to an advected, contraction-triggering chemical. Our minimal two-component model system describing cortex and chemical dynamics predicts self-sustained contraction waves of wave size

From a dynamical systems point of view, previous work accounted for scaling contraction waves only when also the period of contraction scaled (41), whereas in our mechanism the period barely changes. Also the observation of mode multiplying at factors of up to 8 vastly exceeds previous observations of mode doubling or mode tripling (40). Within our model we find that mode multiplying is determined by the ratio between the scaling limit

The oscillatory nature of fluid flows allowed by periodic boundary conditions or by imposed flow is crucial to generate scaling beyond the intrinsic wavelength **7**). The value of the intrinsic wavelength may vary broadly between different systems. Assuming our representative parameter values, that contractility is of the same order as stiffness

For the system best studied for its cortex-driven cytoplasmic flows, *P. polycephalum*, and with parameter values inferred from related organisms where necessary (*Materials and Methods*), the predicted wave size is *P. polycephalum* forms a network of tubes with more viscous bags pooling fluid at the growing fronts. It is fascinating to speculate how the network morphology impacts the dynamics of contractile waves. It is likely that the contractions of the viscous bags at the growing fronts here do serve as pumps very much similar to the imposed flow boundary conditions we implemented. The growing fronts could thereby also account for the resurrection of scaling contraction waves after contraction stopped due to harmful external stimuli (42). In contrast to *P. polycephalum* to date detailed quantitative data are lacking in other systems to allow for quantitative comparison. However, cortex contractions and oscillatory flows are very general components for many other systems, even beyond the single cell. Thus, the interplay of fluid flows and mechanical oscillations resulting in scaling might be broadly relevant.

In general, our model broadens the budding understanding of the fundamental role of cytoplasmic flows in a large class of biophysical systems (1, 43⇓–45). In very diverse systems, flows appear to be a fundamental part of a self-organized machinery. In our case, their oscillations are crucial to drive and organize patterns of contractions on a large scale, a mechanism likely present in many other biological systems. More fundamentally, our result opens perspectives on how including active advection in classical reaction–diffusion frameworks leads to unexpected observations such as scaling.

## Materials and Methods

### Implementation.

Numerical solutions of the model equations were explored with a custom-written Crank–Nicholson scheme implemented in MATLAB (The Mathworks). Simulations started from the spatially uniform equilibrium value for tube radius and chemical concentration. To perturb the stable state, uncorrelated, Gaussian fluctuations of SD 0.01 were added to the radius. Three kinds of boundary conditions were implemented: periodic, closed, or flow. For flow boundary conditions, the radius and the chemical concentration at the boundaries of the tube are both assumed to be equal to their value at the uniform equilibrium, and fluid flow is imposed on one end of the tube. In growing tubes, linear growth is simulated by changing dynamically the mesh size used for spatial discretization. The mesh is refined when the length of the tube doubled. The growth rate is small compared with the contraction period to decouple the dynamics of the system from growth. When tubes reach their target lengths, simulations are continued for roughly 200 additional contraction periods to ensure that growth has no impact on the simulated pattern.

### Parameters.

Simulations parameters were *P. polycephalum* (13). The resulting flow velocities in our simulations were around ^{−1} to 30 μm⋅s^{−1}, matching cytoplasmic flows for *P. polycephalum* (2).

## Acknowledgments

This work was supported by the Max Planck Society.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: karen.alim{at}ds.mpg.de.

Author contributions: J.-D.J. and K.A. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1805981115/-/DCSupplemental.

Published under the PNAS license.

## References

- ↵
- ↵
- Alim K,
- Andrew N,
- Pringle A,
- Brenner MP

*Physarum polycephalum*. Proc Natl Acad Sci USA 114:5136–5141. - ↵
- Koslover EF,
- Chan CK,
- Theriot JA

- ↵
- ↵
- ↵
- ↵
- ↵
- Yoshida K,
- Soldati T

- ↵
- ↵
- Wolke U,
- Jezuit EA,
- Priess JR

- ↵
- Atwell K, et al.

*C. elegans*germ line suggests feedback on the cell cycle. Development 142:3902–3911. - ↵
- Hecht I,
- Rappel WJ,
- Levine H

- ↵
- Alim K,
- Amselem G,
- Peaudecerf F,
- Brenner MP,
- Pringle A

*Physarum polycephalum*organizes fluid flows across an individual. Proc Natl Acad Sci USA 110:13306–13311. - ↵
- ↵
- ↵
- Taylor DL,
- Condeelis JS,
- Moore PL,
- Allen RD

- ↵
- ↵
- Jäger W,
- Murray JD

- Oster GF,
- Odell GM

- ↵
- ↵
- ↵
- Levasseur M,
- Carroll M,
- Jones KT,
- McDougall A

- ↵
- Antunes M,
- Pereira T,
- Cordeiro JV,
- Almeida L,
- Jacinto A

- ↵
- Glogauer M, et al.

- ↵
- ↵
- Matthews BD,
- Overby DR,
- Mannix R,
- Ingber DE

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Janson LW

- ↵
- ↵
- Zhang S,
- Guy RD,
- Lasheras JC,
- del Álamo JC

*Physarum*fragments. J Phys D Appl Phys 50:204004. - ↵
- Tazawa M

- Kuroda R,
- Hatano S,
- Hiramoto Y,
- Kuroda H

- ↵
- ↵
- ↵
- ↵
- Kuroda S,
- Takagi S,
- Nakagaki T,
- Ueda T

- ↵
- ↵
- Kessler DA,
- Levine H

- ↵
- ↵
- Woodhouse FG,
- Goldstein RE

- ↵
- ↵
- ↵
- Puchkov EO

- ↵
- Naib-Majani W,
- Teplov VA,
- Baranowski Z

*Cell Dynamics*, Protoplasma, ed Tazawa M (Springer, Vienna), Vol 1, pp 57–63. - ↵
- ↵
- ↵
- ↵
- ↵

## Citation Manager Formats

## Article Classifications

- Physical Sciences
- Physics

- Biological Sciences
- Biophysics and Computational Biology