# Flow Induced Symmetry Breaking in a Conceptual Polarity Model

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## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Linear Stability Analysis

#### 3.1. Linearized Dynamics and Basic Results

#### 3.2. Intuition for the Flow-Driven Instability and Upstream Propagation of the Unstable Mode

#### 3.3. Long Wavelength Limit

#### 3.4. Limits of Slow and fast Flow

#### 3.5. Summary and Discussion of Linear Stability

## 4. Pattern Propagation in the Nonlinear Regime

## 5. Flow-Induced Transition from Mesa to Peak Patterns

## 6. Flow-Induced Pattern Formation

## 7. Conclusions and Outlook

## Supplementary Materials

- Growth of a pattern from a homogeneous steady state in the presence of flow. Top: concentration profiles in space; bottom: corresponding density distribution in the phase space. (Parameters: ${D}_{m}=0.1\mathsf{\mu}{\mathrm{m}}^{2}/\mathrm{s},\overline{n}=3\mathsf{\mu}{\mathrm{m}}^{-1}$, $L=20\mathsf{\mu}\mathrm{m}$ and ${v}_{\mathrm{f}}=20\mathsf{\mu}\mathrm{m}/\mathrm{s}$.)
- Simulation with adiabatically increasing flow speed from ${v}_{\mathrm{f}}=0\mathsf{\mu}\mathrm{m}/\mathrm{s}$ to ${v}_{\mathrm{f}}=100\mathsf{\mu}\mathrm{m}/\mathrm{s}$. Note the flattening of the cytosolic concentration profile as the flow speed increases. (Fixed parameters as for Movie 1.)
- Corresponds to the space-time plot in Figure 4A.
- Pattern transformation from a mesa pattern to a peak pattern as flow speed is adiabatically increased from ${v}_{\mathrm{f}}=0$ to ${v}_{\mathrm{f}}=45\mathsf{\mu}\mathrm{m}/\mathrm{s}$. (Fixed parameters as for Movie 1.)
- Pattern formation triggered by mass-redistribution due to a spatially non-uniform flow (parabolic flow profile shown in Figure 6A). After the flow is switched off at $t=200\mathrm{s}$, the pattern is maintained. (Parameters: ${D}_{m}=0.1\mathsf{\mu}{\mathrm{m}}^{2}/\mathrm{s},L=30\mathsf{\mu}\mathrm{m}$, ${v}_{\mathrm{max}}=1\mathsf{\mu}\mathrm{m}/\mathrm{s}$, and $\overline{n}=1\mathsf{\mu}{\mathrm{m}}^{-1}$.)
- As Movie 5, but with lower average mass, $\overline{n}=0.8\mathsf{\mu}{\mathrm{m}}^{-1}$. This mass is not sufficient to maintain a stationary peak in the absence of flow. Therefore, the peak disappears after the flow is switched off ($t>200\mathrm{s}$).

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

McRD | mass-conserving reaction–diffusion |

2cMcRD | two-component mass-conserving reaction–diffusion |

FBS | flux-balance subspace |

## Appendix A. Limit of Slow Flow and Timescale Comparison

## References

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**Figure 1.**One-dimensional two-component system with cytosolic flow into the positive x-direction. The reaction kinetics include (

**1**) attachment, (

**2**) self-recruitment, and (

**3**) enzyme-driven detachment.

**Figure 2.**(

**A**) Sketch of real (solid) and imaginary (dotted) part of a typical dispersion relation with a band $[0,{q}_{\mathrm{max}}]$ of unstable modes. (

**B**) The initial dynamics of a spatially homogeneous state with a small random perturbation (blue thin line). The direction of cytosolic flow is indicated by a blue arrow. The typical wavelength ($\lambda $) of the initial pattern is determined by the fastest growing mode ${q}^{*}$ and the phase velocity is determined by the value of the imaginary part of dispersion relation at the fastest growing mode (${v}_{\mathrm{phase}}=-\mathrm{Im}\sigma \left({q}^{*}\right)/{q}^{*}$). The growth of the pattern is indicated by orange arrows, while the traveling direction is indicated by pink arrows.

**Figure 3.**Sketch of the initial dynamics of an laterally unstable spatially homogeneous steady state. The role of reactions (

**A**), diffusion (

**B**), and advection (

**C**) for a mass-redistribution instability are presented for the membrane (top) and cytosolic (middle) concentration profiles and in phase space (bottom). (

**A**) A small perturbation of the spatially homogeneous membrane concentration (orange dashed lines in top panel) leads to a spatially varying local total density $n\left(x\right)$, with a larger total density at the maximum of the membrane profile (open circle) and a smaller total density at the minimum (open star). These local variations in total density lead to attachment zones (green region) and detachment zones (red region). The reactive flow, indicated by the red and green arrows, points along the reactive subspace (gray lines) in phase space towards the shifted local equilibria (black circles). These reactive flows lead to the solid orange density profiles after a small amount of time. (

**B**) Faster diffusion in the cytosol compared to the membrane (indicated by the large and small blue arrows in the middle and top panel, respectively), lead to net mass transport from the detachment zone to the attachment zone. Again, dashed and solid lines indicate the state before and after a short time interval of diffusive transport. (

**C**) Cytosolic flow shifts the cytosolic concentration with respect to the membrane concentration (orange dashed to orange solid lines), increasing the cytosolic concentration on the upstream side of the pattern and decreasing the cytosolic concentration on the downstream side. In phase space, the trajectory of this density profile forms a ‘loop’.

**Figure 4.**Pattern dynamics far from the spatially homogeneous steady state. (

**A**) Time evolution of the membrane-bound protein concentration. At time ${t}_{0}=240\mathrm{s}$ a constant cytosolic flow with velocity ${v}_{\mathrm{f}}=20\mathsf{\mu}\mathrm{m}/\mathrm{s}$ towards the right is switched on (cf. Movie 3). (

**B**) Relation between the peak speed (${v}_{\mathrm{p}}$) and flow speed (${v}_{\mathrm{f}}$). Results from finite element simulations (black open squares) are compared to the phase velocity of the mode ${q}_{\mathrm{max}}$ obtained from linear stability analysis (green solid line) and to an approximation (orange open circles) of the area enclosed by the density distribution trajectory in phase space (area enclosed by the ‘loop’ in

**D**). (The domain size, $L=10\mathsf{\mu}\mathrm{m}$, is chosen large enough compared to the peak width such that boundary effects are negligible.) (

**C**) A schematic of the phase portrait corresponding to the pattern in

**D**. The density distribution in the absence of flow is embedded in the flux-balance subspace (FBS) (blue straight line). In the presence of flow, the density distribution trajectory forms a ‘loop’ in phase space. The upstream and downstream side of the pattern are highlighted in cyan and magenta, respectively. Red and green arrows indicate the direction of the reactive flow in the attachment and detachment zones, respectively. At intersection points of the density distribution with the nullcline (${c}_{\mathrm{L}}$ and ${c}_{\mathrm{R}}$) the system is at its local reactive equilibrium. (

**D**) Sketch of the membrane (orange solid line, top) and cytosolic (orange dashed line, bottom) concentration profiles for a stationary pattern in the absence of cytosolic flow. Flow shifts the cytosol profile downstream (orange solid line, bottom).

**Figure 5.**Demonstration of the transition from a mesa pattern to a peak pattern. Each panel shows a snapshot from finite element simulations in steady state. Top concentration profiles in real space; bottom: corresponding trajectory (blue solid line) in phase space. (

**A**) Mesa pattern in the case of slow cytosol diffusion and no flow. The two plateaus (blue dots) and the inflection point (gray dot) of the pattern correspond to the intersection points of the FBS (blue dashed line) with the reactive nullcline (black line). (

**B**) For fast cytosol diffusion, the third intersection point between FBS and nullcline lies at much higher membrane concentration such that it no longer limits the pattern amplitude. Therefore, a peak forms whose amplitude is limited by the total protein mass in the system. (

**C**) Slow flow only slightly deforms the mesa pattern, compare to (

**A**). Fast cytosolic flow leads to formation of a peak pattern (

**D**), similarly to fast diffusion. Parameters: $\overline{n}=7\mathsf{\mu}{\mathrm{m}}^{-1},{D}_{m}=0.1\mathsf{\mu}{\mathrm{m}}^{2}/\mathrm{s}$, and $L=20\mathsf{\mu}\mathrm{m}$.

**Figure 6.**Flow-driven protein mass accumulation can induce pattern formation by triggering a regional lateral instability. (

**A**) Top: quadratic flow velocity profile: ${v}_{f}\left(x\right)/{v}_{\mathrm{max}}=1-4{\left(x/L-1/2\right)}^{2}$. Bottom: illustration of the total density profiles at different time points starting from a homogeneous steady state (i) to the final pattern (iv); see Movie 5. Mass redistribution due to the non-uniform flow velocity drives mass towards the right hand side of the system, as indicated by the blue arrows. The range of total densities shaded in orange indicates the laterally unstable regime determined by linear stability analysis. Once the total density reaches this regime locally, a regional lateral instability is triggered resulting in the self-organized formation of a peak (orange arrow). (

**B**) Sketch of the phase space representation corresponding to the profiles shown in A. Note that the concentrations are slaved to the reactive nullcline (black line) until the regional lateral instability is triggered. (

**C**) Schematic representation of the state space of concentration patterns in a case where both the homogeneous steady state and a stationary polarity pattern are stable. Thin trajectories indicate the dynamics in the absence of flow and the pattern’s basin of attraction is shaded in orange. The thick trajectory connecting both steady states shows the flow-induced dynamics, corresponding to the sequence of states (i)–(iv) shown in

**A**and

**B**.

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**MDPI and ACS Style**

Wigbers, M.C.; Brauns, F.; Leung, C.Y.; Frey, E. Flow Induced Symmetry Breaking in a Conceptual Polarity Model. *Cells* **2020**, *9*, 1524.
https://doi.org/10.3390/cells9061524

**AMA Style**

Wigbers MC, Brauns F, Leung CY, Frey E. Flow Induced Symmetry Breaking in a Conceptual Polarity Model. *Cells*. 2020; 9(6):1524.
https://doi.org/10.3390/cells9061524

**Chicago/Turabian Style**

Wigbers, Manon C., Fridtjof Brauns, Ching Yee Leung, and Erwin Frey. 2020. "Flow Induced Symmetry Breaking in a Conceptual Polarity Model" *Cells* 9, no. 6: 1524.
https://doi.org/10.3390/cells9061524