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Oscillations and Bifurcation Structure of Reaction-Diffusion Model for Cell Polarity Formation

We investigate the oscillatory dynamics and bifurcation structure of a reaction-diffusion system with bistable nonlinearity and mass conservation, which was proposed by [Otsuji et al, PLoS Comp. Biol. 3 (2007), e108]. The system is a useful model for understanding cell polarity formation. We show that this model exhibits four different spatiotemporal patterns including two types of oscillatory patterns, which can be regarded as cell polarity oscillations with the reversal and non-reversal of polarity, respectively. The trigger causing these patterns is a diffusion-driven (Turing-like) instability. Moreover, we investigate the effects of extracellular signals on the cell polarity oscillations.


Introduction
Reaction-diffusion systems with bistable nonlinearity are useful for understanding basic biological phenomena such as the population dynamics of competing species and propagating pulses on the nerve axon [18,21]. As for cell polarization, which is a key process in cell division and differentiation, [15] proposed a reaction-diffusion model with bistable nonlinearity for understanding the wave-pinning phenomena: A transient and localized stimulus to an unpolarized cell is spatially amplified to result in a robust subdivision of the cell into two clearly defined regions, front and back. Although the bistability in their model, which is represented by a Hill function, may not explicitly correspond to concrete molecular networks, it is a simple (phenomenological) model and hence allows us to explain a mechanism refereed to as the wave-pinning by a mathematical analysis. Moreover, [16] investigated the bifurcation structure of the model in [15].
The maintenance of cell polarity can be considered as a dynamic process.
In fact, cells in a developing epithelial tissue alternately repeat the establishment and loss of polarity in a cell cycle-dependent manner. This repetition is known as a cell polarity oscillation [3], where the reversal of polarity does not occur in such a way that the cells in an epithelial tissue are organized into layers. Moreover, cell polarity oscillations are also observed in cell migration, which is the directed movement of a single cell or a group of cells in response to chemical and/or mechanical signals. For example, bacteria cells during rippling (accordion-like movements) [4,25] and migrating melanoma cells derived from tumors [24] can reverse their polarity in an oscillatory fashion.
For understanding cell polarity oscillations, some mathematical models have been proposed. For example, [24] proposed an ODE system based on biomolecular dynamics at the front and back lamellipodia in a migration cell to better understand their experimental results concerning the change of cell polarity in melanoma cells derived from tumors. [4] proposed an ODE system based on three proteins to understand cell polarity oscillations with the polarity reversal of bacteria cells during rippling (accordion-like movements) [25]. [12] proposed a delay reaction-diffusion equation with a nonlocal term for studying the effects of a delayed negative feedback on cell polarity oscillations with the aid of numerical simulations. Recently, [28] generalized the ODE system in [4], and proposed an ODE system that generates four different mechanisms for switching cell polarity, including a mechanism to generate the periodic reversal of polarity.
In this paper, we consider a reaction-diffusion model with bistable nonlinearity proposed by [23], which exhibits dynamics similar to that of considerably realistic reaction-diffusion models based on concrete molecular species and networks concerning cell polarity formation [19,23,27]: and which implies 0 < α < 1. We note that u and v should be considered as deviations from the reference values of the concentrations of the chemicals because either u or v can be negative, even if the initial values of u and v are positive as seen in the next section. Moreover, by setting a 2 → a 2 − εa E (x) in (1.2), we obtain the perturbed system of (1.1), i.e., where a E (x) is a smooth function that represents the waveform of an extracellular signal, and ε is a small positive parameter that controls the amplitude of the extracellular signal. We consider (1.1) and (1.4) on an interval I = (−K/2, K/2) under the periodic boundary condition for sufficiently large K.
In general, a reaction-diffusion system (1.1) with a smooth function f is called a reaction-diffusion system with mass conservation because holds for any (smooth) solutions under the periodic or Neumann boundary conditions. For certain functions f and boundary conditions, some mathematical aspects of (1.1) have been studied elsewhere [1,6,8,16,17,22], and one of the typical dynamics of (1.1) is summarized as follows: Solutions with initial values near a homogeneous equilibrium destabilized through the same mechanism as the Turing instability eventually approach a localized unimodal stationary pattern (see, Figure 2). This dynamics is interpreted as cell polarization, and the position of the peak of the localized unimodal pattern determines the direction of the polarity. As for the function f given by (1.2), it was numerically shown in [23] that (1.1) has such dynamics under certain conditions. In addition, [23] performed a formal stability analysis for the final localized unimodal stationary pattern of (1.1), and numerically showed that a solution of (1.4) starting from the final pattern of (1.1) can translationally move to the maximum point of a E (x) when a E (x) is given by with (1.2) has large amplitude limit cycles (see, Figure 3). These limit cycles exhibit an alternating repetition of unimodal and spatially homogeneous patterns, and hence can be interpreted as cell polarity oscillations. Moreover, using a numerical bifurcation analysis, we investigate the bifurcation structure of (1.1) with (1.2), which generates the limit cycles when we vary we examine the position of the peak of the periodically appearing unimodal patterns.
The remainder of this paper is organized as follows. In Section 2, we describe the bistability of an ODE system obtained from (1.1) by dropping the diffusion terms. Our characterization of the bistability is different from that of [15,16]. In Section 3, we investigate spatiotemporal patterns of the limit cycles exhibiting such an alternating repetition, and examine whether the position of the peak of the periodically appearing unimodal pattern can be determined by the maximum point of a E (x). These results are useful for understanding the effects of extracellular signals on cell polarity oscillations.
Section 6 discusses the results of this study.

The dynamics of ODE system
In this section, we investigate the dynamics of an ODE system obtained from (1.1) by dropping the diffusion terms. Our characterization of the bistability of the ODE system is different from that of [15,16].
We consider the dynamics of where a 1 > 0, a 2 > 0 and 0 < α < 1. It is easy to see that f (u, v) = 0, an equation for u with arbitrarily fixed v, has three roots We note that any solution of (2.1) satisfies u + v = ξ for some ξ, and thaṫ has a unique stable equilibrium for ξ = 0 by 0 < α < 1. Hence, we see that (2.1) has a stable equilibrium with ξ = 0, which corresponds to the stable equilibrium of (2.3) for ξ = 0. Therefore, we see that (2.1) has two families of stable equilibria Γ s Thus, we can consider that (2.1) is bistable in the usual sense because it has two equivalent stable states Γ s 1 and Γ s 2 , and one unstable state Γ u , which is a separatrix as seen in Figure 1. Moreover, we note that the dynamics of (2.1) does not qualitatively change when we vary α ∈ (0, 1).

Remark 2.1
Although we cannot apply a standard argument concerning the invariant region of reaction-diffusion systems [26], the phase plane in We note that for each fixed v, the ODEu = f (u, v) is monostable because it has two unstable equilibria h 1 (v) and h 2 (v), and one stable equilibrium −v. This is equivalent to a condition in which for each fixed u, the ODEv = −f (u, v) is bistable because it has two stable equilibria h −1 1 (u) and h −1 2 (u), and one unstable equilibrium −u. The condition in whichu = f (u, v) is monostable for each fixed v is opposite to a condition in [15,16] in whicḣ is bistable for each fixed v, which is used for a standard argument concerning the traveling waves and transition layers of reaction-diffusion systems (cf. [20] and the references therein). In contrast, we are concerned with unimodal patterns of (1.1) with (1.2), which do not need such an argument.

3
Spatiotemporal patterns of unperturbed system In this section, we investigate spatiotemporal patterns of the unperturbed system (1.1) with (1.2), which is useful for understanding spontaneous cell polarization without extracellular signals. In particular, we numerically show limit cycles that exhibit an alternating repetition of unimodal and spatially homogeneous patterns. These limit cycles can be interpreted as cell polarity oscillations.

Concrete expression of localized unimodal stationary pattern
We give a concrete expression of a localized unimodal stationary pattern of (1.1) with (1.2) for sufficiently large K.
We consider stationary solutions of (1.1), i.e., where f (u, v) is given by (1.2). Applying a transformation of variables to (3.1), we have Since q satisfies the periodic boundary condition, it follows from the first equation of (3.3) that q(x) ≡ q 0 . Therefore, by the second equation of (3.3), we have where On the other hand, a direct calculation shows that (3.5) Therefore, noting (3.2), we see that satisfy (3.1). Since sech(bx) = O(e −b|x| ) as |x| → ∞, it is easy to see that (1.1) has a stationary solution (ū(x),v(x)) satisfying the periodic boundary as K → ∞. We consider that (ū(x),v(x)) is a localized unimodal stationary pattern as shown in Figure 2(b) when p 0 > 0 and K is sufficiently large.
Moreover, we note that (ū(x − c),v(x − c)) for some c ∈ [−K/2, K/2) is also a stationary solution of (1.1) because of the periodic boundary condition.
It follows from (3.8) and (3.9) that and integrating (3.10) over I = (−K/2, K/2), we have as K → ∞. This implies that the amplitude of the localized stationary pattern given by (3.8) and (3.9) can be explicitly determined by the total mass of components.

Formation of localized unimodal pattern
We numerically investigate the formation process of a localized unimodal pattern of (1.1) with (1.2). Our numerical simulations show that a solution starting from a spatially homogeneous initial state with a small disturbance conditions, whereū andv are given by (3.8) and (3.9), respectively.
Noting the mass conservation property (1.6) and (3.12), we see that the amplitude of the localized stationary pattern given by (3.8) and (3.9) is determined by for sufficiently large K, where can be considered as a parameter. For each fixed ξ, we numerically solve We choose K = 10.0 in such a way that (ϕ 1 (x), ϕ 2 (x)) can be numerically regarded as a stationary solution satisfying the periodic boundary condition.
The initial value (u(x, 0), v(x, 0)) is given by (ξ/2, ξ/2) with a small random perturbation, which means that the initial states of u and v are almost the same and spatially homogeneous. Notice that these parameter values are the same as those in [23].
We use a numerical scheme in [11] based on the pseudospectral method; we obtain numerical solutions of (1.1) on 0 ≤ x ≤ K, which can be regarded We do not consider the case ξ = 0 that numerical solutions do not move.

Spatiotemporal oscillatory patterns
We cannot observe that numerical solutions converge to the localized uni-

Numerical bifurcation analysis
In this section, we numerically demonstrate that the bifurcation diagram of (1.1) with (1.2) contains a structure that generates spatiotemporal patterns as presented in the last section. By using AUTO [2], a software package used for studying the bifurcation structure of ODE systems, we investigate the bifurcation diagram of an ODE system, which is obtained by the finite Fourier series approximation for (1.1) with (1.2) in the same way as [9]. We investigate the bifurcation diagram of the ODE system with respect to  Unstable branches bifurcating from unstable branches for d 1 ≤ 0.30 are omitted. Notice that the two branches bifurcating from a pitchfork bifurcation point are piled up and are displayed in this bifurcation diagram. The primary branch starting from d 1 = 0.8 represents a family of spatially homogeneous equilibria (4.1), which correspond to spatially homogeneous stationary patterns as shown in Figure 3(d). The secondary branch bifurcating from the primary branch via the supercritical pitchfork bifurcation at d 1 ≈ 0.74 represents a family of (localized) unimodal stationary patterns as shown in Figure 3(a).
represents a family of spatially homogeneous equilibria whereas unstable for 0 < d 1 < d c 1 , where d c 1 is given by It should be noted that d c 1 is independent of ξ. When d 1 = 0.8, we have d c 1 ≈ 0.74, which gives a supercritical pitchfork bifurcation point detected by AUTO in Figure 4. We should consider that this bifurcation is due to a diffusion-driven instability but not a well-known Turing instability. In fact, Remark 4.1 When we consider (1.1) with (1.2) for fixed α, we can similarly verify that a pitchfork bifurcation point where the stability of the equilibria (4.1) changes is given byd In the strict sense, this bifurcation is not due to a Turing instability. In fact, as seen in Section 2, the equilibria (4.1) are stable in the ODE (2.1) obtained from (1.1) by dropping the diffusion terms. However, the linearized operator of the right hand side of (2.1) at (4.1) has zero eigenvalue, which implies that a well-known condition for Turing instability is not satisfied (cf. [18]). Therefore, the bifurcation in this case is sometimes called a Turing-like bifurcation [6].
The secondary branch bifurcates from the primary branch via the supercritical pitchfork bifurcation at d 1 ≈ 0.74 as d 1 decreases. This stable branch represents a family of (localized) unimodal stationary patterns as shown in Figure 3(a). The bifurcation diagram presented in Figure 4 suggests that (1.1) with (1.2) has no limit cycles as shown in Figures 3(b) and (c). This type of bifurcation diagram can be obtained for ξ 0.67.  t)dx, respectively. The white and black squares indicate the pitchfork and Hopf bifurcation points, respectively. The curve represented by a family of the black circles • indicates the stable branch of periodic solutions. The primary branch starting from d 1 = 0.8 represents a family of spatially homogeneous equilibria (4.1), which correspond to spatially homogeneous stationary patterns as shown in Figure 3(d). The secondary branch bifurcating from the primary branch via the supercritical pitchfork bifurcation at d 1 ≈ 0.74 represents a family of (localized) unimodal stationary patterns as shown in Figure 3(a). The third branch (indicated by •) bifurcating from the second branch via the Hopf bifurcations at d 1 ≈ 0.46 and d 1 ≈ 0.21 represents a family of limit cycles as shown in Figure 3(c). Figure 5 shows the bifurcation diagram for ξ = 0.64. As seen in Figure 4, the secondary branch consisting of (localized) unimodal stationary patterns bifurcates from the primary branch consisting of spatially homogeneous equilibria via a supercritical pitchfork bifurcation at d 1 ≈ 0.74 as d 1 decreases.
However, (localized) unimodal stationary patterns on the second branch are unstable for 0.21 d 1 0.46. In fact, the third branch consisting of stable limit cycles as shown in Figure 3(c) bifurcates from the secondary branch via Hopf bifurcations at d 1 ≈ 0.46 and d 1 ≈ 0.21. In this case, we cannot observe limit cycles as shown in Figure 3(b). This type of bifurcation diagram can be obtained for 0.61 ξ 0.66. Figure 6 shows the bifurcation diagram for ξ = 0.60. As seen in Figure 5, the third branch consisting of limit cycles as shown in Figure 3(c) bifurcates from the second branch consisting of (localized) unimodal stationary patterns via Hopf bifurcations at d 1 ≈ 0.53 and d 1 ≈ 0.14. However, the third branch has unstable limit cycles for 0.13 d 1 0.14 and 0.18 d 1 0.31. We can observe limit cycles as shown in Figure 3 , which correspond to spatially homogeneous stationary patterns as shown in Figure 3(d). The secondary branch bifurcating from the primary branch via the supercritical pitchfork bifurcation at d 1 ≈ 0.74 represents a family of (localized) unimodal stationary patterns as shown in Figure 3(a). The third branch (indicated by • and •) bifurcating from the second branch via the Hopf bifurcations at d 1 ≈ 0.53 and d 1 ≈ 0.14 represents a family of limit cycles as shown in Figure 3(c). At present, we cannot trace the fourth branch bifurcating from the third branch via the period doubling bifurcations at d 1 ≈ 0.31 and d 1 ≈ 0.18. This branch that is not displayed here corresponds to a family of limit cycles as shown in Figure 3

Motion of localized unimodal pattern
In the same way as Subsection 3.2, we find that a solution of (1.4) starting from a spatially homogeneous initial state with a small disturbance quickly and temporally approaches ( under certain conditions, whereū andv are given by (3.8) and (3.9), respectively. Then, the solution translationally moves to the maximum point of a E (x), as shown in Figure 8. We derive the equation of motion for this translational movement according to [10]. In this subsection, to avoid confusion, we rewrite (1.1) and (1.4) as respectively, and (1. 2) as Moreover, to obtain a concrete and exact expression of the equation of motion, we consider that where a E (x) = A cos(2πx/K) (5.5) takes its maximum at x = 0. In this subsection, we carry our argument based on [10] by using (5.1)-(5.5).
Remark 5.1 The theory of [10] was built on a mathematical assumption [10,Assumption 3.4], which guarantees that solutions stay in a small neighbor- Although such an assumption is indispensable to build a rigorous theory (cf. [5]), it is technically difficult to check it in many practical problems. Here, we formally apply an argument in [10] to our problem without checking the mathematical assumption.

Dynamics of spatiotemporal oscillatory patterns
The result in the last subsection suggests that the position of the peak of the final localized unimodal stationary pattern of (1.4) can be determined by the maximum point of a E (x). In this subsection, we investigate whether the perturbed system (1.4) has spatiotemporal oscillatory patterns as shown in

Discussion
In this paper, we show that a reaction-diffusion system with bistable nonlinearity and mass conservation, proposed by [23], exhibits four different spatiotemporal patterns including two types of oscillatory patterns, which can be regarded as cell polarity oscillations with the reversal and non-reversal of polarity, respectively. The trigger causing these patterns is a diffusiondriven (Turing-like) instability. Moreover, we show that the periodic reversal is due to a period doubling bifurcation using a numerical bifurcation anal-  (1) can also be obtained from other reaction-diffusion models [6,23]. In contrast, assertion (2) is a characteristic of our results based on the reaction-diffusion model (1.1) with (1.2). When |X| is constant, we can regard K as a bifurcation parameter. Let us suppose a situation in which the parameter K increases whereas |X| is constant, and consider that this situation corresponds to the stage (just) before cell division. We can then see that cell polarity oscillations can be observed (just) before cell division, which is consistent with our naive intuition (cf. [3]). Similarly, when K is constant, we can regard |X| as a bifurcation parameter. We suppose a situation in which the parameter |X| decreases whereas K is constant. We can then see that cell polarity oscillations occur when the amount of chemical X decreases. This implies that chemical X is indispensable for freezing cell polarity.
In simple terms, mathematical models for studying cell polarity oscillations are classified into two types, i.e., ODE models and reaction-diffusion models. In general, ODE models are based on concrete molecular species and networks, and explicitly or implicitly assume the existence of the front and back of a cell. Such models cannot represent cell polarity oscillations with the non-reversal of polarity, as observed in cells in a developing epithelial tissue [3]. In contrast, the reaction-diffusion model (1.1) with (1.2) can represent cell polarity oscillations with both the non-reversal and reversal of polarity. Moreover, we note that it does not assume a priori the existence of the front and back of a cell.
The Gierer-Meinhardt model [14] seems to be the first reaction-diffusion system in history, which can be used to study spatiotemporal patterns con-sisting of (many) localized unimodal patterns. This model has been considered as a reference for studying cell polarity, and [14] noticed that oscillations are necessary for the reorientation of polarity. However, they did not give any concrete information concerning the oscillations. More than twenty years later, [13] proposed an ODE system consisting of many components (cells), which can represent cell polarity oscillations with the periodic reversal of polarity. Consequently, as shown in a review paper [7], reaction-diffusion models have been considered to be undesirable for representing the dynamic maintenance of cell polarity such as cell polarity oscillations. Our result suggests a counter-example for such a common view, i.e., (1.1) with (1.2) can be a simple reaction-diffusion model for studying cell polarity oscillations.
A reaction-diffusion model of the Rho GTPases (Rac, Cdc42, RhoA), which are key regulators for cell polarity, was also proposed by [23]. Moreover, [23] compared this model with reaction-diffusion models proposed by [19,27], which are based on a phosphoinositide cycle that recruits and activates the Rho GTPases (Rac, Cdc42). Consequently, the dynamics of these models share common properties, which can be interpreted as follows: (1) the spontaneous formation of cell polarity, (2) the uniqueness of polarity determining the axis of a cell, (3) the polarized sensitivity wherein a polarized cell responds to a change in the extracellular signal gradient. Although these models based on the concrete molecular species and networks are considerably realistic, they consist of many components and have complicated nonlinear terms, and hence it is difficult to understand a mechanism for generating the dynamics with three properties. Therefore, [23] provided the concept of reaction-diffusion systems with mass conservation, which is given by (1.1). In fact, the realistic models include the structure of (1.1), and it was shown in [23] that (1.1) can exhibit dynamics with three properties for appropriate nonlinear terms f (u, v).
The reaction-diffusion system (1.1) with (1.2) is a phenomenological model, which may not explicitly correspond to concrete molecular species and networks. Therefore, at present, our results may not be applicable to practical biological problems. However, as was mentioned above, the reaction-diffusion system (1.1) with (1.2) can exhibit the dynamics observed in the realistic models proposed by [19,23,27]. Hence, it is important to explore a parameter region where these realistic models exhibit cell polarity oscillations.
Once we find such a region, we can propose physical characteristics or parameters related to mechanisms for switching cell polarity, and can therefore compare our results with those in [28]. Moreover, we might propose experimentally testable hypotheses concerning cell polarity oscillations with the aid of biologists.
Since our results were obtained using a specific reaction-diffusion system with bistable nonlinearity and mass conservation, it is also important to investigate conditions in which general reaction-diffusion systems with bistable nonlinearity and mass conservation can generate cell polarity oscillations.
This investigation will consist of three steps: (1) investigating conditions in which (1.1) with bistable nonlinearity has a localized unimodal stationary pattern, (2) investigating conditions in which the unimodal pattern becomes unstable through a Hopf bifurcation with respect to the ratio of the diffusion coefficients by a normal form analysis, as presented in [22], (3) investigating conditions in which oscillatory patterns bifurcating from the unimodal pattern undergo a period doubling bifurcation by a numerical bifurcation analysis, as presented herein. We expect that the generalization of nonlinear terms will provide a viewpoint for exploring biological phenomena, which can be studied by reaction-diffusion systems with bistable nonlinearity and mass conservation.
Acknowledgments. The authors would like to express their appreciation to the referees for their useful suggestions and comments, which have improved the original manuscript. The first, second, and third authors were supported in part by the JSPS KAKENHI 16K05273, 17K14237, 19H01805, respectively.