An Efficient, Nonlinear Stability Analysis for Detecting Pattern Formation in Reaction Diffusion Systems

Abstract

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, nonlinear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and nonlinear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have nonlinear patterning regimes not previously detected. The real power of this technique, however, is its simplicity and applicability to larger complex systems where other nonlinear methods become intractable. This is demonstrated through analysis of a chemotaxis regulatory network comprised of interacting proteins and phospholipids. In each case, predictions of this method are verified against results of numerical simulation, linear stability, asymptotic, and/or full PDE bifurcation analyses.

Appendices

Appendix A: Schnakenberg Asymptotics

This analysis closely follows (Ward and Wei 2002). Consider the Schnakenberg system (11a), (11b) on the interval [−1,1]. In Ward and Wei (2002), it was shown that this system exhibits stable spike solutions when a=0. That analysis can be extended to show such spikes in fact exist for all values of a under certain asymptotic conditions.

To begin, define

$$ D=\frac{\bar{D}}{\epsilon}, \qquad v=\epsilon\bar{v}, \qquad u= \frac{\bar{u}}{\epsilon}, $$

(20)

and subsequently drop the $\bar{}$ to yield

$$\begin{aligned} u_t(x,t)&=a\epsilon-u+u^2v+ \epsilon u_{xx}, \end{aligned}$$

(21)

$$\begin{aligned} \epsilon v_t(x,t)&=b-\frac{u^2v}{\epsilon} + D v_{xx} , \end{aligned}$$

(22)

Assuming D≫1/ϵ, v=v ₀+ϵv ₁(x)+⋯, and integrating (22), it can be determined that

$$ 2b=\frac{v_0}{\epsilon}\int_{-1}^1 u^2(x,t) \, dx . $$

(23)

A spike solution of the form u(x)=u ₀+u ₁(x/ϵ) is now sought where u ₀ and u ₁ are the outer and inner solutions. It is assumed u ₀ is spatially constant. Collecting terms involving the same powers of ϵ shows the outer solution is u ₀≈aϵ and the inner solution satisfies

$$ u^{\prime\prime}_{1}(z)-u_1(z)+u_1^2(z) v_0=0 $$

(24)

on x/ϵ=z∈[−1,1] with no flux boundary conditions. The solution to this problem is known (see Ward and Wei 2002) yielding

$$ u(x)=u_0+u_1= a \epsilon+ \frac{3}{2 v_0} \operatorname{sech}^2 \biggl(\frac{x}{2\epsilon}\biggr) . $$

(25)

Integrating the square of this expression and substituting into (23) yields v ₀=b/3. Unravelling the change of coordinates yields the approximate spike solution for the original problem (Eqs. (11a), (11b)) on [−1,1]

$$ u(x) \approx a + \frac{b}{2 \epsilon} \operatorname{sech}^2 \biggl( \frac{x}{2\epsilon}\biggr), \qquad v(x) \approx\frac{3 \epsilon}{b}. $$

(26)

So the Schnakenberg system (11a), (11b) in fact produces spike type solutions for all values of a in the limit ϵ→0. This is in agreement with the results of the LPA in Fig. 2a and the progression of the fold bifurcation (where the spike is lost) to ∞ as a→∞ in the bifurcation analysis in Fig. 2c. Further, the maximum value of u in (26) with a=0 compares to good precision with the maximum values shown at a=0 in Fig. 2c, supporting these results.

Appendix B: Proof of Theorem 4.1

To prove Theorem 4.1, first notice the eigenvalues of J _k (18) can be segregated into two regions of the complex plane using the Gershgorin circle theorem (see Fig. 6). Fix a specific wave number k, let a _i,j be the elements of J _k , and define

$$ R_i=\sum_{j \neq i} |a_{i,j}|, \qquad C_i = C(a_{i,i},R_i) $$

(27)

where C(a,r) is the circle with center a and radius r. The Gershgorin circle theorem states that each eigenvalue of J _k lies in at least one of the disks C _i . The structure of J _k is such that the off diagonal entries are O(1) with respect to D. So R=max{R _i } is O(1). The diagonal entries fall into two categories, those that are O(1) (corresponding to the small diffusion entries), and those that are −k ² D+O(1) (corresponding to large diffusion entries). Define Ω _s to be the union of the disks C _i that are characterized by O(1) diagonal entries and Ω _l as the union of disks characterized by O(D) diagonal entries. Since these disks have a maximal radius R independent of D, there exists disks C _l =C(−k ² D,κ _l R) and C _s =C(0,κ _s R) so that for constants κ _l,s independent of D, Ω _s ⊂C _s and Ω _l ⊂C _l . For sufficiently large D, C _s and C _l do not overlap and hence separate {λ ^k} into two sets (see Fig. 6).

Fig. 6

Schematic of the separation of the eigenvalues of J _k in the complex plane. Grey circles indicate the different Gershgorin circles C _i . The larger darker circles indicate C _l and C _s , which contain all eigenvalues of J _k . These circles separate those eigenvalues into two classes with O(1) and O ⁻(D) real part, respectively

So, for each i, either $\operatorname{\mathfrak{Re}}(\lambda ^{k}_{i})=O^{-}(D)$, or $\operatorname{\mathfrak{Re}}(\lambda^{k}_{i})=O(1)$. Since det(J _k )=O(D ^N), $\{ \lambda^{k}_{i} \}_{i=M+1:M+N}$ must have O ⁻(D) real part. Also note that the imaginary parts of all eigenvalues are constrained to be less than max{κ _s ,κ _l }R so that $\operatorname{\mathfrak{Im}}(\lambda_{k}^{i})=O(1)$ for all i as well, so $|\lambda^{k}_{i}|=O(1)$ for i=1:M.

Eigenvalues of J _k are roots of the characteristic polynomial

$$ \begin{vmatrix} J_k - \lambda I \end{vmatrix} = \begin{vmatrix} f_u \bigl(u^s,v^s\bigr) - \bigl(k^2 \epsilon^2 + \lambda\bigr) I & f_v \bigl(u^s,v^s\bigr) \\ g_u\bigl(u^s,v^s\bigr) & g_v \bigl(u^s,v^s\bigr) - \bigl(k^2 D + \lambda \bigr) I \end{vmatrix} = 0, $$

(28)

where I is a properly sized identity matrix. Let P and Q be the unitary matrices that diagonalize f _u (u ^s,v ^s) and g _v (u ^s,v ^s). Then in particular the diagonal entries of P ⁻¹ f _u (u ^s,v ^s)P are $\{ \lambda^{LP}_{j} \}$ and the entries of Q ⁻¹ g _v (u ^s,v ^s)Q are O(1). Define

$$ T = \begin{bmatrix} P & 0 \\ 0 & Q \end{bmatrix} . $$

(29)

Then the eigenvalue problem translates to

$$ \begin{vmatrix} \bigl[\lambda^{LP}\bigr] - k^2 \epsilon^2 I - \lambda I & P^{-1} f_v Q \\ Q^{-1} g_u P & Q^{-1} g_v \bigl(u^s,v^s\bigr) Q -k^2 D I - \lambda I \end{vmatrix} = \begin{vmatrix} A1 & A2 \\ A3 & A4 \end{vmatrix} =0 $$

(30)

where [λ ^LP] is the diagonal form of f _u (u ^s,v ^s). Notice that A1,A4 are diagonal.

Now consider an eigenvalue λ whose real part is O(1). In this case, the diagonal entries of A4 are O ⁻(D) and it is nonsingular. It can thus be used to eliminate A2. After this is done, the eigenvalue problem becomes

$$ \begin{vmatrix} \bigl[\lambda^{LP}\bigr] - k^2 \epsilon^2 I +O\bigl(D^{-1}\bigr)- \lambda I & 0 \\ Q^{-1} g_u P & Q^{-1} g_v \bigl(u^s,v^s\bigr) Q - k^2 D I - \lambda I \end{vmatrix} =0. $$

(31)

Since the bottom right block is nonsingular, it must be true that

$$ \det\bigl( \bigl[\lambda^{LP}\bigr] - k^2 \epsilon^2 I +O\bigl(D^{-1}\bigr)- \lambda I \bigr) =0 . $$

(32)

where O(D ⁻¹) is a properly sized matrix with entries of this size. With D=∞, the roots of this polynomial are simply $\{ \lambda ^{LP}_{j} - k^{2} \epsilon^{2} \}$. It is tempting to view Eq. (32) as a perturbation of this case and apply some form of perturbation bound. However, f _u is not Hermitian, which is usually required for such bounds. Instead, the best we can say is that by continuity of the determinant, the roots of this polynomial satisfy

$$ \lambda=\lambda^{LP}_j -k^2 \epsilon^2 + c(D), $$

(33)

where c(D)→0 as D→∞.

Appendix C: GTPase Model Equations

Figure 5a schematically diagrams interactions between three interacting GTPases and three phosphoinositides. I briefly outline the model equations describing these interactions. Further specifics can be found in Holmes et al. (2012b), Lin et al. (2012). Modifications of the model presented in those references, which are the subject of investigation here, are described in the main text. Each GTPase undergoes conservative cycling between active membrane bound and inactive forms in the cell interior by (un)binding to the membrane. These dynamics are described by

$$\begin{aligned} \begin{aligned}[c] \frac{\partial G}{\partial t} &= I_G \frac{G^c}{G_t} -\delta_G G+D_m G_{xx}, \\ \frac{\partial G^c}{\partial t} &= -I_G \frac{G^c}{G_t}+ \delta_G G+D_{c} G^c_{xx} , \end{aligned} \end{aligned}$$

(34a)

where G=R,ρ,C represents the membrane bound form and G ^c represents an inactive cytosolic form. Phosphoinositides interconvert between three states through the hydrolysis/phosphorylation activity of PI5K, PI3K, PTEN, etc., which are not explicitly modeled. The GTPase activation rate functions encoding the interactions in Fig. 5a are defined by

$$\begin{aligned} \begin{aligned} I_C &= \biggl( \frac{{\hat{I}}_C}{1+ (\rho/a_1 )^n} \biggr),\qquad I_R = \biggl( {\hat{I}}_{R1}+ \frac{\alpha C + \hat{I}_{R2} \frac{P_3}{P_{3b}}}{1+f_2 ( \rho/ a_3 )^n } \biggr), \\ I_{\rho} &= \frac{{\hat{I}}_{\rho}}{1+ (R/a_2 )^n}. \end{aligned} \end{aligned}$$

(34b)

Phosphoinositide kinetics are modeled by linear and mass action kinetics

$$\begin{aligned} & \frac{\partial P_1}{\partial t} = I_{P1}-\delta _{P1}P_1+k_{21}P_2-f_{\mathrm{PI}5\mathrm{K}}(R,C, \rho) P_1 + D_P P_{1xx}, \\ & \begin{aligned}[c] \frac{\partial P_2}{\partial t} &= -k_{21}P_2+ f_{\mathrm{P}I5\mathrm{K}}(R,C,\rho) P_1 - f_{\mathrm{PI}3\mathrm{K}}(R,C, \rho) P_2 \\ &\quad + f_{\mathrm{PTEN}}(R,C,\rho) P_3 + D_P P_{2xx}, \end{aligned} \\ &\frac{\partial P_3}{\partial t} = f_{\mathrm{PI}3\mathrm{K}}(R,C,\rho) P_2 - f_{\mathrm{PTEN}}(R,C,\rho) P_3 + D_P P_{3xx}, \end{aligned}$$

(34c)

with feedback terms

$$\begin{aligned} \begin{aligned}[c] f_{\mathrm{PI}3\mathrm{K}}&=\frac {k_{\mathrm{PI}3\mathrm{K}}}{2} \biggl(1+ \frac{R}{R_t} \biggr), \quad f_{\mathrm{PI}5\mathrm{K}}=\frac {k_{\mathrm{PI}5\mathrm{K}}}{2} \biggl(1+ \frac{R}{R_t} \biggr) , \\ f_{\mathrm{PTEN}}&=\frac{k_{\mathrm{PTEN}}}{2} \biggl(1+\frac{\rho }{\rho_t} \biggr) . \end{aligned} \end{aligned}$$

(34d)

See Table 2 for a base parameter set for this model.