Dynamics of protein–lipid interactions in a three-variable reaction–diffusion model of myristoyl-electrostatic cycle in living cell

Abstract

Protein–lipid interactions dynamics through a three-variable reaction–diffusion model in the living cell are explored. It is proven that the dynamics of such a system is governed by a two-dimensional cubic complex Ginzburg–Landau equation. Linear stability analysis is carried out through modulational instability. It appears that both rate constant of lipid association of unphosphorylated proteins and the diffusion coefficient of proteins on membrane impact the zone of instability and the amplitude of the growth rate. The $(G'/G)$-expansion method is used to construct analytical spiral and periodic solutions. These structures are robust and propagate for large values of the diffusion coefficient of proteins on membrane during protein–lipid interactions.

Appendices

Appendix A: Calculation details of generating the 2D-CGL equation Eq. (4)

Equation (1) becomes:

$$\begin{aligned} \frac{\partial m}{\partial t}&=k_{ad}c(1-m)-k_{de}m-k_{ki}(1-m)\frac{m}{k_{m}+m}+D_{\textrm{m}}\nabla ^2 m\nonumber \\ \frac{\partial c}{\partial t}&=k_{de}m-k_{ad}c(1-m)+k_{ph}p+D_{\textrm{c}}\nabla ^2c,\nonumber \\ \frac{\partial p}{\partial t}&=k_{ki}(1-m)\frac{m}{k_{m}+m}-k_{ph}p+D_{\textrm{c}}\nabla ^2 p, \end{aligned}$$

(A1)

We linearize the set of equations (A1) and obtain:

$$\begin{aligned} (1+\alpha _0m){\dot{m}}&=\alpha _1m+\alpha _2c+\alpha _3m^2 +\alpha _4mc+\alpha _{5}m^2c+D_{\textrm{m}}\nabla ^2 m+\alpha _6m\nabla ^2 m,\nonumber \\ {\dot{c}}&=-\alpha _2c+\beta _1m+\beta _2p+\alpha _2mc+D_{\textrm{c}}\nabla ^2c,\nonumber \\ (1+\alpha _0m){\dot{p}}&=-\beta _2p+\gamma _{1}m-\gamma _{1}m^2+ \gamma _{2}mp+D_{\textrm{c}}\nabla ^2 p+\gamma _{3}m\nabla ^2p, \end{aligned}$$

(A2)

where $\alpha _{0}=\frac{1}{k_{m}}$; $\alpha _{1}=-(k_{de}+\frac{k_{ki}}{k_m})$; $\alpha _{2}=k_{ad}$; $\alpha _{3}=\frac{k_{ki}-k_{de}}{k_{m}}$; $\alpha _{4}=k_{ad}(-1+\frac{1}{k_{m}})$; $\alpha _5=-\frac{k_{ad}}{k_m}$; $\alpha _6=\frac{D_m}{k_m}$; $\beta _{1}=k_{de}$; $\beta _{2}=k_{ph}$; $\gamma _{1}=\frac{k_{ki}}{k_m}$; $\gamma _{2}=-\frac{k_{ph}}{k_m}$; $\gamma _{3}=\frac{D_c}{k_m}$. We derive the first equation of (A2) with respect to t, with the used of the second equation of (A2) and obtain

$$\begin{aligned}&(1+a_0m)\ddot{m}+(a_1+a_2m){\dot{m}}^2+(b_0+b_1m+b_2m^2) {\dot{m}}+d_3m^3+d_2m^2+\Omega _{0}^2m+d_1p+d_0mp+d_4m^2p=\nonumber \\&(e_0+e_1m+e_2m^2)\nabla ^2m+D_c(1+h_1m+h_2m^2)\nabla ^2{\dot{m}} +(g_0+g_1m){\dot{m}}\nabla ^2m+D_m(1+h_3m){\dot{\nabla }}^2m\nonumber \\&-D_cD_m(1+h_1m+h_2m^2)\nabla ^2(\nabla ^2m), \end{aligned}$$

(A3)

where the expressions of $a_i$, $b_i$, $d_i$, $e_i$,$g_i$, $h_i$ and $\Omega _i$,$i=0,1,2,3,4$ are given by:

$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{cl} &{} a_0=\alpha _0; a_1=\alpha _0-\frac{\alpha _4}{\alpha _2} ;a_2=-\frac{2\alpha _5}{\alpha _2}; \\ &{}b_0=-\alpha _1+\alpha _2;b_1=-2\alpha _3+\alpha _4(1+\frac{\alpha _1}{\alpha _2}) ;b_2=\alpha _5-\alpha _4+\frac{2\alpha _1\alpha _5}{\alpha _2}\\ &{} d_0=-\alpha _4\beta _2; d_1=-\alpha _2\beta _2; d_2=-\alpha _4(\alpha _1+\beta _1)+\alpha _1\alpha _2; { d_3=\alpha _1(\alpha _2-\alpha _5)-\alpha _5\beta _1}; d_4=-\alpha _5\beta _2\\ &{}e_0=\alpha _2D_m-\alpha _{1}D_c; e_1=-(D_m(\alpha _2-\alpha _4)+\frac{D_c\alpha _1\alpha _4}{\alpha _2}) ; e_2=-(D_m(\alpha _4-\alpha _5)-\frac{D_c\alpha _1\alpha _5}{\alpha _2})\\ &{}g_0=\alpha _6-\frac{D_m\alpha _4}{\alpha _2}; g_1=-2\frac{D_m\alpha _5}{\alpha _2}\\ &{} h_1=\frac{\alpha _4}{\alpha _2}; h_2=\frac{\alpha _5}{\alpha _2}; h_3=\frac{\alpha _6}{D_m}\\ &{}\Omega _0^2=-\alpha _2(\alpha _1+\beta _1) \end{array}\right. \end{aligned} \end{aligned}$$

which depend on the parameters of the model. From the third equation of (A2) and Eq. (A3), we have

$$\begin{aligned} & (1+a_0m)\ddot{m}+(a_1+a_2m){\dot{m}}^2+(b_0+b_1m+b_2m^2){\dot{m}}+d_3m^3+d_2m^2+\Omega _{0}^2m+d_1p+d_0mp+d_4m^2p\nonumber \\ & \quad =(e_0+e_1m+e_2m^2)\nabla ^2m+D_c(1+h_1m+h_2m^2)\nabla ^2{\dot{m}}+(g_0+g_1m){\dot{m}}\nabla ^2m+D_m(1+h_3m){\dot{\nabla }}^2m\\ & \qquad -D_cD_m(1+h_1m+h_2m^2)\nabla ^2(\nabla ^2m),\nonumber \\ & \qquad (1+a_0m){\dot{p}}=-\beta _2p+\gamma _{1}m-\gamma _{1}m^2+ \gamma _{2}mp+D_{\textrm{c}}\nabla ^2 p+\gamma _{3}m\nabla ^2p. \end{aligned}$$

(A4)

Here, we have the real-valued conditions:

$$\begin{aligned}&m_{n}^{(-l)}=\left( m_{n}^{(l)}\right) ^{*} \nonumber \\&p_{n}^{(-l)}=\left( p_{n}^{(l)}\right) ^{*}. \end{aligned}$$

(A5)

The asterisk denoting complex conjugations. The insertion of Eq. (A5) into the difference-differential Eqs. (A4) at the different orders of $(\epsilon ^n, A^l$ $n=1,2,3)$, we obtain:

− order $(\epsilon ^1,A^0)$

$$\begin{aligned}&\Omega _0^2m_{1}^{(0)}=0\Rightarrow m_{1}^{(0)}=0\nonumber \\&-\alpha _{11}m^{(0}_{1}+\alpha _9p_{1}^{(0)}=0\Rightarrow p_1^{(0)}=0. \end{aligned}$$

(A6)

− order $(\epsilon ^1,A^1)$

$$\begin{aligned}&(\Omega _0^2+e_0k^2-\omega ^2)m_{1}^{(1)}=0\Rightarrow m_{1}^{(1)}\ne 0 \Rightarrow \omega ^2-\Omega _0^2-e_0k^2=0\nonumber \\&(D_ck^2+\beta _{2}-i\omega )p^{(1)}_{1}+\gamma _1m_{1}^{(1)}=0\Rightarrow p_1^{(1)}=\left[ \frac{\gamma _1(D_ck^2+\beta _2)}{(D_ck^2+\beta _2)^2+\omega ^2} +i\frac{\gamma _1\omega }{(D_ck^2+\beta _2)^2+\omega ^2}\right] m_1^{(1)}. \end{aligned}$$

(A7)

and we consider $m_1^{(1)}=\phi$

$$\begin{aligned}&p_1^{(1)}=(A_1+iA_2)\phi \nonumber \\&A_1=\frac{\gamma _1(D_ck^2+\beta _2)}{(D_ck^2+\beta _2)^2+\omega ^2}, \qquad A_2=\frac{\gamma _1\omega }{(D_ck^2+\beta _2)^2+\omega ^2}; \end{aligned}$$

(A8)

− order $(\epsilon ^2, A^0)$

$$\begin{aligned}&m_0^{(2)}=-\frac{2}{\Omega _0^2}\left[ D_mD_ch_1k^4+(a_1-a_0)\omega ^2+ e_1k^2+d_2+d_0A_1+i(D_ch_1+D_mh_3)\omega k^2\right] \vert \phi \vert ^2=(B_1+iB_2)\vert \phi \vert ^2 \nonumber \\&p^{(2)}_0=\frac{1}{\beta _2}[\gamma _1B_1+2(-\gamma _3A_1k^2-a_0A_2\omega +\gamma _2A_1-\gamma _1)+i\gamma _1B_2]\vert \phi \vert ^2=(\Gamma _1+i\Gamma _2)\vert \phi \vert ^2. \end{aligned}$$

(A9)

with,

$$\begin{aligned} \begin{aligned}&B_1=-\frac{2}{\Omega _0^2}[D_mD_ch_1k^4+(a_1-a_0)\omega ^2+e_1k^2+d_2+d_0A_1]\\&B_2=-\frac{2}{\Omega _0^2}[(D_ch_1+D_mh_3)\omega k^2]\\&\Gamma _1=\frac{1}{\beta _2}[\gamma _1B_1-2(\gamma _3A_1k^2+a_0A_2\omega -\gamma _2A_1+\gamma _1)]\\&\Gamma _2=\frac{\gamma _1B_2}{\beta _2}.\\ \end{aligned} \end{aligned}$$

− order $(\epsilon ^2, A^2)$

$$\begin{aligned} m_2^{(2)}&=\frac{D_mD_ch_1k^4+e_1k^2-(a_0+a_1)\omega ^2+d_2+d_0A_1 +i[(D_ch_1+D_mh_3-g_0)\omega k^2-b_1\omega +d_0A_2]}{3\Omega _0^2}\phi ^2\nonumber \\&=(B_{12}+iB_{22})\phi ^2\nonumber \\ p_2^{(2)}&=\frac{-A_1\gamma _3k^2-a_0A_2\omega +A_1\gamma _2+\gamma _1B_{12} -\gamma _1+i(-A_2\gamma _3k^2+a_0A_1\omega +A_2\gamma _2+\gamma _1 B_{22})}{4D_ck^2+\beta _2-2i\omega }\phi ^2\nonumber \\&=(\Gamma _{12}+i\Gamma _{22})\phi ^2, \end{aligned}$$

(A10)

where

$$\begin{aligned} \begin{aligned}&B_{12}=\frac{D_mD_ch_1k^4+e_1k^2-(a_0+a_1)\omega ^2+d_2+d_0A_1}{3\Omega _0^2}, \qquad B_{22}=\frac{(D_ch_1+D_mh_3-g_0)\omega k^2-b_1\omega +d_0A_2}{3\Omega _0^2} \\&\Gamma _{12}=\frac{(4D_ck^2+\beta _2)(-A_1\gamma _3k^2-a_0A_2\omega +A_1\gamma _2+\gamma _1B_{12} -\gamma _1)-2\omega (-A_2\gamma _3k^2+a_0A_1\omega +A_2\gamma _2+\gamma _1 B_{22})}{(4D_ck^2+\beta _2)^2+4\omega ^2}\\&\Gamma _{22}=\frac{(4D_ck^2+\beta _2)(-A_2\gamma _3k^2+a_0A_1\omega +A_2\gamma _2+\gamma _1 B_{22})+2\omega (-A_1\gamma _3k^2-a_0A_2\omega +A_1\gamma _2+\gamma _1B_{12} -\gamma _1)}{(4D_ck^2+\beta _2)^2+4\omega ^2}. \end{aligned} \end{aligned}$$

Appendix B: Calculation details of implementation of $(G'/G)$-expansion method to the two-dimensional cubic complex Ginzburg–Landau equation

Inserting solution Eq. (20) into Eq. (4) and separating the real and imaginary parts lead to

$$\begin{aligned}&\mu _0(P_1u_1^2+P_2v_1^2)FF''+\mu _0(P_1u_1^2+P_2v_1^2)(F')^2 +2(\omega _1+P_1u_1u_2+P_2v_1v_2)FF'+R_1F^2+2Q_2F^4=0,\nonumber \\&(P_1u_1^2+P_2v_1^2)F''-(P_1u_1^2+P_2v_1^2)F(H')^2-2(\omega _1 +P_1u_1u_2+P_2v_1v_2)FH'+2Q_1F^3\nonumber \\&-(R_2+2\omega _2+P_1u_2^2+P_2v_2^2)F=0. \end{aligned}$$

(B1)

We further set

$$\begin{aligned} H(X)=\mu _0\ln \vert F(X)\vert , \end{aligned}$$

(B2)

where $\mu _0$ is another constant to be determined later. Substituting (B2) into (B1), one arrives to the system

$$\begin{aligned}&\mu _0(P_1u_1^2+P_2v_1^2)FF''+\mu _0(P_1u_1^2+P_2v_1^2)(F')^2+2(\omega _1+P_1u_1u_2+P_2v_1v_2)FF'+R_1F^2+2Q_2F^4=0,\nonumber \\&(P_1u_1^2+P_2v_1^2)FF''-\mu _0^2(P_1u_1^2+P_2v_1^2)(F')^2-2\mu _0(\omega _1+P_1u_1u_2+P_2v_1v_2)FF'+2Q_1F^4\nonumber \\&-(R_2+2\omega _2+P_1u_2^2+P_2v_2^2)F^2=0. \end{aligned}$$

(B3)

The factors in the trial solution Eq.(16) can be found by balancing the highest order derivative term $(F')^2$ and the highest nonlinear term $(F^4)$ appearing in (B3). In fact, the weight of the derivative is $2(s+1)$ and that of the nonlinear term is 4s. Equating the two weights, i.e. $2(s+1)=4s$ leads to the result $s=1$, for which we get

$$\begin{aligned} F(X)=q_{-1}\left( \frac{G'(X)}{G(X)}\right) ^{-1}+q_0+q_{1}\left( \frac{G'(X)}{G(X)}\right) . \end{aligned}$$

(B4)

We substitute Eq. (B4) into Eqs. (B3) and, after eliminating the denominator and collecting all terms with the same order of $\left( G'(X)/G(X)\right)$ together, the left-hand sides of Eqs. (B3) are converted into two polynomials in $\left( G'(X)/G(X)\right)$. Setting the coefficients of these polynomials to zero, we derive a set of algebraic equations for $u_1$, $u_2$, $v_1$, $v_2$, $\omega _1$, $\omega _2$, $\mu _0$, $\Lambda$, $\Upsilon$, $q_0$, $q_1$ and $q_{-1}$. The resolution using the computer algebra software MAPLE leads to solutions given in Eq. (22).