Dynamics of biomembranes with active multiple-state inclusions

Nonequilibrium dynamics of biomembranes with active multiple-state inclusions is considered. The inclusions represent protein molecules which perform cyclic internal conformational motions driven by the energy brought with adenosine triphosphate (ATP) ligands. As protein conformations cyclically change, this induces hydrodynamical flows and also directly affects the local curvature of a membrane. On the other hand, variations in the local curvature of the membrane modify the transition rates between conformational states in a protein, leading to a feedback in the considered system. Moreover, active inclusions can move diffusively through the membrane so that their surface concentration varies. The kinetic description of this system is constructed and the stability of the uniform stationary state is analytically investigated. We show that, as the rate of supply of chemical energy is increased above a certain threshold, this uniform state becomes unstable and stationary or traveling waves spontaneously develop in the system. Such waves are accompanied by periodic spatial variations of the membrane curvature and the inclusion density. For typical parameter values, their characteristic wavelengths are of the order of hundreds of nanometers. For traveling waves, the characteristic frequency is of the order of a thousand Hz or less. The predicted instabilities are possible only if at least three internal inclusion states are present.

Figure 1

(Color online) Force centers of the active force dipoles for $\beta \to \alpha$ transitions are distributed on surfaces described by $h+z_{\alpha \beta }^{\left(u\right)}$, and $h-z_{\alpha \beta }^{\left(d\right)}$. These surfaces and $h\left(\mathbf{r}\right)$ are parallel surfaces, the areas of parallel surface elements from top to bottom are $\left(1-z_{\alpha \beta }^{\left(u\right)}{\nabla }^{2}h\right)dA$, $dA$, and $\left(1+z_{\alpha \beta }^{\left(d\right)}{\nabla }^{2}h\right)dA$, respectively.

Figure 2

Schematics of conformational transitions of an inclusion with five internal states.

Figure 3

(Color online) Nonequilibrium phase diagrams for the membranes with active multistate inclusions. Here $\gamma ={10}^{-4}{k}_{B}T/{\text{nm}}^2$, $k_2=250k_{B}T\cdot {\text{nm}}^2$, ${\varphi }_{t0}=0.1$, $M_{eff}{\chi }_{eff}=1\mu {\text{m}}^2/\text{s}$, and $\eta ={10}^{-3}\text{kg}/\text{m}\cdot \text{s}$. The long-dashed curve indicates the onset of the wave instability, the short-dashed curve indicates the onset of the Turing instability, and the solid line indicates the onset of the long-wavelength instability. (a) $p_A/a=1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=1\text{pN}\cdot \text{m}\text{s}$. (b) $p_A/a=1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=-1\text{pN}\cdot \text{m}\text{s}$. (c) $p_A/a=-1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=1\text{pN}\cdot \text{m}\text{s}$. (d) $p_A/a=-1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=-1\text{pN}\cdot \text{m}\text{s}$.

Figure 4

(Color online) The Turing-type instability: (i) the inclusions are attracted to the regions with the preferred membrane curvature, (ii) the curvature-dependent active force quadrupoles generate membrane curvature that attracts the inclusions.

Figure 5

(Color online) The characteristic wavelengths at the onset of instabilities plotted as functions of $l_K/a$, the dimensionless coefficient for the curvature-dependent pumping rate. $\gamma ={10}^{-4}{k}_{B}T/{\text{nm}}^2$, $k_2=250k_{B}T\cdot {\text{nm}}^2$, ${\varphi }_{t0}=0.1$, $M_{eff}{\chi }_{eff}=1\mu {\text{m}}^2/\text{s}$, $\eta ={10}^{-3}\text{kg}/\text{m}\cdot \text{s}$. ${\kappa }_{eff}=5k_{B}T$, and $c_{eff}^h=c_{eff}^{\varphi }=0.1{\text{nm}}^{-1}$. (a) thin curves: $p_A/a=1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=1\text{pN}\cdot \text{m}\text{s}$ (dashed curve: wave instability; solid curve: Turing-type instability); thick curves: $p_A/a=0.1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=1\text{pN}\cdot \text{m}\text{s}$ (dashed curve: wave instability, solid curve: Turing-type instability). (b) dashed curve: $p_A/a=-1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=1\text{pN}\cdot \text{m}\text{s}$; and solid curve: $p_A/a=-1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=-1\text{pN}\cdot \text{m}\text{s}$.

Figure 6

(Color online) The wave instability: (i) as indicated by the horizontal arrows, inclusions are attracted to the regions with the preferred membrane curvature, this leads to (ii) as indicated by the vertical arrows, in the inclusion-rich domains, the active force quadrupoles produce the membrane curvature that repels the inclusions, therefore the system evolves back to (i).

Figure 7

(Color online) The characteristic time scales at the onset of the wave instability for $\gamma ={10}^{-4}{k}_{B}T/{\text{nm}}^2$, $k_2=250k_{B}T\cdot {\text{nm}}^2$, ${\varphi }_{t0}=0.1$, $M_{eff}{\chi }_{eff}=1\mu {\text{m}}^2/\text{s}$, $\eta ={10}^{-3}\text{kg}/\text{m}\cdot \text{s}$. ${\kappa }_{eff}=5k_{B}T$, and $c_{eff}^h=c_{eff}^{\varphi }=0.1{\text{nm}}^{-1}$. Dashed curve: $p_A/a=1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=1\text{pN}\cdot \text{m}\text{s}$; solid curve: $p_A/a=0.1\text{pN}\cdot \text{m}\text{s}$, $Q_A/a^2=1\text{pN}\cdot \text{m}\text{s}$. The magnitude of $l_K/a$ for the solid curve is indicated at the top of the frame, and the magnitude of $l_K/a$ for the dashed curve is indicated at the bottom.

Figure 8

A conical inclusion in state $\alpha$: $l_{\alpha }$ is the thickness of the inclusion, $r_{\alpha }^{out}$ is the outer radius, and $r_{\alpha }^{in}$ is the inner radius of the inclusion.

Figure 9

(Color online) Regions rich in the inclusions with positive $c_{\alpha }$ tend to have ${\nabla }^{2}h<0$.