Figure 1

Elements of the “polymorphic four-block tube model.” (a) Tube contour with the internal and external frames. (b) A short tube section with intrinsic twist $q_0$ with four blocks at each section. (c) The blocks can fluctuate cooperatively between two discrete states. The outwards curved $\sigma =1$ (arrow) state is energetically preferred over the straight state $\sigma =0$ (circle) with an energy gain $E=-\Delta G$. The junction between straight and curved states along the longitudinal block axis are penalized by a coupling constant $E=+J.$ (d) Competition between block switching energy and elastic lattice strain energy leads to spontaneous symmetry breaking: the tube bends to a randomly chosen direction and assumes a nonzero vectorial polymorphic order parameter $\vec{p}$ (“conformational polarization”). (e) The four-block model can be decoupled into a pair of two-block models which can be mapped onto a ladder type Ising model with coupling constants $-\tilde{B}$ and $+\tilde{J}$ and a bias field $\tilde{\gamma }$.Reuse & Permissions

Figure 2

The curved state density $M=\frac{1}{N}{\sum }_{i=1}^N\langle {\sigma }_i^k\rangle$ vs the state bias parameter $\tilde{\gamma }$ for different values of coupling constant$\tilde{J}$ and fixed stiffness $\tilde{B}=1.$ Note the broad $\tilde{\gamma }$ independent plateau region for $\tilde{J}=1.5$, where $M=1/2.$ For large values of $|\tilde{\gamma }|,$ $M$ is either 0 or 1 corresponding to straight unstrained or straight strained lattice states, respectively.Reuse & Permissions

Figure 3

Plot of the mean polymorphic polarization $P$ of the reduced two-block model vs the conformational bias field $\tilde{\gamma }$ for different values of coupling constant $\tilde{J}$ and effective tube stiffness $\tilde{B}=1.$ For $\tilde{J}⩾1.5$ the tube's curvature is at maximum for $|\tilde{\gamma }|<1,$ as $P\approx 1$ in this region. For larger $|\tilde{\gamma }|$ the tube is in a straight state and $P$ goes to zero.Reuse & Permissions

Figure 4

Representative plots of the angular persistence length $l_p$ vs distance along contour $s$ for intrinsic elastic bending stiffnes $l_B=10\mu \text{m}$ (chosen arbitrary and small for pictorial convenience) and different values of scaled correlation length $\xi q_0.$Reuse & Permissions

Figure 5

Polymorphic entropies $\Sigma$ (four-block model) vs the microtubule's length $L$ for different values of conformational bias $\tilde{\gamma }$ and stiffness $\tilde{B}=1.$Reuse & Permissions

Figure 6

Plots of the polymorphic polarization $P$ (four-block model) vs the microtubule's length $L$ for different values of $\tilde{\gamma }$ and with $\tilde{B}=1.$ We observe that microtubule curvature can be build up progressively with growing MTs.Reuse & Permissions

Figure 7

Log-log plot of the polymorphic Fourier mode distribution $G_{\text{pol}}\left(q\right)$ for different values of $\xi q_0.$ Besides a characteristic peak at $q_0$ one observes a crossover between a small $q$ regime with scaling $q^{-2}$ and a large $q$ regime with a $q^{-4}$ scaling.Reuse & Permissions