Fig. 4.

An energy-based mathematical model of tension sensing in a single microtubule. The model is based on a one-dimensional two-state mechanical model of tubulin protofilament alignment, through GTP hydrolysis a or external pulling force b, as illustrated. State variables are: the real-valued actual length $l\in \mathbb{R}$ of a stretchable segment of MT (e.g., anchor point to  plus end); a binary-valued indicator variable s ∈ {0,1} for the mechanical state of the lengthwise protofilaments at the  plus end cap (s = 0⇒splayed, s = 1⇒ aligned); and optionally a binary-valued indicator variable σ ∈ {0,1} for internal biochemical sensing of the mechanical state s. Principal exogenous parameters are $\lambda \in {\mathbb{R}}^{\ge 0}$, the length of the splayable subregion; $l_0\in {\mathbb{R}}^{\ge \lambda }$, the segment resting length when aligned (so l₀–λ is the resting length when splayed); τ = externally applied tension; μ_s = energy bias in favor of (or, if negative, against) alignment s = 1; μ_σ= energy bias in favor of σ = 1; α = energetic reward for agreement of s = 1 and σ = 1. Given this notation, a Hooke’s law mechanical spring energy with two states can be written as: E_mech = (k/2)[s(l–l₀)²+(1–s)(l–(l₀–λ))²]–τ(l–(l₀–λ)). Additional energy terms specific to discrete end cap state and sensing are: $E_{discrete}=-{\mu }_{s}s-{\mu }_{\sigma }\sigma -\alpha s\sigma$; then the total energy is E(l,s,σ) = E_mech+E_discrete. State probability follows the Boltzmann distribution, exp(–βE)/Z(β,params) where Z normalizes the distribution. Even ignoring σ (case α small) one obtains a double-well potential in the free energy F(τ) = −(1/β)logZ with two minima as a function of tension, one of them near τ = 0. This indicates that nonzero tension can be stabilized by the s = 1 mechanical protofilament alignment state which is in turn correlated (for α ≠ 0) with σ = 1 tension sensing. The readout state σ = 1 could in turn be amplified biochemically by, e.g., a phosphorylation/dephosphorylation cycle as in ref. ⁹¹, assuming that σ affects such enzymatic activity