Table 1. The transition rates between all possible states of a crosslinker ${\displaystyle U\rightleftharpoons (S_A,S_B)\rightleftharpoons D}$.

${\displaystyle \left(S_A,S_B\right)}$ means either head $A$ or $B$ is bound but the other is unbound. All binding rates account for the linear binding density ${\displaystyle ϵ}$ is the length of filament $i$ with center-of-mass position ${\displaystyle {\mathit{x}}_i}$ and orientation ${\displaystyle {\mathit{p}}_i}$ inside the capture sphere with cutoff radius ${\displaystyle r_{c,S}}$ relative to position of motor/crosslinker ${\displaystyle \mathit{x}}$. The sum is over all possible candidate filaments $i$. The unbound-singly bound transition $U\rightleftharpoons (S_A,S_B)$ is determined by the association constant $K_{\mathrm{a}}$ and the force-independent off rate ${\displaystyle k_{\mathrm{o},S}}$. Similarly, the singly bound-doubly bound transition $(S_A,S_B)\rightleftharpoons D$ is determined by the association constant $K_{\mathrm{e}}$ and force-independent off rate ${\displaystyle k_{\mathrm{o},D}}$ is the Boltzmann factor. $E(\mathrm{\ell })$ in the in the $(S_A,S_B)\rightleftharpoons D$ transition rates refers to the tether energy of a motor ${\displaystyle E(\ell )={\displaystyle \frac{1}{2}}{\kappa }_{\mathrm{x}\mathrm{l}}{\left({\ell }_f-{\ell }_0\right)}^2}$. ${\displaystyle {\ell }_0}$ is the free length of a motor, while ${\mathrm{\ell }}_f$ is the length for computing the force when attached to filaments $i$ and $j$ at locations s_i and s_j: ${\displaystyle {\ell }_f(s_i,s_j,{\mathit{x}}_i,{\mathit{p}}_i,{\mathit{x}}_j,{\mathit{p}}_j}$). The dimensionless factor λ determines the energy dependence in the unbinding rate. Both binding and unbinding rates must depend on λ and $k_{o,d}$ such that the equilibrium constant recovers the Boltzmann factor $\exp [-\beta E({\mathrm{\ell }}_f)]$ For force-dependent binding models, the $E(\mathrm{\ell })$ can be simply replaced by the tether force $F(\mathrm{\ell })$. This is not used for results shown in this work, but implemented in the code.

Process	Rate	Value
$U\to (S_A,S_B)$	${\displaystyle R_{\mathrm{o}\mathrm{n},S}(\mathit{x})}$	${\displaystyle {\displaystyle k_{\mathrm{o},S}\frac{3ϵK_{\mathrm{a}}}{4\pi r_{c,S}^3}\sum _i{L}_{\mathrm{i}\mathrm{n},i}(\mathit{x})}}$
$(S_A,S_B)\to U$	${\displaystyle R_{\mathrm{o}\mathrm{f}\mathrm{f},S}}$	${\displaystyle {\displaystyle k_{\mathrm{o},S}}}$
$(S_A,S_B)\to D$	${\displaystyle R_{\mathrm{o}\mathrm{n},D}(s_i)}$	${\displaystyle {\displaystyle k_{\mathrm{o},D}ϵK_{\mathrm{e}}\sum _j{\int }_{L_j}d{s}_j\exp \left[-(1-\lambda )\beta E({\ell }_f(s))\right]}}$
$D\to (S_A,S_B)$	${\displaystyle R_{\mathrm{o}\mathrm{f}\mathrm{f},D}(s_i,s_j)}$	${\displaystyle {\displaystyle k_{\mathrm{o},D}\exp \left[\lambda E({\ell }_f)\right]}}$