Disentangling Entropic, Active, and Frictional Forces in Cytoskeletal Crosslinking

TL;DR

Disentangling Entropic, Active, and Frictional Forces in Cytoskeletal Crosslinking develops a thermodynamically grounded framework to predict the forces between two filaments linked by mixtures of passive and motorized crosslinkers. It decomposes inter-filament forces into entropic, active, and frictional contributions that can be computed from measurable crosslinker and filament properties, and validates the framework across multiple experimental paradigms. By connecting to Hill theory in the low-density limit and showing negligible external drag, the work provides a unified language to compare crosslinker compositions, predict pattern formation, and guide engineering of polymer networks. The approach offers quantitative tools to understand cytoskeletal mechanics and to design crosslink mixtures with tailored mechanical responses.

Abstract

The forces that mixtures of motorized and passive crosslinking proteins collectively generate between cytoskeletal filaments within our cells are the key drivers of active cellular mechanics. Despite their importance, a unified theory to describe such crosslinking forces has so far been missing. In this paper, we derive a theory that predicts the forces generated collectively by crosslinking proteins linking two biopolymer filaments from measurable filament and crosslinker properties, using out-of-equilibrium thermodynamics. Our framework allows us to decompose the forces generated by crosslinkers into three separate components: entropic, active, and frictional. In doing so, it offers a clear physical interpretation of the fundamental mechanisms by which crosslinking proteins self-organize and collectively generate forces. We demonstrate the robustness and utility of this framework by applying it to four different experiments that probe the combined roles of passive and motorized crosslinkers. For each experiment, our theoretical approach allows us to disentangle the relative contributions of entropic, active, and frictional forces, clarifying how different physical processes underpin collective force production. In turn, this makes it possible to quantitatively compare and predict how various crosslinker combinations influence force generation between filaments, pattern formation along filaments, and the dynamics of filament pairs.

Figures (6)

• Figure 1: Sketch and continuous description of crosslinkers and filament pair. a) Chemical reaction network: we consider binding reactions between $a, b, ab$, and $u$ states and hydrolysis of ATP to ADP + P. b) Sketch of the system. The filaments $A$ and $B$ are decorated with a-bound, b-bound or ab-bound crosslinkers. Each filament has a polarity vector $p^{(A)}_\alpha$ or $p^{(B)}_\alpha$. The vector perpendicular to the filaments direction pointing from A to B is $S_\alpha$. c) Sketch of the continuous description in one dimension. The filaments ($A, B$) and crosslinkers ($a,b,ab,u$) are described by continuous fields, $n^{(A)}$, $n^{(B)}$, $n^{(a)}$, $n^{(b)}$ and $n^{(ab)}$, respectively. The maximum number density of crosslinkers on filament is $n^S$.
• Figure 2: Passive crosslinkers induce entropic contractility in crosslinked bundles. a) Sketch of Model 1 of doubly bound crosslinkers between two filaments. b) Force-Time curves of Model 1 (legend in the gray frame). c) Force-Overlap curves of Model 1. d) Sketch of Model 2 with singly and doubly bound crosslinkers. e) Force-Time curves of Model 2. f) Force-Overlap curves for Model 2. g) Force-Time curves used in e) and for the next four pulling event, done with an independent simulation with the same parameters as in e).
• Figure 3: Passive crosslinkers generate inter-filament friction. a) Sketch of the model used in the simulation. b) Normalized Force-Overlap curve obtained in simulation with experimental parameters and V=0.1 $\mu$m-s-1, for N=50 crosslinkers in the overlap. The total force is in black, the entropic and friction contribution are in red and blue, respectively. c) Normalized Force-Overlap curve with V=0.01 $\mu$m-s-1 and N=50. d) Normalized Force-Overlap curve with V=0.01 $\mu$m-s-1 and N=150. e) Heat map of the normalized total force difference varying the V and N. Crosses indicate the locations of the simulations shown. f-h) At fixed velocity (V=0.1 $\mu$m.s-1), we change the parameter $\mathcal{D}/\mathcal{D}_{\text{exp}}$ and the density. f) Force-overlap curve $\mathcal{D}/\mathcal{D}_{\text{exp}}=0.2$. g) Force-overlap curve $\mathcal{D}/\mathcal{D}_{\text{exp}}=10$. g) Heat map of the normalized total force difference varying the Damköhler number $\mathcal{D}$ and the density.
• Figure 4: Motorized crosslinkers patterning on a single filament and the active force generated between two filaments a) Sketch of a single motor walking along a filament under an external load. This represents the low-density limit of our framework, consistent with Hill’s formalism. b) Sketch of a group of motors walking along a single filament. c) Kymograph of the density profile. Color lines correspond to the times at which density profiles are plotted. White dotted curve is the model for the size of traffic jam $(L-a)(1-\exp(-b t))$ with a=3.85 $\mu$m and b=0.003 s-1 as measured in Leduc2012. d) Density profiles at different times. Color lines are the density of motor, black dashed line is the density of filament A, which is the available attachment space for motors. e) Kymograph of the simulation with motorized crosslinkers between two filaments. The filaments are in red and the motorized crosslinkers in green. White dotted lines represent the change of external ATP load. f) Sketch of the filaments and crosslinkers states during the three phases of the simulation. g) Entropic (red), friction (blue), and active (green) forces during the simulation.
• Figure 5: Motorized and passive crosslinkers compete to define the overall force balance and sliding movement. a) Kymograph of the simulation. Template filament is in dark red, sliding filament in red and passive crosslinkers in green. Motorized crosslinkers are not shown. White plain curve is the outline of the experimental observation of Braun2011adaptive. White Dashed lines separate the phase I, II, and III. b) Sketches of the system for each phase. c) Plot of the entropic, active and friction forces.