A Theory that Predicts the Behaviors of Disordered Cytoskeletal Networks

Julio M Belmonte, Maria Leptin and Francois Nédélec
Molecular Systems Biology, 2017

Abstract

Morphogenesis in animal tissues is largely driven by actomyosin networks, through tensions generated by an active contractile process. Although the network components and their properties are known, and networks can be reconstituted in vitro, the requirements for contractility are still poorly understood. Here, we describe a theory that predicts whether an isotropic network will contract, expand, or conserve its dimensions. This analytical theory correctly predicts the behavior of simulated networks consisting of filaments with varying combinations of connectors, and reveals conditions under which networks of rigid filaments are either contractile or expansile. Our results suggest that pulsatility is an intrinsic behavior of contractile networks if the filaments are not stable but turn over. The theory offers a unifying framework to think about mechanisms of contractions or expansion. It provides the foundation for studying a broad range of processes involving cytoskeletal networks, and a basis for designing synthetic networks.


Theory
Actomyosin Networks
Contractile/Expansile System with Rigid Filaments
Other Recipes
Heterogeneous Networks
Pulsatile System

Theory

NetworkConfigurations

To predict the behavior of a network, previous theories have considered a pair of filaments with a single connector between them, while the theory presented here is based on the effects that two connectors bound to a single filament have on the rest of the network.

Pairs of connectors may generate local stress in the network depending on how the subunits move relative to one another on the filament. If the initial distance a0 between the subunits is maintained, the network does not deform. This occurs if the connectors do not move (in static configurations) or if they move in the same direction at the same speed (in neutral configuration). Local contraction is expected for contractile configuration in which the connectors move towards each other (aa0). If the filament is flexible, however, the expansile stress can be reduced if the filament buckles.

Theory

The behavior of a disorganized network of filaments can be predicted analytically following a three-step procedure.

(A) A list of all possible configurations involving one filament and two connectors is compiled. For each configuration, the separation between the connectors a_i, the speed v_i at which the they move in relation to one another, and the likelihood p_i of finding the configuration within the network are noted.

(B) These quantities are combined into a scalar χ, using a function Φ, depending on the nature of the filaments. For rigid filaments that do not buckle, all contributions are added (Φ=1). For flexible filaments that buckle readily under compression, only contractile configurations (v_i<0) are retained. For a network made of semi-flexible filaments, expansile configurations above the buckling threshold b are discarded.

(C) The scalar χ predicts the contraction rate of the network, depending on its dimensionality, as indicated. The sign of χ indicates if the network is contractile (χ<0) or expansile (χ>0).


Actomyosin Networks

TableOfConfigurations

Table of configurations

A system composed of flexible filaments and two types of connectors: crosslinkers and bifunctional motors. The table lists the four possible configurations for two connectors bound to a filament, the relative movement of the connectors (da/dt), and the likelihood and the mechanical nature of each configuration. The likelihoods are combinations of P_M and P_C, i.e. the probabilities of having at least one motor or one crosslinker at an intersection of two filaments

ContractileSystem

Contractile system

Contraction of a network of 1000 filaments of length 5 µm (red) distributed over a disc of radius 10 µm, with 8000 motors (blue) and 8393 crosslinkers (gray). This movie is similar to the network shown on Figure 2B, but the filament rigidity is 0.05 pNµm2, and the segmentation is 50nm.

moviemovie (zoom)config file
TheoryAndSimulations

Quantitative prediction of contractile rates

The contraction rate of a simulated network as a function of the ratio of crosslinkers to motors, with the total number of connectors kept constant. Each symbol indicates the result of one simulation. The broken line indicates the analytical prediction made by the theory

config file (template)
PhaseDiagram

Phase diagram

Evolution over time of 16 examples of networks of flexible filaments shown in Figure 2D. The number of crosslinkers is varied from left to right (750, 3000, 12000 and 48000), and the number of motors is varied from bottom to up (750, 3000, 12000 and 48000). All simulations start with 1500 straight filaments distributed over a circular area of radius 15 μm. The timespan covered is 30s.

movieconfig file (template)

Contractile/Expansile System with Rigid Filaments

TableOfConfigurations

Table of configurations

A system composed of rigid filaments and two types of connectors. One connector consists of a plus-end directed motor combined with a minus-end binder, the other is a plus-end directed motor combined with a plus-end binder. There are four possible configurations involving these two connectors.

ContractileSystem

Contractile system

Movie of a contractile network composed of 1500 straight (i.e. with infinite rigidity) filaments (white), 48000 motor-plus-end-binders (red) and 750 motor-minus-end-binders (blue) connectors distributed over a circular area of radius 15μm. The higher proportion of motor-plus-end-binders connectors leads to a higher probability of contractile configurations where filaments are pulled together towards their plus ends. The timespan covered is 60s.

movieconfig file
ExpansileSystem

Expansile system

Movie of an expansile network composed of 1500 straight (i.e. with infinite rigidity) filaments (white), 750 motor-plus-end-binders (red) and 48000 motor-minus-end-binders (blue) connectors, distributed over a circular area of radius 15μm. The higher proportion of motor-minus-end-binders connectors leads to a higher probability of extensile configurations where filaments are pushed apart. The timespan covered is 30s.

movieconfig file
TheoryAndSimulations

Quantitative prediction of contractile/expansile rates

The contraction rate of a network as a function of the numbers of two types of connectors, which are inversely varied. Each symbol represents a simulated random network of 4000 straight filaments initially distributed over a circular area of radius 25μm. The broken line indicates the analytical prediction made by the theory

config file (template)
PhaseDiagram

Phase diagram

Evolution over time of different networks of straight filaments (i.e. with infinite rigidity) as shown in Figure 3E. The number of motor-minus-end-binders is varied from bottom to top (750, 3000, 12000 and 48000), and the number of motor-plus-end-binders per filament is varied from left to right (750, 3000, 12000 and 48000). All simulations start with 1500 straight filaments and a varying number of connectors randomly distributed over a circular area of radius 15 µm. The timespan covered is 30s.

movieconfig file (template)

Other Recipes

Flexible_Xlinkers Rigid_Xlinkers

Network with only crosslinkers

(left) A system composed of flexible filaments and a connectors made of two binder subunits, which constitutes a crosslinker. This system has only one configuration that is static and the theory predicts that the system will neither expand or contract.

(right) A system composed of rigid filaments and a connectors made of two binder subunits, which constitutes a crosslinker. This system has only one configuration that is static and the theory predicts that the system will neither expand or contract.

config file (flexible)config file (rigid)
Flexible_Xlinkers Rigid_Xlinkers

Network with only motors

(left) A system composed of flexible filaments and a connectors made of two motor subunits. This system has only one configuration that is neutral and the theory predicts that the system will neither expand or contract.

(right) A system composed of rigid filaments and a connectors made of two motor subunits. This system has only one configuration that is neutral and the theory predicts that the system will neither expand or contract.

config file (flexible)config file (rigid)
Flexible_Xlinkers Rigid_Xlinkers

Network with binders and motors

(left) A system composed of flexible filaments and a connectors made of a binder and a motor subunit. This system has a contractile and expansile configuration, but due to buckling of the flexible filaments, most of the expansile configurations are spoiled and the theory predicts that the system will contract.

(right) A system composed of rigid filaments and a connectors made of a binder and a motor subunit. This system has a contractile and expansile configuration that balances out and the theory predicts that the system will neither expand or contract.

config file (flexible)config file (rigid)
Flexible_Xlinkers Rigid_Xlinkers

Network with motors and plus-end binders

(left) A system composed of flexible filaments and a connectors made of a (plus-end directed) motor and a plus-end binder subunit. This system has only neutral and contractile configurations the theory predicts that the system will contract.

(right) A system composed of rigid filaments and a connectors made of a (plus-end directed) motor and a plus-end binder subunit. This system has only neutral and contractile configurations the theory predicts that the system will contract.

config file (flexible)config file (rigid)
Flexible_Xlinkers Rigid_Xlinkers

Network with motors and minus-end binders

(left) A system composed of flexible filaments and a connectors made of a (plus-end directed) motor and a minus-end binder subunit. This system has only neutral and expansile configurations that are mostly lost due to filament buckling and the theory predicts that the system will neither expand nor contract.

(right) A system composed of rigid filaments and a connectors made of a (plus-end directed) motor and a minus-end binder subunit. This system has only neutral and expansile configurations the theory predicts that the system will expand.

config file (flexible)config file (rigid)

Heterogeneous Networks

TableOfConfigurations

Table of configurations

Configurations present in a heterogeneous network containing rigid minifilaments and flexible actin-like filaments. The motors are permanently attached at the extremities of the minifilaments, so as to represent myosin minifilaments. The system is predicted to be contractile in the presence of passive crosslinkers connecting actin filaments directly, and neutral without crosslinkers.

OnlyMinifilaments

Heteronegeous system with only myosin minifilaments

Simulation of a heterogeneous network containing rigid minifilaments and flexible actin-like filaments. In the absence of crosslnkers the sytem only have static and neutral configurations and the thepry predicts that the network will neither expand or contract.

movie - config file
MinifilamentsAndXlinkers

Heteronegeous system with myosin minifilaments and crosslinkers

Simulation of a heterogeneous network containing rigid minifilaments and flexible actin-like filaments. In the presence of crosslnkers the sytem have both contractile and expansile configurations, but due to buckling of the flexible filaments, most of the expansile configurations are spoiled and the theory predicts that the system will contract.

movie - config file

Pulsatile System

PulsatileSimulation

Pulsatile system

Network with filament turnover (lifetime ~11.11s) and periodic boundary conditions (size 29x29 µm). The network contains 10000 flexible filaments of length 2.5µm (rigidity 0.075 pN µm2), 80000 motors, 40000 crosslinkers. Foci move and fuse with each other.

movie - config file - config file (smaller system)
DensityXtime

Periodicity scaling with lifetime

The local density of filaments in an arbitrarily chosen region covering ~6% of the simulated space as a function of time. The network continues to redistribute, showing irregular variations of the local filament density, and does not contract into one spot.

config files (zipped)

Please, check our work on
cytosim.org.