# How crosslink numbers shape the large-scale physics of cytoskeletal
materials

Sebastian Fürthauer  Center for Computational Biology, Flatiron Institute, New
York, NY 10010, USA  Michael J. Shelley  Center for Computational Biology,
Flatiron Institute, New York, NY 10010, USA  Courant Institute, New York
University, New York, NY 10012, USA

###### Abstract

Cytoskeletal networks are the main actuators of cellular mechanics, and a
foundational example for active matter physics. In cytoskeletal networks,
motion is generated on small scales by filaments that push and pull on each
other via molecular-scale motors. These local actuations give rise to large
scale stresses and motion. To understand how microscopic processes can give
rise to self-organized behavior on larger scales it is important to consider
what mechanisms mediate long-ranged mechanical interactions in the systems.
Two scenarios have been considered in the recent literature. The first are
systems which are relatively sparse, in which most of the large scale momentum
transfer is mediated by the solvent in which cytoskeletal filaments are
suspended. The second, are systems in which filaments are coupled via
crosslink molecules throughout. Here, we review the differences and
commonalities between the physics of these two regimes. We also survey the
literature for the numbers that allow us to place a material within either of
these two classes.

##  I Introduction

Living systems move. On cellular scales this motion is actuated by networks of
polymer filaments crosslinked by molecular-scale motors that exert forces
between them. This system is collectively referred to as the cytoskeleton
howard01 ; alberts07 . Understanding how the material properties of the
cytoskeleton emerge from the properties of cytoskeletal components is one of
the great challenges for soft condensed matter physics and cell biology
foster2019cytoskeletal . Solving it will allow biologists to predict the
effects of molecular scale perturbations on cellular organelles and enable
physicists and engineers to pursue strategies for developing similarly complex
and robust materials in the lab. Here, we take measure of the current state of
theory for understanding the emergent physics of cytoskeletal systems based on
filament scale interactions.

Symmetry based theories have helped clarify the fields, such as densities,
velocities and order parameters, through which the dynamics of active
materials can be described on long length and time scales kruse2005generic ;
furthauer2012active ; julicher2018hydrodynamic . These theories have also
provided means for generating a more or less complete list of material
properties (or phenomenological coefficients), such as viscosities, elastic
moduli, and activity coefficients which characterize a material. While this
type of information can be used to relate the effects of molecular
perturbations to material properties naganathan2018morphogenetic , they do not
address the question how micro-scale processes set the large-scale physics.

To approach this challenge physicists and mathematicians have generalized the
methods developed for polymeric and liquid crystalline systems doi1988theory ;
degennes1995 , and Boltzmann’s statistical physics for gases, to incorporate
the effects of the microscopic activity of molecular-scale motors. These
efforts have led to a set of theories which describe relatively sparsely
connected polymeric assemblies whose large scale behavior is dominated by
momentum transfer via solvent flows liverpool2003instabilities ;
aranson2005pattern ; gao2015multiscalepolar . In parallel, other groups have
extended the physics of elastic networks to obtain theories for the physics of
crosslinked materials on time scales that are short compared to crosslinker
dynamics broedersz2014modeling .

However, recent work has increased the awareness that many cytoskeletal
systems, belong to a third class of active materials. They are highly
crosslinked gels, in which filaments are transiently coupled by a large number
of motors and crosslinks, which bind and unbind on time scales that are fast
compared to the long time, large scale dynamics of the system. This highly
crosslinked regime is set apart by new physical phenomena. Most strikingly,
filaments in highly crosslinked systems can slide through the gel at speeds
that are independent of the structure of their local surrounding. This
phenomenon has been predicted theoretically and observed in vitro
furthauer2019self , and in vivo mitchison1989polewards ; yang2008regional ;
dalton2021gelation . Theories for highly crosslinked systems take inspiration
from early work on actin bundles kruse2000actively ; kruse2003self  and
describe larger networks that are crosslinked by molecular motors with
different symmetries and force velocity relations furthauer2021design .

The goal of this review is to give a perspective on our current understanding
of the physics of cytoskeletal materials. We highlight the differences between
highly and sparsely crosslinked cytoskeletal networks, and their models. Where
possible from published data, we will classify known examples along these
lines. Along the way we will identify and highlight some open challenges that
the field needs to address in order to enable a quantitative and predictive
physics of these living materials.

In this review, we focus on continuum models and the microscopic models
underlying their derivations. We do however emphasize that symmetry based
theories marchetti2013hydrodynamics , continuum models obtained by coarse
graining microscopic models saintillan2013active ; maryshev2018kinetic , and
agent based models for the same systems janson2007crosslinkers ; head14 ;
popov16 ; freedman17  have been extensively studied in numerical simulations
as well.

In section II we review the key constituents of cytoskeletal Then, in section
III we propose a classification of cytoskeletal structures as either highly or
sparsely crosslinked. To guide us in this, we review the conditions under
which a materials long-range momentum transport is dominated by crosslink
interactions or by solvent flow, respectively. We then classify studied
cytoskeletal materials given published data. After that, in Section IV, we
review the theoretical descriptions of both highly and sparsely crosslinked
materials, highlighting and contrasting their different physics. We point out
how these differences can be used both as a predictive and diagnostic tools.

##  II Key constituents of the cytoskeletal networks

The cytoskeleton is the cellular machinery which enables cells to do
mechanical work on their organelles and on their surroundings.

Figure 1: Polar cytoskeletal filaments are coupled and actuated by molecular
scale motors (left). Collectively they form the cytoskeletal networks (middle)
which are the materials that make up many important biological structures
(right).

The cytoskeleton consists of polymer filaments, proteins that crosslink and
actuate these filaments, and regulators which help organize and coordinate
their functions in time and space. All of these constituents are suspended in
cytoplasm, which is the aqueous slurry that makes up a cell’s interior.

###  II.1 Cytoskeletal Filaments

Several types of cytoskeletal filaments exist. The two most studied and most
abundant ones are filamentous actin and microtubules alberts07 ; see Fig. (1).
Both are structurally polar, meaning that their molecular structure
differentiates one direction from the other. They are both transient since
they are constantly assembling and disassembling. They are active since
filament assembly and disassembly are driven by energy derived from GTP. While
for our purposes here, it is sufficient to think of cytoskeletal filaments as
polar rods, we want to emphasize that in typical in vivo systems the lifetime
of actin filaments and microtubule is often much shorter than the time scale
of large scale rearrangements of the system. One typical example is
gastrulation in fruitfly. This process, that is driven by actomyosin
contractions, takes about 2 hours and is driven by actomyosin, which turns
over within minutes collinet2021programmed . Another such process is the
separation of chromosomes during cell division in the nematode worm C.
elegans, which takes tens of minutes and is achieved by microtubules that have
lifetimes of tens of seconds redemann2017celegans . There are additional
cytoskeletal filaments beyond actin and microtubules, largely called
intermediate filaments. Examples include keratins, which are a large family of
proteins that encompass vimentins, which play a role in cell motility, and
lamins, which act as a mechanical scaffolding for the cell nucleus. These
intermediate filaments - so-called since their thickness in the electron
tomographs where they where first characterized was between that of thin actin
filaments and that of thick Myosin mini-filaments - are typically longer-
lived, apolar, and are hypothesized to provide cells with long time
elasticity. Kinesins, myosins and dyneins that link intermediate filaments to
the microtubule and actin cytoskeleton have been identified, but so far their
role remains under-explored.

###  II.2 Active and passive crosslinkers

The second class of molecules that are important for the cytoskeleton are
crosslinking proteins which connect cytoskeletal filaments to each other and
to other structures such as chromosomes and membranes. Too many types exist to
do them justice in a review focussed on the physics of the cytoskeleton, so we
refer the reader to pollard2016actin  and alfaro2015building  for recent
perspectives on actin and microtubule associated crosslinkers. Many of these
crosslinkers are molecular scale motors. This means they have access to a
chemical fuel reservoir - in general Adenosine-Tri-Phosphate (ATP) - from
which they can draw power to do mechanical work upon the structures they
crosslink. Motors can walk along cytoskeletal filaments, acting as moving
crosslinks, which slides filaments past each other. Or, they can carry a
cargo, like a vesicle or a mitochondrion, along filaments. The most abundant
actin-associated molecular motors are myosins. The most abundant microtubule-
associated ones are dyneins and kinesins. Typically motors can exert forces of
up to several pico-Newtons, and their unloaded walking speeds on filaments can
vary from nanometers to microns per second. Finally, motors can be more or
less processive. A motor’s processivity measures the expected number of steps
a motor typically takes, before it unbinds from the filament to which it is
attached howard01 ; alberts07 .

###  II.3 Cytoplasm

The last important actor in understanding the physics of the cytoskeleton is
the aqueous slurry in which it is immersed. While the rheology of the
cytoplasm is most certainly complex luby1999cytoarchitecture , most current
physical theories for the cytoskeleton ignore this fact - mainly because the
detailed characteristics of cytoplasmic rheology are poorly understood and
difficult to model. The theories that we review, thus approximate it as a very
viscous Newtonian fluid with a viscosity between 100 and 1000 times that of
water; see for example garzon2016force .

###  II.4 Forces and torques in cytoskeletal networks

Figure 2:  (a) Interaction between two cytoskeletal filament $i$ and $j$ via a
molecular motor. Filaments are characterized by their positions
$\mathbf{x}_{i},\mathbf{x}_{j}$, their orientations
$\mathbf{p}_{i},\mathbf{p}_{j}$, and connect by a motor between arc-length
position $s_{i},s_{j}$. A motor consist of two heads that can be different
(circle, pentagon) and are connected by a linker (black zig-zag) of lengt $R$
(b) The total force on filament $i$ is given by the sum of the forces exerted
by all $a$ (circle) and $b$ (pentagon) heads, which connect the filament into
the network. The shaded area shows all geometrically accessible positions that
can be crosslinked to the central (black) filament.

The goal of a physical theory for the cytoskeleton is to predict how filaments
in a cytoskeletal network rearrange in response to the forces that they
experience and produce. The larger goal is a quantitative understanding of the
mechanical processes that allow cytoskeletal structures to self-organize and
perform their complex tasks within cells.

The elementary starting point is characterizing the total force
$\mathbf{F}_{i}$ that acts on the $i$-th filament in the network. We assume
that it has the form

$\mathbf{F}_{i}=\sum_{j}\mathbf{F}_{ij}^{\times}+\mathbf{F}_{i}^{s}=0,$ (1)

which vanishes since cytoskeletal systems are overdamped; see Fig. 1 (middle).
The total force on each filament consists of crosslink mediated forces
$\sum_{j}\mathbf{F}_{ij}^{\times}$ and a contribution $\mathbf{F}_{i}^{s}$,
which comes from the drag between filaments and a moving solvent. The
crosslink mediated force between filaments $i$ and $j$ is given by

$\mathbf{F}_{ij}^{\times}=\int_{-L/2}^{L/2}ds_{i}\int_{-L/2}^{L/2}ds_{j}\int_{\Omega(\mathbf{x}_{i}+s_{i}\mathbf{p}_{i})}d^{3}x~{}\delta(\mathbf{x}-\mathbf{x}_{j}-s_{j}\mathbf{p}_{j})\mathbf{f}_{ij},$
(2)

where $\mathbf{f}_{ij}$ is the force density exerted by crosslinkers between
the arclength positions $s_{i}$ and $s_{j}$ on the two filaments $i,j$,
respectively; see Fig. 2. Here, $\Omega(\mathbf{x})$ is a sphere centered
around the point $\mathbf{x}$ and whose radius $R$ is the size (maximal
extension) of the cross-linker. Here all filaments are taken to have the same
length $L$. Analogous expressions for the torques and total torque on filament
$i$ hold.

In the following we will argue that modeling of cytoskeletal networks has
focussed on two different regimes: (i) a highly crosslinked regime in which
the long range momentum transport through the gel is dominated by crosslinking
forces and (ii) a regime in which momentum transport through the solvent
dominates. Theories for the two limits make very different predictions for the
scaling of important material properties and dynamics.

##  III Which cytoskeletal materials are highly crosslinked?

Figure 3: Illustration of the key differences between highly and sparsely
crosslinked networks, emphasizing the dominant mechanisms of large scale
momentum transport.

We now discuss the crossover between the highly and sparsely crosslinked
regimes and try to classify a few well studied cytoskeletal systems along
these lines. For this we define the density of cytoskeletal filaments
$\rho=\sum_{i}\delta(\mathbf{x}_{i}-\mathbf{x})$ and the force density
$f=\sum_{i}\delta(\mathbf{x}_{i}-\mathbf{x})\mathbf{F}_{i}$. Since crosslink
mediated interaction are short ranged we can write

$\mathbf{f}=\nabla\cdot\mathbf{\Sigma}-\Gamma\rho(\mathbf{v}-\mathbf{v}^{s})+\mathcal{O}\left(L^{4}\right)=0,$
(3)

where $L$ is assumed small compared to the system scale furthauer2021design .
In the following we use the word gel to signify the filaments and crosslinks
and solvent for the cytoplasm in which they are immersed. The gel stress
$\mathbf{\Sigma}$ encodes the momentum flux through the crosslinks which
connect filaments, and $\Gamma\rho$ is the friction coefficient between the
gel and the solvent. Finally, $\mathbf{v}$ and $\mathbf{v}^{s}$ are the gel
and solvent center of mass velocities, respectively. If we take the solvent to
be a Newtonian fluid with viscosity $\mu$ it obeys the Stokes equation

$\mu\Delta\mathbf{v}^{s}-\nabla\mathbf{\Pi}^{s}+\Gamma\rho(\mathbf{v}-\mathbf{v}^{s})=0,$
(4)

where $\mathbf{\Pi}^{s}$ is the hydrostatic pressure. The characteristic
length scale over which momentum is transported through the solvent, the
permeation length, is thus given by

$\ell^{s}=\sqrt{\frac{\mu}{\rho\Gamma}},$ (5)

which decreases as the density $\rho$ of filaments in the gel increases.
Conversely, crosslinks generate friction between the filaments they connect,
which leads to a viscous contribution $\mu^{g}\rho^{2}\nabla\mathbf{v}$ to
$\mathbf{\Sigma}$, i.e. proportional the strain rate of the gel and to the
number of filament-filament interactions mediated by crosslinks
furthauer2019self ; furthauer2021design . Here $\mu^{g}\rho^{2}$ is the gel
viscosity. Thus we expect the typical length scale over which momentum is
transported through the gel to be

$\ell^{g}=\sqrt{\frac{\mu^{g}\rho^{2}}{\Gamma\rho}},$ (6)

which increases with filament density. The implied assumption of this
calculation is that the number of realized crosslink connections is limited by
the number of geometrically possible filament-filament interactions and not
the number of crosslinking molecules available. It can however easily be
generalized.

In the following we consider a system highly crosslinked if
$\ell^{g}\gg\ell^{s}$ and sparsely crosslinked if $\ell^{s}\gg\ell^{g}$. We
next review the literature and overview which cytoskeletal systems are highly
or sparsely crosslinked, respectively; see Fig. 3.

###  III.1 The cell cortex

Figure 4: Biological and artificial cytoskeletal networks: (A) SIM-TIRF super-
resolution microscopy image of the actin cortex from a kidney culture cell.
With permission from li2015extended . (B) Electron tomography reconstruction
of the inner spindle microtubules of a embryonic C. elegans mitotic spindle.
Taken from redemann2017celegans ; (C): spinning disk fluorescence microscopy
image of microtubules and actin in a Xenopus laevis interphase aster in egg
extract. Courtesy of James Pelletier; (D) snapshot from an experiment an
active microtubule nematic synthesized from engineered multimeric kinesin-1
motors and stabilized microtubules. Adapted with permission from
wu2017transition . Throughout the figure actin filaments a represented in
orange and microtubules in blue.

The cell cortex is a thin layer of actin filaments that localize at the cell
membrane; see Fig. 4 (A). It is actuated by non-muscle myosin 2 and held
together by a variety of crosslinkers and proteins which regulate assembly and
disassembly of both thick (myosin) filaments and thin (actin) filaments
pollard2016actin . Typically the thickness of the cortical layer is 300-1000nm
salbreux2012actin . Flow of actomyosin is vital for cell polarity
establishment (the process by which cells break the symmetry between two
daughter cells) mayer2010anisotropies ; gross2019guiding , cell migration and
shape changes callan2016cortical , cell division turlier2014furrow , as well
as regulating cellular surface tension chugh2017actin . On time-scales longer
than filament lifetimes, the cell cortex is well described as an active
fluidic material mayer2010anisotropies ; naganathan2014active ;
naganathan2018morphogenetic .

Comprehensive estimates of the numbers of crosslinks in the actin cell cortex
are hard to come by. That said, many actin filaments are nucleated by the
nucleator Arp2/3 which is a crosslinker itself. In fritzsche2016actin  the
authors estimate that roughly nine out of ten actin filaments are nucleated by
Arp2/3, while about one out of ten is nucleated by formins. This finding
suggests that the actin cortex is highly crosslinked starting from its
nucleation. Moreover, actin filaments are actively crosslinked by myosin
motors and other crosslinkers. In wu2005counting  the authors count the
numbers of the most important actin binding partners in fission yeast cells.
They find that the number of crosslinkers and motors acting on actin filaments
is comparable (or larger) than the number of nucleators (Arp2/3 and formins)
in the system. Thus, if one estimates that the number of filaments is roughly
proportional to the number of available nucleators, one concludes that there
are several crosslinking interactions per filament exist at any give time.
That is, the actin cortex is highly crosslinked.

###  III.2 The spindle

The spindle is the cellular apparatus which segregates chromosomes during cell
division. Its main constituents are microtubules which are crosslinked and
actuated by dynein and kinesin molecular motors; see Fig. 4 (B). Despite
having common constituents and function, spindle structure can vary widely,
even in a single organism.

One spindle whose physics has been studied extensively is the meiotic spindle
of Xenopus laevis. This spindle can be reconstituted in frog egg extract,
which makes it particularly attractive to study ex vivo. These spindles are
large - about 50 microns long - and consist of microtubules whose lengths and
lifetimes are exponentially distributed around an average length and lifetime
of about 6 microns and 20 seconds, respectively brugues2012nucleation . They
have been shown to obey the fluctuation spectra expected for active nematic
gels brugues2014physical . In these spindles, microtubules nucleate by an
auto-catalytic mechanism in which pre-existing microtubules recruit growth
factors that initiate the formation of new microtubules petry2011augmin ;
petry2013branching ; oh2016spatial ; kaye2018measuring ;
decker2018autocatalytic . This mechanism implies that filaments are created
crosslinked to the spindle gel. The details of the pathway controlling
nucleation have been investigated thawani2018xmap215 ;
thawani2019spatiotemporal . The motor proteins dynein and Eg-5 kinesin
crosslink the spindle network further. While direct measurements of the number
of Eg-5 kinesin crosslinks are not available, observations of microtubule
motion in spindles allow an estimate. In dalton2021gelation  the authors
quantify the motion and polarity of microtubules in Xenopus spindles. They
estimate that two or motors per filament are required to explain experimental
observation that velocity fluctuation are highly correlated throughout the
whole spindle. In our language this means that $\ell^{g}$ is comparable to
system size and thus that the Xenopus spindle is highly crosslinked.

Another well studied spindle is the embryonic mitotic spindle of the nematode
worm C. elegans. These spindles are much smaller - $10\mu$m pole to pole - and
have been fully reconstructed using electron tomography redemann2017celegans ;
see Fig. 4 (C). Using this structural information, together with light
microscopy and mathematical modelling, the authors in redemann2019current
argue that the connection between spindle poles and chromosomes is indirectly
maintained by microtubules crosslinking to other microtubules. While the
evidence is indirect it is suggestive of a highly crosslinked network.

The third example that we want to highlight here is the female meiotic
spindle, also in C . elegans. This spindle is about half the size of the
mitotic one and has also been reconstructed in tomography. The findings from
tomography studies redemann2018switch ; lantzsch2020microtubule  suggest that
it consists of short and short-lived microtubules and is structurally quite
similar to the Xenopus laevis meiotic spindle. It is thus tempting to
speculate that this spindle too is a highly crosslinked network.

###  III.3 Microtubule asters

In cells, spindles are positioned for cell division by microtubule asters
which emanate from the spindle poles and grow out towards the cortex; see Fig.
4 (C). In smaller cells, microtubules interacting with cortex are important
for spindle positioning pavin2012positioning ; garzon2016force ;
nazockdast2017cytoplasmic . Recent work in C. elegans embryos suggests that
dynein mediated pulling forces from cortically bound dynein are crucial
farhadifar2020stoichiometric .

In larger cells, such as the Xenopus zygote, the situation is somewhat more
complicated, since cytoskeletal filament are in general much shorter than the
typical cell radii sulerud2020microtubule . Thus it has been proposed that in
these cells spindle are positioned by cytoplasmic pulling, that is by dynein
motors which carry a cargo from the aster periphery towards its center. This
would cause an active drag against the cytoplasmic fluid, which would
ultimately cause spindle centering meaders2020microtubule ;
xie2020cytoskeleton . To test this hypothesis the authors of pelletier2020co
co-imaged the cytoskeletal actin, microtubules, and membranous networks. They
find that these networks move against each other near aster boundaries, but
move together near the asters center. It is thus tempting to ask what explains
these different behaviors. One appealing hypothesis is that these asters are
highly crosslinked near their centers, and less crosslinked near aster cores.

###  III.4 Artificial systems

Given the complexity and the sheer number of different constituents of
cytoskeletal networks in living cells, biophysicists have developed simplified
in vitro systems made from cytoskeletal components; see Fig. 4 (C). Here, the
physics of cytoskeletal networks can be studied at a reduced complexity and in
a more controllable environment. Beyond that, they have provided foundational
examples of out-of-equilibrium materials.

One prominent system by sanchez2012spontaneous  is made from stable
microtubules and kinesin-1 motors joined by engineered linkers. This same
mixture has been used to study active materials in bulk and on 2d interfaces
sanchez2012spontaneous , in vesicles keber14 , rigid confinements
opathalage2019self ; chandrakar2020confinement  and droplets
vcopar2019topology . The same system has also been adapted to study
disclination loops in 3d active nematics duclos2020topological . It is
surprising, given the importance of these systems, that the structure and
number of crosslinks that bind microtubules remain relatively unexplored. More
studies will be needed to solidly classify these particular systems as either
highly or sparsely crosslinked.

Somewhat more recently, actin-based active networks have been synthesized.
Many of the key findings of this line of research a summarized in
banerjee2020actin . These systems are in general driven by the molecular motor
myosin and require the admixture of passive crosslinkers to generate active
contractions. This peculiarity has been attributed to a liquid-gel transition,
which may imply that the contractile states of these actomyosin networks are
in the highly crosslinked regime.

There is a growing list of in vitro systems, using different motors and
filaments assembled in cell extract or fully reconstituted such as in
foster2015active ; foster2017connecting  or the ones reviewed in
needleman2017active ; banerjee2020actin . For most of these systems it remains
unclear where they fall on the range from sparsely to highly connected
systems.

##  IV Theoretical descriptions of cytoskeletal systems

Figure 5: Top: Illustration of the important hydrodynamic variables. Bottom:
Illustration of externally strained and self-strained materials.

Much of our current theoretical understanding of cytoskeletal active matter
derives from the remarkable success of extending the broken-symmetry
hydrodynamics of liquid crystal system degennes1995  to active materials where
micro-scale non-equilibrium processes can generate force and torque dipoles
kruse2005generic ; furthauer2012active ; julicher2018hydrodynamic . These
theories have clarified the fields (densities, velocities, order parameters)
through which to describe the dynamics of a cytoskeletal material on long-wave
hydrodynamic scales; see Fig. 5, and which material properties, such as the
viscosities and activity coefficients, are needed to characterize these
materials. More microscopic theories derive the values of material properties
and their microscopic origins.

We start by reviewing the fields and material properties that we use to
characterize cytoskeletal materials. We then review theoretical predictions
for how these material properties are determined in highly and sparsely
crosslinked networks.

###  IV.1 Hydrodynamic variables

We seek to describe the evolution of a cytoskeletal system with a small number
of continuous fields. At large length and time scales the fields needed to
sufficiently characterize a material are those associated with conserved
quantities and those associated to continuous broken symmetries
chaikin2000principles ; see Fig. 5 (top). We first introduce the fields
associated to conserved quantities.

The first is the total mass density $\rho^{\mathrm{tot}}(\mathbf{x},t)$ of the
material. Since mass can be neither created nor destroyed,

$\partial_{t}\rho^{\mathrm{tot}}=-\nabla\cdot\mathbf{g}^{\mathrm{tot}},$ (7)

where $\mathbf{g}^{\mathrm{tot}}(\mathbf{x},t)$, denotes the total mass flux -
or momentum density. In the cytoskeletal systems of interest here the total
mass density consists of solvent and gel contributions, such that
$\rho^{\mathrm{tot}}=\rho+\rho^{\mathrm{s}}$, where $\rho$ is the gel
contribution, by which we mean the mass of filaments and proteins attached to
filaments. In general, chemical processes, such as filament nucleation and
polymerization, and the binding and unbinding of proteins, can convert solvent
mass into gel mass and vice versa. Thus we can write

$\displaystyle\partial_{t}\rho$ $\displaystyle=$
$\displaystyle-\nabla\cdot\rho\mathbf{v}+R$ (8)
$\displaystyle\partial_{t}\rho^{\mathrm{s}}$ $\displaystyle=$
$\displaystyle-\nabla\cdot\rho^{\mathrm{s}}\mathbf{v}^{\mathrm{s}}-R,$ (9)

where $R$ is the mass conversion rate from chemical reactions and we have
defined the gel and solvent velocities by
$\mathbf{g}^{\mathrm{tot}}=\rho\mathbf{v}+\mathbf{\rho}^{\mathrm{s}}\mathbf{v}^{\mathrm{s}}$.
In living materials the dominant contribution to $R$ is filament
polymerization and depolymerization, which in general is highly regulated. For
instance in Xenopus’s meiotic spindles the dominant mode of filament creation
is autocatalytic, and new filaments nucleate in a branching process from
preexisting ones petry2011augmin ; petry2013branching ; oh2016spatial ;
kaye2018measuring ; decker2018autocatalytic ; thawani2018xmap215 ;
thawani2019spatiotemporal .

The second field that arises from conservation considerations is the momentum
density itself. In the absence of external forces, the momentum density
$\mathbf{g}^{\mathrm{tot}}(\mathbf{x},t)$ can only change by a momentum flux

$\partial_{t}\mathbf{g}^{\mathrm{tot}}=\nabla\cdot\mathbf{\Sigma}^{tot},$ (10)

where $\mathbf{\Sigma}^{tot}(\mathbf{x},t)$ is total stress in the material.
Like the mass flux we can write
$\mathbf{\Sigma}^{tot}=\mathbf{\Sigma}+\mathbf{\Sigma}^{s}$, and force balance
equations for the gel and solvent components of the material separately. Under
the assumptions that the solvent is Newtonian, that inertial terms can be
neglected, and that the force exerted between the gel and fluid phases is
linear in the gel density and their velocity difference, one recovers Eqns.
(3,4). Thus, the fields that we need to keep track of, since they are
associated to conserved quantities, are $\rho,\rho^{\mathrm{s}},\mathbf{v}$,
and $\mathbf{v}^{\mathrm{s}}$. We next introduce the fields that are
associated with continuous broken symmetries.

Order parameter fields characterize the conformational state of the material.
For the purposes here we assume that cytoskeletal filaments are rod-like and
rigid. Then the tensor orientation order parameter $\mathcal{Q}(\mathbf{x},t)$
naturally arises. From $\mathcal{Q}$ devolves the scalar orientation order
parameter and director field. Unlike purely nematic systems, cytoskeletal
filaments like microtubules or actin are polar, as are the motor-mediated
forces acting upon them. Hence, we will also need a polar order parameter
$\mathbf{P}(\mathbf{x},t)$. If the system were characterized at the
microscopic level by a particle distribution function
$\psi(\mathbf{x},\mathbf{p},t)$ doi1988theory  with $\mathbf{p}$ being the
particle orientation vector on the 2-sphere, then these order parameters arise
as its $\mathbf{p}$-moments $\mathbf{P}=\left<\mathbf{p}\right>/\rho$ and
$\mathcal{Q}=\left<\mathbf{p}\mathbf{p}\right>/\rho$, where the angular
brackets denote a distributional averaging over the $\mathbf{p}$-sphere. In
general higher order symmetry fields exist, but we ignore them here for
simplicity.

In summary, we seek to describe the cytoskeletal systems in terms of just a
few coarse grained continuous fields. They are the densities $\rho$ and
$\rho^{s}$, the velocity fields $\mathbf{v}$ and $\mathbf{v}^{s}$, and the
order parameter fields $\mathcal{Q}$ and $\mathbf{P}$. Thus, the next task is
to arrive at expression for the stresses $\mathbf{\Sigma}^{\mathrm{tot}}$ and
$\mathbf{\Sigma}$, and the transport operators for the order parameter fields
$\mathcal{Q}$ and $\mathbf{P}$, and the chemical reaction rate $R$, in terms
of hydrodynamic variables. In this review we are concerned with the
expressions for the stresses first and foremost.

###  IV.2 Material properties of active cytoskeletal networks

Given the set of hydrodynamic variables, symmetry-based theories postulate
material laws for all unknown quantities by writing down all symmetry allowed
terms that respect the invariances of the systems, such as translation and
rotation invariance. How to obtain symmetry based material equations from
symmetry considerations or from non-equilibrium thermodynamics considerations
is reviewed elsewhere in great detail marchetti2013hydrodynamics ;
julicher2018hydrodynamic .

For our purposes here, it is sufficient to express the total stress by the
equation,

$\mathbf{\Sigma}^{\mathrm{tot}}=\chi^{v}:\nabla\mathbf{v}+\chi^{s}:\nabla\mathbf{v}^{s}+\chi^{a}:\nabla\mathbf{P}+\Pi^{a}\mathcal{I}+\mathcal{A}^{(\mathbf{P})}\mathbf{P}\mathbf{P}-\mathcal{A}^{(\mathcal{Q})}\mathcal{Q}+\bar{\mathbf{\Sigma}},$
(11)

which obeys the symmetry constraints discussed in marchetti2013hydrodynamics ;
julicher2018hydrodynamic . Here, $\chi^{v}$ is the fourth-rank gel viscosity
tensor, $\chi^{s}$ is a fourth-rank solvent viscosity tensor, and $\chi^{a}$
is a fourth-rank tensor that quantifies active polar stresses proportional to
gradients in gel polarity $\mathbf{P}$. The coefficients
$\mathcal{A}^{(\mathbf{P})}$ and $\mathcal{A}^{(\mathcal{Q})}$ quantify the
magnitude of active polar and active nematic stresses in the material,
respectively. Furthermore $\Pi^{a}$ acts as an active pressure and
$\bar{\mathbf{\Sigma}}$ denotes other stresses in the system, which stem from
hydrostatic and steric exclusion and alignment effects. For an in-depth
discussion see degennes1995  and julicher2018hydrodynamic . Together,
$\chi^{v},\chi^{a},\chi^{s},\Pi^{a},\mathcal{A}^{(\mathbf{P})}$, and
$\mathcal{A}^{(\mathcal{Q})}$ are the material properties of the cytoskeletal
filament network. They in turn depend on the state of the network, that is the
hydrodynamic variables, and need either be measured in an experiment, or
derived from more microscopic considerations to allow a complete description
of the material. Symmetry-based theories also elucidate the form of the
transport equations for $\mathbf{P}$ and $Q$. We refer to
marchetti2013hydrodynamics ; julicher2018hydrodynamic  for a more complete
discussion.

In this review we are concerned with theories that derive expressions for the
phenomenological coefficients in (Eq. (11)) from micro-scale considerations of
motors and filaments. In particular we want to contrast theories for highly
crosslinked and sparsely crosslinked systems. We will highlight that the
functional dependencies of
$\chi^{v},\chi^{a},\chi^{s},\Pi^{a},\mathcal{A}^{(\mathbf{P})}$ and
$\mathcal{A}^{(\mathcal{Q})}$ change between these two regimes.

###  IV.3 The highly crosslinked regime of cytoskeletal networks

We first review theories for the hydrodynamic limit of the highly crosslinked
regime of cytoskeletal networks. We consider a cytoskeletal material to be
highly crosslinked if $\ell^{g}\gg\ell^{s}$. The hydrodynamic limit of such a
material is dominated by crosslink interactions, and thus solvent mediated
interactions can be ignored, since the length scale of momentum transport
associated with $\mathbf{\Sigma}$ is much larger than that associated with
$\mathbf{\Sigma}^{\mathrm{s}}$; see Fig. 3. Thus these theories assume that
$\mathbf{\Sigma}^{\mathrm{tot}}=\mathbf{\Sigma}$ which implies
$\mathbf{v}^{s}=\mathbf{v}$ and that $\mathbf{\Sigma}^{\mathrm{s}}$ is
negligible.

To our knowledge this limit of active gels was first discussed in the context
of studying actin bundles kruse2003self ; kruse2003continuum  where the
authors proposed a 1d model for describing actin structures in the cortex.
Only recently have these ideas been extended to larger three-dimensional
systems. In furthauer2019self ; furthauer2021design  a phenomenological model
for the force density $\mathbf{f}_{ij}$, generated by crosslinks between
filaments, was proposed. In this model the force density is given by

$\displaystyle\mathbf{f}_{ij}$ $\displaystyle=$ $\displaystyle
K(s_{i},s_{j})\left(\mathbf{x}_{i}+s_{i}\mathbf{p}_{i}-\mathbf{x}_{j}-s_{j}\mathbf{p}_{j}\right)$
(12) $\displaystyle+$
$\displaystyle\gamma(s_{i},s_{j})\left(\mathbf{v}_{i}+s_{i}\dot{\mathbf{p}}_{i}-\mathbf{v}_{j}-s_{j}\dot{\mathbf{p}}_{j}\right)$
$\displaystyle+$
$\displaystyle\left[\sigma(s_{i},s_{j})\mathbf{p}_{i}-\sigma(s_{j},s_{i})\mathbf{p}_{j}\right].$

The functions $K(s_{i},s_{j}),~{}\gamma(s_{i},s_{j}),~{}\sigma(s_{i},s_{j})$
characterize the types of forces exerted by crosslink populations between the
arclength positions $s_{i},s_{j}$ on filaments $i$, and $j$. The coefficient
$K$ captures effects of crosslink elasticity, the coefficient $\gamma$ acts as
an effective frictional coupling between filaments, and $\sigma$ characterizes
the active forcing from motor stepping; see Fig. 2. The main conceptual
advance between this phenomenological model and the ones proposed in earlier
work, is that here $\mathbf{f}_{ij}$ explicitly depends on the velocities and
rates of rotation of filaments $i,j$. With this dependence, in a dense system
the dynamics of all filaments in the system becomes tightly coupled. In
contrast, earlier work kruse2000actively ; aranson2005pattern ; LM2006 ;
maryshev2018kinetic  had posited motor interactions that depend only on the
relative orientations and center of mass positions of filaments, effectively
constraining themselves to pairwise interactions.

From Eq. (12) one calculates the stress in the network using

$\displaystyle\mathbf{\Sigma}=-\frac{1}{2}\sum_{i,j}\left\lfloor\begin{array}[]{c}\delta(\mathbf{x}-\mathbf{x}_{i})\delta(\mathbf{x}^{\prime}-\mathbf{x}_{j})\\\
\left(\mathbf{x}_{i}-\mathbf{x}_{j}+s_{i}\mathbf{p}_{i}-s_{j}\mathbf{p}_{j}\right)\mathbf{f}_{ij}\end{array}\right\rceil_{\Omega(\mathbf{x}_{i})}^{ij}$
(15)

where the brackets $\lfloor\cdots\rceil^{ij}_{\Omega(\mathbf{x}_{i})}$ denotes
the coarse graining operation

$\lfloor\phi\rceil_{\Omega(\mathbf{x}_{i})}^{ij}=\int\limits_{-\frac{L}{2}}^{\frac{L}{2}}ds_{i}\int\limits_{-\frac{L}{2}}^{\frac{L}{2}}ds_{j}\int\limits_{\Omega(\mathbf{x}_{i})}d\mathbf{x}^{3}\phi.$
(16)

Here $\Omega(\mathbf{x})$ is a sphere centered around the point $\mathbf{x}$
and whose radius $R$ is the size (maximal extension) of the cross-linker. In
this version of the theory, all filaments are taken to have the same length
$L$. Thus, the operation
$\lfloor\cdots\rceil^{ij}_{\Omega(\mathbf{x}_{i}+s_{i}\mathbf{p}_{i})}$
integrates its argument over all geometrically possible crosslink induced
interactions.

The emergent material properties turn out to depend on a small set of
$s$-moments of the phenomenological coefficients $K,~{}\gamma,~{}\sigma$.
These moments are given by

$X_{nm}(\mathbf{x})=\lfloor
X(s_{i},s_{j})s_{i}^{n}s_{j}^{m}\rceil_{\Omega(\mathbf{x})}^{ij},$ (17)

where $X$ can be $K,~{}\gamma,$ or $\sigma$. More explicitly,

$\displaystyle K_{0}$ $\displaystyle=$ $\displaystyle K_{00}=\left\lfloor
K(s_{i},s_{j})\right\rceil_{\Omega(\mathbf{x}_{i})}^{ij},$ (18) $\displaystyle
K_{1}$ $\displaystyle=$ $\displaystyle K_{10}=K_{01}=\left\lfloor
s_{i}K(s_{i},s_{j})\right\rceil_{\Omega(\mathbf{x}_{i})}^{ij},$ (19)
$\displaystyle\gamma_{0}$ $\displaystyle=$
$\displaystyle\gamma_{00}=\left\lfloor\gamma(s_{i},s_{j})\right\rceil_{\Omega(\mathbf{x}_{i})}^{ij},$
(20) $\displaystyle\gamma_{1}$ $\displaystyle=$
$\displaystyle\gamma_{10}=\gamma_{01}=\left\lfloor
s_{i}\gamma(s_{i},s_{j})\right\rceil_{\Omega(\mathbf{x}_{i})}^{ij},$ (21)
$\displaystyle\sigma_{0}$ $\displaystyle=$
$\displaystyle\sigma_{00}=\left\lfloor\sigma(s_{i},s_{j})\right\rceil_{\Omega(\mathbf{x}_{i})}^{ij},$
(22) $\displaystyle\sigma_{10}$ $\displaystyle=$ $\displaystyle\left\lfloor
s_{i}\sigma(s_{i},s_{j})\right\rceil_{\Omega(\mathbf{x}_{i})}^{ij},$ (23)
$\displaystyle\sigma_{01}$ $\displaystyle=$ $\displaystyle\left\lfloor
s_{j}\sigma(s_{i},s_{j})\right\rceil_{\Omega(\mathbf{x}_{i})}^{ij}.$ (24)

With these definitions, one finds for the viscosity tensor

$\chi_{\alpha\beta\gamma\mu}^{v}=\rho^{2}\gamma_{0}\left(\frac{3R^{2}}{10}\delta_{\alpha\gamma}\delta_{\beta\mu}+\frac{L^{2}}{12}\gamma_{0}\left(\mathcal{Q}_{\alpha\gamma}\delta_{\beta\mu}\mathcal{-}Q_{\alpha\beta}\mathcal{Q}_{\gamma\mu}\right)\right),$
(25)

and, very importantly, that

$\chi^{a}_{\alpha\beta\gamma\mu}=\frac{\sigma_{0}}{\gamma_{0}}\chi_{\alpha\beta\gamma\mu}^{v}.$
(26)

for the coefficient quantifying the active polar stress.

The relation between Eqs. (25) and (26) implies that we can combine the first
two terms of Eq. 11 into

$\mathbf{\Sigma}^{ss}=\chi^{v}:\left(\nabla\mathbf{v}+\frac{\sigma_{0}}{\gamma_{0}}\nabla\mathbf{P}\right)$
(27)

which we call the self-straining stress. In this type of material
$\sigma_{0}/\gamma_{0}$ is essentially the free velocity of the motors and
crosslinks coupling filaments. Thus the state of zero self-straining stress is
one where the viscous stress exactly balances the active polar stress, which
can be achieved when all filaments in the material move at the motor speed in
the direction in which they point furthauer2019self . This self-straining
state is a consequence of the fact that the same active crosslinks which drive
filament motion also couple the dynamics over long length scales. In essence,
in a material where the crosslinking is provided by active crosslinks,
filaments can move through the material at the preferred speed of the motor
without stressing the material; see Fig. 5.

The same calculation also makes predictions for other active stresses in the
material furthauer2021design . One finds

$\displaystyle A^{(\mathcal{Q})}$ $\displaystyle=$
$\displaystyle\rho^{2}(\sigma_{10}-\sigma_{0}\frac{\gamma_{1}}{\gamma_{0}}+\frac{L^{2}}{12}K_{0}),$
(28) $\displaystyle A^{(\mathbf{P})}$ $\displaystyle=$
$\displaystyle\rho^{2}(\sigma_{01}-\sigma_{0}\frac{\gamma_{1}}{\gamma_{0}}),$
(29) $\displaystyle\Pi^{a}$ $\displaystyle=$
$\displaystyle\rho^{2}\frac{3R^{2}}{10}K_{0}.$ (30)

These expressions generalize to the continuum limit the predictions made by
belmonte2017theory  in the context of a discrete model.

An analogous calculation provides expressions for the filament speed
$\mathbf{v}_{i}$ and rotation rate $\dot{\mathbf{p}}_{i}$ in the lab frame:

$\mathbf{v}_{i}-\mathbf{v}=\frac{\sigma_{0}}{\gamma_{0}}(\mathbf{p}_{i}-\mathbf{P})+\mathcal{O}{(L^{2})}$
(31)

and

$\displaystyle\dot{\mathbf{p}}_{i}$ $\displaystyle=$
$\displaystyle\left(\mathcal{I}-\mathbf{p}_{i}\mathbf{p}_{i}\right)\cdot\left\\{\begin{array}[]{c}(\nabla\mathbf{v}+\frac{\sigma_{0}}{\gamma_{0}}\nabla\mathbf{P})\cdot\mathbf{p}_{i}\\\
+\frac{12}{\gamma_{0}L^{2}\rho^{2}}\mathcal{E}\cdot\mathbf{p}_{i}\\\
+\frac{12}{\gamma_{0}L^{2}}A^{(\mathbf{P})}\mathbf{P}\end{array}\right\\},$
(35)

where $\mathcal{E}$ is the steric distortion field degennes1995 . We refer the
reader to furthauer2021design  for additional detail.

###  IV.4 The sparsely crosslinked regime of cytoskeletal networks

We now review theories of the sparsely crosslinked regime of cytoskeletal
networks. We consider a cytoskeletal material to be sparsely crosslinked if
$\ell^{s}\gg\ell^{g}$, so that the hydrodynamic (long-wave) limit is dominated
by fluid mediated interactions and the viscous transport of momentum though
the gel is negligible; see Fig. 3. Thus, these approaches proceed from Eq.
(4). Given a microscopic model of how the gel deforms in the fluid, these
models proceed to calculate the active stresses $\mathbf{\Sigma}^{a}$,
typically reflecting force dipoles acting on the solvent. Thus,
$\Gamma\rho(\mathbf{v}-\mathbf{v}^{s})\simeq\nabla\cdot\mathbf{\Sigma}^{a}$.
This implies
$\mathbf{\Sigma}^{\mathrm{tot}}=\mathbf{\Sigma}^{\mathrm{s}}+\mathbf{\Sigma}^{\mathrm{a}}$.
The motion of the gel generally takes the form
$\mathbf{v}^{s}-\mathbf{v}\propto\mathbf{P}$, which allows polar pieces of gel
to ’swim’ through the solvent.

One such line of models devolves from the mechanics of swimming suspensions.
To study the dynamics of suspensions of hydrodynamically interacting swimming
microorganisms, Saintillan & Shelley saintillan2008instabilitiesPRL ;
saintillan2008instabilitiesrPHYSFLUID  developed a kinetic theory coupling a
Smoluchowski equation for particle position and orientation to a forced Stokes
equation; see also SK2009 ; BM2009 ; saintillan2013active . At the particle
level, free swimmers (which are force dipoles in the solvent) are treated as
rod-like particles swimming in a locally linear flow, propelled by a
prescribed active surface stress. The swimmer velocity, rotation rate, and
entire surface stress, can be calculated in terms of the background flow, with
the background flow itself determined through solution of a forced Stokes
equation. The forcing is through an “extra stress” found using Batchelor’s
adaptation Batchelor1970 , for dilute suspensions, of Kirkwood theory. In its
simplest formulation, Saintillan & Shelley find the coupled Smoluchowski
equation for the particle distribution function
$\psi(\mathbf{x},\mathbf{p},t)$

$\partial_{t}\psi+\nabla_{x}\cdot(\dot{\mathbf{x}}\psi)+\nabla_{p}\cdot(\dot{\mathbf{p}}\psi)=0,\textrm{
where }\dot{\mathbf{x}}=\mathbf{v}^{s}(\mathbf{x})+U_{0}\mathbf{p}\textrm{ and
}\dot{\mathbf{p}}=(\mathbf{I}-\mathbf{p}\mathbf{p})\nabla\mathbf{v}^{s}(\mathbf{x})\mathbf{p},$
(36)

where $\mathbf{p}$ is swimmer orientation and $U_{0}$ is its undisturbed
speed, and the last expression is Jeffrey’s equation (cf. Eq. (31,35)). The
solvent velocity $\mathbf{v}^{s}$ is found through solution of a Stokes
equation driven by an active dipolar stress $\mathbf{\Sigma}^{a}$:

$-\nabla_{x}\Pi^{s}+\mu\Delta\mathbf{v}^{s}=-\nabla_{x}\cdot\mathbf{\Sigma}^{a}\textrm{
and }\nabla_{x}\cdot\mathbf{v}^{s}=0,\textrm{ with
}\mathbf{\Sigma}^{a}=\alpha_{0}\rho\mathcal{Q}.$ (37)

Here, $\alpha_{0}$ is the so-called dipole strength whose sign reflects
whether the swimmer force dipole is extensile ($\alpha_{0}<0$) or contractile
($\alpha_{0}>0$). This categorization depends upon the detailed placement of
thrust and no-slip regions on the effective swimmer body. This theory predicts
both the instability of aligned suspensions SR2002 ; ESS2013 , and the
instability of isotropic extensile suspensions when swimmers exceed a critical
concentration saintillan2008instabilitiesrPHYSFLUID ; HS2010 ; ESS2013 . In
later work ESS2013 , inter-particle aligning torques and consequent stresses,
based on Maier-Saupe theory doi1988theory , were introduced to capture
sterically-induced concentration effects; see SS2013  for a review.

Gao et al. gao2015multiscalepolar ; GBGBS2015b  adapted this approach to
treating immersed assemblies of microtubules undergoing polarity sorting by
multimeric motor complexes such as the engineered kinesin-1 complexes used in
sanchez2012spontaneous . To do this, as the basic unit they considered
nematically ordered local clusters of microtubules, say with $n$ of them
pointing rightwards and $m$ pointing leftwards (hence,
$\mathbf{P}=(n-m)/(n+m)\hat{\mathbf{x}}$); see Figure 1 of
gao2015multiscalepolar . It is assumed that every microtubule pair in the
cluster is connected by active, plus-end directed cross-linkers moving at
speed $v_{w}$ on each microtubule. The coupling between the anti-aligned
populations induces a minus-end directed relative sliding for each. By using
Stokesian slender-body theory KR1976  and assuming that the solvent drag thus
calculated is the only force resisting microtubule motion they calculated the
velocities of the left- and rightward pointing MTs to be $v_{L}=(2n/n+m)v_{w}$
and $v_{R}=-(2m/n+m)v_{w}$. This yields $v_{R}-v_{L}=-2v_{w}$. Slender-body
theory again yields the forces each rod exerts upon the fluid, and hence the
induced “extra stress” tensor arising from polarity sorting within the bundle
can be calculated and is found to be proportional to
$v_{w}mn/(m+n)\hat{\mathbf{x}}\hat{\mathbf{x}}$.

Thus, in a cluster with $m=n$, the two populations are pulled past each other
with equal speed $v_{w}$, while producing a maximal polarity-sorting stress.
If most microtubules point rightwards, so that $m\approx 0$, then
$v_{L}\approx 2v_{w}$ and $v_{r}\approx 0$, and the polarity sorting extra
stress is small (with a like statement if $n\approx 0$). That the microtubule
speeds depends upon the polarity is a consequence of the micro-mechanical
model wherein only solvent drag resists microtubule translocation.

Adapting this cluster picture to a more general setting where the cluster also
moves with the background fluid velocity, Gao et al. give the analogous result
to Eqs. (36,37) where the microtubule flux and active stress tensor, induced
by polarity sorting of anti-aligned microtubules ($aa$), are given by

$\dot{\mathbf{x}}=\mathbf{v}^{s}(\mathbf{x})+v_{w}(\mathbf{P}-\mathbf{p}),\textrm{
and }\mathbf{\Sigma}^{aa}=\alpha_{aa}\rho(\mathcal{Q}-\mathbf{P}\mathbf{P}).$
(38)

These forms for $\dot{\mathbf{x}}$ and $\mathbf{\Sigma}^{aa}$ reproduce
exactly the nematic cluster results above, and show the dependencepolarity
sorting of the microtubule velocity upon the local polarity
$\mathbf{P}(\mathbf{x})$.

This theory was further informed by detailed Brownian/Monte-Carlo simulations
of nematic assemblies of rigid filaments interacting through multimeric
kinesin-1 motors, thermal noise, and steric interactions. From these
simulations were gleaned estimates of the activity coefficient $\alpha_{aa}$
giving that $\alpha_{aa}<0$, i.e. that polarity sorting dipolar stresses were
extensile gao2015multiscalepolar ; GBGBS2015b . The stochastic simulations
also showed the presence of an additional subdominant, active and extensile
stress, having the form $\alpha_{pa}\rho(\mathcal{Q}+\mathbf{P}\mathbf{P})$,
arising from the relaxation of cross-link tethers of multimeric motors
connecting polar-aligned ($pa$) microtubules blackwell2016microscopic . With
this continuum model in hand, they analyzed and simulated the dynamics of a
thin layer of active material at the interface between two fluids. Their
analysis demonstrated the existence of a characteristic finite length-scale of
instability due to the external drag of the outer fluids, and of orientational
instabilities for aligned states. Their simulations of the full kinetic theory
showed a nonlinear dynamics of fluid and material flows, and of
nucleating/annihilating disclination defect pairs, with a structure
qualitatively similar to that observed experimentally in
sanchez2012spontaneous .

To make comparisons to the highly cross-linked regime, the total stress in the
Gao et al. model has the form

$\mathbf{\Sigma}^{tot}=\mathbf{\Sigma}^{s}+\mathbf{\Sigma}^{a}=-\Pi^{s}\mathcal{I}+\mu\left(\nabla\mathbf{v}^{s}+\nabla\mathbf{v}^{s~{}T}\right)+(\alpha_{pa}+\alpha_{aa})\rho\mathcal{Q}+(\alpha_{pa}-\alpha_{aa})\rho\mathbf{P}\mathbf{P}+\tilde{\mathbf{\Sigma}}^{s}.$
(39)

where $\tilde{\mathbf{\Sigma}}^{s}$ contains stresses arising from steric
interactions (from Maier-Saupe theory) and particle rigidity; compare to Eq.
(11). Here, the microtubule flux in Eq. (38) is of the same form as in the
highly cross-linked case but with the solvent velocity rather than the
material velocity $\mathbf{v}$ furthauer2019self ; compare Eq. (31). Unlike
the stress in the highly crosslinked regime, the expression in Eq. (39) does
not depend upon $\nabla\mathbf{P}$. It is the relation between
$\nabla\mathbf{P}$ and $\nabla\mathbf{v}$ revealed by Eq. (26) that underlays
the self-straining state of the highly cross-linked state; see Fig. 5.

The cluster picture in gao2015multiscalepolar  can be generalized by allowing
’cluster-cluster’ interactions which can lead to additional stresses
proportional to gradient in the local polarity. Efforts along these lines
where first made in LM2006 ; liverpool2008hydrodynamics . There the key
assumption is that pairs of filaments interact via sparse coupling of constant
velocity motors. This theory also provided an initial understanding of the
origin of contractile behaviors found in many early experiments of
filament/motor mixtures.

##  V Synthesis and Open Challenges

Cytoskeletal networks are the drivers of basic biological functions like cell
division and cell motility. The constituents of cytoskeletal networks also
provide foundational examples in the field of active matter. In this review we
highlight the differences between the physics of highly crosslinked and
sparsely crosslinked cytoskeletal networks. In highly crosslinked systems the
long-ranged transport of momentum is mediated through the crosslinks
themselves ($\ell^{g}\gg\ell^{s}$), whereas in sparsely crosslinked systems it
is the solvent that plays that role ($\ell^{s}\gg\ell^{g}$); see Fig. 3

This has important implications for the resulting material properties. In
highly crosslinked materials, the same crosslinkers which generate the driving
forces between filaments also generate the frictional coupling that keeps the
networks coherent. Consequently, active stresses and the viscosity of the
material are intimately linked. In particular, the polar active stress
proportional to polarity gradients is linked to the viscosity tensor by Eq.
(LABEL:eq:xi_related). For these materials, the viscous and polar stresses
exactly balance when all filaments in the material move at the preferred motor
velocity $\sigma_{0}/\gamma_{0}$ in the direction to which they point
furthauer2019self . Since $\chi^{v}$ and $\chi^{a}$ vary together the
properties of this state are not sensitive to protein concentrations. This
self-straining state is not just a theoretical curiosity; it has been observed
in vitro furthauer2019self  and in spindles yang08 ; dalton2021gelation .

In contrast, in the sparsely crosslinked regime the viscosity tensor
$\chi^{v}$ of the material is largely set by the viscosity of the solvent in
which filaments are suspended and is thus independent of concentrations and
properties of the crosslinking molecules, while the values of active stresses
depend on protein concentrations. This provides a possible diagnostic for
differentiating highly crosslinked and sparsely crosslinked active materials
by modulating crosslink concentrations. Moreover, in the self-straining state
of active materials filaments can move through the material at speeds that are
independent of the local polar and nematic order furthauer2019self , whereas
in more sparsely crosslinked systems gao2015multiscalepolar  the motion of
filaments throughout the system depends on the local polarity.

A second striking difference between the highly crosslinked and sparsely
crosslinked materials is the form of the other active stresses
$\mathcal{A}^{(\mathcal{Q})},\mathcal{A}^{(\mathbf{P})}$. It has long been
known that contractile (or extensile) stresses can be generated only if motors
act in a way that breaks the symmetry between extending and contracting single
filament pairs. This could for instance be achieved by end-clustering or end-
binding affinities of motor proteins kruse2000actively ;
liverpool2008hydrodynamics . On top of that, the mathematics of the highly
crosslinked regime requires that
$\sigma_{1}\neq\gamma_{1}/\gamma_{0}\sigma_{0}$; see Eqns. (28,29). This means
that the anisotropy of friction ($\gamma_{1}/\gamma_{0}$) along filaments
needs to be different from the anisotropy of motor drive
($\sigma_{1}/\sigma_{0}$) to generate active contractions. In practice this
can be achieved by either mixing several types of crosslinks or by creating a
crosslink with two different motor heads furthauer2021design . This prediction
might shed insight into the well known - but so far poorly understood -
observation that in many actomyosin systems contractions only occur if a small
amount of passive crosslinker is added to the system ennomani2016architecture
.

A third important difference between the highly and sparsely crosslinked
regimes comes from the fact that the solvent is incompressible, while the
network itself can be compacted by active processes. As a consequence sparsely
crosslinked theories predict incompressible material flow fields, while highly
crosslinked theories can predict active bulk contraction. Thus it is tempting
to speculate that contractile systems like foster2015active ;
foster2017connecting  are highly crosslinked. In contrast many classic active
nematic experiments show no clear signs of compaction sanchez2012spontaneous ;
wu2017transition ; chandrakar2020confinement .

In section III we sought to classify biological and experimental active matter
systems as being either highly or sparsely crosslinked. We believe that many
important systems, such as the cell cortex and many spindles, are highly
crosslinked. For most systems however, this assertion remains an educated
guess rather than an experimental certainty. Given the increasing awareness
that the physics of active cytoskeletal networks can be drastically different
for different numbers of crosslinks, we hope that this review will serve to
highlight the need for experiments that will answer this question more
definitively.

Finally, we want to conclude by remarking that many active systems are likely
to live in the intermediate regime between highly and sparsely crosslinked.
One example are the microtubule asters studied in extract pelletier2020co ,
which are probably highly crosslinked in their bulk, but more sparsely
crosslinked near their boundaries. These structures are biologically important
and revelatory of new physics in an intermediate crosslinking regime that is
so far barely explored.

In summary, we firmly believe that the science of biologically active
materials will benefit greatly by simply enumerating the number of crosslinks
active in the materials of interest. This enumeration will help identify
important and distinct physical regimes, and help make theories quantitative
and living systems engineerable.

Acknowledgements: We thank James Pelletier, Steffanie Redemann, and Kun-Ta Wu
for providing us with experimental images. We thank Lucy Reading-Ikkanda for
preparation of figures. MJS acknowledges support from NSF grants DMR-2004469
and DMR-1420073 (NYU-MRSEC). We thank our many collaborators who worked with
us in this area.

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