of 35
Suresh, Galstyan
et al
. eLife 2022;11:e79558. DOI: https://doi.org/10.7554/eLife.79558
1 of 35
Modeling and mechanical perturbations
reveal how spatially regulated anchorage
gives rise to spatially distinct mechanics
across the mammalian spindle
Pooja Suresh
1,2†
, Vahe Galstyan
3,4†
, Rob Phillips
5,6,7
*, Sophie Dumont
1,2,7,8
*
1
Biophysics Graduate Program, University of California, San Francisco, San Francisco,
United States;
2
Department of Bioengineering and Therapeutic Sciences, University
of California, San Francisco, San Francisco, United States;
3
Biochemistry and
Molecular Biophysics Option, California Institute of Technology, Pasadena, United
States;
4
A. Alikhanyan National Laboratory (Yerevan Physics Institute), Yerevan,
Armenia;
5
Division of Biology and Biological Engineering, California Institute of
Technology, Pasadena, United States;
6
Department of Physics, California Institute of
Technology, Pasadena, United States;
7
Chan Zuckerberg Biohub, San Francisco, San
Francisco, United States;
8
Department of Biochemistry and Biophysics, University of
California, San Francisco, San Francisco, United States
Abstract
During cell division, the spindle generates force to move chromosomes. In mammals,
microtubule bundles called kinetochore-
fibers (k-
fibers) attach to and segregate chromosomes.
To do so, k-
fibers must be robustly anchored to the dynamic spindle. We previously developed
microneedle manipulation to mechanically challenge k-
fiber anchorage, and observed spatially
distinct response features revealing the presence of heterogeneous anchorage (Suresh et al.,
2020). How anchorage is precisely spatially regulated, and what forces are necessary and sufficient
to recapitulate the k-
fiber’s response to force remain unclear. Here, we develop a coarse-
grained
k- fiber model and combine with manipulation experiments to infer underlying anchorage using
shape analysis. By systematically testing different anchorage schemes, we find that forces solely at
k- fiber ends are sufficient to recapitulate unmanipulated k-
fiber shapes, but not manipulated ones
for which lateral anchorage over a 3 μm length scale near chromosomes is also essential. Such
anchorage robustly preserves k-
fiber orientation near chromosomes while allowing pivoting around
poles. Anchorage over a shorter length scale cannot robustly restrict pivoting near chromosomes,
while anchorage throughout the spindle obstructs pivoting at poles. Together, this work reveals how
spatially regulated anchorage gives rise to spatially distinct mechanics in the mammalian spindle,
which we propose are key for function.
Editor's evaluation
In this elegant and technically sophisticated study, the authors study the mechanical proper
-
ties of the mitotic spindle by combining various experimental biophysical approaches, including
microneedle manipulation and quantitative imaging, with theoretical modeling. By systematically
exploring shapes of unmanipulated and manipulated kinetochore fibers, they provide compelling
evidence for a lateral anchor near the chromosomes. These important findings further our under
-
standing of the balance of forces in the entire mitotic spindle. The work should appeal broadly to
cell biologists and biophysicists who are interested in the cytoskeleton and cell division.
RESEARCh ADv
ANCE
*For correspondence:
phillips@pboc.caltech.edu (RP);
sophie.dumont@ucsf.edu (SD)
These authors contributed
equally to this work
Competing interest:
The authors
declare that no competing
interests exist.
Funding:
See page 20
Preprinted:
08 April 2022
Received:
26 April 2022
Accepted:
12 October 2022
Published:
08 November 2022
Reviewing Editor:
Thomas
Surrey, Centre for Genomic
Regulation (CRG), Spain
Copyright Suresh, Galstyan
et al
. This article is distributed
under the terms of the Creative
Commons Attribution License,
which permits unrestricted use
and redistribution provided that
the original author and source
are credited.
Research advance
Cell Biology | Physics of Living Systems
Suresh, Galstyan
et al
. eLife 2022;11:e79558. DOI: https://doi.org/10.7554/eLife.79558
2 of 35
Introduction
Cell division is essential to all life. The accurate segregation of chromosomes during cell division is
achieved by the spindle, a macromolecular machine that distributes chromosomes equally to each
new daughter cell. To perform this mechanical task, the spindle must be dynamic yet structurally
robust: it must remodel itself and be flexible, yet robustly generate and respond to force to move
chromosomes and maintain its mechanical integrity. How this is achieved remains an open question.
Indeed, while much is known about the architecture (
McDonald et al., 1992
;
Mastronarde et al.,
1993
) and dynamics (
Mitchison, 1989
) of the mammalian spindle, and the molecules essential to its
function (
Hutchins et al., 2010
;
Neumann et al., 2010
), our understanding of how they collectively
give rise to robust mechanics and function lags behind.
In the mammalian spindle, kinetochore-
fibers (k-
fibers) are bundles of microtubules (
McDonald
et al., 1992
;
O’Toole et al., 2020
,
Kiewisz et al., 2021
) that connect chromosomes to spindles
poles, ultimately moving chromosomes to poles and future daughter cells. To do so, k-
fibers must
maintain their connection to the dynamic spindle. The k-
fiber’s connection (anchorage) to the spindle
is mediated by a dense mesh-
like network of non-
kinetochore microtubules (non-
kMTs) which connect
to k-
fibers along their length (
Mastronarde et al., 1993
;
O’Toole et al., 2020
) via both motor and
non-
motor proteins. Although this network cannot be easily visualized with light microscopy, phys-
ical perturbations such as laser ablation (
Kajtez et al., 2016
;
Milas and Toli
ć
, 2016
;
Elting et al.,
2017
) and cell compression (
Trupini
ć
et al., 2022
;
Neahring et al., 2021
) have been instrumental in
uncovering how this network anchors k-
fibers. The non-
kMT network bears mechanical load locally
(
Milas and Toli
ć
, 2016
;
Elting et al., 2017
), links sister k-
fibers together (
Kajtez et al., 2016
), and
contributes to k-
fiber and spindle chirality (
Trupini
ć
et al., 2022
;
Neahring et al., 2021
). Recent
advances in microneedle manipulation enabled us to mechanically challenge k-
fiber anchorage with
unprecedented spatiotemporal control (
Long et al., 2020
;
Suresh et al., 2020
). Exerting forces at
different locations along the k-
fiber’s length revealed that anchorage is heterogeneous along the
k- fiber: k-
fibers were restricted from pivoting near kinetochores, mediated by the microtubule cross-
linker PRC1, but not near poles (
Suresh et al., 2020
). Such reinforcement helps robustly preserve the
k- fiber’s orientation in the spindle center, which we speculate forces sister k-
fibers to be parallel and
promotes correct attachment. However, how this reinforcement is enacted over space, namely how
local or global it is, remains unclear. Furthermore, we do not yet understand which connections along
the k-
fiber’s length are necessary and sufficient to give rise to such spatially distinct mechanics.
The precise spatiotemporal control achieved by microneedle manipulation offers rich quantitative
information on the k-
fiber’s anchorage in the spindle (
Suresh et al., 2020
) and demands a quantita-
tive model-
building approach for its full interpretation. Knowledge of the spindle connections from
electron microscopy (
McDonald et al., 1992
;
Mastronarde et al., 1993
;
O’Toole et al., 2020
) is not
sufficient to understand how they collectively reinforce the k-
fiber, and perturbing different regions
of the network to experimentally test their contribution is challenging. Furthermore, while we can
deplete spindle crosslinkers, quantitatively controlling their combined mechanical function over space
is not currently within reach. In turn, a coarse-
grained modeling approach (accounting for the effective
influence of collective molecular actions) can allow us to systematically dissect the spatial regulation
of k-
fiber anchorage in the spindle. Since the bending mechanics of microtubules is well characterized
(
Gittes et al., 1993
), many modeling studies have used shape to infer forces exerted on microtubules.
This approach has been applied to single microtubules (
Gittes et al., 1996
;
Brangwynne et al.,
2006
), microtubule bundles (
Gadêlha et al., 2013
;
Portran et al., 2013
), as well as k-
fibers in the
spindle (
Rubinstein et al., 2009
;
Kajtez et al., 2016
). To date, k-
fiber models used native shapes (in
unperturbed spindles) to infer underlying spindle forces, without focusing on k-
fiber anchorage. This
is mainly because the presence of anchorage is not easily revealed in unperturbed spindles. Using
k- fiber manipulation in mammalian spindles, we are uniquely positioned to probe k-
fiber anchorage
forces previously hard to detect, and to test models for their underlying basis.
Here, we use coarse-
grained modeling and microneedle manipulation experiments to define the
spindle anchorage forces necessary and sufficient for the k-
fiber to robustly restrict pivoting near
kinetochores while allowing pivoting at poles (
Figure 1
, top). We model the k-
fiber using Euler-
Bernoulli beam theory. We systematically increase model complexity and use shape analysis to infer
the minimal set of forces needed to recapitulate experimental k-
fiber shapes. We find that while forces
and moments at k-
fiber ends (end-
point anchorage) alone are sufficient to recapitulate unmanipulated
Research advance
Cell Biology | Physics of Living Systems
Suresh, Galstyan
et al
. eLife 2022;11:e79558. DOI: https://doi.org/10.7554/eLife.79558
3 of 35
shapes, lateral anchorage is needed to preserve k-
fiber orientation in the spindle center in manip-
ulated spindles. We then systematically test different length scales of lateral anchorage. Global
anchorage leads to a loss of mechanical distinction in the pole and kinetochore regions – a predic-
tion confirmed by manipulating spindles with globally increased anchorage. In turn, local anchorage
preserves the mechanical distinction observed in control manipulations, and a length scale of 3 μm
near kinetochores is necessary and sufficient to recapitulate observed manipulated shapes. This
localized anchorage can preserve k-
fiber orientation near kinetochores without significant k-
fiber-
to-
network detachment for a broad range of microneedle forces. Thus, strong local anchorage enacted
locally within 3 μm of kinetochores can ensure that sister k-
fibers remain aligned and bioriented in the
spindle center robustly, while allowing their pivoting and clustering into poles. Together, we demon-
strate how spatially regulated anchorage gives rise to spatially distinct mechanics, which we propose
support different functions across the spindle.
microneedle
THEORETICAL MODELING
EXPERIMENTAL PERTURBATION
pole
model-generated
shape profile
kinetochores
k-fibers
60
s
Euler-Bernoulli formalism
testing models of increasing complexity
native/unmanipulated
Q:
What forces are necessary and sufficient to preserve k-fiber orientation
in the spindle center but allow pivoting at the pole?
manipulated
bending
moment
deformed
undeformed
-
EI
curvature
flexural rigidity
(E: elastic modulus
I: areal moment of inertia)
radius of
curvature
R(x)
R(x)
1
=
native
manipulated
pole
kinetochor
e
outer k-fiber deformed by
≈2.5 μm over 60 seconds
microneedle manipulation
pivoting
restricted
pivoting
allowed
?
M(x)
spatially distinct mechanical responses
between pole and kinetochore regions
Figure 1.
Overview of the experiment-
theory interplay used for studying the mechanics of k-
fiber anchorage in the mammalian spindle. Top: Schematic
of the experimental perturbation performed in
Suresh et al., 2020
. Microneedle (blue circle) manipulation of outer k-
fibers revealed that k-
fibers do
not freely pivot near kinetochores, ensuring the maintenance of k-
fiber orientation in the spindle center, and pivot more freely around poles. Bottom:
Coarse-
grained modeling approach of the k-
fiber in the spindle context based on Euler-
Bernoulli beam theory. Model complexity is progressively
increased to identify the minimal set of forces necessary and sufficient to recapitulate (dashed blue lines) k-
fiber shapes in the data. From left to
right: we test models with different forces and moments at k-
fiber ends (pole and kinetochore) to recapitulate native k-
fibers, and, then test models
of increasing complexity (first with x- and y- force components and just a moment at the pole, then a moment at the kinetochore and finally lateral
anchorage over different length scales along the k-
fiber (purple arrows)) to recapitulate manipulated k-
fibers. Here, forces (represented as straight
arrows) and moments (represented as curved arrows) together define the bending moment M(x) along the k-
fiber, while k-
fiber shape is determined via
curvature
κ
(x).
Research advance
Cell Biology | Physics of Living Systems
Suresh, Galstyan
et al
. eLife 2022;11:e79558. DOI: https://doi.org/10.7554/eLife.79558
4 of 35
Results
Forces and moments acting on k-fiber ends alone can capture native
mammalian k-fiber shapes
To determine the spindle forces necessary and sufficient to recapitulate k-
fiber shapes, we use the
Euler-
Bernoulli formalism of beam deformation (
Landau and Lifshitz, 1984
). Through the equation
κ
(x)=M(x)/EI (
Figure 1
, bottom), this formalism relates the curvature
κ
(x) of the beam at a given posi-
tion (x) to the local bending moment M(x) (the moment of internal stresses that arise from forces
exerted) and the flexural rigidity EI (a measure of resistance to bending that depends on the elastic
modulus E and the areal moment of inertia I of the beam). We treat the k-
fiber as a single homoge-
neous beam (
Rubinstein et al., 2009
;
Kajtez et al., 2016
) that bends elastically in response to force
(see Materials and methods).
In the mammalian metaphase spindle, native k-
fibers appear in a variety of curved shapes which
arise from the molecular force generators that maintain the spindle (
Elting et al., 2018
;
Nazockdast
and Redemann, 2020
;
Toli
ć
and Pavin, 2021
). To obtain a minimal description of native k-
fiber shape
generation, we considered point forces and moments acting only on the pole and kinetochore ends
of the k-
fiber (
Rubinstein et al., 2009
). These could arise from motor and non-
motor microtubule-
associated proteins that exert force on and anchor k-
fiber ends, for example from dynein and NuMA
at poles (
Heald et al., 1996
;
Merdes et al., 1996
), and NDC80 at the kinetochore (
DeLuca et al.,
2006
). In our minimal description, we coarse-
grained the kinetochore-
proximal forces (a tensile force
at the kinetochore (
McNeill and Berns, 1981
;
Waters et al., 1996
) and a compressive force near the
kinetochore (
Rubinstein et al., 2009
;
Kajtez et al., 2016
) to an effective point force (see Materials
and methods)). Using a fine-
grained junction model with explicit tensile and compressive forces did
not significantly change the model outcomes (
Figure 2—figure supplement 1
), thereby justifying
our coarse-
grained approach. We considered a coordinate system where the pole is at the origin (x=y
= 0) and the kinetochore lies along the x-
axis at position x=L (
Figure 2a
). In this system, a force at the
pole (
F
with components F
x
and F
y
), an equal and opposite force at the kinetochore (at equilibrium),
and a moment at the pole (M
p
) and at the kinetochore (M
k
) together define the shape of the k-
fiber
at every position
r
(x) via the moment balance condition (
M
(x)=
M
p
+
r
(x)×
F
, with M
k
=M(x=L)). The
relatively small deflection of native k-
fibers allowed us to solve for their shape profiles analytically and
gain insight into how these forces and moments uniquely contribute to shape (see Appendix 1). We
found that a purely axial force F
x
generates a symmetric shape profile with the peak (position where
the deflection y(x) is the largest) located in the middle of the pole-
kinetochore axis (
Figure 2b
, top).
In the absence of an axial force F
x
, the moment at the pole M
p
and corresponding force F
y
generate
an asymmetric shape profile with the peak shifted towards the pole (
Figure 2b
, middle); conversely,
the moment at the kinetochore M
k
and corresponding force F
y
generate an asymmetric shape profile
with the peak shifted towards the kinetochore (
Figure 2b
, bottom). This finding is consistent with
the idea that each force and moment component acting on the ends uniquely contributes to k-
fiber
shape.
To determine which subset of force components (
Figure 2b
) is necessary and sufficient to capture
native k-
fiber shapes, we imaged native k-
fibers in PtK2 GFP-
tubulin cells at metaphase (m=26 cells,
n=83 k-
fibers) and extracted the distribution of peak locations along their length (
Figure 2c
). Most
peaks are located closer to the pole or in the middle of the k-
fiber (
Figure 2d
), suggesting that the
moment M
k
is not essential for their shape generation. We then fit different combinations of force
components in our model to the shape profiles extracted from the data (see Materials and methods).
We evaluated the quality of model fits based on two metrics: fitting error (measured by calculating
the root mean square error,
Figure 2e
) and comparison of peak locations between the model fit and
data shape profiles (
Figure 2f
). The combination of F
x
, F
y
and M
p
together produced the lowest fitting
error (
Figure 2e
), and accurately predicted the peak locations (
Figure 2f
, example fits in
Figure 2g
),
while the other subsets of force components performed significantly worse on both metrics. The inclu-
sion of M
k
along with F
x
, F
y
and M
p
did not significantly improve the quality of fits (
Figure 2—figure
supplement 2
), revealing that M
k
is indeed not necessary to recapitulate native k-
fiber shapes. Taken
together, while native k-
fiber shapes are diverse, the consistent shift in peaks toward the pole reveals
a key mechanical role for the moment at the pole. This indicates that forces at the k-
fiber ends and a
moment at the pole (
F
, M
p
), but not at the kinetochore (M
k
= 0), are alone necessary and sufficient to
recapitulate native k-
fiber shapes.