1. Introduction

Mechanobiology is an area of intense research focus nowadays. In cellular length, it tries to describe how mechanical forces can modulate the structure and function of cells. These forces are transmitted to the nucleus, through direct physical connections to the cytoskeleton via LINC [1], and deform the nucleus, which in turn triggers different biochemical pathways. Therefore, nuclear deformations are critical structural and biophysical parameters in mechanobiology [1].

Recent studies have highlighted the variations of cellular responses with the surrounding stiffness [2]. Inspired by this, we decided to use the finite element method to investigate variations of nuclear deformations with different mechanical environments. For this aim, we modeled the realistic geometry of a single cell in Abaqus^® to measure to deformation applied to the nucleus after the contraction of the acto-myosin units.

2. Materials and Methods

We created a 3-D model of a cell with a volume of 537 nm³ including the membrane, nucleus, nuclear cap, dorsal and ventral f-actins, LINCs, and cytoplasm. We assumed that the dorsal and ventral f-actins are tied to the cell membrane in both extremities. Therefore, fibers contraction would deform the cell membrane and the nucleus as well due to, first, the contact forces between fibers and the nuclear cap, and second, the forces coming from the LINCs. Moreover, to take into account the interaction of the call with its surrounding environment, we attached it to the top surface of a substrate to simulate a traction force microscopy test. As shown in figure 1, the modeled cell has been tied to the substrate in 3 focal adhesion sites [1].

3. Results

We measured the deformation applied to the nucleus for a substrate with 3 different Young’s moduli: 2, 4, and 8 kPa. The average values of the maximum principal strain in the nucleus are 5.3%, 5.6%, and 6.0%, respectively. Figure 2 also shows the distributions of the maximum principal strain for these 3 cases. These results suggest that there is a direct relationship between strain values in the nucleus and the stiffness of the surrounding environment.

4. Discussion and Conclusions

We developed the first finite-element model of a realistic and complex single cell to quantify deformations of the nucleus after the contraction of the acto-myosin units. We showed that finite element modeling can also be regarded as a strong tool to measure the interactions between different cellular components. The simplicity of the proposed model is a promising asset for further quantifications of intracellular mechanics in 2-D and 3-D environments.

5. References

1. Ghosh S et al., Cell reports, 27(5): 1607-1620 (2019).

2. Petit C et al., Biomechanics and Modeling in Mechanobiology, 20(2):717-731 (2021).

The cortex is a thin layer located beneath the membrane of animal cells. It governs important cell mechanical properties, including cell shape. It is composed of a dense network of actin filaments and actin-binding proteins, in particular myosin motors. Myosins generate active stresses in the actin network and make it behave as an active material. The spatiotemporal organization of the cortex is tightly coupled to that of signaling molecules, in particular Rho GTPases. Indeed, the interaction between actin, myosin and Rho GTPases can give rise to self-organized patterns. Thus far, studies have focused on spatially extended patterns, like actin polymerization waves. However, spatially localized patterns are also observed in the cortex, whereby isolated spots enriched in actin, myosin and signaling molecules play a role in crucial processes, including cancer invasion. What is the origin of such spatially localized patterns? We use a simple physical theory to show that they can emerge from the coupling between active cortical mechanics and signaling reactions, through an instability called "slanted snaking". Beyond single cells, our theory could also explain the origin of spatially localized, mechanochemical cues in tissue morphogenesis.

Cytoskeletal networks play a key role in multiple mechanical and dynamical processes in cells. Recently, a continuum theory has been developed [1], allowing for the prediction of the material properties of highly crosslinked cytoskeletal networks from a phenomenological modelling of the microscopic interactions between the cytoskeleton filaments. We extend this theoretical framework to account for external forces, allowing us to explore how the properties of cytoskeletal networks are affected by the presence of various interfaces. This extended theory allows to study the interplay between cytoskeletal networks and the organelles that it interacts with,such as the centrosomes in spindles, vesicles embedded in intracellular actin networks, or even cortex-membrane interactions at the cell periphery.
[1] S. Fürthauer, D. J. Needleman, M. J. Shelley, NJP 23, 013012 (2021).

Eukaryotic cells exhibit a complicated rheology in response to mechanical stimuli, including an elastic response due to the cell cytoskeleton (a network of crosslinked filamentous proteins) and energy dissipation resulting from transport of cytosol through this network as well as transient crosslink dynamics [1]. Existing models of cytoskeletal mechanics fall into two categories: discrete (microscale) models enable inclusion of detailed biophysics but are typically computationally challenging due to the large number of discrete elements and their interactions, while continuum (macroscale) models are easier to solve but the manner in which microscale parameters and processes manifest themselves at the macroscale is usually unclear. Mathematical modelling efforts involving more rational and rigorous mathematical methods (such as discrete-to-continuum upscaling or homogenization) to systematically bridge between these two approaches are still largely missing. Our aim is to develop a multiscale framework providing such a bridge.

We introduce a discrete mathematical model for the mechanical behaviour of the eukaryotic cell cytoskeleton during intracellular transport. The model involves an initially regular array of pre-stretched protein filaments (e.g. actin, vimentin) which exhibit resistance to enthalpic stretching, joined at crosslinks to form a planar network. To mimic inertialess motion of a small object placed in the domain, we impose a quasi-static displacement of a set of crosslinks in the centre of the domain, solve for the remaining nodes through a local force balance, and calculate the net force on the object. Assuming that the inter-crosslink distance is much shorter than the lengthscale of the cell, we rationally upscale the force balance equations using the discrete-to-continuum method based on Taylor expansions to form a continuum system of governing equations, inferring the corresponding macroscopic stress tensor and strain energy.

We solve these discrete and continuum models numerically and infer force-displacement curves, which show good quantitative agreement across the parameter space. The force-displacement curves have weak dependence on the pulling angle (with respect to the initial filament orientation). The net force acting on the object increases with increasing pre-stress and larger objects. Furthermore, we linearize the continuum model to construct analytical approximations for the stress and strain fields in the neighbourhood of the moving object, and explicitly compute the net force required to generate a given deformation as a function of key parameters, such as the object and mesh sizes, the pulling angle and the network pre-stress.

Our mathematical formulation allows us to make explicit predictions of the force-displacement curve in optical-tweezers experiments [1], analytically characterizing the (linearized) rheology of the cytoskeleton. Future work will also incorporate nonlinear effects in polymer elasticity and dynamic aspects of cell rheology (poro-visco-elasticity). We also plan to apply similar ideas to other biopolymer networks, namely collagen fiber networks forming extra-cellular matrices.

Acknowledgements:

This research was funded by EPSRC grant EP/S030875/1. We thank Profs. Ming Guo and
Roger Kamm (MIT) for valuable discussions.

References:

[1] Hu, J. et al. “Size-and speed-dependent mechanical behavior in living mammalian
cytoplasm.” Proceedings of the National Academy of Sciences (2017): 9529-9534.