Waleed Mirza defended his PhD thesis A theoretical and computational study of the active self-organization of nematic patterns in thin cytoskeletal layers and their effect on curvature, supervised by Marino Arroyo on 16 February 2023 within the UPC doctoral program in Applied Mathematics. Currently he is in the group of Alejandro Torres-Sánchez at the European Molecular Biology Laboratory in Barcelona, Spain.
Thesis summary
D’Arcy Thompson’s renowned book «On Growth and Form» (1917) [1] dismisses vitalism, arguing that the growth and forms of living organisms are governed by physical laws and can be mathematically analyzed. Thompson uses analogies between living and non-living matter, like the similarities in the shapes of soap bubbles, cells, and corals, to argue that physical laws dictate pattern formation without pre-planned design Fig. 2[(a)Top]. This perspective refutes teleology, focusing instead on causality. Over a century later, Thompson’s ideas continue to impact the discipline of developmental biology, highlighting the crucial interplay between physics and biology in the self-organizing processes of morphogenesis.
Self-organization refers to the spontaneous creation of structured systems through local interactions, characterizing systems far from thermodynamic equilibrium. In contrast to passive systems that increase entropy until equilibrium, driven systems maintain high entropy and energy fluxes by exchanging matter or energy with their environment. This energy input facilitates spatio-temporal organization in various scales, evident in phenomena like ocean circulation, coastline formation [2], to skin patterns on animals [3] Fig. 2[(a)Bottom], and cellular structures.
In the first half of the thesis, we aim to investigate the mechanisms behind the self-organization of active nematic bundles in the actin cytoskeleton [4] Fig. 2[(b)Top]. To this end, we have developed a comprehensive suite of continuum theoretical and computational models tailored to these systems. The initial phase of our thesis focuses on studying these gels on two-dimensional, flat surfaces. Central to our methodology is the development of a systematic modeling approach designed to accurately simulate the dynamics of the actin cytoskeleton. This approach is grounded in Onsager’s variational formalism [5], which postulates that the behavior of these systems arises from the interplay of energy release, energy dissipation, and active motion. To enhance our research further, we have introduced an innovative numerical method employing finite element analysis. This method builds upon Onsager’s formalism by adding a temporal dimension, ensuring our simulations are not only stable but also in line with the fundamental principles of thermodynamics. Employing this combination of theoretical and numerical frameworks, our study investigates the active self-assembly of nematic patterns, starting from a state where the gel is uniform and quiescent. We use both linearized theory and advanced nonlinear simulations to identify specific conditions required for the formation of nematic bundles from these initial states. Additionally, we examine how various parameters in our theoretical models affect the structure and dynamics of the resulting self-organized nematic patterns Fig. 2[(b) Middle and top]. To reinforce the credibility of our theoretical discoveries, we employ discrete network simulations. This method enables us to corroborate the essential conditions for the self-organization of active nematic patterns, which aligns with the outcomes we previously identified through continuum simulations.
We then explore the functional significance of these self-organized nematic patterns, particularly in cellular processes like wound healing. Our simulations reveal that following a cellular injury, nematic patterns spontaneously align around the wound area. This alignment interacts with the curvature of the wound, playing a crucial role in facilitating the healing process. We validate these findings by comparing our simulation results with in-vitro experiments, finding substantial agreement [6]. Although our primary focus is on sub-cellular patterns within the cytoskeleton, we also venture into the application of our model at a supracellular level. An example of this is the examination of a confluent monolayer of MDCK cells, where similar nematic patterns are observed [7]. Our simulations in this context demonstrate that these self-organized nematic patterns can induce stresses within cell monolayers, leading to the generation of spontaneous hydrodynamic flows [8]. This extension of our model to different scales not only demonstrates its versatility but also provides a deeper understanding of the role of nematic patterns in various biological contexts, from individual cells to complex cellular assemblies.
In the second half of this thesis, our exploration shifts to developing a comprehensive model for self-organized nematic patterns on curved deformable surfaces. Here, we revisit Onsager’s nonlinear formalism, a method that balances robustness with simplicity, enabling us to create complex models that are both clear and precise. Our novel model integrates the dynamics of shape changes on curved surfaces, nematic ordering, density variations, and fluid dynamics at the surface level, forming a comprehensive system to investigate their interplay. One of the critical applications of this model is to investigate the self-organization processes of the cytokinetic ring in the actin cytoskeleton, a vital structure in cellular division and migration. The numerical solution of our theoretical model recapitulates the formation of nematic contractile rings, mirroring their significant roles in both cell division Fig.2(c) [9] and migration [10]. Through these simulations, we gain valuable insights into the functionality of these structures, particularly their contribution to the efficiency of morphogenetic processes. This part of our research not only enhances our understanding of nematic structures in the actin cytoskeleton but also underscores the broader significance of these patterns in complex biological systems.
Mirza_Fig1Figure 2: (a) Examples of Self-Organization in Nature (b) Nematic Bundles in Actin Cytoskeleton. Top: Nematic bundles observed in the actin cytoskeleton. Middle: Nematic bundles generated through the numerical solution of the proposed model in a bi-periodic domain. Bottom: Nematic bundles in a circular domain of the actin cytoskeleton. (c) Self-Organized Nematic Bundles Facilitating Cell Division.
Building on our research on nematic models in the actin cytoskeleton, we investigate the application of our model to different surfaces, particularly focusing on vesicles. Vesicles are small, spherical compartments, enclosed by a lipid bilayer, found within cells [11]. They play crucial roles in various cellular processes, including transport, metabolism, and communication between different parts of a cell and with its environment. In this expansion of our study, we apply our nematic model to vesicles, treating them as active, deformable liquid crystalline surfaces. Such surfaces present a unique environment where the irregularities in nematic architecture create out-of-plane deformation. Our theoretical and computational models allow us to meticulously observe and analyze how nematic irregularities in the structured order move and interact with the dynamic shapes of these vesicular surfaces [11].
In this thesis, we achieve a significant advancement in comprehending morphogenesis within biological systems, which echoes and reinforces D’Arcy Thompson’s early 20th-century concepts regarding the intricate interplay of biological processes and physical laws in shaping morphogenesis. By developing a novel theoretical and computational framework based on Onsager’s variational formalism, we successfully model the interplay of biology and laws of physics. This approach explains the self-organization of prevalent active nematic patterns and their role in morphogenetic events such as cell wound healing, cell division, and cell migration, as well as in other supracellular systems such as patterns in colonies of cells or deformations in vesicles. In summary, these achievements offer profound insights into the fundamental processes of cellular morphology and the universal principles guiding life’s development and adaptation.
References
[1] D. W. Thompson, On Growth and Form, Cambridge University Press, 1917.
[2] G. Coco and A. B. Murray, «Patterns in the sand: From forcing templates to self-organization,» Geomorphology, 91, 271-278, 2007.
[3] A. M. Turing, «The chemical basis of morphogenesis,» Bulletin of Mathematical Biology, 52, 153-197, 1952.
[4] G. Salbreux, G. Charras, and E. Paluch, «Actin cortex mechanics and cellular morphogenesis,» Trends in Cell Biology, 22, 536-545, 2012.
[5] M. Arroyo et al., «Onsager’s Variational Principle in Soft Matter,» in The Role of Mechanics in the Study of Lipid Bilayers, D. J. Steigmann (Ed.), Springer, 2018, pp. 287-332.
[6] C. A. Mandato and W. M. Bement, «Contraction and polymerization cooperate to assemble actomyosin rings,» J. Cell Biology, 154, 785-795, 2001.
[7] G. Duclos et al., «Perfect nematic order in confined monolayers of spindle-shaped cells,» Soft Matter, 10, 2346-2353, 2014.
[8] M. M. Norton et al., «Insensitivity of active nematic liquid crystal dynamics to topological constraints,» Phys. Rev. E, 97, 012702, 2018.
[9] A.-C. Reymann et al., «Cortical flow aligns actin filaments to form a furrow,» eLife, 5, e17807, 2016.
[10] V. Ruprecht et al., «Cortical Contractility Triggers a Stochastic Switch to Fast Amoeboid Cell Motility,» Cell, 160, 673-685, 2015.
[11] F. C. Keber et al., «Topology and dynamics of active nematic vesicles,» Science, 345, 1135-1139, 2014.