Balancing reaction-diffusion network for cell polarization pattern with stability and asymmetry

Joint Public Review:

The polarisation phenomenon describes how proteins within a signalling network segregate into different spatial domains. This phenomenon holds fundamental importance in biology, contributing to various cellular processes such as cell migration, cell division, and symmetry breaking in embryonic morphogenesis. In this manuscript, the authors assess the robustness of stable asymmetric patterns using both a previously proposed minimal model of a 2-node network and a more realistic 5-node network based on the C. elegans cell polarisation network, which exhibits anterior-posterior asymmetry. They introduce a computational pipeline for numerically exploring the dynamics of a given reaction-diffusion network and evaluate the stability of a polarisation pattern. Typically, the establishment of polarisation requires the mutual inhibition of two groups of proteins, forming a 2-node antagonistic network. Through a reaction-diffusion formulation, the authors initially demonstrate that the widely-used 2-node antagonistic network for creating polarised patterns fails to maintain the polarised pattern in the face of simple modifications. However, the collapsed polarisation can be restored by combining two or more opposing regulations. The position of the interface can be adjusted with spatially varied kinetic parameters. Furthermore, the authors show that the 5-node network utilised by C. elegans is the most stable for maintaining polarisation against parameter changes, identifying key parameters that impact the position of the interface. While the results offer novel and insightful perspectives on the network's robustness for cell polarisation, the manuscript lacks comprehensive validation against experimental data, justified node-node network interactions, and proper estimation of model parameters (based on quantitative measurements or molecular intensity distributions). These limitations significantly restrict the utility of the model in making meaningful predictions or advancing our understanding of cell polarisation and pattern formation in natural systems, such as the C. elegans embryo.
In more detail, the authors demonstrate that the simplified 2-node model requires precise parameter fine-tuning to maintain stable polarisation. Any single modification to this 2-node network disrupts the polarisation pattern, highlighting the model's lack of robustness. However, stability is achieved when two opposite modifications are applied, which also increases the number of parameter sets that sustain the pattern. This robustness is contingent on monotonic correlations between all system parameters.

The study extends its significance by examining how cells maintain pattern stability amid spatial parameter variations, which are common in natural systems due to extracellular and intracellular fluctuations. The authors found that in the 2-node network, varying individual parameters spatially disrupt the pattern, but stability is restored with compensatory variations. Additionally, the polarisation interface stabilises around the step transition between parameter values, making its localisation tunable. This suggests a potential biological mechanism where localisation might be regulated through signalling perception.

Focusing on the C. elegans cell polarisation network, the authors propose a 5-node network based on an exhaustive literature review, summarised in a supplementary table. Using their computational pipeline, they identify several parameter sets capable of achieving stable polarisation and claim that their model replicates experimental behaviour, even when simulating mutants. They also found that among 34 possible network structures, the wild-type network with mutual inhibition is the only one that proves viable in the computational pipeline. Compared with previous studies, which typically considered only 2- or 3-node networks, this analysis provides a more complete and realistic picture of the signalling network behind polarisation in the C. elegans embryo. In particular, the model for C. elegans cell polarisation paves the way for further in silico experiments to investigate the role of the network structure over the polarisation dynamics. The authors suggest that the natural 5-node network of C. elegans is optimised for maintaining cell polarisation, demonstrating the elegance of evolution in finding the optimal network structure to achieve certain functions.

Noteworthy limitations are also found in this work. To simplify the model for numerical exploration, the authors assume several reactions have equivalent dynamics, reducing the parameter space to three independent dimensions. While the authors briefly acknowledge this limitation in the "Discussion and Conclusion" section, further analysis might be required to understand the implications. For instance, it is not clear how the results depend on the particular choice of parameters. The authors showed that adding additional regulation might disrupt the polarised pattern, with the conclusion apparently depending on the strength of the regulation. Even for the 5-node wild-type network, which is the most robust, adding a strong enough self-activation of [A], as done in the 2-node network, will probably cause the polarised pattern to collapse as well.

Additionally, the authors utilise parameter values that are unrealistic, fail to provide units for some of them, and assume unknown parameter values without justification. The model appears to have non-dimensionalised length but not time, resulting in a mix of dimensional and non-dimensional variables that can be confusing. Furthermore, they assume equal values for Hill coefficients and many parameters associated with activation and inhibition pathways, while setting inhibition intensity parameters to 1. These arbitrary choices raise concerns about the fidelity of the proposed model in representing the real system, as their selected values could potentially differ by many orders of magnitude from the actual parameters.

The definition of stability and its evaluation in the proposed pipeline might also be too narrow. Throughout the paper, the authors discuss the stability of the polarised pattern, checked by an exhaustive search of the parameter space where the system reaches a steady state with a polarised pattern instead of a homogeneous pattern. It is not clear if the stability is related to the linear stability analysis of the reaction terms, as conducted in Goehring et al. (Science, 2011), which could indicate if a homogeneous state exists and whether it is stable or unstable. The stability test is performed through a pipeline procedure where they always start from a polarised pattern described by their model and observe how it evolves over time. It is unclear if the conclusions depend on the chosen initial conditions. Particularly, it is unclear what would happen if the initial distribution of posterior molecules is not exactly symmetric with respect to the anterior molecules, or if the initial polarisation is not strong.

Regarding the biological interpretation and relevance of the model, it overlooks some important aspects of the C. elegans polarisation system. The authors focus solely on a reaction-diffusion formulation to reproduce the polarisation pattern. However, the polarisation of the C. elegans zygote consists of two distinct phases: establishment and maintenance, with actomyosin dynamics playing a crucial role in both phases (see Munro et al., Dev Cell 2004; Shivas & Skop, MBoC 2012; Liu et al., Dev Biol 2010; Wang et al., Nat Cell Biol 2017). Both myosin and actin are crucial to maintaining the localisation of PAR proteins during cell polarisation, yet the authors neglect cortical flows during the establishment phase and any effects driven by myosin and actin in their model, failing to capture the system's complexity. How this affects the proposed model and conclusions about the establishment of the polarisation pattern needs careful discussion. Additionally, they assume that diffusion in the cytoplasm is infinitely fast and that cytoplasmic flows do not play any role in cell polarity. Finite cytoplasmic diffusion combined with cytoplasmic flows could compromise the stability of the anterior-posterior molecular distributions. The authors claim that cytoplasmic diffusion coefficients are two orders of magnitude higher than membrane diffusion coefficients, but they seem to differ by only one order of magnitude (Petrášek et al., Biophys. J. 2008). The strength of cytoplasmic flows has been quantified by a few studies, including Cheeks et al., and Curr Biol 2004.

Although the authors compare their model predictions to experimental observations, particularly in reproducing mutant behaviours, they do not explicitly show or discuss these comparisons in detail. Diffusion coefficients and off-rates for some PAR proteins have been measured (Goehring et al., JCB 2011), but the authors seem to use parameter values that differ by many orders of magnitude, perhaps due to applied scaling. To ensure meaningful predictions, whether their proposed model captures the extensive published data should be evaluated. Various cellular/genetic perturbations have been studied to understand their effects on anterior-posterior boundary positioning. Testing these perturbations' responses in the model would be important. For example, comparing the intensity distribution of PAR-6 and PAR-2 with measurements during the maintenance phase by Goehring et al., JCB 2011, or comparing the normalised intensity of PAR-3 and PKC-3 from the model with those measured by Wang et al., Nat Cell Biol 2017, during establishment and maintenance phases (in both wild-type and cdc-42 (RNAi) zygotes) could provide insightful validation. Additionally, in the presence of active CDC-42, it has been observed that PAR-6 extends further into the posterior side (Aceto et al., Dev Biol 2006). Conducting such validation tests is essential to convince readers that the model accurately represents the actual system and provides insights into pattern formation during cell polarisation.

A clear justification, with references, for each network interaction between nodes in the five-node model is needed. Some of the activatory/inhibitory signals proposed by the authors have not been demonstrated (e.g. CDC-42 directly inhibiting CHIN-1). Table S2 provided by the authors is insufficient to justify each node-node interaction, requiring additional explanations. (See the review by Gubieda et al., Phil. Trans. R. Soc. B 2020, for a similar node network that differs from the authors' model.) Additionally, the intensity distributions of cortical PAR-3 and PKC-3 seem to vary significantly during both establishment and maintenance phases (Wang et al., Nat Cell Biol 2017), yet the authors consider the PAR-3/PAR-6/PKC-3 as a single complex. The choices in the model should be justified, as the presence or absence of clustering of these PAR proteins can be crucial during cell polarisation (Wang et al., Nat Cell Biol 2017; Dawes & Munro, Biophys J 2011).

In summary, the authors successfully demonstrate the importance of compensatory actions in maintaining polarisation robustness. Their computational pipeline offers valuable insights into the dynamics of reaction-diffusion networks. However, the lack of detailed experimental validation and realistic parameter estimation limits the model's applicability to real biological systems. While the study provides a solid foundation, further work is needed to fully characterise and validate the model in natural contexts. This work has the potential to significantly impact the field by providing a new perspective on the robustness of cell polarisation networks.

The computational pipeline developed could be a valuable tool for further in silico experiments, allowing researchers to explore the dynamics of more complex networks. To maximise its utility, the model needs comprehensive validation and refinement to ensure it accurately represents biological systems. Addressing these limitations, particularly the need for more detailed experimental validation and realistic parameter choices, will enhance the model's predictive power and its applicability to understanding cell polarisation in natural systems.