Joint Public Review:
In this manuscript, the authors proposed an approach to systematically characterise how heterogeneity in a protein signalling network affects its emergent dynamics, with particular emphasis on drug-response signalling dynamics in cancer treatments. They named this approach Meta Dynamic Network (MDN) modelling, as it aims to consider the potential dynamic responses globally, varying both initial conditions (i.e., expression levels) and biophysical parameters (i.e., protein interaction parameters). By characterising the "meta" response of the network, the authors propose that the method can provide insights not only into the possible dynamic behaviours of the system of interest but also into the likelihood and frequency of observing these dynamic behaviours in the natural system.
The authors study the Early Cell Cycle (ECC) network as a proof of concept, focusing on pathways involving PI3K, EGFR, and CDK4/6 with the aim of identifying mechanisms that may underlie resistance to CDK4/6 inhibition in cancer. The biochemical reaction model comprises 50 state variables and 94 kinetic parameters, implemented in SBML and simulated in Matlab. A central component of the study is the generation of large ensembles of model instances, including 100,000 randomly sampled parameter sets intended to represent intra-tumour heterogeneity. On the basis of these simulations, the authors conclude that heterogeneity in kinetic rate parameters plays a stronger role in driving adaptive resistance than variation in baseline protein expression levels, and that resistance emerges as a network-level property rather than from individual components alone. The revised manuscript provides additional clarification regarding aspects of the simulation and filtering procedures and frames the comparison with experimental data as qualitative. Nonetheless, the study is best interpreted as a theoretical and exploratory analysis of the model's behaviour under heterogeneous conditions. Consequently, questions remain regarding the biological grounding of the sampled parameter regimes and the extent to which the reported frequencies of resistance-associated behaviours can be directly interpreted in physiological terms.
While the authors propose a potentially useful computational framework to explore how heterogeneity shapes dynamic responses to drug perturbation, a number of important conceptual and methodological concerns remain to be addressed:
(1) The sampling of kinetic parameters constitutes the backbone of the manuscript, yet important concerns remain regarding its biological grounding and transparency. Although the revised version provides additional clarification on the exploration of "model instances", it is still not sufficiently clear how parameter values and initial conditions are generated, nor how the chosen ranges relate to biological measurements. The kinetic rates are sampled over broad intervals without explicit justification in terms of experimentally measured bounds or inferred distributions. As a consequence, it remains uncertain whether the ensemble of simulated behaviours reflects physiologically plausible cellular regimes or primarily the properties of the assumed parameter space. In this context, the large-scale sampling (100,000 parameter sets) resembles a Monte Carlo exploration of the model rather than a biologically calibrated representation of tumour heterogeneity.
Furthermore, the adequacy of the sampling strategy in such a high-dimensional space (94 free parameters) remains open to question. In the absence of biologically informed constraints, the combinatorial space of possible parameter configurations is vast, and it is unclear to what extent the sampled ensembles can be considered representative. This issue is particularly relevant because the manuscript interprets the frequency of resistance-associated behaviours as indicative of their likelihood.
The validation presented in Figure 7 does not fully resolve these concerns. The comparison with experimental data is qualitative, and the simulations are performed in arbitrary time units, which complicates direct interpretation alongside time-resolved experimental measurements. Moreover, certain qualitative discrepancies between simulated and experimental trends (e.g., persistent versus decreasing CDK4/6 activity) are not thoroughly discussed. As this figure represents the primary empirical reference point in the manuscript, the extent to which the model captures experimentally observed dynamics remains uncertain.
Finally, aspects of presentation continue to limit transparency. Parameter ranges are described at different points in the manuscript but are not consolidated clearly in the Methods, and the definition of initial conditions remains ambiguous - particularly whether these correspond to conserved quantities or to the dynamic variables used to initialise simulations. In addition, the exact number of model instances underlying specific analyses and figures is not always explicit. Greater clarity on these issues is essential for assessing reproducibility and for interpreting the quantitative claims of the study.
(2) A central conclusion of the manuscript is that heterogeneity in protein-protein interaction kinetics is a stronger driver of adaptive resistance than heterogeneity in protein expression levels. To assess the latter, the authors fix a nominal set of kinetic parameters and generate 100,000 random initial concentrations for the 50 model species. However, according to the simulation protocol described in the manuscript, each trajectory includes three phases: (i) simulation under starvation conditions to equilibrium, (ii) mitogenic stimulation to a second ("fed") equilibrium, and (iii) application of drug treatment. The equilibrium concentrations reached in phases (i) and (ii) are determined by the kinetic parameters of the model and are independent of the initial concentrations, provided the system converges to a stable steady state. In dynamical systems terms, stable equilibria are defined by the parameter set and attract all initial conditions within their basin of attraction. Since the kinetic parameters are fixed in this experiment, the pre-treatment equilibrium that serves as the starting point for drug application should likewise be fixed. Under these conditions, it is therefore not unexpected that sampling a large number of initial concentrations has limited influence on the treated dynamics.
This raises conceptual questions about the interpretation of the comparison between kinetic and expression heterogeneity. If the system converges to a unique stable steady state prior to treatment, then variability in initial concentrations does not propagate into variability in drug response, and the observed dominance of kinetic heterogeneity may partly reflect this structural property of the model rather than a biological principle. Clarification is needed regarding whether multiple steady states exist under the nominal parameter set, and if so, how basins of attraction are explored.
More broadly, it remains unclear why initial protein concentrations can be sampled independently of the kinetic parameters. In biological systems, steady-state expression levels are typically determined by the underlying kinetic rates. A more consistent approach might require constraining initial concentrations to correspond to equilibrium states of the chosen parameter set, thereby introducing relationships between at least some of the 50 initial conditions and the 94 kinetic parameters. Finally, the manuscript employs a non-standard terminology regarding "initial conditions," which may further obscure interpretation of these results and would benefit from clarification.
(3) The technical implementation of the modelling and simulation framework remains difficult to evaluate due to insufficient methodological detail. Although the authors state that kinetic parameters are randomly sampled, the manuscript does not specify the distributions from which parameters are drawn, nor whether potential correlations between parameters are considered or explicitly ignored. Without this information, it is not possible to assess how implicit modelling assumptions shape the ensemble of simulated behaviours. Given that the conclusions rely on frequency-based interpretations across sampled parameter sets, greater transparency regarding the sampling procedure is essential.
A further concern relates to the parameter filtering step. The authors report that the "vast majority" of sampled parameter sets produced systems that were "too stiff," and that these were excluded on the grounds that stiff dynamics are not biologically plausible. However, the manuscript does not clearly define how stiffness is assessed, nor why stiffness is interpreted as biologically unrealistic rather than as a numerical property of the formulation. In standard practice, stiff systems are typically handled using appropriate implicit solvers rather than being discarded. Similarly, parameter sets that produce negative state values are excluded, yet such behaviour may arise from numerical artefacts rather than from intrinsic model inconsistency. The rationale for excluding these parameter sets, rather than adapting the numerical scheme, is not sufficiently justified.
The reported rejection rate - approximately 90% of sampled parameter sets - is substantial and raises questions regarding the interplay between model structure, parameter ranges, and numerical methods. As currently described, the filtering step appears to select parameter sets based primarily on computational tractability rather than on experimentally motivated biological criteria. The manuscript would be strengthened by clarifying whether the retained parameter sets are representative of biologically meaningful regimes, and by distinguishing clearly between exclusions based on biological plausibility and those arising from numerical considerations.
Finally, important aspects of the simulation protocol require clarification. The model is simulated under "fasted" and "fed" conditions until equilibrium is reached, yet the criterion used to determine convergence is not specified. It would be important to describe how equilibrium is assessed (e.g., based on the norm of the time derivatives). Additionally, it remains unclear whether the mitogenic stimulus applied in the "fed" phase is assumed to be constant over time and, if so, how this assumption relates to biological experimental conditions. Greater detail on these implementation choices is necessary to ensure interpretability and reproducibility.
(4) The manuscript states that the modelling conclusions are strongly supported by existing literature; however, the validation presented does not fully substantiate this claim. As noted above, the comparison with CDK2 and CDK4/6 experimental data remains qualitative, and the use of arbitrary simulation time units complicates interpretation of temporal agreement. The extent to which the model quantitatively or mechanistically recapitulates experimentally observed dynamics therefore remains uncertain.
The claim that the model reproduces known resistance mechanisms is also difficult to assess in light of Figure S10, where a large fraction of network nodes (~80%) appear implicated in resistance under some conditions. If most components of the network can, in at least some parameter regimes, be associated with resistance phenotypes, the resulting lack of selectivity weakens the strength of model-based validation. It becomes challenging to distinguish specific mechanistic insights from generic consequences of network connectivity.<br />
In addition, the Supplementary Information notes that certain components of the mitogenic and cell-cycle pathways were abstracted or excluded in order to maintain computational tractability. While such abstraction is understandable in a large ODE framework, it raises interpretative questions. Proteins identified as potential resistance drivers within the model may, in some cases, represent aggregated or simplified pathway effects. Clarifying in the main text how such abstractions may influence the attribution of resistance mechanisms would strengthen the biological interpretation of the results.
Drug inhibition is central to the manuscript's conclusions. The revised version clarifies that inhibition is implemented as a fixed fractional modification of specific kinetic rate laws. This abstraction is appropriate for exploring network-level responses, but it represents a stylised perturbation rather than a pharmacologically calibrated model of drug action. For full interpretability and reproducibility, the mathematical form of the modified rate laws, as well as the timing of inhibition relative to network equilibration, should be specified unambiguously. The biological implications of the findings depend critically on understanding this modelling choice.
The one-at-a-time perturbation analysis presented in Figure 5 provides an interpretable ranking of first-order control points across the ensemble and offers mechanistic insight into primary sensitivities of the network. However, many targeted therapies act on multiple components, and resistance frequently arises through combinatorial mechanisms. The reported rankings should therefore be interpreted as identifying primary influences under isolated perturbations, rather than as a comprehensive account of multi-target drug behaviour.
Overall, the manuscript succeeds in presenting a conceptual and exploratory framework for analysing how signalling network topology can shape the qualitative landscape of adaptive responses under heterogeneous kinetic conditions. Its principal contribution lies in establishing a systematic platform for large-scale in silico exploration. At the same time, the current limitations in biological calibration, parameter grounding, and validation constrain the extent to which the conclusions can be interpreted as predictive or quantitatively representative of specific tumour contexts. Addressing these issues would further strengthen the connection between the theoretical landscape described here and experimentally observed resistance dynamics.