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Response to Reviewers
We thank the Reviewers for their appreciative comments (Reviewer 1: “first time that a well-established existing mathematical model of signaling response extended and applied to heterogeneous ligand mixtures”)and constructive suggestions for improvement. In this extensive revision, we have not only addressed the suggestions comprehensively but also extended our analysis of signaling antagonism to all doses and at the single-cell level using novel computational workflows. This resulted in the discovery of several mechanismsof antagonism and synergy that are dose-dependent, and dependent on the cell-specific state of the signaling network, thereby manifesting in only a subset of cells.
We have addressed Reviewer comments: we have made substantial revisions to improve clarity, rigor, and biological interpretation. Below we briefly summarize the main concerns raised by Reviewers 1-3 and how we have addressed them.
- We have rewritten the Methods section to clarify our approaches. We have also added the explanation of methodology and the rationale in the main text to improve readability and comprehensiveness (Addressing Reviewer #1 comments). This includes explaining and justifying the signaling codon approaches (Reviewer 1), our core-module parameter matching methodology and discussion (Reviewer #1, point 11, Reviewer #2, point 1), and the model schematic (Reviewer #1, point 5).
- For one of our major conclusions – that macrophages may distinguish stimuli in the context of ligand mixtures – we have validated these results with experiments, which increases confidence in this conclusion (Reviewer #2, point 3, Reviewer #3, point 2).
- We have updated the model for CpG-pIC competition using Michaelis–Menten kinetics without any additional parameters, rather than introducing new free parameters. This change removes parameter freedom for fitting combinatorial conditions, leading to a more constrained and mechanistically grounded model whose predictions align better with experimental data (Updated Figures 2 and S2; Reviewer #2, point 2).
- We have addressed all other editorial and clarification-related concerns as well, as detailed in our point-by-point response below.
In addition, we have extended the scope of the manuscript. We have extended our analysis of ligand combinations across a broad dose range, from non-responsive to saturated conditions. This led to several additional discoveries. For example, we show that ultrasensitive IKK activation can underlie synergistic combinations of ligands at low doses. In contrast, beyond the CpG-poly(I:C) antagonism, we identify that competition for CD14 uptake by LPS and Pam can generate antagonism between these ligands within specific dose ranges.
Importantly, such antagonism or synergy is not evident in all cells in the population. It may also not be picked up by studies of the mean behavior. With our new computational workflow that allows for single-cell resolution we identify the conditions that must be met by the signaling network state, for antagonism or synergy to take place.
Further, we examine the hypothesis that such signaling pathway interactions affect stimulus-response specificity in combinatorial stimulus conditions. By comparing models with and without this antagonism, we demonstrate that antagonistic interactions can improve stimulus-response specificity in complex ligand mixtures.
These additional analyses provide a new mechanistic understanding of cellular information processing and elucidate how synergy and antagonism can mechanistically shape signaling fidelity in response to complex ligand mixtures.
Point-by-Point Response
Reviewer #1
Evidence, reproducibility and clarity
The authors extend an existing mathematical model of NFkB signalling under stimulation of various single receptors, to model that describes responses to stimulation of multiple receptors simultaneously. They compare this model to experimental data derived from live-cell imaging of mouse macrophages, and modify the model to account for potential antagonism between TLR3 and TLR9 response due to competition for endosomal transport. Using this framework they show that, despite distinguishability decreasing with increasing numbers of heterogenous stimuli, macrophages are still able in principle to distinguish these to a statistically significant degree. I congratulate the authors on an interesting approach that extends and validates an existing mathematical model, and also provides valuable information regarding macrophage response.
Response: We thank the reviewer for this appreciative assessment and for the careful reading of our work. The constructive comments helped us substantially improve the rigor and clarity of the manuscript.
In addition to revising the text for clarity, we have extended our analysis to systematically investigate dose-response behavior for each pair of ligand combination. Using the experimentally validated model, we explored 10 ligand pairs across a range of doses from non-responsive to saturating. This allowed us to identify mechanistic regimes in which synergy and antagonism arise at the single-cell level. In particular, we found that low-dose synergy can be explained by ultrasensitive IKK activation (Figure 4 and corresponding supplementary figures), while antagonism can emerge from competition for shared components such as CD14 (Figure 5 and corresponding supplementary figures). We further show that antagonism can enhance condition distinguishability in ligand mixtures, thereby contributing to stimulus-response specificity (Figure 5 and corresponding supplementary figures).
There are no major issues affecting the scientific conclusions of the paper, however the lack of detail surrounding the mathematical model and the 'signaling codons' that are used throughout the paper make it difficult to read. This is exacerbated by the fact that I was unable to find Ref 25 which apparently describes the model, however I was able to piece together the essential components from the description in Ref 8 and the supplementary material.
Response: This comment helped us to improve the writing. We apologize that the key reference 25 was still not publicly available. It is now published in Nature Communications. In addition, we have added more details to clarify the mathematical model as well as the signaling codons, in results and in methods. Please see below for details.
Lots of the minor comments below stem from this, however there are also a few other places that could benefit from some additional clarification and explanation.
Significance:
1. '...it remains unclear complex...' -> '...it remains unclear whether complex...'
Response: We have rewritten the Significance (now it is Synopsis).
Introduction:
2. 'temporal dynamics of NFkB' - it would be good to be more concrete regarding the temporal dynamics of what aspect of this (expression, binding, conformation, etc), if possible.
Response: It refers to the presence of NFκB into nucleus, which represents active NFκB capable of activating gene expression. We have clarified this (Lines 59-61 in introduction paragraph 2). “Upon stimulation, NFκB translocates into the nucleus, … activating immune gene expression (10, 15–19).”
'signaling codons' - the behaviour of these is key to the entire paper, so even if they are well described in the reference, it would be good to have a short description as early as possible so that the reader can get an idea in their mind what exactly is being discussed here. Later, it would be good to have concrete description of exactly what these capture.
Response: We thank the reviewer for this comment. We have added one whole paragraph in the early introduction to describe the concept of Signaling Codons which allow quantitative characterization of NFkB stimulus-response-specific dynamics (Lines 60-67). We have also added more concrete description of Signaling Codons in the results as well as adding an illustration for the signaling codons (Lines 169-175, Figure S2B).
'This challenge...population of macrophages' - this seems a bit out of place, and is a bit of a run on sentence, so I suggest moving this to the next paragraph and working it into the first sentence there '...regulatory mechanisms, and this challenge could be addressed with a model parameterised to account for heterogeneous...Early models ...', or something similar.
Response: We thank the reviewer for this suggestion, we have revised this as suggested. This improves the logic flow (Lines 87-88).
Ref 25: I can't find a paper with this title anywhere, so if it's an accepted preprint then it would be good to have this available as well. That said, I still think it would be difficult to grasp the work done in this paper without some description of the mathematical model here, at least schematically, if not the full set of ODEs. For example, there are numerous references to how this incorporates heterogeneous responses, the 'core module', etc, and the reader has no context of these if they aren't familiar with the structure of the model.
Response: We apologize that Ref 25 was not on PubMed. Now it’s published, and we have updated the corresponding information. This comment also helped us to improve the writing by adding a description of the mathematical model in the Introduction (Lines 95-105), the results (Lines 129-141), and a detailed description of the model in the Methods (Simulation of heterogenous NFκB dynamical responses.)
We have also added the schematic of the model topology in Figure S1 (adapted from previous publications Guo et al 2025, Adelaja et al 2021) to make sure the paper is self-contained.
'A key challenge which is...' -> 'A key challenge is...'
Response: We have revised the Introduction and removed this sentence.
'With model simulation ...' -> a bit of a run on sentence, I suggest breaking after 'conditions'.
Response: We have revised the introduction and removed this sentence.
Results:
- This section would benefit from a more in-depth description of the model and experimental setup. In particular for the experiment, the reader never really knows what this workflow for this is, nor what the model ingests as input, and what the predictions are of.
Response: This comment helped us to improve clarity by adding an in-depth description of the model and experimental setup. We have revised the Results as suggested (Lines 129-141). We also appended the corresponding revision here for reviewer reference.
“This mechanistic model was trained on single-ligand response experimental datasets, capturing the single-ligand stimulus-response specificity of the population of macrophages while accounting for cellular heterogeneity. Specifically, quantitative NFκB dynamic trajectory data from hundreds of single macrophages responding to five single ligands (TNF, pIC, Pam, CpG, LPS) at 3-5 doses was obtained from live cell imaging experiments. The mathematical model (Figure S1) consists of a 52-dimensional system of ordinary differential equations, including 52 intracellular species, 101 reactions and 133 parameters, and is divided into five receptor modules, which respond to the corresponding ligands respectively, and the IKK-NFκB core module that contains the prominent IκBα negative feedback loop. By fitting the single-cell experimental data set with a non-linear mixed effect statistical model (coupling with 52-dimensional NFκB ODE model), the parameter distributions for the single-cell population were inferred. Analyzing the resulting simulated NFκB trajectories with Information theoretic and machine learning classification analyses confirmed that the virtual cell model simulations reproduced key SRS performance characteristics of live macrophages.”
'..mechanistic model was trained...' - trained in this study, or in the previous referenced study?
Response: The mechanistic model was trained in a previous study (Guo et al 2025 Nature Comm), and we have clarified this in the revision (Lines 127 - 129).
- 'determined parameter distributions' - this is where it would be good to have more background on the model. What parameters are these, and what do they correspond to biologically? It would also be nice to see in the methods or supplementary material how this is done (maximum likelihood, etc).
Response: This comment helps us to clarify the predetermined parameter distributions. We have revised the methods to include this information (Simulation of heterogenous NFκB dynamical responses, paragraph 3). We have appended the corresponding text here for reviewer’s convenience.
“The ODE model was then fitted to the population of single-cell trajectories to recapitulate the cell-to-cell heterogeneity in the experimental data (2). This is achieved by solving the non-linear mixed effects model (NLME) through stochastic approximation of expectation maximation algorithm (SAEM) (3–6). Seventeen parameters were estimated. Within the core module, the estimated parameters included the rates governing TAK1 activation (k52, k65), the time delays of IκBα transcription regulated by NFκB (k99, k101), and the total cellular NFκB abundance (tot NFκB). Within the receptor module, receptor synthesis rates (k54 for TNF, k68 for Pam, k85 for CpG, k35 for LPS, k77 for pIC), degradation rates of the receptor–ligand complexes (k56, k61, k64 for TNF; k75 for Pam; k93 for CpG; k44 for LPS; k83 for pIC), and endosomal uptake rates (k87 for CpG; k36 and k40 for LPS; k79 for pIC) were fitted. All remaining parameters were fixed at literature-suggested values (1). The single-cell parameters inferred from experimental individual‐cell trajectories then served as empirical distributions for generating the new dataset (see SupplementaryDataset2).”
'matching cells with similar core model...' - it's difficult to follow the logic as to why this is done, so I think this needs to be a little clearer. My guess would be that the assumption is that simulated cells with similar 'core' parameters have a similar downstream signalling response, and therefore the receptors can be 'transplanted'. So it would be nice to see exactly what these distributions are and what the effect of a bad match would be.
Response: We thank the reviewer for this comment. In the revision, we have explained the rationale for matching cells with similar core module (Lines 145-152).
“Previous work determined parameter distributions for only the cognate receptor module (and the core module) that provided the best fit for the relevant single ligand experimental data (Figure 1A, Step 1), but other receptor modules’ parameter values were not determined. To simulate stimulus responses to more than two ligands, we imputed the other ligand-receptor module parameters using shared core-module parameters as common variables and employing nearest-neighbor hot-deck imputation (35). In this setup, the core module functions as an “anchor” to harmonize two or more receptor-specific parameter distributions.”
This nearest-neighbor hot-deck imputation approach (the core module matching method) was shown to outperform other approaches, including random matching and rescaled-similarity matching (Guo et al. 2025, Supplementary Figure S11). For the reviewer’s convenience, we have also appended the corresponding figure below.
Figure S11 from (Guo et al., 2025). Assessment of matching techniques for predicting single-cell responses to various ligand stimuli (a-d). Heatmaps illustrating the Wasserstein distance between the signaling codon distributions predicted by the model and those observed in experiments. The analysis employs four distinct matching methods to align the five ligand-receptor module parameters: (a) “Random Matching”, (b) “Similarity Matching” (the method used in our study), (c) “Rescaled-Similarity Matching”, and (d) “Sampling Approximated Distribution”. In the heatmaps, rows represent signaling codons, columns denote ligands, and the color intensity indicates the Wasserstein distance, providing a visual metric of similarity between model predictions and experimental data. e-f. Histogram of the average Wasserstein distance between the model-predicted and experimentally observed signaling codon distributions, summarized across signaling codons (e) and ligands (f).
Some explanation of how this relates to the experimental data the parameters are fit on would also be useful. (a) Is there a correspondence between individual simulated cells and the experimental data for the single ligand stimulation, and then the smallest set of these is taken? Is there also a matching from the simulated multi-receptor modules and the multi-receptor data, and if so, is this done in the same way?
Response: This comment to help us clarify the correspondence relationship between model simulations and experimental data.
Yes—there is a correspondence between individual simulated cells and the previously published experimental data (Guo et al., 2025b) for single-ligand stimulation. We have revised the first paragraph of the Results (Lines 136–148) and the Methods (Lines 544-557) to clarify how the model simulations were fit to the previous experimental dataset. See Reviewer 1, Comments 10 for the updates in Methods. We have pasted in the revised Results section below for the reviewer’s reference.
“By fitting the single-cell experimental data set with a non-linear mixed effect statistical model (coupling with 52-dimensional NFκB ODE model), the parameter distributions for the single cell population were inferred.”
'six signaling codons' - here it would be good to recapitulate what these represent, but also what the 'strength' and 'activity' correspond to (total integrated value, maximum value, etc)
Response: We thank the reviewer for the suggestion and have clarified this point (Lines 169-175, Figure S2B).
'pre-defined thresholds' - no need to state these numerically in the text (although giving some sense of how/why these were chosen would give some context), but I couldn't find the values of these, nor values corresponding to the signaling codons.
Response: We appreciate the reviewer’s comment. We have added this information in the figure legend (Figure 1B-C) and Method -- “Responder fraction” (Lines 666-672). Specifically, for the model simulation data, the integral thresholds are 0.4 (µM·h), 0.5 (µM·h), and 0.6 (µM·h). The peak thresholds are 0.12 (µM), 0.14 (µM), and 0.16 (µM). For the experimental data, the integral thresholds are 0.2 (A.U.·h), 0.3 (A.U.·h), and 0.4 (A.U.·h). The peak thresholds are 0.14 (A.U.), 0.18 (A.U.), and 0.22 (A.U.). Thresholds were selected so that the medium threshold yields 50% responder cells under single-ligand conditions, while the responder ratio remains unsaturated under three-ligand stimulation.
'non-responder cells are likely a result of cellular heterogeneity in receptor modules rather than the core module' - is this the 'ill health' referenced earlier? If so make this clear.
Response: Yes, this is the ‘ill health’ referenced earlier, and we have clarified this (Lines 198-199).
It's also very difficult to follow this chain of logic, given that the reader at this point doesn't have any knowledge of what the 'core' module is, nor the significance of the thresholds on the signaling codons. I would suggest making this much clearer, with reference to each of these.
Response: We apologize for the poor explanation. We have now explained in the Introduction (Lines 95-106) and the results (Lines 129-141) how the model is structured into receptor-proximal modules that converge on the common core module. We have also added a schematic for clarity (Figure S1). For further clarification of the math models, we have significantly revised the Methods (Simulation of heterogenous NFκB dynamical responses). The defined thresholds are clarified in the Methods -- “Responder fraction”.
'...but the model represented these as independent mass action reactions' - the significance of this may not be clear to someone not familiar with biophysical models, so probably better to make it explicit.
Response: We thank the reviewer for this reminder, and we have added a description of the significance of this point (Lines 225-227).
'...we trained a random forest classifier...' - is this trained on the 'raw' experimental time series data, or on the signaling codons?
Response: It is trained on the signaling codons calculated from model simulations of NFκB trajectories. We have clarified this (Lines 260-261).
'We also applied a Long Short-Term Memory (LSTM) machine learning model...' - it might be good to reference these three approaches at the beginning of this section, otherwise they seem to come out of the blue a little.
Response: We have added the references of these three approaches in the beginning of this section (Lines 242-246).
'We then used machine learning classifiers...' - random forests, LSTMs, or a different model?
Response: We have clarified that this as random forest classifier (Line 276).
Discussion:
- '...over statistical models...' - suggest maybe 'purely statistical models'
Response: We thank the reviewer for this suggestion. We have rewritten the whole Discussion to include the new insights of antagonism and synergy and their roles in maintaining unexpectedly high SRS performance. Thus, this sentence was removed.
'We found that endosomal transport...' - A paper by Huang, et. al. (https://www.jneurosci.org/content/40/33/6428) observed a synergistic phagocytic response between CpC and pIC stimulation in microglia. This is still consistent with a saturation effect dependent on dose, but may be worth a mention.
Response: We thank the reviewer for referring this interesting paper to us, and this comment helps us to improve the Discussion of inflammatory signaling pathways besides NFκB. This paper demonstratessynergistic effects between CpG and pIC in inhibiting tumor growth and promoting cytokine production(Huang et al., 2020), such as IFN-β and TNF-α, whose expression is also regulated by the IRF and MAPK signaling pathways (Luecke et al., 2021; Sheu et al., 2023). This finding does not contradict our findings that CpG and pIC act antagonistically in the NFκB signaling pathway because of the combinatorial pathways that act on gene expression: CpG can activate the MAPK signaling pathway (Luecke et al., 2024) but not the IRF signaling pathway, whereas pIC activates the IRF signaling pathway (Akira and Takeda, 2004) but only weakly the MAPK pathway. Therefore, their combination can synergistically regulate inflammatory responses. We have added this to the discussion (Lines 515-522).
'...features termed...' -> 'features, termed'
Response: We thank the reviewer for their carefully reading, and we have rewritten the Discussion.
'...we applied a Long Short-Term Memory (LSTM) machine learning model..' - maybe make clear that this is on the time-series data (also LSTM has already been defined).
Response: We thank the reviewer for their carefully reading, and we have rewritten the Discussion.
Materials and methods:
- The descriptions in this section are quite vague, so I would suggest expanding this with more detail from the supplementary material, where things are quite well explained.
Response: We thank the reviewer for this suggestion, and we have rewritten the whole Methods as suggested.
'sampling distribution' - not clear what this refers to in this context
Response: We have clarified this in the revision (Methods -- Simulation of heterogenous NFκB dynamical responses, paragraph 3).
“The single-cell signaling-pathway parameter values used for bootstrapping sampling to generate model simulations are given in Supplementary dataset 2.”
'RelA-mVenus mouse strain' - it would be good to mention the relevance of the reporter for NFkB signaling
Response: We have added the relevance of the reporter for NFkB signaling (Methods, Lines 624-626).
'...A random forest classifier...' -> a random forest classifier
Response: We have rewritten the methods.
Significance
This study provides mechanistically interpretable insight on the important question of how immune cells perform target recognition in realistic scenarios, and also provides validation of existing mathematical models by extending these beyond their original domain. The paper uses 'signaling codons' as a proxy for information processing, however in this instance it is cross-validated with an LSTM model that is applied directly to the time series data. Nevertheless, the scope of the paper is such that it does not deal with the question of how these signals are transmitted or used in a downstream immune response. To my knowledge, this is the first time that a well established existing mathematical model of signalling response has been extended and applied to heterogeneous ligand mixtures. These results will be of interest to those studying immune cell responses, and to those interested in basic research on mathematical models of signaling and cellular information processing more generally.
My background is in biophysical models, machine learning, and signaling in cancer. I have a basic understanding of immunology, but no experience in experimental cell biology.
Response: We thank the reviewer for highlighting the novelty of our study. We appreciate the reviewer’s recognition that our work advances the understanding of cellular information processing in the context of ligand mixtures, particularly as the first to extend computational models to investigate signaling fidelity under mixed-ligand conditions.
We agree that this work will interest computational biologists focused on signaling network modeling and information processing. In addition, we believe it will also be valuable for all signaling biologists, as we provide fundamental insights. For experimental biologists in particular, our model provides an efficient, quantitative framework for exploring and generating testable hypotheses.
We would also like to gently emphasize that evaluating specificity within signaling pathways is as essential as studying downstream functional responses. While immune function outcomes are certainly important, they rely on the upstream signaling pathways that first respond to environmental cues. Understanding how these signaling pathways achieve specificity and discriminability is therefore crucial. For example, this is particularly relevant for drug development targeting pathways such as NFκB, where assessing the direct signaling output—NFκB activation dynamics—can provide valuable insight into the effects of pharmacological interventions.
Reviewer #2
Evidence, reproducibility and clarity
Guo et al. developed a heterogeneous, single-cell ODE model of NFκB signaling parameterized on five individual ligands (TNF, Pam, LPS, CpG, pIC) and extended it, via core-module parameter matching, to predict responses to all 31 combinations of up to five ligands. They found that simulated responder fractions and signaling codon features generally agreed with live-cell imaging data. A notable discrepancy emerged for the CpG (TLR9) + pIC (TLR3) pair: experiments exhibited non-integrative antagonism unpredicted by the original model. This issue was resolved by incorporating a Hill-type term for competitive, limited endosomal trafficking of these ligands. Finally, by decomposing NFκB trajectories into six "signaling codons" and applying Wasserstein distances plus random-forest and LSTM classifiers, the authors showed that stimulus-response specificity (SRS) declines with ligand complexity but remains statistically significant even for quintuple mixtures.
This is a well written and scientifically sound manuscript about complexities of cellular signaling, especially considering the limitations of in vitro experiments in recapitulating in vivo dynamics.
Response: We thank the reviewer for carefully reading the manuscript and for this endorsement. We have significantly improved the manuscript thanks to the reviewer’s insightful comments (see below for point-to-point responses).
Besides addressing the reviewer’s questions, we have further extended our work to investigate how ligand pairs interact across all doses and how those interactions affect stimulus-response specificity. As the reviewer pointed out, experimental studies are limited in recapitulating the multitude of complex physiological contexts. The model is helpful to explore more complex scenarios beyond the feasibility of in-vitro experimental setups. Using computational simulations, we have further explored 360 conditions generated from 10 ligand pairs, each evaluated at 6 doses spanning non-responsive to saturating levels, and with each condition considered 1000 cells to capture the heterogeneity of the population.
From this extended analysis, we identified the mechanistic bases for observations of both synergy and antagonism. Synergy for certain low-dose ligand combinations can be explained by ultrasensitive IKK activation (Figure 4), while antagonism between LPS and Pam arises from competition for the cofactor CD14 (Figure 5). We show that these phenomena are dependent on the signaling network state and therefore are not observed in all cells of the population. We define the network conditions that must be met for antagonism and synergy to occur. Importantly, we then show that antagonism can contribute to stimulus-response specificity in ligand mixtures (Figure 5).
Here are a few comments and recommendations:
- The modeling approach used in this manuscript, while interesting, might need further validation. Inferring multi-ligand receptor parameters by matching single-ligand cells on core-module similarity may not capture true co-variation in receptor expression or adaptor availability. Single cell measurements of receptor expressions could be done (e.g. via flow cytometry) to ground this assumption in real data. If the authors think this is out of scope for this manuscript, they could fit core-matched single cell models with two receptor modules from scratch to the two-ligand experimental data. Would this fitted model produce similar receptor parameters compared to the presented approach? At least the authors should add a bit more explanation for why their modeling approach is better (or valid) than fitting the models with 2/3/4/5 receptor modules from scratch to the experimental data.
Response: We thank the reviewer for this comment, this helped us improve the explanation of the methodology, the rationale, and the validation. The methodology is based on the well-established statistical method of nearest-neighbor hot-deck imputation (Andridge and Little, 2010). In this implementation, the core module functions as a stabilizing “anchor” (common variables) to harmonize various receptor-specific parameter distributions. Similar methodologies have been successfully applied to correct batch effects or integrate single-cell RNAseq datasets using anchor cell types (Stuart et al., 2019). Our workflow has been validated on single-ligand stimuli conditions in a previous study (Guo et al., 2025) (See below 3rdparagraph). Here, we used this method to generate predictions for ligand mixtures and have validated them with experimental studies of the dual-ligand stimuli, and we found that our predictions align well with the experimental data. As the reviewer suggested in point 3, in the revision, we also added experimental validation on the binary classifiers of macrophage determines whether specific stimuli are presented in the ligand mixture. The question we are interested in in this work is how macrophage process ligand-specific information in the context of ligand mixtures. For this question, the experimental results align with the model predictions, reaching consistent conclusions.
In the revision, we have explained the rationale for using the nearest-neighbor hot-deck imputation by matching cells with similar core module (Lines 143-150).
“Previous work determined parameter distributions for only the cognate receptor module (and the core module) that provided the best fit for the single ligand experimental data (Figure 1A, Step 1), and other receptor modules parameter information is missing. To simulate stimulus responses to more than two ligands, we imputed the other ligand–receptor module parameters using shared core-module parameters as common variables and employing nearest-neighbor hot-deck imputation (35). In this setup, the core module functions as an “anchor” to harmonize two or more receptor-specific parameter distributions. This was achieved by by minimizing Euclidean distance between the core module parameters associated with the independently parameterized single-ligand models (Figure 1A, Step 2). ”
In Guo et al. (2025) (see Supplementary Figure S11), the nearest-neighbor hot-deck imputation approach (core module similarity matching method) was compared with other approaches, including random matching and rescaled-similarity matching. The results show that, after matching, the core module method best preserves the single-ligand stimulus signaling codon distributions. For the reviewer’s convenience, we have also appended the figure in the response to Reviewer 1, Comment 11.
The advantage of our workflow is that it does not need to be fit to new experimental data and still gives reliable predictions on signaling dynamics. For the reviewer’s interest, we have tried to fit core-matched single cell models with two receptor modules. As fitting parameters require sufficiently large and high-quality datasets, single-ligand stimulation data with more than 1,000 cells can be adequate to estimate 6~7 parameters (Guo et al., 2025) (approx. 1400 cells to 2000 cells per ligand). However, our current experimental dataset for combinatorial-ligand conditions contains only 500~1,000 cells, and we have tested these datasets but results show a poor fit of heterogeneous signaling dynamics. This is due to an insufficient number of cells for estimating 8~10 parameters. We estimate that at least ~1,500 cells would be needed for reliable parameter estimation under dual-ligand stimulation (and more cells may be needed for combinatorial ligand stimuli involving more ligands). This is currently not feasible to obtain for mixed ligands given the large number of combinatorial conditions.
Overall, in this paper, the nearest-neighbor hot-deck imputation approach is presented as a feasible and acceptable approach that best reflects our current understanding of the signaling network. Importantly, it helps identify potential gaps by highlighting discrepancies between model predictions and experimental observations.
(a) The refined model posits competitive, saturable endosomal transport for CpG and pIC, but no direct measurements of endosomal uptake rates or compartmental saturation thresholds are provided, leaving the Hill parameters under-constrained. The authors could produce dose-response curves for CpG and pIC individually and in combination across a range of concentrations to fit the Hill parameters for competitive uptake. (b) If this is out of scope for this paper, the authors should at least comment on why the endosome hypothesis is better than others e.g. crosstalks and other parallel pathway activations. Especially given that even the refined model simulations with Hill equations for CpG and pIC do not quite match with the experimental data (Fig 2 B,E).
Response: (a) The reviewer’s comments helped us to improve our work by employing the Michaelis-Menten Kinetics for substrate competition reactions, which increases the mathematic rigor of the CpG-pIC competition model. In this updated model, there is no free parameters to tune, as all the Vmax, Kd, should be consistent with the single-ligand scenario. And the Hill is same as single-ligand case, equal to 1.
The comments on examining dose-response curves for CpG and pIC inspired us to extend the dose-response curves for all ligand pair combination, allowing us to identify the synergy in low-dose ligand pairs and antagonism for high-dose LPS-Pam, besides CpG-pIC (new Figure 4 & 5).
(b) Regarding alternative hypotheses for antagonism—such as crosstalk or parallel-pathway activation: any antagonistic effect would have to arise from negative regulation acting within the first 30 min. However, IκBα-mediated feedback only becomes appreciable after ~30 min (Hoffmann et al., 2002), and A20-dependent attenuation requires ≥2 h (Werner et al., 2005). Beyond these delayed feedback, NFκB activation depends primarily on phosphorylation and K63-linked ubiquitination, for which no mechanism produces true antagonism; at most, combinatorial inputs saturate the response to the level of the strongest single ligand. We have added this rationale to the Discussion to explain why we favor the endosome saturation hypothesis over other mechanisms (Lines 459-465). While this may not capture every nuance, it represents the simplest model extension capable of reproducing the observed antagonism.
Authors asses the distinguishability of single-ligand stimuli and combinatorial ligands stimuli using the simulations from the refined model. While this is informative, the simulated data could propagate deviations from the experimental data to the classifiers. How would the classifiers fare when the experimental data is used to assess the single-stimulus distinguishability? The authors could use the experimental data they already have and confirm their main claim of the paper, that cells retain stimulus-response specificity even with multiple ligand exposure. In short, how would Fig 3E look when trained/validated on available experimental data?
Response: We thank the reviewer’s valuable comments, and they helped us strengthen the rigor of our analysis by incorporating cross-model testing. Specifically, we refined our analysis of ligand presence/absence classification by including ROC AUC and balanced accuracy metrics. This adjustment accounts for the fact that the experimental data did not cover all combinatorial conditions, thereby mitigating potential biases from data imbalance and threshold choice. The experimental results are qualitatively consistent with the simulations, though—as expected—they show somewhat lower ligand distinguishability compared to the noise-free simulated dataset. We have updated Figures 3E–F (previously Figure 3E), added Figure S8, and revised the manuscript accordingly (Lines 292–301). For the reviewer’s convenience, we have also pasted in the revised manuscript text below.
“Classifiers trained to distinguish TNF-present from TNF-absent conditions achieved a Receiver Operating Characteristic-Area Under the Curve (ROC AUC) of 0.96, significantly above the 0.5 baseline (Figure 3D, Figure S8A). Extending this analysis to other ligands, cells detected LPS (0.85), Pam (0.84), pIC (0.73), and CpG (0.63) in mixtures (Figure 3D, S8A). Using experimental data from double- and triple-ligand stimuli (Figure 1D), ROC AUC values were TNF 0.74, LPS 0.74, Pam 0.66, pIC 0.75, and CpG 0.66 (Figure 3E, S8B). Classifier accuracies yielded consistent results (Figure S8C-D). These results indicated a remarkable capability of preserving ligand-specific dynamic features within complex NFκB signal trajectories that enable nuclear detection of extracellular ligands even in complex stimulus mixtures.”
While the approach of presented here with multiple simultaneous ligand exposures is a major step towards the in vivo-like conditions, the temporal aspect is still missing. That is, temporal phasing i.e. sequential exposure to multiple ligands as one would expect in vivo rather than all at once. This is probably out of scope for this paper but the authors could comment how how their work could be taken forward in such direction and would the SRS be better or worse in such conditions.
Response: We thank the reviewer for this insightful comment. We have added “the temporal aspect of multiple ligand exposures” to the discussion (Lines 503-510), and we pasted the corresponding paragraph here for reviewer’s references (black fonts are previous version, and blue fonts is the revised new texts):
“Cells may be expected to interpret not only the combination of signals but also their timing and duration to mount appropriate transcriptional responses (58, 59). For example, acute inflammation integrates pathogen-derived cues with pro- and anti-inflammatory signals over a timeframe of hours to days (58), to coordinate the pathogen removal and tissue repairing process. Investigating sequential stimulus combinations in our model is therefore crucial for understanding how cells process complex physiological inputs. Simulations that account for longer timescales may require additional feedback mechanisms, as described in some of our previous studies for NFκB (15, 60). ** ”
There is no caption for Figure 3F in the figure legend nor a reference in the main text.
Response: In the revised manuscript we actually removed Figure 3F.
Significance
General assessment: This is a good manuscript in it's present form which could get better with revision. There needs more supporting data and validation to back the main claim presented in the manuscript.
Significance/impact/readership: When revised this manuscript could be of interest to a broad community involving single cells biology, cell and immune signaling, and mathematical modeling. Especially the models presented here could be used a starting point to more complex and detailed modeling approaches.
Response: We thank the reviewer for this endorsement. The reviewer’s constructive suggestion helped us significantly improve the clarity and rigor of our main conclusion.
In summary, we have strengthened the computational framework in several ways. We improved the model’s fit to experimental single-ligand training data and reformulated the antagonistic CpG-pIC model using Michaelis–Menten kinetics, thereby reducing parameter arbitrariness and increasing mechanistic interpretability. These changes led to better agreement between model predictions and experimental observations for combinatorial ligand responses (Updated Figure 2 and Figure S2), which we hope will further increase experimentalists’ confidence in the modeling results. We have also validated one key conclusion (“cells retain stimulus-response specificity even with multiple ligand exposure”) using the experimental dataset, and it aligns with the model predictions.
In addition, we have further extended our analysis and the scope. Inspired by the reviewer’s advice (and Reviewer 3’s comment 1b) on dose-combination study for CpG-pIC pair, we expanded our research to dose-response relationships for all dual-ligand combinations (Lines 302-406, Figure 4-5). This additional comprehensive analysis allowed us to identify the mechanism of synergistic and antagonistic effects in single-cell responses and to pinpoint the corresponding dose ranges among different ligand pairs.
Interestingly, we found that IKK ultrasensitive activation may lead to low-dose ligand combinations synergistic response for single cells. We also found that CD14 uptake competition between LPS and Pam may lead to antagonistic/non-integrative combination. Our simulation-based finding of non-integrative combination of LPS-Pam stimuli aligns with previous independent experimental finding of non-integrative response for LPS and Pam combination (Kellogg et al., 2017), and this independent experimental study validated our model prediction.
We further analyzed stimulus-response specificity under conditions predicted to exhibit synergy or antagonism. Our results indicate that antagonistic combinations of ligands can increase stimulus-response specificity in the context of ligand mixtures.
Reviewer #3
Evidence, reproducibility and clarity
The authors investigate experimentally single macrophages' NF-kB responses to five ligands, separately and to 3 pairs of ligands. Using the single ligand stimulations, they train an existing mathematical model to replicate single-cell NF-kB nuclear trajectories. From what I understand, for each single cell trajectory in response to a given ligand, the best fit parameters of the core module and the receptor module (specific for the given ligand) are found.
Then (again, from what I understand), single ligand models are used to generate responses to combinations of ligands. The parametrizations of single ligand models (to be combined) are chosen to have the most similar core modules. It is not described how the responses to more than one ligand are calculated - I expect that respective receptor modules work in parallel, providing signals to the core module. After observing that the response to CpG+pIC is lower (in terms of duration and total) than for CpG alone, the model is modified to account for competition for endosomal transport required by both ligands.
Having the trained model, simulations of responses to all 31 combinations of ligands are performed, and each NF-κB trajectory is described by six signaling codons-Speed, Peak, Duration, Total, Early vs. Late, and Oscillations. Next, these codons are used to reconstruct (using a random forest model) the stimuli (which may be the combination of ligands). The single and even the two ligand stimuli are relatively well recognized, which is interpreted as the ability of macrophages to distinguish ligands even if present in combination.
We thank the reviewer for careful reading of the manuscript.
Major comments
1) The demonstrated ability to recognize stimuli is based on several key assumptions that can hardly be met in reality.
Response: We thank the reviewer for this comment, which prompted us to carefully reflect on the rigor of our work, inspired us to extend our analysis to a broad range of ligand-dose combinations, and helped us improve clarifying the limitations of our approach. Please see our detailed responses below.
a) The cell knows the stimulation time, and then it can use speed as a codon. Look on fig. S4A: The trajectories in response to plC are similar to those in response to TNF, but just delayed.
Response: We thank the reviewer for this comment. We updated the model parameterization to better fit to the single-ligand pIC condition (Lines 557-559). In the updated model, the simulated responses to TNF and pIC are quite different (Fig. S2A-B, Fig. S5A-B). Specifically, the Peak, Duration, EarlyVsLate, and Total signaling codons have different values. In addition, the literature suggests that timing difference of NFκB activation are sufficient to elicit differences in downstream gene expression responses, especially for the early response genes (ERG) and intermediate response genes (ING) (Figure 1 in Ando, et al, 2021). For reviewer’s convenience, we have also appended the figures. Specifically, within the first 60 minutes, ctrl exhibit higher Speed of NFκB activation, and the NFκB regulated ERG and ING show differences in the first 60 minutes (Below Fig 1a,b). Ando et al then identified the gene regulatory mechanism that is able to distinguish between differences in the Speed codon. Importantly, this mechanism does not require knowledge of t=0, i.e. when the timer was started.
The signaling codon Speed, which is based on derivatives, is one way to quantify such timing differences in activation. It was selected from a library of more than 900 different dynamic features using an information maximizing algorithm (Adelaja et al., 2021). It is possible that other ways of measuring time, e.g. time to half-max, might not be distinguished that well by these regulatory mechanisms.
b) The increase of stimulus concentration typically increases Peak, Duration, and Total, so a similar effect can be achieved by changing the ligand or concentration.
Response: This (“the increase of stimulus concentration typically increases Peak, Duration, and Total”) is not an assumption. What the reviewer described (“a similar effect can be achieved by changing the ligand or concentration”) may occur or may not. The six informative signaling codons can vary under different ligands or doses. For example, with increasing doses of Pam, the NFκB response shows a higher peak, potentially making it appear more like LPS stimulation. However, as the Pam dose increases, the response duration decreases, which distinguishes it from LPS stimulation (See experimental data shown in Figure 4A, second row, and Figure 3A, second row in Luecke et al., (2024), we also pasted the corresponding figure below for reviewer’s convenience).
Figure 4A and Figure 3A from Luecke et al., (2024). Figure 4A: NFκB activity dynamics in the single cells in response to 0, 0.01, 0.1, 1, 10, and 100 ng/ml P3C4 stimulation. Eight hours were measured by fluorescence microscopy of reporter hMPDMs. Each row of the heatmap represents the p38 or NFκB signaling trajectory of one cell. Trajectories are sorted by the maximum amplitude of p38 activity. Data from two pooled biological replicates are depicted. Total # of cells: 898, 834, 827, 787, 778, and 923. Figure 3A: NFκB activity dynamics in the single cells in response to 100 ng/ml LPS stimulation. Eight hours were measured by fluorescence microscopy of reporter hMPDMs. Each row of the heatmap represents the NFκB signaling trajectory of one cell (with p38 measured shown in the original paper). Trajectories are sorted by the maximum amplitude of p38 activity. Data from two pooled biological replicates are depicted.
Inspired by the reviewer’s comment (and also Reviewer 2’s comments), in the revision, we expanded our research to dose-response relationships for all dual-ligand combinations (Lines 302-406, Figure 4-5). This additional comprehensive analysis allowed us to identify the mechanism of synergistic and antagonistic effects in single-cell responses and to pinpoint the corresponding dose ranges among different ligand pairs.
Interestingly, we found that IKK ultrasensitive activation may lead to synergistic responses to low-dose ligand combinations but only in a subset of single cells. We also found that CD14 uptake competition between LPS and Pam may lead to antagonistic/non-integrative combination. Our simulation-based finding of non-integrative combination of LPS-Pam stimuli aligns with previous independent experimental findings of non-integrative response for LPS and Pam combination (Kellogg et al., 2017).
c) Distinguishing a given ligand in the presence of some others, even stronger bases, on the assumption that these ligands were given at the same time, which is hardly justified.
Response: We agree with the reviewer that ligands could be given at different times. Considering time delays between ligands (the inset and also removal) dramatically adds to the combinatorial complexity. Some initial studies by the Tay lab are beginning to explore some scenarios of time-shifted ligand pairs (Wang et al 2025). Here we focus on a systematic exploration of all ligand combinations at 6 different doses. The fact that we do not consider time delays is not an assumption but admittedly a limitation that may well be addressed in future studies. We have included a brief discussion of this issue in the discussion (Lines 503-514). We’ve appended here for reviewer’s convenience.
“Cells may be expected to interpret not only the combination of signals but also their timing and duration to mount appropriate transcriptional responses (Kumar et al., 2004; Son et al., 2023). For example, acute inflammation integrates pathogen-derived cues with pro- and anti-inflammatory signals over a timeframe of hours to days (Kumar et al., 2004), to coordinate the pathogen removal and tissue repairing process. Investigating sequential stimulus combinations in our model is therefore crucial for understanding how cells process complex physiological inputs. Simulations that account for longer timescales may require additional feedback mechanisms, as described in some of our previous studies for NFκB (Werner et al., 2008, 2005). ”
We would like to suggest that despite (or maybe because) limiting our study to coincident stimuli, we made some noteworthy discoveries.
2) For single ligands, it would be nice to see how the random forest classifier works on experimental data, not only on in silico data (even if generated by a fitted model).
Response: This comment and Reviewer 2 comment 3 have helped us strengthen the rigor of our analysis by incorporating cross-model testing. We pasted the response below.
Specifically, we refined our analysis of ligand presence/absence classification by including ROC AUC and balanced accuracy metrics. This adjustment accounts for the fact that the experimental data did not cover all combinatorial conditions, thereby mitigating potential biases from data imbalance and threshold choice. The experimental results are qualitatively consistent with the simulations, though—as expected—they show somewhat lower ligand distinguishability compared to the noise-free simulated dataset. We have updated Figures 3E–F (previously Figure 3E), added Figure S8, and revised the manuscript accordingly (Lines 292–301). For the reviewer’s convenience, we have also included the revised manuscript text below.
“Classifiers trained to distinguish TNF-present from TNF-absent conditions achieved a Receiver Operating Characteristic-Area Under the Curve (ROC AUC) of 0.96, significantly above the 0.5 baseline (Figure 3D, Figure S8A). Extending this analysis to other ligands, cells detected LPS (0.85), Pam (0.84), pIC (0.73), and CpG (0.63) in mixtures (Figure 3D, S8A). Using experimental data from double- and triple-ligand stimuli (Figure 1D), ROC AUC values were TNF 0.74, LPS 0.74, Pam 0.66, pIC 0.75, and CpG 0.66 (Figure 3E, S8B). Classifier accuracies yielded consistent results (Figure S8C-D). These results indicated a remarkable capability of preserving ligand-specific dynamic features within complex NFκB signal trajectories that enable nuclear detection of extracelular ligands even in complex stimulus mixtures.”
3) My understanding of ligand discrimination is such that it is rather based on a combination of pathways triggered than solely on a single transcription factor response trajectory, which varies with ligand concentration and ligand concentration time profile (no reason to assume it is OFF-ON-OFF). For example, some of the considered ligands (plC and CpG) activate IRF3/IRF7 in addition to NF-kB, which leads to IFN production and activation of STATs. This should at least be discussed.
Response: We thank the reviewer for this comment and fully agree. In the previous version, we discussed different signaling pathways combinatorically distinguishing stimulus. In the revision, we have extended this discussion to include the example of pIC and CpG activation, as suggested (Lines 515-522). We pasted the corresponding text below.
“Furthermore, innate immune responses do not solely rely on NFκB but also involve the critical functions of AP1, p38, and the IRF3-ISGF3 axis. The additional pathways are likely activated in a coordinated manner and provide additional information (Luecke et al., 2021). This is exemplified by the studies demonstrating synergistic effects between CpG and pIC in inhibiting tumor growth and promoting cytokine production (Huang et al., 2020), such as IFNβ and TNFα, whose expression is also regulated by the IRF and MAPK signaling pathways (Luecke et al., 2021; Sheu et al., 2023). Therefore the inclusion of parallel pathways of AP1 and MAPK, as well as the type I interferon network (Cheng et al., 2015; Davies et al., 2020; Hanson and Batchelor, 2022; Luecke et al., 2024; Paek et al., 2016; Peterson et al., 2022) are next steps for expanding the mathematical models presented here.”
Technical comments
1) Reference 25: X. Guo, A. Adelaja, A. Singh, W. Roy, A. Hoffmann, Modeling single-cell heterogeneity in signaling dynamics of macrophages reveals principles of information transmission. Nature Communications (2025)
does not lead to any paper with the same or a similar title and author list. This Ref is given as a reference to the model. Fortunately, Ref 8 is helpful. Nevertheless, authors should include a schematic of the model.
Response: We apologize for the paper not being accessible on time. It is now. We have also added a schematic of the model as suggested (Figure S1) and have added detailed description of the model and simulations in introduction (Lines 95-106), results (Lines 129-141), and methods (Simulation of heterogenous NFκB dynamical responses).
2) Also Mendeley Data DOI:10.17632/bv957x6frk.1 and GitHub https://github.com/Xiaolu-Guo/Combinatorial_ligand_NFkB lead to nowhere.
Response: We thank the reviewer for this comment, and we have made the GitHub codes public. Mendeley Data DOI:10.17632/bv957x6frk.1 can be accessed via the shared link: https://data.mendeley.com/preview/bv957x6frk?a=6d56e079-d7b0-482e-951f-8a8e06ee8797
and will be public once the paper accepted.
3) Dataset 1 is not described. Possibly it contains sets of parameters of receptor modules (different numbers of sets for each module, why?), but the names of parameters never appear in the text, which makes it impossible to reproduce the data.
Response: We thank the reviewer for this comment, and we have added the description of the dataset (S3 SupplementaryDataset2_NFkB_network_single_cell_parameter_distribution.xlsx) and added the parameter names in the methods (Simulation of heterogenous NFκB dynamical responses).
4) It is difficult to understand how the simulations in response to more than one ligand are performed.
Response: We thank the reviewer for this comment, and we have improved the explanation of the methods (Results, Lines 145-152) and included a detailed description of the model and simulations for combinatorial ligands (Methods, Predicting heterogeneous single-cell responses to combinatorial-ligand stimulation).
Significance
A lot of work has been done, the methodology is interesting, but the biological conclusions are overstated.
Response: We thank the reviewer for their interest in the methodology. We have revised the title, the abstract, and added the discussion about our finding to more accurately document what we have found. In the revision, we have increased the clarity and rigor of the work. For the key conclusion that macrophages maintain some level of NFκB signaling fidelity in response to ligand mixtures, we have validated the binary classifier results on experimental data as reviewer suggested.
In the revision, we have also extended our methodology to explore further, the dose-response curves for different dosage combination for ligand pairs. This further work allowing us identified the synergistic and antagonistic regimes. By comparing the stimulus response specificity for antagonistic model vs the non-antagonistic model, we demonstrated that signaling antagonism may increase the distinguishability of presence or absence of specific ligands within complex ligand mixtures. This provides a mechanism of how signaling fidelity is maintained to the surprising degree we reported.
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