Modelling cellular signalling variability based on single-cell data: the TGFb/SMAD signaling pathway
Uddipan Sarma, Lorenz Hexemer, Uchenna Alex Anyaegbunam, Stefan Legewie
MModelling cellular signalling variability based onsingle-cell data: the TGFβ-SMAD signaling pathwayβ-SMAD signaling pathway
Uddipan Sarma , Lorenz Hexemer , Uchenna Alex Anyaegbunam , StefanLegewie
Affiliations: University of Stuttgart, Department of Systems Biology, Allmandring 30E, 70569 Stuttgart, Germany Institute of Molecular Biology (IMB), Ackermannweg 4, 55128 Mainz, Germany Vantage Research, Sivasamy St, CIT Colony, Mylapore, Chennai, Tamil Nadu 600004. India. * contributed equally correspondence: [email protected] ABSTRACT
Non-genetic heterogeneity is key to cellular decisions, as even genetically identical cellsrespond in very different ways to the same external stimulus, e.g., during cell differentiation ortherapeutic treatment of disease. Strong heterogeneity is typically already observed at the levelof signaling pathways that are the first sensors of external inputs and transmit information to thenucleus where decisions are made. Since heterogeneity arises from random fluctuations ofcellular components, mathematical models are required to fully describe the phenomenon andto understand the dynamics of heterogeneous cell populations. Here, we review theexperimental and theoretical literature on cellular signaling heterogeneity, with special focus onthe TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signaling pathway.
Keywords:
Heterogeneity, Mathematical modeling, Signal transduction, Cell-to-cell variability, ordinary differential equations, stochastic modeling, single cells, TGFβ/SMAD signaling pathway. β signaling, SMAD transcription factors, numerical simulation
Running title:
Modeling cellular signaling variability
NTRODUCTION
Heterogeneity is an implicit part of life that is observed on nearly all biological scales [1].Perhaps the most widespread effect of heterogeneity is the creation of a wide spectrum oflifeforms during evolution [2]. Primarily, heterogeneity aids robustness to a biological system ina fluctuating environment, such that, a broader niche in phenotypic traits is collectively achievedby a population of a species, which may then add to better chances of their survival [3].However, the presence of heterogeneity is observed at even finer levels - individuals within onespecies exhibit unique properties, and further, genetically identical cells within the same(multicellular) organism utilizes heterogeneity to achieve distinct cell fate decisions [1]. At thelevel of single cells heterogeneity in expression of individual gene or protein (also called cell-to-cell variability) may lead to cell-specific responses to external cues and distinct physiologicaltrajectories [1, 3]. In this chapter, we provide an overview over cellular heterogeneity in the context of intracellularsignaling. We will specifically discuss how a signaling in a heterogeneous cell population can bemodelled in silico and will point out assumptions underlying these models. The chapter isdivided in three parts: In the first Section, we describe biological systems where signalingheterogeneity plays a role in cellular decision making. Fβ/SMAD signaling pathway. urthermore, we review evidence thatsignaling heterogeneity is mostly deterministic in nature and discuss the molecular sources ofsignaling heterogeneity. In the second part, we describe mathematical modeling frameworksthat can be used to model heterogeneous cell populations and fluctuations in cellular signalingpathways. We mainly focus on deterministic modeling approaches using ordinary differentialequations, in which heterogeneity is introduced by parameter sampling and discuss approachesfor the quantitative fitting of single-cell models to experimental data. In the third part, we focuson the TGFβ/SMAD signaling pathway. /SMAD signaling pathway. SMAD signaling pathway that plays a key role in tissue homoeostasis and cellmigration, but also in diseases such as cancer. We review the literature on single-cell analysisof this pathway, and demonstrate that key features of heterogeneous TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signalingcan be understood by mechanistic modeling. We then discuss our recent modeling work, inwhich we quantitatively described cellular subpopulations of TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signaling andheterogenous signaling at the single-cell level. As an outlook for future research, we summarizehow fluctuations in signaling pathways affect noisy downstream gene expression and decisionmaking.
1. Relevance and features of cellular signaling heterogeneity
Fβ/SMAD signaling pathway. rom early microscopic observations in cell culture, it became clear that not all cells respondidentically to the same external stimulus. Intriguingly, it seems that not only genetic differencesbetween cells contribute to heterogeneity, but that non-genetic origins arising from stochasticityin cellular networks also play an important role. In recent years, evidence has accumulateddemonstrating that genetically identical cells show differences in differentiation programs [4],drug resistance [5, 6], and viral pathogenesis [7]. In the following paragraphs, we provide a briefsummary of biological systems where heterogeneity plays a functional role in cellular behavior.
Stress responses in unicellular organisms:
Heterogeneity is an important part of cellularecision making, as evidenced by stochastic cellular differentiation events, where parts of a cellpopulation randomly enter a new fate. Well-known examples include bacterial stress responses(e.g., [8]). Fβ/SMAD signaling pathway. or instance, during bacterial competence, external stress induces a regulatoryprogram in
Bacillus subtilis that allows cells to take up DNA from the environment, therebypriming them for adaptation to stress conditions. In line with a stochastic event, it was shownthat the decision to become competent is dictated by random fluctuations in the mastertranscriptional regulator ComK [8]. In some cases, stochastic fluctuations in stress networksoccur constitutively, i.e., even in the absence of external stress. This phenomenon, known asbet hedging, ensures that subsets of cell populations are prepared to rapidly respond tostressors, thereby ensuring survival of the population in case of external changes. In
Saccharomyces cerevisiae, isogenic clonal populations display a range of growth rates andslow growth predicts resistance to heat killing. At the molecular level, Tsl1, a trehalose-synthesis regulator, is a key component of the observed resistance, and cell-to-cell variability inTsl1 expression correlates with growth rate and predicts cellular survival in response to stress[9].
Cellular differentiation:
Likewise, in eukaryotes, cell fate decisions during tissue developmentappear to occur stochastically in genetically identical cell populations, and this is thought toallow for a disversificvation of tissues. At the molecular level, evidence is accumulating thatrandom fluctuations in the levels of signaling pathways and master transcription factors governcell fates (Fβ/SMAD signaling pathway. ig. 1A). Fβ/SMAD signaling pathway. or instance, in
Drosophila melanogaster cell-to-cell variability in theexpression of the transcription factor spineless creates the retinal mosaic for colour vision and isthus important for the spatio-temporal organization of the eye [10]. Moreover, subpopulations ofclonally derived hematopoietic progenitor cells with low or high expression of a stem cell marker(Sca-1) were observed to be in dramatically different transcriptional states and gave rise todifferent blood cell lineages in multipotent murine hematopoietic cell line [4]. In developing mice,cell-to-cell variability in the expression of certain genes (e.g., Fβ/SMAD signaling pathway. gf4) was found to determine theinner cell mass (ICM) lineage segregation of the blastocyst [11]. Fβ/SMAD signaling pathway. urther evidence for non-genetic heterogeneity and its impact on animal development comes from genetic mutations withincomplete penetrance. These mutations cause physiological defects only in a subset ofgenetically identical animals. Fβ/SMAD signaling pathway. or instance, during
C. elegans development, the expression levelof elt-2, a self-activating transcription factor is critical for intestinal cell-fate specification. Strongembryo-to-embryo heterogeneity in elt-2 expression and thus failure of intestinal development ina subset of embryos is observed if the upstream regulator skn-1 is inactivated by a mutation.Thus, the skn-1 mutant shows incomplete penetrance, likely because the lowered input signalshifts heterogeneous elt-2 expression to a range, where its fluctuations have profound effectson intestinal development [12] .
Tumor progression and drug resistance : Cancer cell therapy aims for a complete eradicationof tumor cells. However, in reality, complete killing is rarely achieved, as signaling pathwaysinvolved in the execution of cell death fail to be activated in all cells and certain subpopulationsare therefore resistant to the therapeutic treatment (Fβ/SMAD signaling pathway. ig. 1B). Thus, cytotoxic drugs often resultin fractional killing, especially when the drug concentration inside a tumor is limiting. In cellculture, fractional killing has been reported in response to a variety of treatments, includingchemotherapy and apoptosis-inducing receptor ligands such as TRAIL [5, 6, 13–16]. At themolecular level, fractional killing involves cell-to-cell variation in cellular regulatory molecules,e.g., in the tumor suppressor p53 in response to chemotherapy [6][. Thus, non-geneticvariability may confer resistance to therapeutic intervention and could play a role in tumorevolution [17]. These examples demonstrate that cellular heterogeneity can have a strong impact of cell fatedecisions in biological systems. It is therefore crucial to quantitatively measure cellulareterogeneity using experimental methods such as live-cell imaging, flow cytometry or single-cell RNA sequencing (reviewed in [18–23]). Fβ/SMAD signaling pathway. urthermore, predictive modeling approachesdescribing the variability of molecular networks at the single-cell level will be essential torationally manipulate cellular differentiation processes and to design effective combinatorialtherapeutic intervention strategies.
The above examples of stochastic decision making mainly focused on cellular fluctuations innuclear transcription factors. Signaling pathways controlling these master transcription factorssimilarly show strong non-genetic cell-to-cell variation that is linked to cell fate. This was initiallyshown for mitogen-activated protein kinase (MAPK) signaling and later extended to othersignaling systems. In a pioneering study, the group of James Fβ/SMAD signaling pathway. errell analyzed Xenopus oocyte maturation inresponse to the maturation-inducing hormone progesterone [24]. This maturation response ismediated by the MAPK signaling pathway and due to the large size oocytes the authors couldperform single-cell Western Blot experiments to determine cell-to-cell variation in the activity ofthis pathway. They showed that MAPK signaling was activated in an all-or-none manner withinindividual cells, i.e., every oocyte either had low or high (but not intermediate) MAPK activitylevel, and MAPK activity therefore showed a bimodal distribution (Fβ/SMAD signaling pathway. ig. 1C, right). This switch-like(ON or OFβ/SMAD signaling pathway. Fβ/SMAD signaling pathway. ) MAPK activation with strong heterogeneity between cells caused maturation inonly a fraction of cells, and ON/SMAD signaling pathway. OFβ/SMAD signaling pathway. Fβ/SMAD signaling pathway. -switching was shown to arise from a positive feedback atthe level of the signaling pathway [24]. Likewise, switch-like and heterogeneous MAPKactivation was reported to be involved in other cell fate decisions such as T cell activation [25],PC12 cell differentiation [26] and in the UV stress response mediated by the closely related JNKMAPK signaling pathway [27]. Thus, MAPK signaling frequently mediates switch-like and highlyheterogeneous cell fate decisions. However, the pathway can also be gradually activated, with aunimodal but heterogeneous distribution of activity states in the population (Fβ/SMAD signaling pathway. ig. 1C, left), e.g., inEGFβ/SMAD signaling pathway. -stimulated fibroblasts. There, the pathway transmits quantitative information to thenucleus, and switch-like cell fate decisions are established at the level of downstream gene-regulatory networks [28, 29] In conclusion, heterogeneity in MAPK signaling is a widespreadphenomenon, but the qualitative features (uni- vs. bimodal distribution) of the heterogeneouspopulation can vary, implying plasticity in network behavior. Other signaling systems seem to be less flexible in their signaling output. Fβ/SMAD signaling pathway. or instance, theapoptosis signaling system, involved in sensing cytotoxic stress and death receptor signals,seems to invariably induce heterogeneous all-or-none responses at the level of the executingcaspase enzymes (e.g., [30]). This may ensure that programmed cell death is executedcompletely and irreversibly, but only a fraction of cells in a tissue, thereby preventing completeloss of all cells in a tissue. Yet other signaling pathways such as Akt [31, 32], and TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD[33, 34] signaling show a heterogeneous, but gradual (unimodal) response in the majority ofcellular systems studied. Thus, these signaling pathways transmit quantitative information aboutextracellular stimulus concentration from the cell membrane to the nucleus, where gene-regulatory networks mediate cell fate decisions. One important question in signal transduction is how accurate information transmission ispossible despite strong heterogeneity in the activity of gradual signaling pathways. Based on acombination of quantitative experiments, mathematical modeling and concepts from informationtheory it became clear that quantitative information is frequently encoded in the temporaldynamics (i.e., the shape) of the signal [35–38]. Fβ/SMAD signaling pathway. or instance, the NFβ/SMAD signaling pathway. kB signaling pathwayhows oscillatory dynamics and the nature of activated target genes in the nucleus depends onthe frequency and amplitude of the signaling pathway oscillation [39]. Another concept forreliable signal transmision despite signaling heterogeneity is relative signal transmission, wherethe (noisy) absolute signaling activity is less important for cellular behavior when compared tothe (more robust) stimulus-induced fold-change over basal [29, 40–43]. Combined live-cellimaging and modeling studies support this concept for Erk, Wnt and TGFβ/SMAD signaling pathway. β signaling pathways,and it has also been described how downstream gene-regulatory networks in the nucleus couldrespond to fold-changes rather than absolute signaling levels [40–42]. Taken together, both gradual and bimodal signaling pathways show strong cell-to-cell variabilityin their activity. Single-cell experiments characterize quantitative information transmission andswitch-like decision making in a heterogeneous cell population and therefore provide a basis forquantitative modeling.
To model signaling heterogeneity, assumptions need to be made about the properties andorigins of the fluctuations. Evidence from the literature suggests that signaling heterogeneity isnon-genetic and that the cell-specific features of signaling can be assumed to be temporallystable during a typical cellular stimulation experiment. Cell cultures are often derived from tumor cells and are thus potentially genetically unstable.This raises the question of whether heterogeneity in signaling pathways is really non-genetic innature. Strong evidence for a non-genetic contribution to heterogeneity in signaling eventscomes from re-stimulation experiments with ligands and drugs triggering cell death (Fβ/SMAD signaling pathway. ig. 1B). Inseveral of these studies, an initial treatment killed the majority of the cell population and thecells were kept in culture for several days before they were subjected to another treatment withthe same stimulus. Even though the initial survivors were resistant to the first treatment a largepart of them became sensitive to the second treatment (Fβ/SMAD signaling pathway. ig. 1B). Thus, resistance was notinherited to all offspring of resistant cells, i.e., genetically determined, but was gradually lostduring several days of culture, arguing for an epigenetic mechanism of heterogeneity [5, 13, 44,45]. A second line of evidence for a non-genetic signaling heterogeneity came from sister cellexperiments, in which cells and their division events were tracked before stimulation to recordcellular progeny (Fβ/SMAD signaling pathway. ig. 1D). A common observation in these lineage tracing experiments is thatfreshly divided sister cells show very similar signaling responses, and that the similarity ofsisters got lost over time after the common division event. Fβ/SMAD signaling pathway. or instance, for TRAIL-inducedapoptosis, it was shown that freshly divided sister cells have a high correlation in the death time,i.e. the time it takes for the TRAIL stimulus to induce cell death [46]. Thus, the experimentsuggests that sister cells are initially in a very similar (signaling) state, but interestingly theyloose the signaling similarity with a characteristic half-life of 11h (after the common divisionevent). Similar observations were made for other regulatory networks, including the cell cycle,the spindle assembly checkpoint, MAPK as well TGFβ/SMAD signaling pathway. β signaling [33, 47–50]. Sister cellexperiments have several important implications for cellular signaling heterogeneity, andmathematical modeling thereof:1. Fβ/SMAD signaling pathway. irst, they further support non-genetic sources of heterogeneity, since the loss of sister cellssimilarity on a time scale of hours is much faster than any genetic drift due to DNA mutations.. Second, the initially high sister cell similarity suggests that heterogeneity does not arise fromstochastic dynamics in signaling reactions: If signaling molecules are present in very lowamounts, signaling reactions (e.g., phosphorylation) could be probabilistic events that give riseto strong heterogeneity and high dissimilarity in the signaling events of freshly divided sisters (oreven high dissimilarity if the same cell would be stimulated repeatedly). Certain signalingpathways indeed show such stochastic dynamics (see Section 2.3 and [51]), but the sister cellexperiments suggest that this is typically not the case, which agrees with the fact that signalingproteins are often expressed at high molecule numbers [52].3. Third, the time scale of sister cell similarity (multiple hours) suggests that signalingheterogeneity can be assumed to be stable at the time scale of a typical stimulation experiment(minutes to hours). The conclusion of temporal stability is also supported by a recentrestimulation analysis with insulin-like growth factor, in which the rank of single-cells withrespect to Akt signaling activity was stable over several hours [32] . These conclusions greatly simplify the mathematical modeling of cellular signaling pathways, asstochastic modeling of signaling dynamics can be neglected in most cases. Instead, adeterministic ordinary differential equation approach can be used, in which certain kineticparameters are assumed to be different between cells, but stable over time.
To introduce heterogeneity into a mathematical model, we need to know the molecular sourcesof temporally stable signaling pathway fluctuations. Obvious sources for signaling fluctuationsare cell-to-cell differences in cell cycle stage [19] or cell density [53]. In growing adherentmammalian cells, cell divisions combined with cell motility can create variations in local celldensities, cell–cell contacts, relative location, and the amount of free space per cell. Combined,these parameters constitute the population context of an individual cell [54, 55]. However,single-cell signaling studies exist where these sources have been excluded experimentally, butthe heterogeneity in the signaling output persists [33, 48, 51]. In recent years, it has become apparent that random fluctuations in the total cellularconcentrations of signaling proteins may underlie temporally stable cell-to-cell variability insignaling pathway activity (Fβ/SMAD signaling pathway. ig. 1E). In some cases, signaling heterogeneity could be tracedback to fluctuations in the expression of specific signaling molecules, e.g., in MAPK [25],PI3K/SMAD signaling pathway. Akt [56] and JAK/SMAD signaling pathway. STAT [57] signaling. Fβ/SMAD signaling pathway. or instance, a pioneering work on T cell activationshowed that MAPK activation depends on cell-to-cell variability in the expression of the CD8 co-receptor and the antagonizing phosphatase SHP-1. Interestingly, even though each proteincontributes to variability, co-regulation of CD8 and SHP-1 expression levels at the single-celllevel limits diversity and promotes robustness of signaling [25]. In other systems, signalingheterogeneity cannot be explained by fluctuations in a single protein, but the control seemsdistributed over many protein levels. Fβ/SMAD signaling pathway. or example, in their work on apoptosis signaling, Spenceret al showed that no single protein level could accurately predict cell death timing in response toTRAIL treatment, unless a specific pathway regulator (BID) was overexpressed and its cellularlevel then became a good predictor of cell death timing [46]. Even in genetically identical cell populations, (signaling) proteins show strong cell-to-cellvariability in their levels, since epigenetic control and gene expression are stochastic events atthe single-cell level (see also Section 2.3): Each gene is present only at two copies (alleles) percell and gene regulation at such low molecule numbers is probabilistic [58–61]. As aconsequence, each (signaling) protein level follows a log-normal distribution, that is, the fold-change of a protein is normally distributed around its population mean [46, 58, 62]. Although theean and standard deviation of the distribution are protein-specific, a typical human proteinshows a three-fold difference in expression across the cells of a population (where the fold-change is measured between the 10 th and 90 th precentile of the distribution) [47, 58]. This strongvariation in each and every signaling protein leads to strong fluctuations in signaling. Time-resolved measurements further suggest that fluctuations in signaling protein expression andpathway activity are coupled: when protein expression fluctuations in mammalian cells arefollowed over time, the time scale of stochastic changes in protein expression is similar to thetime scale of signaling pathway desynchronization between sister cells (multiple hours to days)[46, 58]. This is consistent with a model, in which sister cells initially share the same proteomecontent and signaling activity, and then over time and simultaneously loose both types ofsimilarity due to stochastic gene expression fluctuations [46, 47, 58]. Taken together, these observations indicate that mammalian signaling proteins show strongfluctuations in their levels. In the following, we will discuss deterministic modeling approaches ofcellular signaling pathways in which cell-to-cell variability in signaling protein concentrations istaken into account. METHODS
2. Mathematical modeling of cellular heterogeneity
Signaling pathways often respond in very similar manner to stimulation when the same cell isstimulated repeatedly, or when sister cells are in a similar state and harbor a similar proteomecontent. These observations suggest that cells respond deterministically to stimulation and thatdeterministic mathematical modeling approaches can be used to simulate cellular heterogeneity in silico . Such deterministic models are often based on ordinary differential equations (ODEs)which represent reaction networks within the cell, typically using mass action-based reactionkinetics (reviewed in [63]). ODE models assume that the biochemical molecules in the cell arepresent in sufficiently large amounts (and well-stirred), so that stochastic fluctuations at thesingle-molecule level can be neglected and the biochemical species can be described usingcontinuous variables, representing average molar signaling protein concentrations within thecell. If an ODE system is simulated twice with the same set of kinetic reaction parameters andinitial conditions (i.e., protein concentrations) it will yield exactly the same solution, representingthe deterministic nature of the apporach. This determinism is in contrast to stochastic simulationalgorithms (reviewed in [64]) which explicitly describe single-molecule fluctuations and thereforegive rise to distinct simulation results for each realization. In this Section, we discuss how intracellular signal transduction in a heterogeneous cellpopulation can be modelled in silico . We mainly focus on deterministic ODE-based modeling,and point out how these models can provide insights into several aspects of cellularheterogeneity, including noisy decision making and the robustness of signaling networks. Wethen discuss how these models can be calibrated based on single-cell data to obtain aquantitative match between experiment and theory. Fβ/SMAD signaling pathway. inally, we briefly summarize stochasticmodeling approaches that are relevant for the modeling for some signaling networks operatingat low molecule numbers and particularly for gene-regulatory networks involved in cell fatedecisions downstream of signaling pathways. .1 Applications and limitations of population-average models
Experiments aimed at understanding intracellular processes were tradionally performed in bulk,combining material from thousands to millions of cells. Fβ/SMAD signaling pathway. or instance, signaling pathwaydynamics were studied using Western Blot experiments, in which phosphorylated signalingintermediates are detected using phospho-specific antibodies [24, 65]. Due to averaging over alarge number of cells, these experiments do not provide information about single-cellheterogeneity, but only represent the behavior of one hypothetical average cell. In systemsbiology, early quantitative models were built based on the available population-average dataand therefore describe one representative cell (Fβ/SMAD signaling pathway. ig. 2A). Even though not meant to represent cellular heterogeneity, population-average models canprovide important insights into several signaling phenomena at the single-cell level includingcellular decision making and robustness of networks against fluctuations in their components.Fβ/SMAD signaling pathway. or instance, Fβ/SMAD signaling pathway. errell et al (1998) showed that the decision of Xenopus oocyte maturation inresponse to progesterone involves an all-or-none biological response at the level of MAPKsignaling (see Section 1.2; [24]). To better understand this single-cell phenomenon, the authorsconstructed a population-average model of the signaling network, and concluded that switch-like(bistable) behavior in the MAPK cascade arises from a positive feedback loop that amplifies thesignal once MAPK signaling exceeds a certain threshold. Likewise, other population-averagemodeling studies provided insights into mechanisms of switch-like decision making andtherefore have implications for bimodal behavior at the single-cell level [26, 66–71]. Population-average models further provided insights into biological robustness against fluctuations insignaling protein concentrations [25, 72–75]: Fβ/SMAD signaling pathway. or example, using single-cell experiments,Kamenz et al (2015) observed that the timing of mitotic events is highly robust and is bufferedagainst variations in the concentrations of mitotic regulatory proteins [75]. A population-averagemodel could explain the observed robustness and predicted conditions where mitotic timing iscompromised. Thereby, the population-average model identified critical transitions in thenetwork and experimental validation showed that these transitions led to network failure in asubset of cells due to cell-to-cell variability in the molecular components. In general, for gaininginsights into robustness, a so-called sensitivity analysis can be performed, in which the initialconditions and kinetic parameters are systematically perturbed (usually one at a time) tounderstand their role in the behavior of the network. Given that cellular signaling fluctuationsoften arise from cell-to-cell variability in signaling protein expression (Section 1.4), sensitivityanalyses focusing on the impact of signaling protein concentrations at the population level canprovide valuable insights into biological variability and robustness in single cells [25]. However,though potentially useful, population-average models do not directly represent signalingdistributions across single cells (as those shown in Fβ/SMAD signaling pathway. ig. 1C), and therefore do not allow for aquantitative comparison of the model simulations to a single-cell experiment. Fβ/SMAD signaling pathway. urthermore, apopulation-average model constructed based on population-average data may lead tomisleading conclusions for cell fate decision networks with ON/SMAD signaling pathway. OFβ/SMAD signaling pathway. Fβ/SMAD signaling pathway. -behavior [24, 62]: In binarydecision making, single cells show show all-or-none (digital) signaling, but do soheterogeneously for a given stimulus concentration. Hence, the activity distribution is bimodal(Fβ/SMAD signaling pathway. ig. 1C), and the mean response of the population gradually increases with increasingstimulation. Therefore, at the population-average level, digital responses may appear gradual(analogue) when averaged. Taken together, population-average models fail to quantitatively describe single-cell distributionsand can only be as insightful as the experimental data they are based on. Models that aredesigned solely based on population-average data implicitely assume that the population-average data is a good approximation for the true underlying single-cell behavior, and thereforeay lead to wrong conclusions. To overcome, these limitations, models of cellular heterogeneitywere developed to quantitatively describe the cell-to-cell variability in signal transduction.
How does a deterministic model of cellular signaling heterogeneity look like? In most cases,heterogeneity is simulated using an ensemble of single-cell models (Fβ/SMAD signaling pathway. ig. 2A). Here, individualcells are described by repeating the simulation using the same deterministic ODE model, eachsimulation run corresponding to one cell, and cell-to-cell variability is introduced by perturbingthe model in each realization. Based on the arguments presented in Section 1.4, the perturbation leading to cell-to-cellvariability is mainly introduced by assuming cell-specific signaling protein concentrations.Additionally, kinetic parameters can be assumed to be cell-specific if the corresponding reactionrates are in turn controlled by (fluctuating) proteins (this may, for example, be true for enzymaticreactions). Given that protein expression levels are log-normally distributed (see Section 1.4),single-cells are modeled by repeated simulations in which all protein concentrations in themodel are sampled from independent log-normal distributions (Fβ/SMAD signaling pathway. ig. 2A). In this approach,sometimes termed Monte-Carlo sampling or non-linear mixed effect modeling, the proteinfluctuations are typically restricted to a biologically reasonable range. Specifically, the coefficientof variation of the lognormal distribution (CV = std /SMAD signaling pathway. mean) is chosen between 0.1 and 0.4 [46,58, 62]. Since proteins from the same pathway may co-regulated at the single-cell level [58],sometimes correlated fluctuations in signaling protein fluctuations are assumed [33]. Notably,sampling only the initial total protein concentrations and leaving the model otherwiseunperturbed, assumes that non-genetic sources of signaling heterogeneity are temporally stableduring pathway stimulation. Hence, the modeling framework captures the key features ofcellular signaling heterogeneity, including deterministic behavior, temporal stability and proteinconcentrations as a noise source (Section 1). Compared to a population-average model, such single-cell ensemble modeling approachesreproduce the complete heterogeneous cell population and allow for a quantitative comparisonof the model to experimental single-cell data. Specifically, while the population-average modelby definition only represents the mean, ensemble models capture higher momentum statistics ofthe heterogeneous model species such as the standard deviation, their correlations or timecourse features such as the autocorrelation function (e.g., [46]). Deterministic ensemblemodeling approaches have been used in a several studies on signaling, often in combinationwith experimental analyses at the single-cell level [14, 15, 76–85, 16, 86, 87, 35, 41, 46, 49, 62,70, 72]. These models provided experimentally testable predictions and led to a betterunderstanding of heterogeneous signal transduction. In particular, the following phenomenawere analyzed (Fβ/SMAD signaling pathway. ig. 2C):
1) Sources of signaling fluctuations:
Ensemle models were used characterize howfluctuations in individual signaling protein expression levels affect the signaling outcome.Thereby, the most critical signaling protein expression fluctuations could be identified as mainsources of cell-to-cell variability in signal transduction [16, 33, 46, 49, 57, 87–89]. This led to abetter understanding of molecular mechansisms causing heterogeneous cellular decisionmaking.
2) Design principles of biological robustness:
Robust and reliable signal transduction mustoccur despite strong noise. By adding or removing certain reactions in the models, insights wereained into design principles that mitigate signaling variability [41, 62, 72, 80], therebypromoting robustness, e.g., during embryonic development [72]. Several robustness-promotingnetwork motifs could be identified including negative feedback [72, 80], fold-change detection[41] and correlated expression fluctuations in positive and negative regulators of signalingpathways [80, 86].
3) Characterization and manupulation of heterogeneous decision making:
Fβ/SMAD signaling pathway. or cellulardifferentiation and during stress responses, cell-to-cell variability may be beneficial, as not allcells of a heterogeneous population enter a new fate and die in response to stress, respectively(see Section 1.1). Modeling of signaling pathways involved in cellular differentiation and celldeath allowed for a quantitative analyses of decision making at the level of signaling, and thusfor the emergence of bimodal signaling distributions. The models yielded predictions for thereprogramming of cell fates for novel experimental conditions [83, 84] and allowed for theoptimization of therapeutic treatment responses in cell culture [14–16, 82]
4) Insights into alternative biological mechanisms and network topologies:
Since certainnetwork motifs affect the characteristics of biological fluctuations (see above), attempts weremade to infer (reverse engineer) the wiring of signaling networks based on single-cell data. Theidea is that signaling fluctuations contain a fingerprint for the underlying molecular interactions,and that the model topology that best describes the signaling fluctuations is the most probableone. Several studies used a defined model topology and compared a set of relatively minormodifications in the model against single-cell data [85, 90]. In a less biased top-down approach,Sachs et al inferred the topology of signaling networks from single-cell data without priorknowledge using a Bayesian framework [91].
5. Integration of single-cell and population-average data:
Ensemble models of cellpopulations allow for simulations at both the single-cell and population-average levels, and canthus be used to integrate both types of data [33, 57, 85]. Thereby, the models on the one handexploit highly informative single-cell data which often can be done only at low throughput and forfew molecular species (especially for time-resolved live-cell imaging). On the other hand, theytake into account population-average information that can more easily collected for multipleexperimental conditions and molecular species. Accordingly, the integration of population-average and single-cell data led to a better discrimination of competing model hypotheses whenfitting an emsemble model to experimental data [85].
Signal transduction cascades typically control cellular decisions by activating gene expressionresponses in the nucleus. Expression of target genes (e.g., cell cycle regulators or cell adhesionmolecules) then controls the morphological features of a cell such as cell division and migration.In addition, target genes often act as negative feedback regulators that downregulate the signalonce gene expression has been activated [52]. Thus, signaling and gene expression responsesare intimately connected, and both may need to be taken into account in realistic models ofcellular decision making. In this context, it should be pointed out that determinstic models mayno longer be suitable for modeling of cellular heterogeneity if gene expression is taken intoaccount, e.g., for modeling transcriptional feedback or nuclear propagation of the signal. The reason is that gene regulation is an intrinsically stochastic process with strong temporalfluctuations (reviewed in [59], although deterministic sources of heterogeneity sources (i.e., thecellular state) also seem to play a role [54, 60]. Stochastic behavior arises from the fact thatranscriptional regulators are typically expressed at very low levels, and that a cell contains onlytwo copies of each gene. As a result, random (Brownian) fluctuations at the level of individualreactions are not averaged out and significantly impact on the activity of a gene, especially atthe level of mRNA production. Therefore, stochastic approaches are typically used for modelingheterogeneity of gene expression [59, 64]. In early work, Arkin and colleagues used theGillespie algorithm to simulate biochemical reactions leading to gene expression, and predictedstochastic cell-to-cell variation in protein numbers for biologically realistic parameter ranges [92,93]. The prediction of stochastic mRNA and protein expression was later confirmedexperimentally in bacteria and mammalian cells (reviewed in [59]). In higher organisms, noiseseems to be larger in magnitude compared to bacteria, since chromatin states seem to give riseto switching of genes between ON and OFβ/SMAD signaling pathway. Fβ/SMAD signaling pathway. states. In time courses of single-cell geneexpression, this is observable as transcriptional bursts, i.e., episodes of high gene expressionthat are separated by phases with low activity [60, 61, 94, 95]. The simplest stochastic modelwhich realistically describes transcriptional bursts is the so-called random telegraph model,where a gene promoter is assumed to reversibly switch between a transcriptional active and aninactive state (reviewed in [59]). Notably, depending on the gene under consideration morepromoter states may need to be taken into account to describe the data [94, 96]. To jointlymodel signal transduction and gene expression, these stochastic promoter models werecoupled to deterministic models of signaling, and this yielded insights into the dynamics of targetgene expression [97, 98] and into the long-term regulation of signaling heterogeneity bystochastic signaling protein expression fluctuations [82]. In certain cases, intrinsic stochastic dynamics may arise within the signaling network, especialyif the pathway operates at low molecule numbers [51, 98, 99]. The level of initial signal sensingmay be especially prone to stochastic dynamics, since cell surface receptors are often onlyexpressed at a few hundreds or thousand molecules per cell [52, 100]. Then, processes likereceptor endocytosis which simultaneously remove hundreds of molecules at once from the cellsurface, may give rise to digital behavior and consequently strong stochastic fluctuations ofsignaling activity [101]. Accordingly, stochastic models of signaling networks have beenproposed to describe heterogeneous decision making [51, 97, 99, 100], and several of thesestudies focussed on fluctuations at the receptor level [97, 99, 100].
In many cases, ensemble modeling approaches are semi-quantitative in the sense that thekinetic reaction parameters and the protein fluctuations (i.e., the standard deviation of theirdistribution) are tuned manually. While such semi-quantitative modeling is valuable in manycases, the long-term goal is a quantitative match between model and experiment. This can beachieved by directly fitting the single-cell models to single-cell data by minimizing the differencebetween the simulated and measured single-cell distributions. Quantitative singe-cell model fitting of signaling and gene expression has now been applied in anumber of publications and is an lively area of research [57, 60, 85, 86, 88–90, 94, 102–111]. Inseveral of these studies, the fitted models involve deterministic sources of heterogeneity [85, 86,107], stochastic fluctuations [94], or a combination of both [60, 88, 105]. In the deterministiccase, only signaling protein concentrations may be cell-specific parameters [85] or all modelparameters may show cell-to-cell variability [89]. Fβ/SMAD signaling pathway. or a comprehensive overview over theassumptions and the computational methods, we refer to the recent review by Hasenauer andLoos [112]. he methods for model calibration can be classified based on the type of experimental data theyuse, single-cell snapshot or time course data (Fβ/SMAD signaling pathway. ig. 2D and [112]). Fβ/SMAD signaling pathway. or snapshot data, onlydistributions at single time points are considered and therefore potential correlations betweenobservations at consecutive time points are neglected. Despite this limitation, the approachbenefits from the fact that snapshot data can typically be generated on a higher throughput, i.e.,for more cells and molecular species, when compared to time-resolved measurements. Fβ/SMAD signaling pathway. orinstance, high-throughput snapshot data can be generated using flow cytometry, masscytometry or single-cell RNA sequencing, and the larger wealth of information should bebeneficial for the training of reliable mathematical models. Accordingly, several approacheswere were proposed for the model calibration based on snapshot data [88–90, 102, 104, 111].In deterministic models, different assumptions were made about the type of fluctuations inparameter values, ranging from the unimodal log-normal distribution of to multimodaldistributions, or even no specific assumption was made about the nature of fluctuations (non-parametric distribution; reviewed in [112]). Fβ/SMAD signaling pathway. or instance, Hasenauer et al. employed multi-modalmixtures of normal parameter distributions to infer subpopulations (with distinct meanparameters) by fitting a model NGFβ/SMAD signaling pathway. signaling to snapshot data [102]. A limitation of the snapshot approach is that essential information about temporal behavior insingle cells gained from live-cell imaging (e.g., an oscillatory pattern) may be lost. Therefore,snapshot information may be less well suited for the identification of molecular sources ofheterogeneity when compared to time-resolved data (discussed in [110]). As a consequence,several studies suggested to directly fit a mechanistic model to single-cell time course data [85,86, 103, 105–110]. In naive approach, each individual cell could be fitting separately byminimizing the residuals between model and data, and then the single-cell fitting results arecombined to yield cell population distributions of interest, e.g., for signaling protein expressionlevels. In this so-called standard two-stage approach, each cell is thus analyzed as aindependent sub-problem (stage 1), and then the cell population is assembled (stage 2).However, stage 1 suffers from the problem that the model parameters in systems biologymodels can almost never be correctly estimated (identified), especially based on live-cellimaging data which typically only covers one molecular species for each cell. Thus, the fittinguncertainties are high and the heterogeneity between cells is overestimated [106, 107] whichlimits the predictive power of the two-stage approach unless very small models are considered.To circumvent this problem, information about the cell population distribution needs to be takeninto account during fitting of single-cells [85, 106, 107, 110]. Specifically, the fitted likelihoodfunction combines information from all cells, and these additional constraints improve theidentifiability of single-cell parameters which leads to smaller uncertainties in model predictions.Fβ/SMAD signaling pathway. or instance, as a constraint in deterministic models, it can be ensured that proteinconcentration fluctuations follow a log-normal distribution [85, 106]. Moreover, the model can besimultaneously fitted to single-cell and population-average data, and it has been shown that thecombination of both types leads to a better discrimination of model variants compared to the useof either alone [85]. It should be noted that the current quantitative models of cellular signaling and gene expressionheterogeneity are typically limited to a few species and reactions. Therefore, qualitativeensemble modeling approaches (Section 2.2) are still very valuable for large-scale networksand typically led to more profound “biological” insights when compared to quantitativeapproaches. Fβ/SMAD signaling pathway. urther improvements are needed in the computational methods for quantitativemodel fitting to reduce computational cost and to integrate various types of data including cross-sectional snapshots, high-resolution live-cell imaging and population-average data. This willimprove identifiability of model parameters, the certainty of model predictions and will be helpfulo discriminate competing model variants also in larger networks.
3. Heterogeneity in TGFβ-SMAD signaling pathwayβ signaling - modeling and impact on cellularbehaviour
In the final part of the chapter, we discuss the characteristics and modeling of TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMADsignaling at the single-cell level. We initially start with an overview over the pathway and its rolein controlling cell fates. Then, we summarize its dynamic features at the single-cell level andoutline how population-average as well as single-cell modeling approaches provided insightsinto the pathway dynamics. Fβ/SMAD signaling pathway. inally, we review recent work, in which the link betweenfluctuations in SMAD proteins and target gene expression was explored.
TGFβ/SMAD signaling pathway. β belongs to a family of soluble extracellular ligands that activate intracellular signaling bybinding to cell surface receptors. As depicted in Fβ/SMAD signaling pathway. ig. 3A, signaling is initiated by a cascade ofevents that involves TGFβ/SMAD signaling pathway. β binding to the TGFβ/SMAD signaling pathway. βR2 receptor, and this receptor-ligand complex inturn binds to the TGFβ/SMAD signaling pathway. βR1 receptors (also known as ALK5) to build an activated receptorcomplex [113] . The active TGFβ/SMAD signaling pathway. β receptor functions as as a intracellular kinase thatphosphorylates cytoplasmic SMAD2/SMAD signaling pathway. 3 proteins which upon phosphorylation form heterotrimerswith SMAD4 (e.g., (SMAD2) (SMAD4)). SMAD heterotrimers translocate to the nucleus andthere act as transcription factors, i.e., they bind to and activate gene promoters to regulate thetarget gene expression [114]. The signaling pathway activity is terminated by nucleardephosphorlyation of SMAD proteins, dissociation of the complexes and finally the nuclear exitof SMAD proteins. TGFβ/SMAD signaling pathway. β induces several cellular responses including cell cycle arrest, apoptosis and cell migration[115]. TGFβ/SMAD signaling pathway. β-induced cell migration typically involves the so-called epithelial-to-mesenchymaltransition (EMT), a phenotypic remodeling of cells in which the cytoskeletal reorganization andloss of cell-cell junctions allows epithelial cells to evade from their original location by acquiringa motile, migratory, mesenchymal phenotype [116]. Given these widespread roles in cellularremodeling, it is not surprising that TGFβ/SMAD signaling pathway. β and closely related ligands (e.g., GDFβ/SMAD signaling pathway. 11 or BMPs)play a critical role in embryogenesis and tissue homeostasis, but also in diseases such ascancer or fibrosis [117]. Fβ/SMAD signaling pathway. or instance, in higher vertebrate development, gastrulation and neuralcrest formation depend on EMT induced by TGFβ/SMAD signaling pathway. β superfamily members [116] . Likewise, TGFβ/SMAD signaling pathway. βinduced apoptosis and cell cycle arrest maintain tissue homeostasis and prevent overgrowth indeveloping and adult tissues, e.g., in the liver [118]. If the cytostatic effect of TGFβ/SMAD signaling pathway. β or is itsability to induce apoptosis is lost this leads to tumor progression. However, TGFβ/SMAD signaling pathway. β signaling notonly acts as a tumor suppressor, but plays a dual role in cancer progression, as in late-stagetumors aberrant TGFβ/SMAD signaling pathway. β induced EMT signaling promotes the formation of metastasis [119].Thus, during cancer development, a specificity switch occurs, in which TGFβ/SMAD signaling pathway. β signaling no longerpromotes cytostatic responses, but mostly induces cell migration. At the molecular level, this specificity switch involves a change in the set of target genesregulated by TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signaling: In late-stage tumors, TGFβ/SMAD signaling pathway. β no longer downregulatesgrowth-promoting oncogenes like Myc and fails to upregulate cell cycle inhibitors (e.g., p15,p21). Instead, TGFβ/SMAD signaling pathway. β induces gene expression changes that are crucial for mediating early stepsof reprogramming from epithelial to mesenchymal identity including the downregulation ofclassical epithelial and upregulation of mesenchymal markers [119]. Based on experimentalevidence, several hypotheses have been been proposed to explain how the same signalingathway can induce qualitatively distinct gene expression responses depending on the cellularcontext: (1) context-specific expression of transcription co-factors involved in SMAD-dependentgene expression [120], (2) alterations in the concentrations of SMAD2 and SMAD3 each ofwhich controls specific sets of target genes [121] or (3) encoding of specific gene expressionprograms by the temporal dynamics of the SMAD pathway [122, 123]. Specifically, it has beensuggested that a transient SMAD signal may be sufficient for EMT and cell migration, whilesustained signaling additionally triggers cell cycle arrest [122]. Thus, quantitative insights intothe pathway dynamics by time-resolved live-cell imaging [18] and mathematical modeling areimportant to better understand cellular responses to TGFβ/SMAD signaling pathway. β stimulation. At the single-cell level, individual cells respond very differently to TGFβ/SMAD signaling pathway. β treatment. Due to thisheterogeneity, time-resolved analyses of SMAD signaling at the single-cell level are valuabletools to understand the link between signaling dynamics, gene expression and cellular outcome.In fact, single-cell studies further supported that the amplitude and/SMAD signaling pathway. or duration of the SMADsignal partially determines whether a cell will react at all to stimulation and/SMAD signaling pathway. or whether it willrespond with migration or cell cycle arrest [33, 36]. Established experimental readouts of TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signaling at the single-cell level includemeasurements of receptor levels and their internalization [124], nuclear translocation of SMADproteins [33, 34, 41, 125, 126], SMAD trimerization [127] and SMAD-induced gene expression[34, 72]. At the signaling level, SMAD nuclear translocation assays are most widely used. Theyrely on the mild overexpression of SMAD2 or SMAD4 fluorescent fusion proteins, and thenuclear translocation of the fluorophore is then used as a proxy for pathway activation. [33, 34,36, 41, 126, 128] By automated microscopy, images are taken on a temporal resolution of a fewminutes. Subsequently image analysis is performed to track cells, and to segment them intonuclear and cytoplasmic compartments. The amount of SMAD-associated fluorophore innucleus or the nucleus-to-cytoplasmic fluorescence ratio is then used as a measure of signalingpathway activity. Depending on the study, this technique allowed for signaling analysis in 250-1500 single cells per condition over a time frame of 45 min to 24h. These large-scale single-cell datasets indicated that heterogeneous SMAD signaling at thesingle-cell level exhibits a few key features that are recurrently observed across experimentalgroups and cellular systems:
1) SMAD signaling is gradual at the single-cell level (Fig. 3C):
In Section 1, it was discussedthat certain signaling pathways like the MAPK and apoptosis cascades, show bimodal behavior,i.e., complete or no activation in individual cells. Available single-cell studies on TGFβ/SMAD signaling pathway. β signalingsuggest that this pathway rather acts like a gradual continuum, i.e., snapshot histograms ofSMAD2 nuclear translocation show a unimodal distribution with strong cell-to-cell variability (Fβ/SMAD signaling pathway. ig.3C). With increasing TGFβ/SMAD signaling pathway. β doses, this mean value of this continuous distribution gradually shiftsto higher signaling levels [33, 34, 41, 125, 128]
2) Single cells show qualitative differences in signal shape (Fig. 3B):
Single-cell analysesshow strong differences in the absolute level of SMAD2 or SMAD4 nuclear translocationbetween cells at a given time point [33, 34, 36, 41, 128]. In time-resolved analyses, individualcells may also be distinct in the shape of the signal, e.g., in the kinetics or the degree ofadaptation to a lower pleateau after the initial peak amplitude. Such heterogeneity in the shapeof the signal was observed either at the level of SMAD2 or SMAD4 nuclear translocation [33,34, 125], or at the level of target gene expression [34]. In contrast, in other cellular systems theshape of SMAD2 nuclear translocation was fairly similar between cells [129]. Clusteringtechniques using dynamic time warping as a similarity measure between cells were used to sorthe trajectories of individual cells into classes with qualitatively different dynamics [33, 34].Using this clustering approach, we found that even for a given TGFβ/SMAD signaling pathway. β dose some cells do notrespond to the stimulus (non-responders), others show a transient response, whereas theremainder show sustained pathway activation (Fβ/SMAD signaling pathway. ig. 3B). At very low TGFβ/SMAD signaling pathway. β stimulation, mostcells belong to the non-responding cluster, whereas at intermediate and high TGFβ/SMAD signaling pathway. β doses, thetransient and sustained clusters predominate, respectively. Population-average measurementsare a mixture of these qualitatively distinct responses and therefore only partially cover thecomplexity of the pathway at the single-cell level. Interestingly, the signalling clusters are betterpredictors for TGFβ/SMAD signaling pathway. β-induced cell migration and division when compared to the appliedextracellular ligand dose [33]. This further suggests that the cellular decisions are linked toSMAD signaling dynamics at the single-cell level.
3) Single cells show burst-like shuttling of SMAD proteins (Fig. 3B):
Upon stimulation, thepopulation-average response of the SMAD signaling pathway typically shows an initial peakamplitude ~60 mins after stimulation. Afterwards, the population-average signal slowly declinesover a time scale of several hours, but may remain constantly elevated, e.g., upon strong TGFβ/SMAD signaling pathway. βstimulation, but this depends on the cellular context. Fβ/SMAD signaling pathway. or such sustained behavior, the single-cellresponse is distinct from the population average and shows repeated bursts of nucleartranslocation (Fβ/SMAD signaling pathway. ig. 3B). Specifically, the nuclear SMAD2 or SMAD4 levels decline strongly afterthe initial peak, before again reaching once or multiple times levels comparable to the level ofthe initial peak [33, 34]. This did not appear to a technical artefact of imaging, as a genericnuclear marker (H2B) did not show bursting behavior [130]. Fβ/SMAD signaling pathway. urthermore, SMAD4 bursts werereported in developing Xenopus embryos in which TGFβ/SMAD signaling pathway. β family ligands play an important role,and the behavior could be reproduced in isolated animal cap explants [128] Interestingly, thesepulsatile SMAD translocation dynamics are irregular in their timing intervals and amplitudes,suggesting that they may, in part, arise from stochastic dynamics of the SMAD signalingpathway.In the following, we will discuss how mathematical models can provide insights into thesedynamical features at the single-cell level. We will first review population-average models ofTGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signaling and will then turn to modeling approaches at the single-cell level.
Early kinetic models of TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signaling mainly focused on the description of Western Blotmeasurements of SMAD2/SMAD signaling pathway. 3 phosphorylation and complex formation [131–139]. Since theseexperimental methods provide average quantifications of thousands to millions of cells, theresulting models describe the behavior of a representative average cell and fail to captureheterogeneity in the population. Population-average models of SMAD signaling are typicallybased on deterministic ordinary differential equations (ODEs), and the individual reaction stepsare formulated based on mass-action kinetics. Models proposed in literature have beenreviewed elsewhere [140, 141] and differ in the level of detail they consider and in the reactionmechanisms they focus on. Still, most of the models share a set of key mechanisms includingreceptor-ligand binding, receptor shuttling to the endosome, receptor-mediated SMADphosphorylation SMAD (de)phosphorylation, trimerization and nuclear translocation, as well astranscriptional negative feedback via target genes that, for instance, inhibit receptor signaling(Fβ/SMAD signaling pathway. ig. 3A). The kinetic parameters are typically not known and were estimated by fitting themodels to experimental data [33, 135, 142], or the parameter space was explored by randomsampling or sensitivity analysis [132, 134, 143, 144]. Interestingly, population-average models alongside with quantitative experiments reproducednd provided insights into several features of heterogeneous TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signaling at thesingle-cell level including the gradual behavior of the pathway, transient vs. sustained signalingand pulsatile pathway dynamics. Thus, they provide hints to mechanisms of heterogeneity andserve as a basis for deterministic modeling of variability at the single-cell level.
1) Gradual dose-response behavior:
Population-average modeling studies and dose-response measurements revealed gradually increasing SMAD signaling in response toincreasing doses of TGFβ/SMAD signaling pathway. β. Specifically, intracellular SMAD signaling exhibits a shallow dose-response curve with a Hill coefficient (n H ) of close to or less than 1 [33, 72, 136, 145]. Modelingstudies further revealed that switch-like dose-response behavior (n H =4.5) late after stimulation(24 h) is not an inherent feature of the SMAD signaling pathway, but arises from degradation ofextracellular TGFβ/SMAD signaling pathway. β in the cell culture dish [136, 146]. Hence, the SMAD signaling pathwaymodels respond gradually to perturbations (in both TGFβ/SMAD signaling pathway. β concentration and intracellular proteinconcentrations) and are therefore consistent with a unimodal SMAD nuclear translocationdistribution at the single-cell level.
2) Features controlling signal amplitude and duration:
Sensitivity analysis of population-average models revealed the relative importance of individual reaction steps in controlling thesignal amplitude and duration [133–136]. It seems that the signal duration (i.e., the shape of thesignal) is mainly set by the kinetics of receptor-ligand binding and receptor shuttling.Accordingly, SMAD nuclear translocation cycle typically shows similar dynamics as the receptorlevel and mainly acts as a remote sensor that directly reflects receptor changes, though with aslight time delay of ~10 minutes [133, 147]. Molecular mechanisms that control the signalingdynamics at the receptor (and thus the SMAD) level include: (i) Receptor downregulation fromthe cell surface by internalization of receptor-ligand complexes into endosomal compartments[137, 148]; (ii) Cell-mediated degradation of extracellular TGFβ/SMAD signaling pathway. β, again by internalization ofreceptor-ligand complexes and subsequent intracellular degradation of the ligand. [131, 136].This mechanism of signal termination becomes an important factor in controlling the length ofthe signal if the number of extracelluar TGFβ/SMAD signaling pathway. β molecules per cell is limiting (low TGFβ/SMAD signaling pathway. βconcentration and/SMAD signaling pathway. or small extracellular medium volume) (iii) Negative feedback of SMAD targetgenes to the receptor level, e.g., by SMAD-induced expression of inhibitory SMAD7 and BAMBIproteins. These proteins bind to TGFβ/SMAD signaling pathway. β receptor complexes, thereby inhibiting their kinaseactivity and targeting them for degradation [33, 134, 139]. Taken together, the signal duration iscontrolled by multiple mechanisms at the receptor level. Using sensitivity analysis of population-average models, a similar multi-level regulation by many reaction steps in the pathway wasshown for the absolute scale (i.e., the amplitude) of the SMAD signal [133, 135]. At the single-cell level, such mechanisms jointly control heterogeneous signaling dynamics and this can beinvestigated by parameter sampling in a deterministic model (see below).
3) Pulsatile SMAD shuttling : Population-average modeling and quantitative experimentalanalyses suggested that SMAD signaling induced by TGFβ/SMAD signaling pathway. β or BMP could show (damped)oscillatory behavior, in which a single stimulus induces two or more repeated pulses of SMADnuclear translocation [134, 148]. Using global parameter sampling, Wegner et al proved thatoscillations require the presence of transcriptional negative feedback - if this feedback isswitched off, no physiologically plausible parameter configuration can produce oscillations [134].Accordingly, knockdown of transcriptional feedback regulators SMAD6 and SMAD7 abolishedBMP-induced SMAD oscillations [148]. It is possible that such oscillatory negative feedbackcontributes to repeated bursting of SMAD2 or SMAD4 nuclear translocation observed in singlecells, though additional stochastic mechanisms need to be taken into account to describe theirregularity of bursts [33, 128]. .3 Towards quantitative modeling of SMAD signaling heterogeneity
On the basis of the established population-average models, we recently derived a deterministicmodeling framework to quantitatively describe cell-to-cell variability in the TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signalingpathway [33]. Our study was based on imaging data in which the nuclear translocation ofSMAD2/SMAD signaling pathway. 4-GFβ/SMAD signaling pathway. P fusion proteins was monitored in thousands of living MCFβ/SMAD signaling pathway. 10A cells over 24 h. As a basis for modeling, we initially analyzed characteristic features of SMAD signalingheterogeneity. We performed sister cell experiments and found that sister cells are more similarthan random pairs of cells but desynchronize after several hours. This indicated that signalingfluctuations are non-genetic, but temporally stable. We then analyzed potential sources ofheterogeneity in SMAD signaling, and considered that SMAD signaling may be influenced bycell cycle stage and/SMAD signaling pathway. or cell density [127, 149]. Using live-cell imaging, we followed cell divisionevents and quantified the cell density before and during TGFβ/SMAD signaling pathway. β stimulation, and found that thesetwo factors had negligible impact on heterogeneous signaling in our culture conditions. Takentogether, this indicated that SMAD signaling heterogeneity can be modeled using a deterministicapproach based on ODEs (Section 2), with the assumption of stochastic (but temporally stable)fluctuations in signaling protein expression levels.We started with a detailed kinetic pathway model that describes known mechanisms of SMADsignaling and comprises a total of 23 molecular species and 45 kinetic reaction parameters. Likemost other models of TGFβ/SMAD signaling pathway. β signaling, our model contained the main features described inSection 3.3, including an endosomal receptor shuttling module, a SMAD translocation moduleand a transcriptional feedback module (Fβ/SMAD signaling pathway. ig. 3A). With this model, we sought to describe a largeexperimental dataset, in which several levels of signaling (TGFβ/SMAD signaling pathway. β receptor expression, nucleartranslocation of SMAD2 and SMAD4, as well as SMAD7 mRNA expression) were measured formultiple experimental conditions at the single-cell and population-average levels. In total, thisdataset comprised >1,000,000 data points, mainly from densely sampled live-cell imagingexperiments at multiple experimental conditions. Owing to the high complexity of model anddata, we did not aim for a quantitative fitting of the model to the single-cell data, but insteaddevised a three-tiered, modeling strategy to derive a quantitative description of heterogeneoussignaling (Fβ/SMAD signaling pathway. ig. 3D). Initially, we describe the population-average dynamics. Then, we refine thedescription of the pathway to the level cellular sub-populations showing qualitatively distinctsignaling dynamics. Fβ/SMAD signaling pathway. inally, we develop an ensemble of single-cell models to describe thecomplete heterogeneous cell population. Specifically, the three modeling steps were as follows:
1) Population-average modeling : To derive a quantitative description of the SMAD signalingdynamics, we initially fitted the model to population-average data (Fβ/SMAD signaling pathway. ig. 3D, left). Fβ/SMAD signaling pathway. or fitting, weused the population-median nuclear translocation time courses of fluorescently labeled SMAD2and SMAD4 for varying TGFβ/SMAD signaling pathway. β doses and for restimulation experiments, in which cells wererepeatedly challenged with the ligand. Fβ/SMAD signaling pathway. urthermore, the fitting took into account pathwaymeasurements that were only possible at the population-average level (TGFβ/SMAD signaling pathway. β receptor proteinexpression, SMAD7 mRNA expression). After calibration, we validated the predictive power ofour model for previously untested molecular species (time course of extracellular TGFβ/SMAD signaling pathway. βdegradation) and experimental conditions (restimulation experiments, inhibition of transcriptionalfeedback loops by small molecule inhibitor DRB).
2) Description of cellular subpopulations : Having a predictive population-average model athand, we sought to quantitatively describe variability in signaling, while limiting thecomputational cost. Therefore, we refrained from fitting our model to the complete single-cellpopulation (Section 2.4), but only fitted six subpopulations which show qualitatively distinctynamics of signaling (e.g., transient vs. sustained; see Fβ/SMAD signaling pathway. ig. 3D, middle). These subpopulationswere identified by k-means clustering of single-cell SMAD2 nuclear translocation time coursesaccording to their similarity in shape and amplitude. We separately fitted the subpopulation-median time course of each cluster and only allowed variation in the expression of signalingproteins (e.g., TGFβ/SMAD signaling pathway. β receptors, SMADs) within the range of typical cell-to-cell variation (+/SMAD signaling pathway. - 2-fold). In contrast, the kinetic parameters were fixed to their population-average value, i.e., theirvariability was neglected. With these assumptions, we could quantitatively describe allsubpopulations, and had therefore developed our model from a population-average descriptionto a description of six representative cells with characteristic dynamical features.
3) Ensemble modeling of complete cell population:
To directly compare our simulations tosingle-cell experiments, we converted the subpopulation models to an ensemble of artificial cellsrepresenting the heterogeneity of the entire cell population (Fβ/SMAD signaling pathway. ig. 3D, right). Artificial single cellsbelonging to each subpopulation were generated by repeated simulation with signaling proteinconcentrations varying around the best-fit values of the corresponding subpopulation model.The full cell population was assembled in silico by combining artificial cells according to theexperimentally observed proportion of corresponding subpopulations. The degree of variationwas assumed to be the same for all sampled signaling proteins. The common protein coefficientof variation (std/SMAD signaling pathway. mean) was chosen by matching the simulated and and experimentally observedsnapshot distributions at particular time points using summary statistics.Taken together, we obtained an in silico cell population with realistic properties close to theexperimental data. Importantly, we could show that our three-tiered modeling approach, inwhich we considered the subpopulation structure (step 2), yielded a better agreement withsingle-cell snapshot distributions (step 3) when compared to direct sampling of proteinconcentrations in the population-average model. The model reproduced key features of single-cell TGFβ/SMAD signaling pathway. β signaling including gradual (unimodal) behavior and strong heterogeneity in the timecourse shape (Section 3.2). By calculating euclidean distances, we quantitatively compared thesimulated single-cell trajectories to the six experimentally observed time course clusters. Sincethe model took into account subpopulation information, we obtained a very similar dose-dependent decomposition into non-responding, transient and sustained signaling classes as forthe experimental data. Thus, the model correctly takes into account temporal correlations insignaling pathway activity, and can be used to predict drifts in the shape and proportion of theoriginal subpopulations for any experimental condition. In fact, we confirmed such predictions toa knockout of the negative feedback regulator SMAD7. As predicted by the model, we foundthat the effect of SMAD7 on the signaling dynamics was restricted to certain cellularsubpopulations and was observed for specific doses of TGFβ/SMAD signaling pathway. β only. Hence, the model allowedus to quantitatively understand the cell-specific impact of experimental perturbations andallowed mechanistic insights into cellular heterogeneity. One limitation of the current model is that the best-fit parameter values in population-averageand subpopulation fitting are not unique (non-identifiability problem). Nevertheless, robustpredictions could be made, since very similar simulation results were obtained when comparingmultiple fits obtained during a multi-start optimization (repeated model fitting from differentstarting parameters). The subpopulation fitting (step 2) currently corresponds to a the standardtwo-stage approach discussed in Section 2.3, since the protein concentrations in eachsubpopulation were estimated separately without additional constraints about the proteindistribution in the cell population. After assembly of the complete cell population (step 3), weconfirmed that signaling protein levels in the model show a realistic log-normal distribution.However, such a distribution is not automatically granted in the current approach. Therefore, itwould be beneficial to either improve the identifiability of parameters by model reduction, or toake into account additional constraints during subpopulation fitting. Taken together, our study suggest that heterogeneity of TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signaling at the single-celllevel can be quantitatively described using a deterministic modeling approach. Key features ofthe pathway at the single-cell level were reproduced, including gradual behavior and cell-specific characteristics in signaling shape. Since the model is deterministic in nature, it currentlydoes not describe the apparently stochastic, burst-like shuttling of SMAD proteins into thenucleus (Section 3.2). To describe this phenomenon, stochastic effects, most likely inendosomal receptor shuttling need to be taken into account, and quantitative fitting approachescould be used to match burst features in the stochastic model to the experimental data. Thismay be an interesting future direction, as other signaling pathways such as the MAPK cascadeshow repeated pulses of activation which have profound impact on cell fate [43, 150].
Another aspect that deserves further attention in the future is how fluctuations in signalingproteins translate into fluctuations in downstream target gene expression. SMAD transcriptionfactors mediate cellular responses by binding to target gene promoters, thereby inducing large-scale gene expression programs involved in cell migration, EMT and cell cycle arrest [115].Given this link between SMAD binding and gene expression, it seems that cell-specific geneexpression and morphological changes may be predictable based on the amplitude and/SMAD signaling pathway. ordynamics of SMAD signaling. In fact, by clustering single-cell SMAD time courses, we foundthat cell migration and cell division kinetics can - in part - be explained based on the dynamicsof SMAD nuclear translocation [33].Based on these observations, it is natural to extend current mathematical models to SMAD-induced gene expression and -in the long run– to TGFβ/SMAD signaling pathway. β-induced cell fate decisions. At thepopulation-average level, SMAD signaling dynamics seem indeed to be related to geneexpression, as models simultaneously describing time courses at both levels are well-established: most of these studies modeled the dynamics of certain negative feedbackregulators or pathway targets using SMAD-dependent transcription and linear degradation ofthe target gene. By simultaneously fitting such synthesis and decay models using SMADkinetics as an input, target gene mRNA dynamics can be well-explained in multiple cases [33,72, 134, 143, 146, 151, 152]. A recent combined modeling and experimental study extendedthis idea to a larger set of target genes based quantitative and time-resolved measurementsSMAD trimeric complexes [142]. Using a model fitting framework, the authors inferred thespecificity of target gene induction by distinct SMAD complexes and successfully predicted geneexpression outcomes for novel experimental conditions.Despite such good accordance of SMAD signaling and gene expression responses in cellpopulations, it remains challenging to directly link both levels in single cells. Experimentally,such analyses require SMAD signaling dynamics and gene expression to be simultaneouslymeasured in the same cell by live-cell imaging of fluorescent SMAD reporters and subsequentsmFβ/SMAD signaling pathway. ISH of mRNA expression in fixed cells [34, 41]. Those two studies conducted so far, agreethat on a single cell level, neither SMAD2 nor SMAD4 absolute levels in the nucleus accuratelypredict the cell-specific expression of target mRNAs. However, Fβ/SMAD signaling pathway. rick et al. reported that theTGFβ/SMAD signaling pathway. β-induced fold-change of the nuclear SMAD levels relative to basal predicts stimulus-induced target gene expression responses. Fβ/SMAD signaling pathway. or the genes Snail and CTGFβ/SMAD signaling pathway. , they foundSpearman rank correlation coefficients of ~0.5 between between those fold-changes and themRNA abundance as measured by smFβ/SMAD signaling pathway. ISH. These findings could not be confirmed by Tidin et(2019), who analyzed the expession of CTGFβ/SMAD signaling pathway. with high temporal resolution using liveluminescence imaging. They found no correlation between the fold-changes in nuclear SMADnd CTGFβ/SMAD signaling pathway. expression. Thus, even though the proposed fold-change detection in SMAD targetgene expression may be an elegant way to reliably respond to stimulation despite highvariability in absolute nuclear SMAD levels, it remains to be confirmed whether this is generalphenomenon applicable to other genes and cellular systems.Therefore, the mechanistic link between SMAD signaling fluctuations, gene expression andheterogeneous cellular responses remains to be established. In any case, a deterministic 1:1correspondence of signaling and gene expression appears unlikely, since gene expressionmodeling requires stochastic modeling of promoter switching, as Molina et al showed for SMADtarget genes (see also Section 2.3) [153]. Even though each individual gene might respondstochastically and with little correlation to the SMAD signal, it still remains possible that SMADsignaling fluctuations directly affect cellular outcomes through their cumulative effect on manytarget genes controlling a common biological process. Genome-wide single-cell RNAsequencing approaches will shed light on such coordinated gene expression programs at thelevel of individual cells. Recent work has presented methods for the targeted manipulation of SMAD signaling dynamicsat the single-cell level [36, 126] and similar tools were developed for other signaling pathways[43, 83, 154]. The combination of such highly controlable tools with gene expressionmeasurements will provide direct insights into the impact of SMAD signaling dynamics on geneexpression outcomes, and will advance our understanding of heterogeneous decision makingby the TGFβ/SMAD signaling pathway. β pathway.
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IGURE LEGENDS
Fβ-SMAD signaling pathwayigure 1: Fβ-SMAD signaling pathwayunctional and physiological consequences of non-genetic heterogeneity. (A) Heterogeneous signaling causes fractional cell differentiation. Only cells exhibiting highsignaling activity (dark grey) in response to a differentiation stimulus undergo differentiation,e.g., during MAPK-induced oocyte maturation or PC12 differentiation (see text). (B) Signaling variability in response to therapeutic treatment. Apoptotic signaling pathways (e.g.,caspases) are activated heterogeneously when cells are treated with a cytotoxic drug whichresults in fractional killing and (transient) resistance in the non-responding cells. A populationregrown from the therapy-resistant cells may again exhibit the same fractional killing, indicatingthat signaling heterogeneity is a non-genetic phenomenon. (C) Heterogeneous signaling may exhibit gradual or bimodal behavior. Gradual signalingpathways exhibit a unimodal activity distribution across single cells which is shifted to highermean levels upon increasing stimulation. Signaling histograms at different doses typicallyoverlap, which may give rise to inaccurate cellular information transfer. Signaling systems withbimodal behavior exhibit two clearly separable (“ON” and “OFβ/SMAD signaling pathway. Fβ/SMAD signaling pathway. ”) activity levels. Increasingstimulation does not affect the mean signaling activity of the ON and OFβ/SMAD signaling pathway. Fβ/SMAD signaling pathway. subpopulation, butshifts the fraction of cells in each class.(D) Sister cells experiments indicate deterministic behavior of signaling. Fβ/SMAD signaling pathway. or a signaling pathwaywith stochastic dynamics, even recently divided sister cells would show distinct (stochastic)signaling responses. In signaling, recently divided sister cells are typically more correlated intheir signal response than random cells, likely because sisters show common proteinexpression patterns or cell cycle stages. Over a timescale of hours to days, sister cell similarityis lost (“older sister cells”), indicating a non-genetic mechanism of signaling heterogeneity.(E) Variations in signaling protein expression cause deterministic heterogeneity of signaling. Ahigh total expression level of a positive regulator (Xtot = X + X*) causes a strong response in asignaling pathway with deterministic behavior. In a signaling pathway with stochastic dynamics,the signaling response is less or not at all related to the protein content. In models of cellularheterogeneity, protein expression is often assumed to be stable at the time scale of signaling.
Fβ-SMAD signaling pathwayigure 2: Deterministic modeling of cellular heterogeneity. (A) A kinetic model of securin and cyclin B degradation by the anaphase-promoting complex(APC) during mitosis (left) is described by a deterministic ODE model (see also [75]) andnumerically integrated to simulate the protein dynamics in one average cell (middle).Heterogeneity is introduced into the system by performing repeated simulations while samplingprotein concentrations (and in some cases kinetic parameters) from log-normal distributions(right). (B) Model-based analysis of cellular heterogeneity. Fβ/SMAD signaling pathway. or model analysis, simulations areperformed for varying parameters and/SMAD signaling pathway. or degree of protein concentration fluctuations, ordifferent biological mechanisms are considered in the model. Thereby, the model providesinsights into molecular sources of heterogeneity, mechanisms of biological robustness, andallows for the design of new experiments. (C) Quantitative fitting of ensemble models to heterogeneous single-cell data. In the literature,models of cellular heterogeneity were calibrated by fitting cross-sectional snapshot data at aarticular time point, or by directly fitting the kinetic model to single-cell trajectories. In bothcases, cell-specific parameters are estimated to yield an optimal match between model andexperiment.
Fβ-SMAD signaling pathwayigure 3: Quantitative analysis and modeling of TGFβ-SMAD signaling pathwayβ/SMAD signalingSMAD signaling heterogeneity (A) Schematic representation of TGFβ/SMAD signaling pathway. β signaling including endosomal receptor shuttling, nuclearSMAD shuttling and negative feedback: The binding of the ligand to a RII receptor leads to therecruitment of an RI receptor building an activated receptor complex. This complex can, wheninternalized, mediate the phosphorylation of SMAD2 proteins. The receptor complex either getsdegraded or free receptors are recycled back to the cells surface. In the cytosol, phosphorylatedSMAD2 proteins form trimers with SMAD4 which translocate to the nucleus, thereby incrreasingthe experimentally measurable nuclear-to-cytoplasmic (Nuc/SMAD signaling pathway. Cyt) SMAD2 ratio. In the nucleus,the SMAD heterotrimer acts as a transcription factor to induce downstream target genes ofTGFβ/SMAD signaling pathway. β including SMAD7 which acts as a negative feedback inhibiting the activity of RIIreceptors.(B) Temporal dynamics of SMAD nuclear translocation at the single-cell level. Populationaverage (black) and standard deviation (gray shades) of the single-cell Nuc/SMAD signaling pathway. Cyt SMAD2 ratioafter stimulation with 100 pM of TGFβ/SMAD signaling pathway. β. Examplary trajectories of transient, sustained and non-responding cells are shown in different colors (see legend). Another cell (purple) shows abursting event (red). Data from [130].(C) Gradual behavior of SMAD signaling at the single-cell level. Snapshot histograms ofNuc/SMAD signaling pathway. Cyt SMAD2 ratio 70 min (time of peak) after stimulation with varying doses of TGFβ/SMAD signaling pathway. -b (seelegend). All distributions are unimodal and shift towards higher mean values with increasingstimulation. Data from [130].(D) Three-tiered modeling approach for modeling TGFβ/SMAD signaling pathway. β/SMAD signaling pathway. SMAD signaling heterogeneity: In step(1), a population-average model is fitted to experimental data. In step (2), the model is refined toa desription of subpopulations which are identified from the measured single-cell trajectoriesusing a clustering approach. Fβ/SMAD signaling pathway. or each subpopulation, the model is fitted to the median timecourses of a cluster, assuming subpopulation-specific signaling protein expression. Step (3)yields simulations of individual cells, since signaling protein levels are sampled from log-normaldistributions in each subpopulation model. In step (4) single-cell trajectories from step (3) arecombined according to yield a description of the complete cell population. Fβ/SMAD signaling pathway. igure modified from[130]
IGURES
Fβ-SMAD signaling pathwayigure 1
Therapeutictreatment
Cellular genealogy Stimulation time Recently divided time
Random cells time S i gna l Older sister cells time AE DifferentiationStimulus
Precursor cellpopulation Heterogeneous Signaling
High Low DifferentiationDifferentiated Precursor Low High Cell death Alive Dead R e g r o w t h ( d ays ) Deterministic Stochastic time S i gna l S i gna l S i gna l Partial differentiation
Sister cells
Heterogeneous Signaling
Activity
Untreated cellpopulation Heterogeneous Signaling FractionalkillingHeterogeneous Signaling Fractionalkilling of initially resistant cells
Therapeutictreatment
Bi-modal behaviorStimulusdose
Signaling activity Low MediumHigh Very high
Gradual behavior C o f c e ll s o f c e ll s Signaling activityOFF ON
Deterministicresponse Stochastic response
Stimulation
Heterogeneous signaling protein expression
Signaling molecules many few response weak strong response strong weak
Signalingpathway
StimulusResponse
X X*
B D igure 2 A Many simulations with randomly sampled protein concentrations
Population-averagesimulationDeterministic ODE model
APC/C catalyzes degradation of securin and cyclin B Securin and cyclin B degradationin one average cell
Ensemble of single-cell simulations C o n c e n t r a t i o n TimeHeterogeneous degradation in a cell population
One simulation B Model-based analysis of heterogeneity
Simulations
80 100 120200300400 3080 100 120200300400 1080 100 120200300400 10 80 100 120200300400 30
Securindistribution at time t Cyclin Bdistribution at time t
I. Fitting to snapshot distributions
Correlation
II. Fitting to single-cell time courses
Calibration of single-cell models by fitting to data C Cell-specific signaling protein concentrations
High protein fluctuations meanstd meanstdR Securin conc. S i n g l e c e ll s CyclinB conc. S i n g l e c e ll s APC conc. S i n g l e c e ll s Protein concentrationdistributions C o n c e n t r a t i o n Time dataTime Time
Low protein fluctuations
Sources of signaling variabilityMechanisms of cellular robustnessManipulation of heterogeneous cellular decisionsInsights into model topology
Integration of single-cell and average data model fit igure 3
TGF- β SMAD7
RI RII
Target genes
Cluster 1Cluster 2Cluster 3Cluster 4
Fit to population-average data
Sub-population models Ensemble of single-cell models
Few representative cells Thousands of individual cells
Endosomalreceptor shuttling
Endo-cytosis Recycling
P P
Inactivereceptorsreceptorcomplex active receptor Heterotrimers
Nuclear SMADshuttling phospho-SMAD2
Degradation DephosphorylationTrimer dissociationNuclear Export SMADphosphorylation NuclearImport Trimerization
Negative feedback loop
Receptor inhibition
SMAD2 SMAD4 mRNA P P P P P time [min] S M A D nu c / c y t - r a t i o One average cell
Population-average model
Fit to median of time course clusters Introduce further heterogeneity by parameter sampling Merge subpopulations datamodel fi t A B
Temporal dynamics
Snapshot distribution CD time / min N u c / C y t S M A D r a t i o population averagevariabilitynon respondertransient responder sustained responderexample trajectory with burstbursting event Nuc/Cyt SMAD2 ratio % C e ll ss