## Abstract

The role of cell-cell communication in cell fate decision-making has not been well-characterized through a dynamical systems perspective. To do so, here we develop multiscale models that couple cell-cell communication with cell-internal gene regulatory network dynamics. This allows us to study the influence of external signaling on cell fate decision-making at the resolution of single cells. We study the granulocyte-monocyte vs. megakaryocyte-erythrocyte fate decision, dictated by the GATA1-PU.1 network, as an exemplary bistable cell fate system, modeling the cell-internal dynamic with ordinary differential equations and the cell-cell communication via a Poisson process. We show that, for a wide range of cell communication topologies, subtle changes in signaling can lead to dramatic changes in cell fate. We find that cell-cell coupling can explain how populations of heterogeneous cell types can arise. Analysis of intrinsic and extrinsic cell-cell communication noise demonstrates that noise alone can alter the cell fate decision-making boundaries. These results illustrate how external signals alter transcriptional dynamics, and provide insight into hematopoietic cell fate decision-making.

## Introduction

The production of mature cell types from stem cells and progenitors is essential for development and organ homeostasis. Nevertheless, in few cases are we able to fully specify the conditions necessary to drive stem cell differentiation towards a particular cell lineage. Stem cell differentiation is controlled by cell-internal and external signals that, in turn, control the transcriptional state of the cell and specify its eventual fate [1–3]. These changes in transcriptional state are often irreversible and involve binary choices. Thus, multipotent cells, through successive binary lineage specification choices, eventually become committed to a specific lineage and cell fate. Significant unanswered questions remain regarding the cell fate decisions that dictate lineage specification of stem/progenitor cells: how large is the role of extracellular signaling in regulating cell differentiation? What mechanisms allow cells from initially homogeneous or clonal populations to converge to different lineages? How does one population of progenitor cells maintain stable heterogeneous subpopulations of committed cell types?

During lineage specification, changes in gene expression are controlled by gene regulatory networks (GRNs), consisting of genes and their protein products (transcription factors), which are able to regulate the expression of various genes, including other transcription factors, creating feedback loops. Codified through mathematical models, GRNs can be studied in light of their steady states, where each steady state can represent a committed cell fate. Certain GRN topologies permit bistability, that is, more than one steady state can be reached for a single set of biological conditions [4]: such topologies are frequently observed in networks that control lineage decisions [5]. The GRN topology that dictates a particular lineage decision is generally conserved across cells, thereby providing insight into the intracellular dynamics that occur during cell differentiation. Crucially, the GRNs that instigate or reinforce lineage decisions are not only controlled by cell-intrinsic networks, but also by cell-extrinsic signals.

The GRN topology considered in this work consists of two mutually repressive genes; a topology frequently observed among gene networks mediating lineage decisions [6]. One such decision point occurs during hematopoiesis: myeloid progenitor cells make a binary choice between commitment to the megakaryocyte-erythroid (ME) lineage or the granulocyte-monocyte (GM) lineage. *GATA1* and *PU.1* (*SPI1*) mutually inhibit one another; *GATA1* is expressed in the ME lineage and *PU.1* is expressed in the GM lineage. This mutually repressive GRN has been extensively studied and characterized in models, mostly consisting of ordinary differential equations that permit bistability, thus enabling investigation into the dynamics of this myeloid lineage decision [7–11]

Given the *GATA1-PU.1* mutual inhibitory loop that leads to bistability, changing the initial conditions (gene expression levels) within a bistable region is sufficient to change the cell fate. It has thus been proposed that random fluctuations of *GATA1* and *PU.1* levels are primarily responsible for determining cell fate in the bipotent progenitor population that has dual ME and GM lineage potential [10]. More recently this notion was challenged by a study that used a double reporter mouse (PU.1^{eGFP}Gata1^{mCherry}) to show that that random fluctuations of *PU.1* and *GATA1* are insufficient to initiate the cell fate decision between ME and GM lineages [12]. Hoppe et al. also provide strong evidence that ME v. GM lineage specification cannot be determined solely from initial ratios of *PU.1* to *GATA1* expression.

As noted by Hoppe et al. [12], the role of extracellular signaling could resolve this controversy. Dynamic extracellular signaling however has been understudied in investigations into gene regulatory networks and their stability. Extracellular signaling through paracrine factors enables populations of cells to share information and coordinate behaviors over short distances, and is implicated in a range of cell fate behaviors from developmental branching [13, 14] to patterning [15] and migration [16]. We begin this study with a simple question: how does short distance intercellular signaling impact myeloid lineage specification controlled by the *GATA1-PU.1* gene regulatory network?

Several studies have characterized the impact on cell fate of cell-cell communication through extracellular signals. One approach is to model the internal gene dynamics with differential equations and model cell-cell communication through allowing for molecular diffusion of proteins between neighboring cells [17–19]. These methods however must neglect anisotropic effects, and assume that signaling between cells occurs on the same timescale as intracellular dynamics: these processes can occur on vastly different timescales. Other approaches have relaxed this assumption, permitting different timescales by modeling the response time of a cell to a signal received as a random variable [20]. This approach however omits any detail of the intracellular dynamics, which are often crucial to the response; GRNs are the enforcing mechanisms of the phenotypic switches. Therefore, in order to model the behaviors of a population of communicating cells, the gap between intracellular dynamics and extracellular signaling must be bridged. A model of cell fate decision-making is needed that permits cell-cell communication while allowing for both description of the cell-internal dynamics and of the extracellular signaling dynamics without making diffusion-like approximations.

We present here a multiscale model that bridges this gap. We will assume deterministic dynamics within each cell and thus model cell-internal GRNs with ordinary differential equations (ODEs). We assume that signals sent between cells can be described by a Poisson process. Signals received by cells can then alter the internal GRN dynamics by parameter modifications. We test this model in a large range of different intercellular signaling topologies. We find that the addition of cell-cell communication to GRN dynamics leads to model outcomes (cell fate distributions) becoming probabilistic, with cell fate choice probabilities that depend on the cell’s position in a particular signaling topology. We discover that the model can intuitively characterize cell-cell coupling, changes to which impact the cell fate decisions made, which can lead to mixtures of heterogeneous cell types within a population. Finally, we study how noise propagates across various signaling topologies, and show that although both intrinsic and extrinsic noise alter the cell fate decision-making boundaries, extrinsic noise is the dominant driver of cell fate variability.

## Methods

### A multiscale model of cell-cell communication between single cells

To investigate how external signaling impacts intracellular dynamics during cell fate decisions, we first must select an internal GRN topology to model. We choose to model a mutual inhibitory GRN: such network topologies arise frequently in cell fate decision making and developmental biology [21, 22]. Within certain parameter regions, such models give rise to two stable steady states and bistability (Fig. 1A). For example, it has been observed that the GRN determining the bifurcation of erythroid/megakaryocyte lineages from granulocyte/monocyte lineages contains a core mutual inhibitory loop topology consisting of mutual antagonism of *GATA1* (*G*) and *PU.1* (*P*). For this genetic switch, high expression of *G* corresponds to commitment to the erythroid/megakaryocyte lineage and high expression of *P* corresponds to commitment to the granulocyte/monocyte lineage. This myeloid progenitor cell lineage decision has been rigorously studied, and there exist robust models of the intracellular GRN dynamics [7, 8, 10, 11, 23, 24]. Here we implement the ODE model defined by Chickarmane et al., given by Eqs. 1-3 [7]. These ODEs give rise to two stable steady states for a region of parameter space with respect to the parameters *A, B* and *C*, where *A, B*, and *C* are parameters summarizing environmental signals (Fig. 1B). The parameters through which the extracellular environment is summarized will be modified below to implement a multiscale model of cell-cell communication. The parameter values used throughout this work can be found in Table S1.

To appropriately incorporate intercellular signaling into the model, we modify the cell-internal ODEs in such a way that reflects the signals received without unrealistically changing the internal dynamics (e.g. a loss of bistability). For simplicity, we choose a single parameter A to summarize the effects of external signaling. We study both signals that recruit nearby cells to commit to the same lineage and the opposite lineage of the sender. For the first case, we consider two cells, cell 1 and cell 2, where cell 1 signals to cell 2. Let *G*_{1}(*t*) and *P*_{1}(*t*) denote the concentrations of *G* and *P* in cell 1 at time *t* respectively and let *A*_{2}(*t*) be the value of the parameter *A* in cell 2 at time *t*. Since cell 1 signals to cell 2 to be of the same lineage, then *A*_{2}(*t*) should increase if the ratio *G*_{1}(*t*): *P*_{1}(*t*) indicates that cell 1 is in, or is moving toward, the ME state where *G* is highly expressed. Conversely, *A*_{2}(*t*) should decrease if the ratio *G*_{1}(*t*): *P*_{1}(*t*) indicates that cell 1 is in, or is moving toward, the GM state where P is highly expressed. Based on these assumptions, we constructed a communication model: we treat signaling as a Poisson process, where the wait time between signals being sent by a cell follows an exponential random variable with mean *μ*. After this wait time, at time *t _{s}*, a signal is sent and we change the value of

*A*

_{2}by:

After a delay *τ*, *A*_{2} returns to it’s original value, *A*_{0} (Fig. 1C). Returning to the same value *A*_{0} between signals ensures that attractor states (Fig. 1B) do not change. In Fig. 1D is a schematic of this signaling model. Fig. S1 gives a sample simulation of a chain of four cells and the values of *A*_{2},*A*_{3}, and *A*_{4} plotted over time. In the case of signals that promote nearby cells to commit to the opposite lineage, we can follow the same principles to define a signal:

These definitions of signaling also allow for multiple cells to signal a single cell at similar times while still having *A*_{2} well defined.

The model developed here shares similarities with models specified by piecewise-deterministic Markov processes (PDMPs), first introduced by Davis [25]. Such models are comprised of continuous dynamics and a discrete Markov process which can alter the continuous dynamics at discrete time points. PDMPs have previously been applied to biological systems for the study of gene expression and genetic networks [26, 27]. In the case of two cells sharing one directed signal between them, our model resembles a PDMP with the single exception that in our case the discrete process is not Markovian.

### Model specification and choice of parameters

To elucidate and justify how we arrived upon the choice of signaling model, we briefly discuss some alternatives. Examples of alternative reasonable signaling model choices include: signals that change more than one ODE parameter, additive rather than multiplicative signaling, or signals that permanently change ODE parameters. We choose to modify a single parameter for interpretability and to constrain the model space: the ODE system exhibits bifurcations with respect to many of its parameters, and we do not seek to explore bifurcations of codimension 2 or greater here. Modeling with additive signals brings complications, such as the possibility of obtaining negative values of *A*. Also, if a large number of committed cells are communicating with a single cell, the value of |*A*| may be unrealistically large and dominate the ODE. This problem is dealt with in the signaling model we present here through the parameter λ. Lastly, permitting signals to cause permanent changes to ODE parameters (rather than over time intervals) can result in the divergence of *A*. Thus, we have selected a signaling model that can capture how changes in the extracellular environment can alter cell internal GRN dynamics while still preserving the overall behaviors (i.e. the attractor states) of the dynamical system.

In this work, we will assume that cells are homogeneous, i.e. all cells share the initial conditions (*G, P, X*) = (20,1, 20). The internal GRNs are identical with the same parameter values, including the same initial value of *A*, denoted *A*_{0}. We set the signaling period to be *τ* = 5, and the mean of the signaling wait time distribution to be *μ* = 50. Changing the values of *μ* and *τ* does not seem to cause qualitative changes as long as *τ* is sufficiently large relative to *μ*. For more insight on the relationship between the wait time distribution and *τ*, see Fig. S2. Unless otherwise specified, we set λ = *K* = 1. We initially chose λ = *K* = 1 to avoid unrealistic behaviors when *G* or *P* ≈ 0. We will provide an in depth discussion on the roles of these parameters in the next section. In any topology schematic, regular arrows correspond to signaling with respect to Eq. 4 and inhibition arrows correspond to signaling with respect to Eq. 5.

As a result of signaling, cell fate decisions became probabilistic rather than deterministic. Then, for each cell in a signaling topology, we can examine the probability distribution of the cell converging to a certain lineage. To do so, for each signaling topology tested, we selected a range of values of *A*_{0} to simulate. The step size between the *A*_{0} values was scaled between each topology depending on the range of the probability distribution. For each value of *A*_{0}, we simulated *N* trajectories, counting the number of times each cell converged to both fates. From these counts, we estimated the probability that each cell converged to the *G* high state and plotted:

A sample of the simulated data points can be found in Fig. S3. Unless stated otherwise, probabilities are estimated by running *N* = 1,000 simulations.

While λ = 1 was an intuitive first choice to avoid divergence in signaling parameters, it relies on the assumption that a cell converging to the *P* high state implies that *P* is more highly expressed than *G* + 1 and vice versa. This turns out to not necessarily be true. Rather, for a given value of *A*_{0}, being in the *P* high state means that *P* is more highly expressed relative to the other stable steady state value of *P*. Recalling Fig. 1B, we see that for some values of *A*_{0}, for example *A*_{0} = 0.8, *G* is always more highly expressed than *P* regardless of which stable steady state a cell converges to. Looking at the bifurcation diagram, we see that in the *P* high state, the maximum steady state value of *G* is approximately 16.97 and the minimum value of *P* is approximately 1.475. Similarly, in the *G* high state, the maximum steady state value of *P* is approximately 0.3529 and the minimum value of *G* is approximately 40.35. From here, we wanted to select values of λ which satisfy the following:

These inequalities are satisfied for λ ∈ [15.495, 39.997]. We will explore how selecting values of λ in or near this range changes the behaviors of both individual cells and populations of cells.

### Intrinsic and extrinsic cell-cell communication noise

In this work, we focus on how intercellular signaling alters cell decision making. Therefore, we only consider noise with respect to signaling (not the internal GRN dynamics). Noise in the signal can arise due to the noisy extracellular environment (extrinsic noise) or can arise during the signal transduction within a cell (intrinsic noise) [19, 20]. Here we define models for both sources of noise.

For extrinsic noise, i.e. noise with respect to the extracellular environment, let *η*_{e} be a random variable such that , where . For Cell *k* in a given topology, at the start of each new wait period, we sample a value of *η*_{e} and update *A _{k}*(

*t*) by

We truncate *A _{k}*(

*t*) at zero to avoid negative values. However, we select values of small enough that the truncation is rarely necessary in simulation. Note that noise in

*A*(

_{k}*t*) results in noise during the signaling period as well by Eqs. (4) and (5).

For intrinsic noise, i.e. noise with respect to signal transduction, let *η _{p}* be a random variable such that , where . For Cell

*k*in a given topology, at the start of each new pulse period, we sample a new value of

*η*and update

*A*(

_{k}*t*).

When the pulse period *τ* is significantly smaller than the mean wait time *μ*, incorporating noise using Eq. (7) has little impact on the cell fate dynamics. Thus, in the section where we study noise, we decrease the mean wait time *μ* in order to distinguish the effects of both noise types.

## Results

### Cell-cell communication over a diverse range of signaling topologies leads to divergent cell fates

We begin by testing the base model behavior for different signaling topologies. We sought to ensure that core assumptions were upheld, namely that all cells eventually reach a steady state, even in the presence of noisy signaling, and that cells exhibit bistability with respect to *A*_{o} (the initial value of *A*). These behaviors were preserved in all signaling topologies tested. Moreover, we found that the addition of cell-cell signaling to a previously deterministic model results in probabilistic cell fate choices.

In Fig. 2, simulated trajectories for a 20-cell loop topology are shown. We plot only the concentration of *GATA1* (*G*) as a proxy for fate choice, since its state is sufficient to determine the steady state of the full system (if *G* is high at steady state, *P* is low, and vice versa); as illustrated by the trajectories of both *G* and *P* (Fig. S4). These trajectories illustrate the variety of cell fate distributions that are observed in the presence of cell-cell communication (fate convergence to either the high or low state, and fate divergence). In Fig. 2C-D, the values of *A*_{0} are equal, demonstrating that *A*_{0} does not determine the proportion of cells in each state. Rather, for each value of *A*_{0}, the probability of each cell converging to a certain state can be computed.

Investigating how signals propagate down a chain of cells, two striking observations are made. First, each additional signal shifts the *P*(*G*) curve (the probability of reaching the high state) to the left, although these shifts are successively smaller down the chain such that eventually the distributions converge; second, a “sharpening” of the *P*(*G*) distributions occurs whereby the region of uncertain fate decreases for cells further down a chain (Fig. 3A). The latter of which implies that accumulated signals by cell-cell communication reinforce cell fate decision-making, leading to smaller regions of uncertainty.

In loops, all cells converge to the same fate for each simulated trajectory, therefore every cell in the model has the same probability distribution. (Below we will see that the coordination of fates in all cells within a loop signaling topology can be broken by changing the cell coupling strength.) However, the probability distribution for a cell loop depends on the number of cells in the loop, and we see that for loops of size *n*, both of the behaviors observed above are recapitulated (Fig. 3B). Moreover, the limiting distributions for the cell chain and the cell loop are the same, highlighting an important underlying symmetry.

If we consider a dissensus signal (the opposite of a consensus signal), the fate probability distribution of cell 2 shifts in the opposite direction than previously (Fig. 3C), relative to the value of *A*_{0} at which cell 1 switches lineages. When an inhibition signal is incorporated into a loop (Fig. 3D), the two cells in the loop no longer coordinate lineages for every trajectory as they did when all signals were regular. Notably, here there exists a large range of *A*_{0} values where cell 1 converges to the *G* high state and cell 2 converges to the *G* low state with probability one. Thus, the inhibition signals significantly changes the resulting behaviors in both chains and loops of cells.

For signaling topologies consisting of three cells, similar patterns due to cell-cell communication emerge. In a feed forward signaling motif (Fig. 3E), multiple promoting signals reinforce the lineage choice of cell 3, whereby it thus chooses the *G* high state at smaller values of *A*_{0} than it would in a cell chain. Other three-cell signaling topologies also display interesting behaviors (Fig. 3F-G and Fig. S5). For an inhibitory feed forward motif, we observe a multiplicative effect between the inhibitory signal from cell 1 and the activating signal received indirectly from cell 1 via cell 2 (Fig. 3F). The addition of the inhibition signal results in the distribution of cell 3 being shifted to the right relative to a corresponding cell in a chain. For a double inhibitory topology (Fig. 3G), we find that for a region of *A*_{0} the cell fates can diverge (two cells in the high state; the third in the low state) with nonzero probability.

While these results pertain to small and somewhat idealized signaling networks, they point to a general conclusion. In all of the above experiments, the addition of cell-cell signaling transforms a population of homogeneous, independent, and deterministic cells into one comprised of heterogeneous cells that choose fates non-deterministically. This model shows that, given cells with identical initial conditions (i.e. transcriptional states), external signaling changes outcomes at the population level. This is corroborated by a major finding of Hoppe et al. [12]: that GMP/MEP cell fate decisions are not predictable from initial transcription factor ratios alone. Our model not only supports this result, but offers an explanation for the missing cell fate determinant: cell-cell communication via external signaling, which acts to reduce the dependence of cell fate choice on cell autonomous conditions.

### Varying the strength of cell-cell coupling results in different ratios of stable subpopulations of heterogeneous cell types

Through deeper investigation of cell-cell communication in the model framework introduced above, we found that divergence of cell fates can occur under the control of the model parameter λ. We thus describe λ as a “cell-cell decoupling” mechanism. Analysis of cell-cell coupling and decoupling led to two significant findings: cell-cell communication can explain how a population of cells maintains stable subpopulations of heterogeneous cell types; and the distribution of cells that converge to each fate depends both on cell-cell coupling and on the external environment.

First, we assessed how different values of λ change the simplest signaling topology: a chain of two cells (Fig. 4A). As λ increases, the probability distribution of the second cell shifts to the right. Note that λ = 22 is close to a critical value of λ, resulting in near-deterministic cell fate decision-making. The value of *A*_{0} where the curve for λ = 22 switches from one lineage to another is extremely close to the value at which a deterministic cell with the same initial conditions switches. Overall, λ determines the range over which the probability curve for the second cell is nonzero.

We next assessed how different values of λ changed coordination of cell fate decisions between lineages. Previously, we observed that for any trajectory of a loop topology of any size, where λ = 1, all cells converged to the same state, regardless of the value of *A*_{0}. Further, although all cells in the topology converge to the same lineage, the probability that they all converge to a given lineage depended on *A*_{0}. To see whether or not this behavior was conserved for other values of λ, we tested a two cell loop topology with a range of λ values. We observed that there exists a value λ* ≈ 22 such that for λ < λ*, all the cells coordinate their lineage just as we had seen before. Interestingly, for λ > λ*, the two cells do not always coordinate their lineage decisions. For example, Fig. S6 gives sample trajectories showing two cells in a loop converging to each combination of states. Fig. 4D shows the two cell loop results for λ = 38, giving the probabilities that each cell converges to the *G* high state as well as the probabilities that the cells are decoupled, converging to opposite lineages. For each λ, we recorded the maximum value of the probability that the cells converge to different lineages. Fig. 4C shows how the maximum probability of cells converging to different lineages increases with λ.

Next, we looked at larger loops of 10 cells. For each value of *A*_{0}, we simulated 500 trajectories and recorded how many cells were in the *G* high state verses the *G* low state. Fig. 4D shows results for λ = 30 and *A*_{0} = 1.0,1.002, and 1.004. We see that the distribution of cells converging to each state changes with the value of *A*_{0}. Further analysis of distributions with different values of λ can be found in Fig. S7.

In this section we have identified an explicit mechanism by which cell-cell communication can break the symmetry of a homogeneous population of progenitor cells, and give rise to stable, heterogeneous populations of lineage-committed cells. The proportion of cells committed to each lineage depends on the external environment, *A*_{0}, and the strength of cell-cell coupling due to signaling, λ. Moreover, these results show that fluctuations in the external environment can lead to shifts in the relative abundances of committed cell types. These results corroborate previous work that studied the generation of heterogeneous cell populations through stem cell differentiation [28], and showed that external signals (e.g. through cell-cell communication) are required both to maintain heterogeneous cell populations and to shift relative cell types abundances in response to environmental perturbations.

### Intrinsic and extrinsic noise alter cell fate decision-making boundaries

We have until now assumed that signals are passed between cells with perfect fidelity. In fact, multiple sources of noise contribute to imperfect communication between cells, and the modeling framework here lends itself well to the investigation of the effects of intrinsic vs. extrinsic noise [29, 30]. We investigated the impact of two different sources of noise in cell-cell communication: due to cell-extrinsic factors, i.e. noise with respect to the extracellular environment (Eq. 6); or due to cell-intrinsic factors, i.e. noise due to signal transduction downstream of a paracrine signaling factor (Eq. 7). These sources of noise are represented in the model by varying either the baseline level of cell-cell communication (extrinsic noise) or the cell signaling pulse level, i.e. the intrinsic signal transduction noise (Fig. 5A).

For simple topologies, as expected, the observed variability in cell fate outcomes increases as the variance of either the extrinsic or the intrinsic noise increases (Fig. S8). This results also holds for larger cell-cell communication topologies, e.g. a ten-cell loop, for both intrinsic (*η _{p}*) and extrinsic (

*η*) noise (Fig. 5B-D): both reduce the sensitivity of the cell fate decision-making boundary. We also observe a striking and unexpected result: not only are the probability distributions flattened by either noise source, but they are shifted to the left, i.e. intrinsic and extrinsic noise directly affect the decision-making boundary (by shifting its mean), as well as the sensitivity of cell-fate decision-making.

_{e}Through comparison of the relative effects of the intrinsic signal transduction noise (Fig. 5B) and the extrinsic extracellular noise (Fig. 5C-D), we see that the impact on the cell fate decision-making boundary is much larger for extrinsic rather than intrinsic noise contributions. Indeed, when , (Fig. 5D), the probability curves for cells 2-10 in the chain flatten to the extent that the sensitivity with which to distinguish cell fate decision-making by cell position along the chain is lost entirely. Moreover, the probability curves for cells ≥ 2 along the chain intersect with the point at which cell one (which is deterministic) switches fates from the low to the high state (black line in Fig. 5D), thus forcing all other cells also into the high state with probability = 1. This coordination of cell fates is influenced by cell-cell coupling (here we use λ =18 < λ*). Recall that for λ values less than the critical value λ*, we observed total fate coordination between cells in loop topologies. The dominant impact of extrinsic over intrinsic noise here is in agreement with previous works, including a study that quantified the contributions of extrinsic and intrinsic noise in the MAPK signaling pathway, and showed that extrinsic noise is the dominant driver of cell-to-cell variability [31]. It has also been shown that explicit extrinsic noise contributions are necessary to explain mRNA abundance distributions [32].

In summary, the effects of intrinsic and extrinsic noise on cell-cell communication topologies are to increase the variance of the resulting cell fate distributions and (surprisingly) to alter the mean values of these probability distributions. In other words, the presence of noise alone can force cells to change lineages. The observed increases in the variability of cell fate decision-making are maintained for large cell-cell communication topologies. A similar result was described in a study of cell fate decision-making during early mouse gastrulation [33]: transcriptional noise is greatest at the point of cell fate decision-making (when epiblasts begin to differentiate). Our results reiterate the same point made by Mohammed et al., that gene expression noise is beneficial during windows of cell fate decision-making as it leads to an increased possible repertoire of cell fates. Our findings go even further in that they suggest a rationale: that the increase in transcriptional noise results from noisy extracellular factors influencing cell-cell signaling during differentiation.

## Discussion

Despite many theoretical and experimental advances in our understanding of gene regulatory network (GRN) dynamics, our ability to use GRN models to explain cell fate decision-making during differentiation of multipotent progenitor cells remains limited. Here, with application to a well-studied cell fate control GRN – the *GATA1-PU.1* mutual inhibition loop [7–11] – we introduced a new model that can simultaneously describe GRN dynamics and single cell-resolved cell-cell communication. Notably, although cell-cell communication is often assumed to be a critical component of cell differentiation, it is rarely incorporated into models. The previous studies that have characterized cell-cell communication in models did not capture the detailed complexity of these dynamics, by making simplifying assumptions either regarding the GRN dynamics [20] or regarding the mechanisms by which cells signal [18, 19]. We found that the model introduced here is able to describe noisy cell fate decision-making at single-cell resolution and reconcile several outstanding controversies in the field.

Over a large domain of possible cell-cell communication topologies, we found that cell-cell communication alters cell fate decision-making boundaries, which become probabilistic in response to the levels of external signaling factors. This helps to reconcile a controversy in the literature: on whether or not transcriptional stochasticity is sufficient to initiate the granulocyte-monocyte vs. megakaryocyte-erythrocyte cell fate decision. Previous models supported the hypothesis, however Hoppe et al. presented compelling evidence to contradict it [12]. Our results agree with Hoppe et al., in that we show that eventual cell fate cannot be inferred from the initial gene expression state alone—rather, it is probabilistic. Moreover, the results presented here offer a plausible explanation of the missing element required to initiate fate decision-making: cell-cell communication that affects GRN dynamics. Through analysis of cell-cell coupling effects, we found that stable distributions of heterogeneous cell types are determined by the external environment. This offers insight into another open question: that of how population-level cell fate behaviors emerge during cell differentiation [28]. Finally, we show how (primarily extrinsic) noise increases the variability of cell fate decision-making, in line with previous analyses of transcriptional noise during development [33].

In this work we sought to constrain model complexity for parsimony and interpretability, yet relaxing these constraints may lead to many further interesting observations. We assumed throughout that cells were initially homogeneous, i.e. they shared the same initial conditions and internal GRN networks. Heterogeneous initial conditions and heterogeneous cell fate decision-making (i.e. different GRNs in different cells) ought to be explored, for example by considering interactions between two progenitor cell types, each controlled by their own GRN. There is also much room for exploration of larger and more varied cell signaling network topologies. Here, future work should be guided by data, as it becomes harder to justify large signaling networks chosen a priori, i.e. spatial transcriptomics could be used to infer cellular networks (e.g. of cells of similar type) that could then be input to our model framework. There could include the incorporation of more dissensus as well as consensus signal types. There are also a variety of well-informed modifications that could be made to the signaling model definition. For example, currently signals only interact with *GATA1*; it would be interesting in future work to explore signals that also interact directly with *PU.1* expression by modifying the model parameter *B*.

A central challenge for the model introduced here is that of fitting to data. Ideally this would require both spatially and temporally resolved single-cell transcriptomic data—at the limits of current technologies. Thus in the current work we rely on comparison of qualitative features arising from the model with previous experimental studies. Further, for Bayesian parameter inference, due to its hybrid deterministic-stochastic formulation resulting in time-dependent signaling parameters, we doubt that it will be possible to derive an appropriate likelihood function. Thus, approximate Bayesian computation will likely be most appropriate. Yet, even here, simulation times may be prohibitive or require further approximations to be made.

Given the stochastic nature of gene expression, a further model development would be to use stochastic rather than deterministic dynamics to model the internal GRN, through discrete stochastic simulation or stochastic differential equations (SDEs). We chose to model the GRN deterministically here again for simplicity, and so that the bifurcation behavior of the model was fully tractable. This also allowed us to better account for the effects of cell signaling variability and noise, without the need to deal with confounding sources. Nonetheless, noise arising from transcription and translation plays a vital role in single cell dynamics. Subsequent studies that seek to faithfully reproduce cell-internal dynamics in the presence of cell-cell communication ought to consider GRN stochasticity, e.g. via use of SDEs. This would be straightforward to accomplish within our model framework, without the need to change the structure of the signaling model used for cell-cell communication. Such modifications would however complicate the analysis of the model bifurcations.

Overall, this model has helped to explain numerous cell fate decision-making phenomena by introduction of a single-cell resolved cell-cell communication model coupled to GRN dynamics. Even with a tightly constrained model, we have shown that modifications to the distribution of cell fates due to cell-cell communication can be broad and varied. More generally, this highlights the importance of considering multiscale effects in light of models of cell dynamics. We anticipate that the application of similar methods to the study of different gene regulatory networks will lead to further advances in our understanding of specific cell fate decision-making points, as well general principles that describe the control of cell differentiation.