Statistical Dynamics of Spatial-Order Formation by Communicating Cells

Summary

Communicating cells can coordinate their gene expressions to form spatial patterns, generating order from disorder. Ubiquitous “secrete-and-sense cells” secrete and sense the same molecule to do so. Here we present a modeling framework—based on cellular automata and mimicking approaches of statistical mechanics—for understanding how secrete-and-sense cells with bistable gene expression, from disordered beginnings, can become spatially ordered by communicating through rapidly diffusing molecules. Classifying lattices of cells by two “macrostate” variables—“spatial index,” measuring degree of order, and average gene-expression level—reveals a conceptual picture: a group of cells behaves as a single particle, in an abstract space, that rolls down on an adhesive “pseudo-energy landscape” whose shape is determined by cell-cell communication and an intracellular gene-regulatory circuit. Particles rolling down the landscape represent cells becoming more spatially ordered. We show how to extend this framework to more complex forms of cellular communication.

Graphical Abstract

Highlights

• •Field of communicating, secrete-and-sense cells form spatial patterns
• •Particle rolling down a “pseudo-energy landscape” models pattern-formation dynamics
• •Equation of motion describes the particle's motion and hence pattern formation
• •This statistical mechanics-type framework applies to other forms of communication

Statistical Physics; Systems Biology; Complex Systems

Introduction

Cells can communicate by secreting signaling molecules, and this often underlies their collective behaviors. A striking example is that of initially uncoordinated cells, through cell-cell communication, coordinating their gene expressions to generate spatial patterns or structures (Gregor et al., 2010, Sawai et al., 2005, Danino et al., 2010, Liu et al., 2011). Many cells partly or completely control such “disorder-to-order” dynamics by simultaneously secreting and sensing the same signaling molecule (Doğaner et al., 2016, Youk and Lim, 2014a). These “secrete-and-sense cells” appear across diverse organisms and include quorum-sensing social ameba, Dictyostelium discoideum, that form fruiting bodies (Gregor et al., 2010, Sawai et al., 2005, Sgro et al., 2015) and autocrine-signaling T cells (Antebi et al., 2013, Sporn and Todaro, 1980, Youk and Lim, 2014b). Based on mounting evidence from studies of various organisms (Gregor et al., 2010, Danino et al., 2010, Youk and Lim, 2014a, Antebi et al., 2013, Mehta et al., 2009, Kamino et al., 2017, Hart et al., 2014, De Monte et al., 2007, Umeda and Inouye, 2004, You et al., 2004, Pai et al., 2012, Coppey et al., 2007, Shvartsman et al., 2001), researchers now suspect that secrete-and-sense cells, many of which are governed by the same type of genetic circuit (Doğaner et al., 2016), are highly suited for spatially coordinating their gene expressions. However, if true, exactly why this is so, whether there are common design principles shared by the different organisms, what the dynamics underlying their disorder-to-order transition is, and how to even quantify their spatial order remain open questions. In this article, we address these questions in the context of initially disordered fields of secrete-and-sense cells that self-organize into spatially ordered fields without any pre-existing morphogens. Specifically, we develop a theoretical framework that takes a simple and ubiquitous class of secrete-and-sense cells, sensibly defines and quantifies the notion of the cells' spatial order, and then elucidates how the spatial order evolves over time. We focus here on analytically describing how spatial correlations among cells' gene-expression levels dynamically emerge rather than on describing the shapes, sizes, and formations of specific spatial patterns (e.g., stripes). To study how these cells generate specific patterns, one often uses exhaustive numerical simulations that are adapted to particular settings (Cotterell and Sharpe, 2010, Cotterell et al., 2015, Chen et al., 2015). Although such simulations provide insights into the dynamics of spatial-order formation, a different modeling framework may provide complementary insights that are difficult to extract from the often large numbers of parameters involved in numerical simulations.

Our main idea is that describing hundreds to thousands of secrete-and-sense cells forming a particular spatial configuration is infeasible without exhaustive numerical simulations but that it is possible to analytically describe how an ensemble of “similar” spatial configurations evolves over time without knowing the state of every single cell. As we will show, we do this by defining quantities that are similar to those found in statistical physics but have meanings and properties that are very different and are adapted for describing cells. Specifically, we will define a “spatial index”—a number whose magnitude is between zero (complete disorder) and one (complete order). Inspired by approaches of statistical mechanics, we will group all lattices of cells that have the same spatial order parameter and average gene-expression level into an ensemble that we will call a “macrostate.” Surprisingly, we find that this macrostate moves like a particle that drifts and diffuses in an abstract, two-dimensional space that we will call a “phase space”—since it describes all possible spatial configurations of the lattice—and whose coordinates denote the cells' spatial order and average gene-expression level. We find that the particle, representing an entire cellular lattice, moves in the phase space by rolling down on a “pseudo-energy landscape,” which is a visual landscape that is shaped by communication among the cells and the intracellular gene-regulatory circuit that controls how the cells secrete and sense the molecule. We will show that the shape of this landscape is quantitatively defined by a function that we will call “pseudo-energy” and show that although it mathematically resembles the Hamiltonian of the Ising model, it has different properties. We will show that the gradient of the pseudo-energy and a “trapping probability,” which quantifies the adhesiveness of the pseudo-energy landscape, together determine the particle's trajectories in the phase space—the particle rolls down along the negative of the gradient of the pseudo-energy, and at locations where the landscape is highly adhesive, it halts. Crucially, we will show that these trapping locations on the pseudo-energy landscape—the locations where the particle halts—correspond to highly ordered spatial configurations such as islands of cells that have the same gene-expression level. A moderate amount of noise can induce the particle to roll down further on the pseudo-energy landscape, and this corresponds to the cells forming patterns with even higher spatial organizations. We thus provide here an intuitive and visual picture, based on experimentally attainable quantities, that is both practical and conceptual for elucidating how a simple class of secrete-and-sense cells spatially coordinate their gene expressions. We will also show that this modeling framework is extendable to more complex forms of cell-cell communication, including those involving more than one type of signaling molecule and multiple cell types.

Results

Cellular Automaton Simulates Secrete-and-Sense Cells that Slowly Respond to a Rapidly Diffusing Signaling Molecule

We used a cellular automaton (Ermentrout and Edelstein-Keshet, 1993) to simulate secrete-and-sense cells. We will compare the results of the cellular automaton with our theory's predictions. We considered a two-dimensional, triangular lattice of N spherical, immobile secrete-and-sense cells of radius R and a lattice spacing a_o. As a proof of principle, we considered “simple” secrete-and-sense cells, which we define to be cells (1) that very slowly respond to their fast diffusing signal and (2) whose gene-expression level, which is determined by the extracellular concentration of the signal and signal-secretion rate, exhibit switch-like (digital) bistability (see Supplemental Information section S1). These two features were motivated by experimentally characterized secrete-and-sense cells. Examples include yeasts that secrete and sense a mating pheromone in a nearly digital manner (diffusion timescale ∼1 s; response timescale ∼30 min) (Youk and Lim, 2014a, Rappaport and Barkai, 2012) and mouse hair follicles, which are secrete-and-sense organs that act as digital secrete-and-sense cells on a triangular lattice (diffusion timescale ∼12 hr; response timescale ∼1.5 days) (Chen et al., 2015, Maire and Youk, 2015a) (see also Table S1). Each cell's gene expression is either “ON” (when its signal-secretion rate is at a maximum) or “OFF” (when its signal-secretion rate is at a minimum, basal level). Each cell senses a steady-state signal concentration c on itself. If c is higher (lower) than a threshold concentration K, which we call an “activation threshold,” then the cell is ON (OFF). When N = 1, the lone ON-cell (OFF-cell) maintains a steady-state concentration C_ON (C_OFF) on itself. We set C_OFF = 1 so that we express all concentrations as multiples of C_OFF. Our cellular automaton computes the concentration on every cell, then synchronously updates each cell's state, and then repeats this process until the cellular lattice reaches a steady-state configuration in which no cell's state requires an update. By running the cellular automaton on randomly distributed ON- and OFF-cells, we observed that initially disordered lattices could indeed evolve into spatially ordered steady-state configurations such as islands of ON-cells (Figure 1A) (Maire and Youk, 2015b).

Figure 1.

Secrete-and-Sense Cells Can Be Classified Into Distinct Behavioral Phases

(A) Snapshots of cellular automaton at different time points, in which an initially disordered cellular lattice becomes more ordered over time. White circle is an OFF-cell, and a black circle is an ON-cell.

(B) (Left column): Phase diagrams for a weak interaction (top panel; f_N (a_o) < 1), a critical interaction (middle panel; f_N (a_o) = 1), and a strong interaction (bottom panel; f_N (a_o) > 1), where the interaction strength is defined as $f_N\left(a_o\right)\equiv {\sum }_{i,j}\frac{e^{R-r_{ij}}}{r_{ij}}\sinh \left(R\right).$ (Right column): Different colors denote distinct behavioral phases.

See also Table S1.

Secrete-and-Sense Cells Can Be Classified Into Distinct Behavioral Phases

To reveal how the disorder-to-order dynamics arises, we will analyze the cellular automaton in each of the cells' “behavioral phases” that we described in a previous work (Figure 1B; details in Supplemental Information section S1) (Maire and Youk, 2015b). As the previous work showed, the behavioral phases represent how one cell turns on/off another cell. They arise from self-communication (i.e., a cell captures its own signal) competing with neighbor communication (i.e., a cell captures the other cells' signal). The communication between two cells, cell-i and cell-j, is quantified by an “interaction term” for that pair, $f\left(r_{ij}\right)\equiv \frac{e^{R-r_{ij}}}{r_{ij}}\text{sinh}\left(R\right)$ (where r_ij is the distance between the centers of cell-i and cell-j and R is both cells' radius). This term is directly proportional to the concentration of the signaling molecule on cell-i that is due to cell-j, and vice versa. We then quantify the competition between the self- and neighbor communication among the N cells with the “interaction strength,” $f_N\left(a_o\right)\equiv {\sum }_{i,j}\frac{e^{R-r_{ij}}}{r_{ij}}\text{sinh}\left(R\right)$, which is the sum of the interaction terms of all cell pairs. It is a function only of the cells' radius R and the lattice spacing a₀. The latter is because all distances between the cells are determined by specifying the lattice spacing. The interaction strength f_N (a_o) measures how much each cell captures the signals from all the other cells (see Supplemental Information section S1) (Maire and Youk, 2015b). For a given interaction strength, the activation threshold K and the C_ON determine the cells' behavioral phase. The values of K, C_ON, and f_N are held fixed, and thus the cells' behavioral phase also remains unchanged over time. We categorize a behavioral phase as either an “insulating phase”—in which no cell can turn on/off the other cells due to dominant self-communication—or a “conducting phase”—in which cells can turn on/off the others due to dominant neighbor communication (Figure 1B). Regardless of the interaction strength, cells can operate in two conducting phases: (1) “activate phase,” in which neighboring ON-cells can turn on an OFF-cell, and (2) “deactivate phase,” in which neighboring OFF-cells can turn off an ON-cell. In addition, when the interaction is weak [i.e., f_N(a₀) < 1], cells can operate in an “autonomy phase,” which is an insulating phase whereby a cell can stay ON/OFF regardless of the other cells' states. On the other hand, when the interaction is strong [i.e., f_N(a₀) > 1], cells can operate in an “activate-deactivate phase,” which is a conducting phase whereby the cells can both activate and deactivate the others depending on their relative locations.

Grouping Multiple Spatial Configurations Into One Macrostate Based on Their Common Spatial Index I and Fraction p of Cells that Are ON

We now present our framework's central ingredient. Let us define two “macrostate” variables: (1) the fraction p of cells that are ON (equivalent to the average gene-expression level) and (2) a “spatial index” I that we define as

$$I=\frac{N}{{\sum }_{i,j\ne i}f\left(r_{ij}\right)}\frac{{\sum }_{i,j\ne i}f\left(r_{ij}\right)\left(X_i-\langle X\rangle \right)\left(X_j-\langle X\rangle \right)}{{\sum }_i{\left(X_i-\langle X\rangle \right)}^2}\text{,}$$ (Equation 1)

where X_i is +1 (−1) for an ON (OFF)-cell and $\langle X\rangle =\frac{1}{N}{\sum }_i{X}_i$ is the average over all the cells. The spatial index I, in fact, belongs to a widely used class of statistical metrics called Moran's I (Moran, 1950). Moran's I is frequently used for spatial analysis in diverse fields, including geographical analysis (Getis and Ord, 1992), ecology (Legendre, 1993), and econometrics (Anselin, 2008). Our spatial index I measures a spatial autocorrelation among the cells by weighing each cell pair by that pair's interaction term f (r_ij) (Maire and Youk, 2015b). Thus, roughly speaking, the spatial index measures the average correlation between the states of any two cells by assigning a higher weight to those cell pairs that communicate more with each other (i.e., more signal is shared between them). By construction, −1 ≤ I ≤ 1 and 0 ≤ p ≤ 1. When I = 0, ON- and OFF-cells are randomly distributed across the lattice, yielding maximally disordered lattices (Figure 2A, top row, and Figure S1). When |I| is large, the cells are more spatially ordered and the lattice consists of large contiguous clusters of ON/OFF-cells (Figure 2A, bottom row, and Figure S1). For I > 0, cells of the same ON/OFF-state tend to cluster together, whereas for I < 0, cells of the same ON/OFF-state tend to avoid each other (Figure S1). As we will see below, we can focus on lattices with a positive spatial index for our purpose. For positive values of I, a key feature that the value of the spatial index tells us is whether the lattice consists of one large, contiguous island of ON/OFF-cells (when I is close to one; Figure 2A, bottom row) or of many fragmented small islands of ON/OFF-cells (when I is close to zero; Figure 2A, top row). Our central idea is to group cellular lattices that have the same (p, I) into a single ensemble (examples in Figure 2A). We then view this ensemble as a particle that moves in an abstract space whose position at time t is (p(t), I(t)). We call this abstract space a “phase space” because each point (p, I) in this space represents an ensemble of all possible spatial configurations that have the same value of p and the same value of I. The procedure of grouping spatial configurations based on their (p, I) is akin to a situation in physics in which many microstates (e.g., the positions and momenta of all particles) are grouped into a single macrostate (e.g., pressure or temperature). Thus, we will call each lattice configuration a “microstate,” and the ensemble of these microstates represented by a given (p, I), a “macrostate” (Figure 2A).

Figure 2.

Spatial Configurations of Secrete-and-Sense Cells (microstates) Can Be Grouped into Macrostates

(A) Examples of microstates that have the same fraction of cells being ON (denoted p) and spatial index I grouped into a single macrostate, denoted by (p, I). For each macrostate (p, I), three microstates are shown as examples.

(B) (Top row): Probability density maps showing the particle's final value of p (denoted p_final) for each initial value of p (denoted p_initial) in the activate phase, deactivate phase, and activate-deactivate phase. Color code for the probability density is shown in the color bar at the bottom. The red dashed line in the activate phase (left panel) and the green dashed line in the activate-deactivate phase (right panel) approximate the lowest value of p_initial that is required to turn on every cell (i.e., reach p_final = 1). The red dashed line in the deactivate phase (middle panel) and in the activate-deactivate phase approximates the highest value of p_initial required to turn off every cell (i.e., reach p_final = 0). (Bottom row): Trajectories (red and green curves) in p-I space (called “phase space”) in the activate phase (left panel), deactivate phase (middle panel), and activate-deactivate phase (right panel). Gray insets show zoomed-in views of some trajectories. Black dots denote the trajectories' endpoints.

See also Figure S1.

Cellular Lattice Is Represented by a Particle Whose Position (p, I) and Trajectory Depend on the Behavioral Phase

By randomly choosing thousands of microstates that all belong to the same disordered macrostate (p = p_initial, I ≈ 0) and then running the cellular automaton on each of these microstates, we observed how the lattices evolved out of disorder. Specifically, we obtained a distribution of their trajectories, and thus also a distribution of their final positions (p = p_final, I = I_final), for every value of p_initial in each behavioral phase (Figures 2B and S3). The fact that we obtained, for a fixed value of p_initial, a distribution of values for p_final (Figure 2B, top row) and a distribution of trajectories (Figure 2B, bottom row) instead of a single trajectory, indicates that the particle moves stochastically in the p-I space. This stochasticity arises from the cellular automaton operating on individual cell's state X_i, a microstate variable, at each time step rather than operating on the macrostate variables, p and I. Also, since, at the macrostate level, we are ignorant of the exact microstate that the cellular automaton is operating on, the macrostate-level description of the particle's motion, once we deduce it, would have to be a stochastic description. We found several promising signs that an analytical, macrostate-level description is possible. First, we observed that particles that started at the same position (p_initial, 0), for the most part, remained close to each other in subsequent times, leading to tightly bundled trajectories in the p-I space despite the stochasticity (Figure 2B, bottom row). Furthermore, we observed other features that were shared by all the trajectories for each behavioral phase. Specifically, in the activate phase, we observed that if the p_initial was above a certain threshold value (red vertical line in Figure 2B, top left panel), then almost all cells were turned on, whereas if it was below the threshold value, then the activation was minimal owing to the cellular automaton not starting with enough ON-cells. In the deactivate phase, we observed that if the p_initial was below a certain threshold value (red vertical line in Figure 2B, top middle panel), then almost all cells turned off, whereas if it was above the threshold, then the deactivation of ON-cells was minimal owing to the cellular automaton not starting with enough OFF-cells. Finally, in the activate-deactivate phase, we observed a threshold value for activation (green vertical line in Figure 2B, top right panel) and a threshold value for deactivation (red vertical line in Figure 2B, top right panel). Between these two thresholds, a particle stops with a value of p that is either only slightly higher (activation) or slightly lower (deactivation) than the value that it started with (giving rise to a slanted “bow tie” shape in Figure 2B, top right panel). We also observed common features in the shapes of the trajectories themselves in the p-I space. Specifically, we observed that in every trajectory, the I initially increased before plateauing at some value, whereas the p either monotonically increased or decreased over time (Figure 2B, bottom row). Then, one of two events occurred in all trajectories: either (A) the particle stopped, and thus the cellular automaton terminated, with the final value of p (i.e., p_final) between zero and one (see black dots that mark the trajectories' endpoints in Figure 2B, bottom row) or (B) the particle kept increasing or decreasing its p until it reached and stopped at either p = 1 (all cells ON) or p = 0 (all cells OFF), and as it did so, its spatial index abruptly dropped to zero (e.g., most of the red trajectories in Figure 2B). Observation (A) corresponds to a situation in which the cells form an ordered spatial configuration that, being a steady state of the cellular automaton, remains unchanged indefinitely. This situation arose most notably but not exclusively in the activate-deactivate phase. Observation (B) corresponds to a situation in which all cells either turn on or off.

To explain observation (B), we first rewrite Equation 1 as (Supplemental Information section S2)

$$I\left(p\right)=\frac{\text{Θ}-{\left(2p-1\right)}^2{f}_N\left(a_O\right)}{4p\left(1-p\right)f_N\left(a_O\right)}\text{,}$$ (Equation 2)

where $\text{Θ}=\frac{1}{N}{\sum }_{i,j\ne i}f\left(r_{ij}\right)X_i{X}_j$. Note that the p and the spatial index I depend on each other. And since Equation 2 enables us to deduce the Θ if we know the I, and vice versa, we have the option of considering (p, Θ) to be a macrostate instead of (p, I). The main disadvantage of this is that the Θ, unlike the spatial index, is not normalized. This makes it difficult to compare the values of Θ for lattices with different values of p. Thus we will work with (p, I) instead of (p, Θ). From a mean-field approximation, in which we calculate the average amount of signal sensed by each cell (Figure S2 and Supplemental Information section S3), we deduced that the particle's spatial index has an upper bound for each value of p. We denote this p-dependent maximal value of I by a function I_max(p) (dashed black curves in Figure 3). The function I_max(p) sharply drops to zero as p nears zero or one. Accordingly, as the particle's p nears zero or one, its spatial index should sharply decrease to zero in accordance with observation (B) (Figures 3B–3D). This makes sense because the spatial index is a measure of whether or not the lattice consists of a large, contiguous island of ON/OFF-cells. As the spatial index approaches zero, the lattice becomes populated with more fragments of smaller islands of ON/OFF-cells. When the p is near zero (one), as is the case when only one cell is ON (OFF), then no clusters of ON-cells (OFF-cells) are possible since there is only one ON-cell (OFF-cell). Owing to this and from a rigorous calculation of how the I changes as the p approaches zero or one (Supplemental Information section S2), we find that the spatial index is indeed zero when the p is either zero or one. To fully explain the particle trajectories along with observations (A) and (B), we next sought an equation of motion for the particles.

Figure 3.

A Cellular Lattice Acts as a Particle that Rolls Down on and Adheres to a Pseudo-Energy Landscape

(A) Pseudo-energy landscape with a height defined by the pseudo-energy function h(p, I). Orange ball is a particle that represents a cellular lattice. The landscape is defined over a position (p, I). A pseudo-energy landscape for (B) activate phase, (C) deactivate phase, and (D) activate-deactivate phase.

(B–D) Trajectories of the same color start from the same position in each landscape. Black curves show maximally allowed value of the spatial index I [i.e., function defined as I_max(p) in the main text; see Supplemental Information section S4].

See also Figures S2 and S3.

Cellular Lattice Acts as a Particle that Rolls Down on and Adheres to a Pseudo-Energy Landscape

We conjectured that if a cellular lattice indeed moves like a particle, then there may be a “landscape” on which the particle rolls down. To explore this idea, we consider a function h that we call a “pseudo-energy” and define it as $h\equiv -{\sum }_i{X}_i\left(Y_i-K\right)/N$, where Y_i is the signal concentration on cell-i. In fact, we can rewrite h entirely in terms of the macrostate variables, p and I (Supplemental Information section S4). Plotting h(p, I) yields a three-dimensional landscape that we call a “pseudo-energy landscape” (Figure 3A). Its shape depends on the cells' behavioral phase (Figures 3B–3D). Importantly, by plotting the trajectories on top of their respective landscapes, we observed that every particle's pseudo-energy (i.e., value of h) monotonically decreased over time until the particle stopped. We could also rigorously prove this (Supplemental Information section S4). The fact that the pseudo-energy is a decreasing function of the spatial index explains why trajectories in general tend toward increasing values of the spatial index (Figures 3B–3D).

To see, at the microstate level, why the cells' states become more spatially correlated over time, we rewrite the h as

$$h=-\alpha \sum _{i,j\ne i}f\left(r_{ij}\right)X_i{X}_j-B\sum _i{X}_i-N\alpha \text{,}$$ (Equation 3)

where α ≡ (C_ON − 1)/(2N) and B is a “signal field” defined as α (1 + f_N(a_O)) − K/N. Equation 3 is strikingly similar to the Hamiltonians of the Hopfield network (Hopfield, 1982) and magnetic spins with long-range interactions (Kirkpatrick and Sherrington, 1975, Tchernyshyov and Chern, 2011). Note that since αf (r_ij) > 0 and the particle's pseudo-energy keeps decreasing over time before the particle stops, the cells must “align” their states with each other rather than "anti-align" (i.e., the pseudo-energy favors the pairing of two ON-cells rather than pairing of an ON-cell with an OFF-cell). In magnetic spin systems, this would be analogous to a ferromagnetic interaction. As in physical systems, we can view the signal field B as a macroscopic knob that we can tune to change the shape of the pseudo-energy landscape for a given cellular lattice. From the phase diagrams (Figure 1B), we can deduce that B > 0 in the activate phase; that B < 0 in the deactivate phase; and that B can be positive, negative, or zero in the activate-deactivate phase (depending on K and C_ON) (Figure S3). Intuitively, increasing the value of Θ, and thus the value of I (by Equation 2), corresponds to the formation of larger clusters of ON-cells and OFF-cells, which would in turn decrease the pseudo-energy because the first term in Equation 3 equals −αNΘ. Despite these similarities, we emphasize that the cellular lattice is not the same as an Ising spin system. For one, there is no real Hamiltonian in our framework that, for instance, gives rise to a Boltzmann distribution. Importantly, we have not used any quantities from physics in our framework, despite some similar properties shared by the framework presented here and those of statistical physics. In the Discussion section, we will elaborate further on these similarities and differences.

Gradient of the Pseudo-Energy and the Trapping Probability P_eq(p, I) Completely Specify the Particle's Motion

To deduce how exactly the shape of the pseudo-energy landscape determines the particle's motion, we compared the gradient field of the pseudo-energy −∇h(p, I) (arrows in Figures 4A–4D) with the particle trajectories produced by the cellular automaton (red curves in Figures 4A–4D). We discovered that the particles closely follow the streamlines that are dictated by the gradient field. From this and the aforementioned observation that the particles move stochastically, we conjectured that the particles may follow Langevin-type dynamics in which the particle drifts (rolls) down the pseudo-energy landscape due to the gradient field and diffuses due to a noise term. We then proposed a phenomenological equation of motion for the particle,

$$(\text{Δ}p\left(t\right),\text{Δ}I\left(t\right))=-\nabla h(p\left(t\right),I\left(t\right))·\delta +({\eta }_p\left(t\right),{\eta }_I\left(t\right))$$ (Equation 4)

Figure 4.

Gradient Field of the Pseudo-Energy, −∇h(p, I), and the Trapping Probability, P_eq(p, I), Completely Specify the Particle's Trajectory

(A–D) Each gray arrow represents the negative gradient of the pseudo-energy, −∇h(p, I), at each position (p, I). Longer arrows indicate gradients of larger magnitudes. Heat maps show the magnitude of the trapping probability P_eq at each location (Supplemental Information section S5). Red trajectories are exact particle trajectories from the cellular automaton. Green trajectories are particle trajectories produced by Monte Carlo simulations that are dictated by the equation of motion (Equation 4) and the trapping probability. The green dots represent the starting points of the trajectories (same for the trajectories produced by the cellular automaton and the equation of motion), and the black crosses represent the endpoints of the green trajectories. (A) Autonomy phase, (B) activate phase, (C) deactivate phase, and (D) activate-deactivate phase.

See also Figures S4–S7.

Here Δp(t) and ΔI(t) are changes in p and I, respectively, between time steps t and t + 1; δ is a constant factor that scales the gradient to account for the discreteness of time in the cellular automaton; and η_p and η_I are Gaussian noise terms that represent our ignorance of the microstates with a mean of zero and standard deviations of σ_p and σ_I, respectively. We determined δ, σ_p, and σ_I by calculating the mean and the variance of Δp, which in turn are set by the distribution of the signal concentrations that each cell senses for a given (p, I) (Supplemental Information sections S5-S6). Although the pseudo-energy determines the direction and the magnitude of changes in p and I, it does not predict where a particle stops on the landscape. As we noted earlier [observation (A)], the particle can stop before its value of p reaches zero or one. This corresponds to stopping at inclined regions of the pseudo-energy landscape. For this reason, we consider the landscape to be “adhesive,” such that the particle can stop moving on its inclined regions. The gradient of the pseudo-energy is non-zero at such inclined locations, but the particle stops because it has adhered to the landscape at that location. Such particle adhesions occur frequently for the activate-deactivate phase and in the autonomy phase (e.g., termination points of the brown trajectories in Figure 3D). Crucially, the particle halts in a stochastic manner, meaning that for two particles that pass through the same location (p, I), one may get stuck there, whereas the other does not. This is because each macrostate (p, I) can include microstates that are steady states of the cellular automaton and microstates that are not. We need a probabilistic description of how likely it is that a particle at a given location halts since we do not know which microstate is represented by the moving particle when we run a Monte Carlo simulation of Equation 4. To obtain a stochastic description, we used a mean-field approach to estimate, for a given macrostate (p, I), the fraction of microstates in it that are steady states of the cellular automaton (Supplemental Information section S5). We call this fraction, which is between zero and one, the “trapping probability” and denote it by P_eq(p, I). It is the probability that a particle at location (p, I) corresponds to a steady state of the cellular automaton and thus halts there. Roughly speaking, the trapping probability P_eq(p, I) represents the “adhesiveness” of the landscape that we discussed earlier. To produce particle trajectories, we ran a Monte Carlo simulation that combines the phenomenological equation of motion (Equation 4) and the condition that the particle halts at location (p, I) with a probability P_eq(p, I) (Supplemental Information section S6). We found that the particle trajectories obtained from these Monte Carlo simulations (green curves in Figures 4A–4D) recapitulated, for a wide range of parameters, the main qualitative features of the particle trajectories that the cellular automaton produces (red curves in Figures 4A–4D), including the general regions where the particles get stuck, despite some deficiencies (Figures S4–S7). We will discuss the limitations of this approach in the Discussion section.

Stochastic Sensing Can Yield Spatial Configurations that Are More Ordered than Those Formed without Noise

Having shown where the particle gets stuck on the pseudo-energy landscape, a natural question is how stably the particle sticks at each location. Biological noise is a sensible context to address this question. To address this question and as a proof of principle for demonstrating how to include stochastic gene expression in our framework (Raj and Van Oudenaarden, 2008, Sagués et al., 2007, García-Ojalvo, 2011, Tkačik and Walczak, 2011, Sanchez and Golding, 2013, Xu et al., 2016, Friedman et al., 2006), we modified the deterministic cellular automaton that we have been using thus far to include stochastic sensing. Specifically, for each cell and at each time step of the cellular automaton, we now pick a new value for the activation threshold, K + δK. Here, K is the same value for every cell at all times and δK is a Gaussian noise term with a mean of zero and a variance of α² (Figure 5A and Supplemental Information section S7). We then define a “noise strength,” ξ = α/K, that helps us determine how much noise is required to liberate an adhered particle and cause a moving particle to significantly deviate from the path that it would have taken if there were no noise. Intuitively, we would expect such deviations to occur if the noise δK is sufficiently large, such that either an ON-cell, on which the average signal concentration $\langle Y_{ON}\rangle$ is larger than the activation threshold without the noise, K, would turn off due to the noise increasing the activation threshold so that it becomes larger than $\langle Y_{ON}\rangle$, or an OFF-cell, on which the average signal concentration $\langle Y_{OFF}\rangle$ is smaller than K, would turn on due to the noise decreasing the activation threshold so that it becomes smaller than $\langle Y_{OFF}\rangle$. Mathematically, this means that we would expect the minimum noise strength ξ_min required to significantly perturb the particle trajectories to be $\min \left(\left|\langle Y_{ON}\rangle -K\right|,\left|\langle Y_{OFF}\rangle -K\right|\right)/\left(K\sqrt{N}\right)$ (Supplemental Information section S7 and Figure S8). Indeed, we found that a very weak noise (i.e., ξ ≪ ξ_min) cannot detach an adhered particle from the landscape (Figure 5B, left column), whereas a very strong noise (i.e., ξ ≫ ξ_min) can detach an adhered particle and thereby cause the particle to roll down the landscape further. After being detached, the particle further changes its p, decreases its pseudo-energy, and increases its spatial index until its p reaches either zero or one (Figure 5B, right column). Moreover, we found that a moderate noise (i.e., ξ ∼ ξ_min) can liberate the adhered particle and push it further down the landscape, beyond the previously allowed region of the landscape (i.e., beyond the region bounded by I_max(p) [black curve in Figure 5C]), until it adheres to the landscape again, but now with a higher spatial index than before and with an intermediate value of p I (i.e., 0 < p < 1) (Figures 5C and S9). Intriguingly, when there is a moderate noise in the activate-deactivate phase, we observed that some of the trapped particles' p, I, and h very slowly changed over time, allowing the particles to remain stuck with an intermediate value of p over hundreds but not thousands of time steps (Figure 5D). As a follow-up study, it would be interesting to examine if this phenomenon is similar to the glass-type dynamics seen in physics.

Figure 5.

Stochastic Sensing Can Yield Spatial Configurations That Are More Ordered than Those Formed without Noise

(A) (Left column) Schematics of secrete-and-sense cells with noisy sensing. Each cell (circle) is colored by a different shade of orange, with a darker shade representing less noise. (Top right panel) Noise in activation threshold K, denoted δK, is normally distributed with a zero mean and a variance α². (Bottom right panel) Range of activation thresholds K + δK for each cell.

(B) Examples of changing fraction p of cells that are ON, spatial index I, and pseudo-energy h for low noise (left column; ξ < ξ_min) and high noise (right column; ξ > ξ_min) in the activate-deactivate phase. ξ = α/K is the noise strength and ξ_min is the minimum noise strength required to detach an adhered particle (Supplemental Information section S7). Both the low noise and the high noise scenarios begin with a spatial configuration that is a steady state of the deterministic cellular automaton.

(C) Particle trajectories (red curves), in activate-deactivate phase, for a deterministic cellular automaton (left column) and cellular automaton with a moderate noise (i.e., ξ < ξ_min) in sensing (right column). All trajectories start at (p = 0.5, I ≈ 0). Black dots show endpoints of trajectories. Calculated maximum I as a function of p when no noise is absent (black curve) and when a moderate noise is present (orange curve) are shown (also see Supplemental Information section S3).

(D) (Top panel) Snapshots at different times of cellular lattice becoming more ordered due to noise in sensing in the activate-deactivate phase. Black circles are ON-cells, and white circles are OFF-cells. (Bottom panel): Fraction p of cells that are ON (red curve), spatial index I (blue curve), and pseudo-energy h (green curve) over time for the pattern formation shown in the top panel. Zoomed-in views (gray boxes) show slowly changing p (red curve), I (blue curve), and h (green curve) that occur while the cellular lattice is in a highly ordered metastable configuration (shown at t = 300 in the top panel).

See also Figures S8–S10.

Discussion

Here we have uncovered a visual landscape for a ubiquitous form of cellular communication, called secreting and sensing, and showed that it underlies how simple secrete-and-sense cells' gene expressions become more spatially correlated over time in the absence of any pre-existing morphogens. Instead of focusing on how specific spatial patterns such as stripes and islands emerge, we focused on the overall spatial order, a statistical measure of cell-cell coordination of gene expressions that we called spatial index. This macrostate-level description has the advantage of making exhaustive, numerical simulations that are typically used for these systems unnecessary but has the disadvantage of being ignorant of the specific spatial patterns that form. The spatial index, however, still allows us to discern what kinds of spatial patterns are formed because fixing its value restricts the spatial patterns that are possible (Figure 2A). Toward showing that our approach may be adapted to other types of cell-cell communication, we show, in the Supplemental Information, how to extend our framework to lattices with multiple cell types and signal types (Supplemental Information section S8 and Figure S10). These include paracrine signaling, in which one cell secretes a signal without sensing it, whereas another type of cell senses without secreting that signal (Doğaner et al., 2016). Despite its wide applicability, there are instances where the current framework would not apply. We now turn to discussing these situations before concluding with a discussion on how our framework is distinct from that of physics and how one can apply our model to experiments.

Our modeling framework for secrete-and-sense cells with a bistable (ON/OFF) gene expression relied on meeting two conditions: (1) every cell adjusting its ON/OFF-state within the same timescale and (2) the concentration of the signaling molecule on each cell reaching a steady state before the cell can switch its ON/OFF-state. The first condition sets the actual time that each discrete time step of the cellular automaton represents and is the reason that the cellular automaton simultaneously updated every cell's state. It is satisfied if the variability among cells in their response times to the signaling molecule (i.e., time taken by each cell to change between ON- and OFF-states) is smaller than the average response time of the cell. The second condition, which states that the typical response time of the cells is larger than the time that the signaling molecule takes to form a steady-state concentration, is satisfied in several biological processes. They include the aforementioned yeasts that secrete and sense the mating pheromone and the regenerating hair follicles in mice (Youk and Lim, 2014a, Chen et al., 2015, Rappaport and Barkai, 2012, Maire and Youk, 2015a). The condition is also satisfied by several quorum-sensing bacteria (e.g., ∼20–30 s to establish a steady-state concentration) (Kaplan and Greenberg, 1985, Pearson et al., 1999). Despite these examples, a major aspect that we have neglected is that signaling molecules are often affected by processes other than diffusion such as active transporting of the molecules and clustering and endocytosis of receptors. Several studies of morphogen gradients in developing embryos, however, have shown that in many cases, one can use a simple diffusion alone to mathematically reproduce the creation dynamics of morphogen gradients even when there are other processes (Lander et al., 2002). Finally, aside from conditions (1) and (2), our model assumes that cells are arranged on a triangular lattice. Indeed, several systems, including the nuclei inside the early Drosophila melanogaster embryo, can be approximated as being arranged on a triangular lattice despite not satisfying both conditions (1) and (2) (Gregor et al., 2007) (see other examples in Table S1). For other regular lattices, one can modify the framework by changing the functional form of the interaction strength f_N(a_o).

Another element in our framework whose validity requires a careful thought is the equation of motion (Equation 4). The equation of motion is a phenomenological equation that recapitulates the main qualitative features of the particle trajectories but does not reproduce the exact location of the particle at every time step of the cellular automaton (Figures 4A–4D). As an example, given any initial value of the fraction p of ON-cells, the equation of motion accurately predicts whether the p will increase, decrease, or stay the same (Figure S6). However, the trajectories produced by the equation of motion do not exactly match those produced by the cellular automaton. In particular, the trajectories produced by the equation of motion are least likely to match those of the cellular automaton at locations where the gradient vector of the pseudo-energy is perfectly horizontal (i.e., parallel to the p-axis) or vertical (i.e., parallel to the I-axis), and most likely to match when the gradient is at 45° with respect to both axes. Since the gradient is neither perfectly horizontal nor vertical (Figures 4A–4D) at most locations, the gradient of the pseudo-energy together with the trapping probability P_eq (p, I) gives a qualitatively accurate description of the particle's motion. We also found that the equation of motion gives a more accurate description of the particle trajectories for strong interactions [i.e., f_N(a₀) > 1] than weak interactions [i.e., f_N(a₀) < 1]. To see why this is, note that we used mean-field approximations, in which we assumed that ON- and OFF-cells are randomly distributed, to determine the values of σ_p, σ_I and δ in the equation of motion (Equation 4) (Supplemental Information section S6). This mean-field approximation breaks down if long-lived, large islands of ON- and OFF-cells form and slowly grow over time. Such islands indeed frequently form when the interaction is weak and lead to the cellular automaton producing higher values of the spatial index I than the equation of motion allows for (Figure S4). In contrast, when the interaction is strong, the particle typically moves faster because the effect of changing the ON/OFF-state of a single cell propagates to the faraway cells. Thus the entire lattice of cells typically turns on or off in a few time steps without clearly forming local domains of ON/OFF-cells that grow over time (Figure S5). Hence the equation of motion is more suitable for strong interactions than for weak interactions. Finally, we note that another source of quantitative disagreements between the equation of motion and the cellular automaton lies in the fact that in computing the gradient of the pseudo-energy, the equation of motion assumes that p and I are continuous variables when in fact they are discrete quantities since the number of cells N is finite. This continuum approximation, however, is valid in the limit of the population size approaching infinity. This is because the spacing between two adjacent values of p is 1/N and the spacing between two adjacent values of I for a fixed value of p scales as 1/N (when p is neither zero nor one; note that there is only one value for I when p is zero or one).

In this article, we have shown that it is possible build a physics-type framework for complex multicellular systems that are governed by chemical signals, gene-regulatory networks, and multiple cells. Many such systems are currently only treated by exhaustive, numerical simulations and lack analytical frameworks of the type that we presented here. This situation has risen because the established metrics from physics, such as energy and momentum, are ill-suited for describing gene expressions and chemical signals in multicellular systems. Researchers have used physics-type frameworks to explain many-body living systems such as birds that flock together (Vicsek et al., 1995) and tissues that are subject to mechanical forces (Graner and Glazier, 1992), whereas multicellular systems of the type that we studied here, which are not governed by mechanical or electrical means, have been difficult to treat by directly applying existing concepts and quantities from physics. Despite the similarities in the approach that we have taken and that of statistical mechanics, our framework should not be interpreted in terms of existing quantities from physics because our model does not use any existing quantities of physics such as energy, force, momentum, or temperature. For example, the pseudo-energy (Equation 3) only mathematically takes the same form as the long-ranged Ising Hamiltonian. However, the particle does not follow the equations of Hamiltonian mechanics. As another example, the concepts of detailed balance and thermal equilibrium do not apply to the particle that is stuck on the pseudo-energy landscape. In other words, there is no state in which the macroscopic variables remain constant, whereas the cellular lattice dynamically transitions between microstates of the same macrostate. The notions of entropy and temperature also do not have straightforward definitions in our system. One can count the total number of microstates for a given (p, I) or the number of steady-state microstates for a given (K, C_ON) (Maire and Youk, 2015b), but neither would be a thermodynamic entropy. In light of these considerations, it would be interesting to explore, in a future work, if the quantities in our framework can be derived from the quantities of physics.

Experimentally, one can measure the two macrostate variables, p and I, in microscopic images [e.g., by tagging fluorescent protein(s) to the output gene(s)]. One may also use the tools of optogenetics to engineer the cells so that shining light on a single cell would cause the cell to secrete a signaling molecule or switch between the ON- and OFF-state (Guglielmi et al., 2016). One can then use light to sculpt a pattern of secreting ON-cells at the beginning of an experiment, in effect initializing the values of p and I, and then observe how the ON- and OFF-states change by recording over time the fluorescence of each cell, which reports whether the cell is ON or OFF. Our model and its extensions may help in understanding such microscope-based time-lapse movies of secrete-and-sense cells that form spatial patterns. Along with studying how specific spatial patterns, such as stripes and islands, are generated, it is useful to focus on statistically describing how certain classes of spatial patterns arise without knowing the exact spatial patterns involved, as we have done here. This is because one often cannot measure all the parameters that are required for constructing detailed numerical models (e.g., gene-expression level of every cell in a tissue). In such situations, our framework and its extensions may help in predicting, based on a limited knowledge of the underlying gene-regulation scheme and an estimate of the system's initial spatial order, how the spatial configuration of the cells evolves over time without revealing the exact location, shape, and size of the resulting spatial pattern. We hope that our work, along with complementary approaches for studying spatial patterns (Cotterell et al., 2015, Surkova et al., 2009, Sokolowski et al., 2012, Tkačik et al., 2008, Hillenbrand et al., 2016, Erdmann et al., 2009, Fancher and Mugler, 2017, Thalmeier et al., 2016), will inform ongoing efforts to establish quantitative frameworks for multicellular gene regulations.

Methods

All methods can be found in the accompanying Transparent Methods supplemental file.

Published: April 27, 2018