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# Inferring epigenetic dynamics from kin correlations

Contributed by Boris I. Shraiman, March 9, 2015 (sent for review September 15, 2014)

## Significance

Different cells in a clonal population can be in different phenotypic states, which persist for a few generations before switching to another state. Dynamics of switching between these states determines the extent of correlations between the phenotypes of related cells. Here we demonstrate—using ideas from statistical physics—that it is possible to infer simple stochastic dynamics along lineages from an instantaneous measurement of phenotypic correlations in a cell population with defined genealogy. The approach is validated using experimental observations on *Pseudomonas aeruginosa* colonies.

## Abstract

Populations of isogenic embryonic stem cells or clonal bacteria often exhibit extensive phenotypic heterogeneity that arises from intrinsic stochastic dynamics of cells. The phenotypic state of a cell can be transmitted epigenetically in cell division, leading to correlations in the states of cells related by descent. The extent of these correlations is determined by the rates of transitions between the phenotypic states. Therefore, a snapshot of the phenotypes of a collection of cells with known genealogical structure contains information on phenotypic dynamics. Here, we use a model of phenotypic dynamics on a genealogical tree to define an inference method that allows extraction of an approximate probabilistic description of the dynamics from observed phenotype correlations as a function of the degree of kinship. The approach is tested and validated on the example of Pyoverdine dynamics in *Pseudomonas aeruginosa* colonies. Interestingly, we find that correlations among pairs and triples of distant relatives have a simple but nontrivial structure indicating that observed phenotypic dynamics on the genealogical tree is approximately conformal—a symmetry characteristic of critical behavior in physical systems. The proposed inference method is sufficiently general to be applied in any system where lineage information is available.

Collectives of nominally isogenic cells, be it a clonal colony of bacteria or a developing multicellular organism, are known to exhibit a great deal of phenotypic diversity and time-dependent physiological variability. While often transient and reversible, phenotypic states of cells can persist on the time scale of the cell cycle and be transmitted from mother to daughter cells. This epigenetic inheritance has been a subject of much recent research and is known to involve a multitude of different molecular mechanisms (1⇓–3), from transcription factor transmission to DNA methylation (4, 5). Stable phenotypic differentiation is at the heart of any animal and plant developmental program (6, 7). The role and extent of phenotypic variability in microbial populations is less well understood, but is coming into focus with the spread of single cell-resolved live imaging (8, 9) and other single-cell phenotyping methods (10). Phenotypic variability within a colony implements the intuitively plausible bet-hedging strategies of survival (11⇓⇓⇓–15), such as persistence (16), sporulation (17), or competence (18). More generally, phenotypic variability may be implementing interesting “separation of labor”-type cooperative behavior within colonies (19), although evolutionary stability of such strategies remains a subject of much theoretical debate (20⇓–22). Phenotypic variation can originate from precisely controlled pattern-forming interactions either from global or local intercellular signaling, as is the case in animal and plant development. For microbes, intracellular stochasticity is seen as playing a leading role in driving transitions between physiologically significant phenotypic states (23⇓–25). It is an open problem to understand the extent to which the phenotypic diversity in a bacterial system is driven by cell-autonomous stochastic processes as opposed to interaction with their neighbors, which could take the form of a feedback through local nutrient availability, secreted factors (26), or direct contact signals (27).

As an example, we consider *Pseudomonas aeruginosa*, a common bacteria that, like all others, requires iron for metabolism, DNA synthesis, and various other enzymatic activities. To absorb iron from its naturally occurring mineral phase, *P. aeruginosa* produces and releases iron-chelating molecules called siderophores (28, 29). Pyoverdine (Pvd) is a type of siderophore that is particularly suited for experimental analysis, because it is naturally fluorescent (30). Pvd concentration varies significantly from one cell to another (31), which is largely due to the fact that Pvd is trafficking between cells that either sip or secrete them (29). Moreover, Pvd concentration along lineages has a correlation time of the order of two to three cell cycles (31). The feeder/recipient phenotypes are epigenetically passed on for a few generations before switching—a recent observation (31) that changes the landscape of the discourse on common goods, cooperation, and cheating.

Dynamics of stochastic phenotypes can be followed through multiple generations using fluorescent time-lapse microscopy and single-cell tracking (8, 9). However, the number of distinct fluorescent reporters of gene expression in a single cell is inherently limited by their spectral overlap. Alternatively, phenotypic heterogeneity can be measured with relative ease using destructive or fixed cell methods [such as immunostaining (32) and fluorescence in situ hybridization (FISH) (10)] that only provide static snapshots. Destructive measurements can, however, be supplemented with lineage information (kinship) that can be collected using phase time-lapse microscopy and single-cell tracking. We ask: How much can one say about dynamics from a static snapshot of heterogeneity and the knowledge of the relatedness of the individuals in a population? Below, we shall take a constructive approach to this question, demonstrating that by adopting a certain plausible and quite general probabilistic description of phenotypic dynamics along lineages, it is indeed possible to infer the dynamics from static snapshots. We shall test the method on the example of Pvd dynamics in *P. aeruginosa*, comparing the inference to direct dynamical measurements.

Thus, our goal here is to provide a tool for the study of epigenetic dynamics within proliferating collectives of cells. We shall focus on the cell-autonomous dynamics and mother-to-daughter transmission and relate the statistical description of phenotypic dynamics along any one lineage to the observable correlations between phenotypic states in a snapshot of cells at any given time, which, as we shall see, explicitly depend on the degree of kinship of the cells. Below, after framing our approach as an inference problem (*Inference Problem for Phenotypic Dynamics*), we shall define a class of models parameterizing phenotypic dynamics on lineages (*The Minimal Model of Stochastic Phenotype Propagation* and *Effective Interactions Between Siblings*) and explicitly calculate the form of “kin correlations” from which the underlying dynamics is to be inferred. In *Kin Correlations in the Poverdine Dynamics* and *Inferring the Interactions in* P. aeruginosa, we shall apply the approach to the data on siderophore production in *P. aeruginosa* colonies (31), which will allow us to compare the inference results with the direct measurement of time-dependent phenotypes of all cells within the colony, validating our approach. *Spatial Interactions* will address the question of kinship and spatial correlations within a bacterial colony. In *Discussion*, we shall explain why kin correlations have a structure similar to that of correlations in conformal field theories known in physics (33, 34) and address possible practical applications of the approach.

## Results

### Inference Problem for Phenotypic Dynamics.

Consider a growing population of asexual individuals. At every generation, each individual gives rise to two daughters that, with some probability, inherit the phenotypic traits of their parent. This growing population is naturally represented as a tree (see Fig. 1): The most current population of cells corresponds to the leaves of the tree, while the branches represent its history back to the founder cell at the root. Phenotypic dynamics unfolds along the lineage linking any one leaf to the root, and correlations between kin arise from the fact that close relatives share more of their history. We shall assume that phenotypic dynamics is stochastic with some well-defined probabilistic rule (e.g., some Markovian random process), so that the state of a cell along its lineage through the genealogical tree is a realization of the random process. Phenotypic variability within cell population defines a distribution of states *i*,

Kin correlations are then defined as correlations between the phenotypic states of pairs, triples or, in general, *m*-tuples of leaves with the same degree of relatedness. More specifically, we characterize kin correlations by the joint distribution describing the probability of different cells being found simultaneously in certain states. For example, for the pair correlator, *u* generations in the past that are in states *m* and *n*,*u*. Because of the possible correlations, this joint probability may not be equal to the product of probabilities, *n* and *m* on their own. These correlations are explicitly captured by

Similarly, the triple distribution is defined by*u* is the number of generations to the common ancestor of the more closely related pair, *v* is the further number of generations back to the common ancestor of all three nodes (see Fig. 1), and

### The Minimal Model of Stochastic Phenotype Propagation.

Let us begin with the simplest possible model. Assume that stochastic dynamics can be approximated by a Markov process, which means that probability to transition from state *n* to a state *m* in the time of a cell cycle depends only on the two states involved; i.e., the dynamics is defined probabilistically by a transition probability matrix *n* and end up in state *m* time *u* generations later is given by the product of the *u* transition matrices obtained by iterating *l*, *u* generations back,

The third-order correlator can be written down in a similar way,

In our minimal model, we assume that phenotypic states effectively form a chain with transitions occurring only between neighboring states (more generally, any graph without loops would behave the same way). In this case, stochastic dynamics satisfies Detailed Balance (33, 35) (see *SI Text*), meaning that, in equilibrium, the forward and backward fluxes between any pair of states balance: *m*, satisfies

We note that thus defined, our minimal model of epigenetic dynamics is mathematically identical to the model studied by Harlow et al. (33) (in a very different context). Following ref. 33, we diagonalize the symmetric matrix,

To take full advantage of the ensuing simplifications, we define correlators in the

In this basis, pair correlators for our “minimal model” of phenotypic dynamics have a very simple form (33),**8**.*Discussion*.

Note that since **9** and **10**. We emphasize that

In fact, it can be shown (see *SI Text*) that all of the higher-order correlators can be expressed completely in terms of

### Kin Correlations in the Pyoverdine Dynamics in *P. aeruginosa*.

In the experiments of Julou et al. (31), the fluorescence of free Pvd in each bacterium was measured using time-lapse fluorescent microscopy, while the growth of the colony was followed with phase microscopy, providing the genealogical tree. For the analysis below (see *Methods*), we used only the final snapshots of Pvd distribution for nine colonies, each with

It is plausible to think of Pvd dynamics in the colony as a stochastic process on a tree subject to interactions that correspond to local exchange of Pvd. We begin by comparing the observed pairwise kin correlations to the prediction of our minimal model given by Eq. **9**. To that end, we construct correlation matrices for pairs of leaves conditioned by their relatedness, *u*, and diagonalize them. Fig. 2*A* depicts the eigenvalues of the two-point correlation matrices *u*. The eigenvalues are taken to the power of *u* (see Eq. **9**). The observed values, however, are significantly different from constant (see *SI Text* for the *P* values), suggesting either a presence of interaction or a deviation from the simple Markovian or detailed balance form of stochastic dynamics.

However, the observed eigenvalues deviate most at *u*, suggesting the minimal model may still provide a good description of correlations among distant relatives. To test that, we examined third-order correlators, for which the minimal model predicts Eq. **10**: an expression defined entirely in terms of the second-order correlators, without any additional parameters. (As noted above, this relation is the consequence of the hidden conformal symmetry of the process.) Fig. 2 *B* and *C* depicts the three-point correlation functions of the data. The eigenvectors of **10**, computed using

The fact that Eq. **10** correctly predicts the three-point correlation function (based on the measured pair correlator) for sufficiently distant relatives demonstrates that the simple minimal model already provides a reasonable approximation for the long-time dynamics, which is quite remarkable, as it confirms approximate validity of the detailed-balance and Markovian process assumptions. We shall next demonstrate that the deviations at short times can be accounted for by existence of interaction between sisters.

### Effective Interactions Between Siblings.

We now generalize our minimal model to allow for interactions between siblings, which can, in effect, be captured in the form of the mother−daughter transmission function *l*.

Similarly, the third-order correlator is modified to

Without any simplifying assumptions on *T*.

In *SI Text*, we show that *u* asymptotic eigenfunctions and eigenvalues of *u* data. With *u* corrections to

Pair correlators, however, do not fully determine the **13** in terms of the eigenvectors *u* (conformal) limit, we find*δ*, one can approximate by truncating the sum over *δ* and proceed to define

### Testing Interaction Inference on Simulated Data.

The above inference algorithm was applied to simulated trees with random sibling interactions *SI Text* for details). The empirical second- and third-order correlators were measured by counting occurrences of pairs and triplets of phenotypic states as a function of relatedness using Eqs. **1** and **3**. Eigenvectors and eigenvalues of **9**−**11**.

We used three parameters to fit the deviations using the interactive form of the second-order correlator (corresponding to the unique nonvanishing terms in the matrix **15**. terminated at the leading order *SI Text*). Correlators at all distances *u* (**16**. Fig. 3 shows the reduction in deviation of the predicted correlators from observed correlators as interactions are introduced into the minimal model. Fitting the three-point correlators clearly improves the inference of the transition matrix *D*).

### Inferring the Interactions in *P. aeruginosa*.

We now return to *P. aeruginosa* and attempt to infer the form of the interactions from the observed kin correlations of Pvd. Following the above recipe, we have fit the free parameters in *u* (Fig. 2*A*, colored curves). The inferred switching rates *A*). The apparent decrease in the probability of conserving the parental phenotype in the bulk dynamics is due to the ambiguity in determining the parent phenotype; Pvd concentration can fluctuate significantly during a cell cycle.

The inferred *B*). This is consistent with nearest neighbor exchange of Pvd, which reduces sharp gradients (1-3 states) between neighboring cells, in particular, siblings. Fig. 4*B* also shows that the change in likelihood of occurrence of certain sibling pairs is independent of the state of the parent.

Moreover, from the calculated decrease in the likelihood of observing 1-1 siblings pairs and the time scale for division (40 min), we can crudely estimate the exchange rate between neighbors. Define

### Spatial Interactions.

In this section, we address the spatial nature of Pvd exchange. First, we argue that a model that only includes interactions between sisters can be used to infer interactions that take place between all neighboring bacteria in the colony. Next, we try to estimate the expected spatial correlations in Pvd concentration of cells in a colony. To do so, we restore the interactions inferred from siblings to all neighboring pairs of bacteria.

Local interactions in *P. aeruginosa* colonies are believed to be due to the exchange of Pvd. While exchange of Pvd between neighbors can correlate concentrations found in sister cells, exchange is not limited to siblings and occurs between all adjacent bacteria regardless of the degree of relatedness. Nevertheless, we shall argue that the effect of local exchange on the distribution of Pvd on the genealogical tree can be effectively represented through interactions between siblings.

Fig. 5 shows the relationship between spatial distance and the degree of relatedness in the colonies followed in the experiments. Each bacterium has, on average, seven neighbors, defined as cells located within 1.5 cell widths. Although it is more likely to find the sister as one of cell’s neighbors (with probability 0.4 compared with 0.03 for any particular seventh cousin), the neighborhood is dominated by distant cousins. This is because the number of cousins grows exponentially for each additional generation back to the common ancestor.

Thus, exchange with near neighbors is dominated by the exchange with distant relatives, which effectively averages over the whole distribution without contributing to kin correlations. In the limit of a well-mixed population, where neighbors are random nodes from the current generation, local exchange would contribute exactly nothing to kin correlations: Any interaction that is not systematically coupled to the topology of the tree is irrelevant. Bacteria on the plate, however, are not that well mixed: The sister cell is systematically a neighbor and couples the exchange interaction to the topology of the tree. Although close relatives are also overrepresented among near neighbors, we found that to a good approximation to account for local Pvd exchange, it suffices to introduce interactions between sisters.

In the absence of direct spatial interactions, it is possible to map kin correlations to spatial correlations—the probability of observing a pair of bacteria in states *m* and *n* at separation distance *d*,*u* at separation distance *d*. This distribution is determined empirically by tracking the growth of the colony and is depicted in Fig. 5*B*. Here, *u*.

In *P. aeruginosa* colonies, however, direct spatial interactions exist. Local exchange of Pvd implies that kin correlations do not capture all of the spatial correlations. We must reintroduce interactions between neighbors that were averaged out when we computed the kin correlators on the lineage tree. A simple way to do so is using the following observation: Progenies of distant ancestors that by chance remain nearest neighbors of closer ancestors are, in effect, more highly correlated than would be expected from degree of relatedness alone. This is because nearest neighbors are more likely to be in the same phenotypic state. Using this observation (see *Methods*) and the empirical measurement of the probability of finding a relative at lineage distance *u* as a nearest neighbor (Fig. 5*C*, *Inset*), we can estimate *D* shows that the prediction is in good agreement with the observed spatial correlations.

## Discussion

In this study, we have systematically related phenotypic correlation as a function of kinship, or kin correlations, to the underlying epigenetic dynamics. Introducing a rather general class of models, we were able to formulate a method for inferring dynamical parameters from static measurements on cell populations supplemented by the lineage information. This method was then applied to the data on the dynamics of Pyoverdine in *P. aeruginosa* colonies, with the result validated by the comparison with the direct measurements of Pvd dynamics along cell lineages.

Our analysis was based on the minimal model of epigenetic dynamics which assumed (*i*) independent transmission of phenotype from mother cell to its two daughters and (*ii*) detailed balance property of stochastic transitions between phenotypic states. The former assumption was subsequently relaxed, replaced by a general probabilistic model of epigenetic transmission that allowed parameterization of interaction between sister cells. The profound advantage of our minimal model as a starting point is the highly constrained form of the correlations that it entails: Higher-order correlators are completely defined in terms of the pair correlators. Exactly the same relation between correlators is known in field theories describing critical phenomena and is associated with conformal symmetry (33, 34)—a fact noticed by Harlow et al. (33) in the general context of Markovian dynamics on Bethe lattices. It is remarkable that the minimal model of epigenetic dynamics on lineages, with its highly constrained correlators, provides a good description of experimentally observed correlations among distant *P. aeruginosa* cells (31).

The relation between pair and higher-order kin correlations follows from the Detailed Balance property of the minimal model (33). The assumption of detailed balance in the dynamics makes forward and reverse time directions indistinguishable: There is no “arrow of time” associated with lineage dynamics, and the tree is effectively unrooted. As a result, correlations can depend only on the genealogical distance along the tree and must be explicitly independent of the position relative to the root. Now, any unrooted tree may be regarded as a finite chunk of a “Bethe lattice,” where each vertex joins three infinite binary trees, and all vertices are equivalent.

However, unlike a regular lattice (such as the square grid example in Fig. 6*A*) where the number of nodes is a polynomial of lattice size, the Bethe lattice grows exponentially in the number of generations. A representation of the Bethe lattice where all of the angles and edge lengths are constant is fundamentally impossible in Euclidean space. It is possible, however, to embed the Bethe lattice in hyperbolic space, where the negative curvature provides exponentially growing room with increasing distance (37). Fig. 6*B* is a representation of a tree in hyperbolic space using the Poincare disk model (37) (see *SI Text* for details). The angles between the edges are the same for all of the nodes in this representation, and the Poincare disk metric makes all branch lengths equal.

There are transformations, such as rotation by 90 degrees and translations by integer multiples of a lattice constant, that leave the square lattice unchanged (Fig. 6*A*). The invariance of the square lattice under these transformations implies that its correlation functions obey rotational and translational symmetries. A lattice in hyperbolic space is invariant under additional transformations. An easy way to see this is to consider the Poincare disk representation of trees (Fig. 6*B*). Conformal transformations of the Poincare disk onto itself (see *SI Text*) are isometries that leave the lattice invariant (37). Since these transformations do not change the relative position of the bulk nodes, correlation functions on the tree must obey conformal symmetry, which accounts for their strongly constrained form (34).

The connection between our Eqs. **9** and **11** and correlators typically computed in conformal field theories (33, 34) is explained in detail in *SI Text*. However, this unexpected connection, while providing interesting context for our findings, adds little computational power, as all of the key results followed directly from the analysis of Markovian dynamics on a tree.

Despite its generality, the proposed approach has a number of obvious limitations. Virtually by definition, it is blind to phenotypic dynamics that occur on a time scale shorter than a cell cycle and phenotypes that are not transmitted from mother to daughter. Such fluctuations do not contribute to kin correlations; furthermore, they would tend to mask the epigenetically heritable phenotypic variation. Another limitation was evident in our analysis of Pvd dynamics. Our focus on epigenetic dynamics along lineages does not allow for easy incorporation of information on spatial proximity. As a result, instead of directly estimating the interactions due to local exchange of Pvd, we estimated the effect of this interaction on kin correlation, which comes about because siblings are more likely to be exchanging with each other than with anyone else. Hence, our inference yields effective interactions, the origin of which must be examined to be properly interpreted. Other limitations of the present approach, such as discretization of the phenotypic state space state and discretization of time (corresponding to synchronously dividing population), are less fundamental. The model can be generalized to relax these assumptions if warranted by the system under consideration and the extent of available data.

Our example of inferring Pvd dynamics should be thought of as a proof of principle. Dynamics of (naturally fluorescent) Pvd can be directly observed using time-lapse fluorescent microscopy. Dynamic reporters in general, however, require nontrivial genome engineering, and, at best, are limited to a few spectrally distinct fluorophores. By contrast, measurements such as FISH and immunostaining do not have these limitations, but only provide static snap shots (10, 32, 38). High throughput technologies can simultaneously measure numerous biomarkers in large populations at a single-cell resolution (10, 39⇓⇓⇓⇓⇓–45), resulting in a snapshot of a high-dimensional phenotypic space. Our approach is ideally suited for these applications.

More specifically, we envision our analysis applied to understanding developmental programs and dynamics of epigenetic states in stem cells. In these systems, lineage information can be obtained from nonintrusive time-lapse microscopy, and fixed cell measurements such as FISH can provide a snapshot of the expression levels of many genes simultaneously (46). Evidence of broken detailed balance in stem cell epigenetic states can potentially shed light on the underlying pluripotency network. Similar analysis on cancer cellular states (32) can elucidate the dynamics of phenotypic switching in cancer cells without a need for dynamic reporters. Moreover, lineage structure of antibody repertoires (47) and tumor cells (48, 49), when supplemented with single-cell phenotyping, are ideally suited for analysis using our framework. Lastly, our approach can be used to disentangle phenotypic correlations due to shared lineage from those due to other factors such as signaling, which is of particular interest for understanding differentiation and reprogramming (50).

## Methods

### Analyzing the Experimental Data.

The experimental data were in the form of a series of images captured from the growth of *P. aeruginosa* microcolonies; for details of the experiments, see ref. 31. Nine microcolonies were analyzed. The boundary was defined to be the population on the last image. The distance of a pair of boundary nodes was calculated by counting the number of divisions from each node to their common ancestor (CA)—determined by tracing back their history in the images. Although the division time of the bacteria was on average 40 min, fluctuations were observed; number of generations to the CA was sometimes not the same for the two nodes. For these cases, we randomly selected the value for one of the nodes as the distance. The same method was used to determine the distances between three boundary points (values of *u* and *v*).

The signal (Pvd concentration) in each image was calculated as follows: The fluorescence intensity in the cell was subtracted from background fluorescence in that image and then normalized by the mean signal of all of the cells in the image. Normalization removes the effect of increase in the total Pvd concentration in the microcolony over time. The resultant signal distribution is stationary (see *SI Text*). For the boundary cells, we discretized the signal into three phenotypes (low, medium, and high Pvd levels; respectively, 1–3) by binning the signal to ensure a uniform distribution (equal numbers) of each phenotype.

In Fig. 2*C*, the statistical error of the experimental data was estimated by simulating the inferred transition matrix for 64,000 iterations of nine trees of nine generations. The eigenvalues and eigenvectors of the inferred transition matrix were obtained from the observed two-point correlation at **11**. **10**. The deviation is calculated using the matrix norm,

The bulk transition rates in Fig. 4 were determined by counting all occurrences of the phenotypic states of parent and daughter cells in the observed trees. The phenotypic state of a bulk node was taken to be the Pvd state at the last time point of the cell cycle. The results were not sensitive to this choice.

### Lineages in Space.

Distance between any pair of bacteria in a colony is defined as the minimum distance between either pole or centroid of one bacterium and either pole or centroid of the other. Nearest neighbors are defined as pairs whose distance is less than 1.5 times the average cell width. Fig. 5 was computed using spatial information from nine colonies of nine generations. The average coordination number is 7.

### Predicting Spatial Correlations.

The descendants of an ancestor at lineage distance *u* have undergone exchange with the latter ancestor for *u*.

We include the contribution of these “effective” ancestors as follows:

where *τ* is the mean waiting time for exchange, which we estimated roughly as two generations using our inference, *u* at spatial distance *r*, and *u* is a nearest neighbor (Fig. 5*C*, *Inset*).

## Acknowledgments

The authors thank Stephen Shenker, Douglas Stanford, Richard Neher, and David Bensimon for stimulating discussions and helpful comments. This research was supported in part by the National Science Foundation under Grant NSF PHY11-25915. B.I.S. also acknowledges support from National Institutes of Health Grant R01-GM086793. N.D. acknowledges support from Grant ANR-2011-JSV5-005-01.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: shraiman{at}kitp.ucsb.edu.

Author contributions: S.H. and B.I.S. designed research; S.H., N.D., and B.I.S. performed research; S.H. and N.D. analyzed data; S.H. and B.I.S. wrote the paper; and N.D. contributed experimental data.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1504407112/-/DCSupplemental.

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