Living cells transition among many physiologically distinct states during development, for tissue maintenance, and in disease. Dysregulation of transition rates between cell states can lead to pathologies, such as developmental disorders and cancer. In these systems and others, we would like to know which transitions can occur between the cellular states, in what sequence, and at what rates. The Hormoz lab aims to understand the dynamics of cellular state transitions and fate decisions during development and in diseases such as cancer. We use engineering of synthetic gene circuits, single-molecule imaging and single-cell analysis to study embryonic stem cells and other systems. We also develop new mathematical frameworks and theoretical tools to gain insight into stochastic processes and collective phenomena.

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**MEMOIR: recording a cell's lineage history in its own DNA**

Two transformative technologies, high throughput single-molecule imaging and new genome editing tools for synthetic biology, are currently emerging. We combine the two to probe the dynamics of biological systems in new and exciting ways. High throughput single-molecule imaging allows us to read out the expression levels of multiple genes in individual cells. However, this technique can only provide snap shots, a detailed view of a cell's final time point, but no direct information about the history of its dynamics. For example, what signaling factors did the ancestors of a cell experience? When did a cell make its fate decision? What happened to a cell’s close relatives, i.e. its sister, cousins, etc.?

A team of researchers and I have used tools from synthetic biology to engineer modules that can record the lineage history of a cell in its own DNA. The recorded information can be read out in situ at a single-cell level along the expression levels of endogenous genes, and provide information about the dynamics experienced by individual cells.

**Publications:**

KL Frieda*, JM Linton*, S Hormoz*, J Choi, KK Chow, ZS Singer, MW Budde, MB Elowitz, and L Cai. Synthetic recording and in situ readout of lineage information in single cells. Nature 541 (7635): 107-111 (2017). * equal contribution.

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**Cell state transition network in embryonic stem cells**

Measuring cellular dynamics is not easy. To follow the dynamics of genes, we need to engineer fluorescent reporters. But cell line engineering is difficult and time-consuming. Simultaneously following the dynamics of more than a handful of genes becomes quickly impractical. Indirect methods for getting at the dynamics require perturbations, for example sorting cells to distinct subpopulation or synchronizing them.

We have developed an experimental platform for inferring cell state transition dynamics without a need for cell line engineering or perturbations. Our platform uses time-lapse microscopy to follow single cells as they divide (obtaining their lineage history) followed by single-molecule FISH on the same cells to determine their expression levels for multiple genes. Dynamics can be inferred from this data using the kin correlation analysis (KCA) framework (described below).

Embryonic stem cells provide an ideal model system in which to study cell state transition dynamics. Multiple molecularly and phenotypically distinct ES cell states co-exist and interconvert in standard culture conditions. These states differ in developmental potential, global epigenetic profiles, and other characteristics. Despite recent advances in characterizing these states, their dynamics has remained largely unknown.

Using our experimental platform (one example tree is shown above), we inferred the first comprehensive quantitative state transition network in ES cells. Our inference framework uncovered the dynamics without cell line engineering, perturbations, or cell sorting. We discovered that ES cells exhibit a distinct and novel type of cell state transition network based on reversible stochastic transitions along a linear chain of states ranging from totipotent to epiblast-like.

**To see an interactive visualization of our complete data set, lineage trees with end-point ES cell states, visit: **http://cellerie.caltech.edu/

**Publications:**

S Hormoz*, ZS Singer*, JM Linton, YE Antebi, BI Shraiman, MB Elowitz. Inferring Cell-State Transition Dynamics from Lineage Trees and Endpoint Single-Cell Measurements. Cell Systems 3(5): 419-433 (2016). Featured on the cover. * equal contribution.

## A theoretical framework for inferring phenotypic dynamics from static measurements and lineage trees

Two cells that have identical genomes can be in very different physiological and functional cellular states. Just compare one of your skin cells to one of your neurons. During development, maintenance of cycling tissues (like the intestinal lining), and in diseases, proliferating cells transition between distinct cell states. Many key biological functions depend critically on the dynamics of these transitions: which transitions are forbidden or permitted and at what rates they occur. We showed that quantitative cell state transition dynamics can be inferred from a static snapshot of single-cell gene expression levels in a population of cells where the degree of relatedness between all the cells (or their pedigree) is known.

How can we infer dynamics from a static measurement? To intuitively understand this, consider the minimal transition network, in which cells switch reversibly between two states at equal rates (see cartoon). Although the population fraction of each state does not change over time, as cells proliferate, they stochastically transition between the two states. When transition rates per cell cycle are high, the states of sister cells rapidly become uncorrelated with one another. By contrast, when transition rates per cell cycle are low and cells remain in the same state for multiple generations, closely related cells are more likely to be observed in the same state. As a result, the degree of correlation of a cell's state with that of its relatives (sisters, cousins, etc.), called kin correlations, contains information about the underlying dynamic. Cell state correlations, computed between all pairs, or more generally all triplets of cells, as a function of their lineage distance, enable quantitative inference of the transition dynamics through an approach we termed Kin Correlation Analysis (KCA).

Using data from our collaborators David Bensimon (ENS and UCLA) and Nicolas Desprat (ENS), we demonstrated the KCA framework by correctly inferring the dynamics of Pyoverdine production and exchange by Pseudomonas aeruginosa (PNAS, 2015).

**Counter-intuitive mathematics for understanding trees.** Strikingly, the kin correlation functions have the same structure as those in a class of extensively studied models in statistical physics, **conform****al fie****ld t****heories**. We can make sense of this by thinking of the binary tree as a lattice in hyperbolic space (hyperbolic spaces have negative curvature, see the MC Escher drawing below). The tree is invariant under Mobious transformations in hyperbolic space (see animation below), implying that its correlation functions must obey conformal symmetry. This is analogous to how invariance of the square lattice in Euclidean space under 90-degree rotations implies that correlation functions on the square lattice obey rotational symmetry. We are currently trying to understand complex dynamics on trees, such as selection or local signalling, in terms of broken symmetries and interpret them geometrically. We are applying this approach to a diverse set of collective dynamics, production and exchange of common goods in colonies of bacteria, state transitions in embryonic stem cells in culture, and in vivo selection and isotype-switching of B-cells.

**Publications:**

S Hormoz, N Desprat, BI Shraiman. Inferring epigenetic dynamics from kin correlations. PNAS 112, (18) E2281-E2289 (2015).