Christoph Zechner Group

Compartmentalized biochemical reactions are a ubiquitous building block of biological systems. The interplay between chemical and compartmental dynamics can drive rich and complex dynamical behaviors that are difficult to analyze mathematically - especially in the presence of stochasticity. We have recently proposed an effective moment equation approach to study the statistical properties of compartmentalized biochemical systems. So far, however, this approach is limited to polynomial rate laws and moreover, it relies on suitable moment closure approximations, which can be difficult to find in practice. In this work we propose a systematic method to derive closed moment dynamics for compartmentalized biochemical systems. We show that for the considered class of systems, the moment equations involve expectations over functions that factorize into two parts, one depending on the molecular content of the compartments and one depending on the compartment number distribution. Our method exploits this structure and approximates each function with suitable polynomial expansions, leading to a closed system of moment equations. We demonstrate the method using three systems inspired by cell populations and organelle networks and study its accuracy across different dynamical regimes.

Studying the role of molecularly distinct lipid species in cell signaling remains challenging due to a scarcity of methods for performing quantitative lipid biochemistry in living cells. We have recently used lipid uncaging to quantify lipid-protein affinities and rates of lipid trans-bilayer movement and turnover in the diacylglycerol signaling pathway. This approach is based on acquiring live-cell dose-response curves requiring light dose titrations and experimental determination of uncaging photoreaction efficiency. We here aimed to develop a methodological approach that allows us to retrieve quantitative kinetic data from uncaging experiments that 1) require only typically available datasets without the need for specialized additional constraints and 2) should in principle be applicable to other types of photoactivation experiments. Our new analysis framework allows us to identify model parameters such as diacylglycerol-protein affinities and trans-bilayer movement rates, together with initial uncaged diacylglycerol levels, using noisy single-cell data for a broad variety of structurally different diacylglycerol species. We find that lipid unsaturation degree and side-chain length generally correlate with faster lipid trans-bilayer movement and turnover and also affect lipid-protein affinities. In summary, our work demonstrates how rate parameters and lipid-protein affinities can be quantified from single-cell signaling trajectories with sufficient sensitivity to resolve the subtle kinetic differences caused by the chemical diversity of cellular signaling lipid pools.

Basic helix-loop-helix genes, particularly proneural genes, are well-described triggers of cell differentiation, yet information on their dynamics is limited, notably in human development. Here, we focus on Neurogenin 3 (NEUROG3), which is crucial for pancreatic endocrine lineage initiation. By monitoring both NEUROG3 gene expression and protein in single cells using a knockin dual reporter in 2D and 3D models of human pancreas development, we show an approximately 2-fold slower expression of human NEUROG3 than that of the mouse. We observe heterogeneous peak levels of NEUROG3 expression and reveal through long-term live imaging that both low and high NEUROG3 peak levels can trigger differentiation into hormone-expressing cells. Based on fluorescence intensity, we statistically integrate single-cell transcriptome with dynamic behaviors of live cells and propose a data-mapping methodology applicable to other contexts. Using this methodology, we identify a role for KLK12 in motility at the onset of NEUROG3 expression.

Building synthetic multicellular systems using non-living molecular components is a grand challenge in the field of bottom-up synthetic biology. Towards this goal, a diverse range of chemistries have been developed to provide mechanisms of intercellular communication and methods to assemble multicellular compartments. However, building bottom-up synthetic multicellular systems is still challenging because it requires the integration of intercellular reaction networks with compatible cellular compartment properties. In this study, we encapsulated cell-free expression systems (CFES) expressing two quorum sensing genetic circuits into droplet interface bilayer (DIB) synthetic cells to demonstrate bidirectional communication. We further develop a method of generating custom DIB multicellular structures by acoustic liquid handling to automatically dispense the CFES droplets and show the potential for multiplexing compartmentalized gene circuits for generating heterogeneous populations of cells. Our work provides a step towards building more complex multicellular systems with intercellular communication from the bottom-up to study and experimentally model biological multiscalar processes.

Correct nervous system development depends on the timely differentiation of progenitor cells into neurons. While the output of progenitor differentiation is well investigated at the population and clonal level, how stereotypic or variable fate decisions are during development is still more elusive. To fill this gap, we here follow the fate outcome of single neurogenic progenitors in the zebrafish retina over time using live imaging. We find that neurogenic progenitor divisions produce two daughter cells, one of deterministic and one of probabilistic fate. Interference with the deterministic branch of the lineage affects lineage progression. In contrast, interference with fate probabilities of the probabilistic branch results in a broader range of fate possibilities than in wild-type and involves the production of any neuronal cell type even at non-canonical developmental stages. Combining the interference data with stochastic modelling of fate probabilities revealed that a simple gene regulatory network is able to predict the observed fate decision probabilities during wild-type development. These findings unveil unexpected lineage flexibility that could ensure robust development of the retina and other tissues.

Animal genomes are folded into loops and topologically associating domains (TADs) by CTCF and loop-extruding cohesins, but the live dynamics of loop formation and stability remain unknown. Here, we directly visualized chromatin looping at the Fbn2 TAD in mouse embryonic stem cells using super-resolution live-cell imaging and quantified looping dynamics by Bayesian inference. Unexpectedly, the Fbn2 loop was both rare and dynamic, with a looped fraction of approximately 3 to 6.5% and a median loop lifetime of approximately 10 to 30 minutes. Our results establish that the Fbn2 TAD is highly dynamic, and about 92% of the time, cohesin-extruded loops exist within the TAD without bridging both CTCF boundaries. This suggests that single CTCF boundaries, rather than the fully CTCF-CTCF looped state, may be the primary regulators of functional interactions.

The ability to build synthetic cellular populations from the bottom-up provides the groundwork to realize minimal living tissues comprising single cells which can communicate and bridge scales into multicellular systems. Engineered systems made of synthetic micron-sized compartments and integrated reaction networks coupled with mathematical modeling can facilitate the design and construction of complex and multiscale chemical systems from the bottom-up. Toward this goal, we generated populations of monodisperse liposomes encapsulating cell-free expression systems (CFESs) using double-emulsion microfluidics and quantified transcription and translation dynamics within individual synthetic cells of the population using a fluorescent Broccoli RNA aptamer and mCherry protein reporter. CFE dynamics in bulk reactions were used to test different coarse-grained resource-limited gene expression models using model selection to obtain transcription and translation rate parameters by likelihood-based parameter estimation. The selected model was then applied to quantify cell-free gene expression dynamics in populations of synthetic cells. In combination, our experimental and theoretical approaches provide a statistically robust analysis of CFE dynamics in bulk and monodisperse synthetic cell populations. We demonstrate that compartmentalization of CFESs leads to different transcription and translation rates compared to bulk CFE and show that this is due to the semipermeable lipid membrane that allows the exchange of materials between the synthetic cells and the external environment.

Basic helix-loop-helix genes, particularly proneural genes, are well-described triggers of cell differentiation, yet limited information exists on their dynamics, notably in human development. Here, we focus on Neurogenin 3 (NEUROG3), which is crucial for pancreatic endocrine lineage initiation. Using a double reporter to monitor endogenous NEUROG3 transcription and protein expression in single cells in 2D and 3D models of human pancreas development, we show peaks of expression for the RNA and protein at 22 and 11 hours respectively, approximately two-fold slower than in mice, and remarkable heterogeneity in peak expression levels all triggering differentiation. We also reveal that some human endocrine progenitors proliferate once, mainly at the onset of differentiation, rather than forming a subpopulation with sustained proliferation. Using reporter index-sorted single-cell RNA-seq data, we statistically map transcriptome to dynamic behaviors of cells in live imaging and uncover transcriptional states associated with variations in motility as NEUROG3 levels change, a method applicable to other contexts.

Many biochemical processes in living organisms take place inside compartments that can interact with each other and remodel over time. In a recent work (Duso and Zechner, 2020), we have shown how the stochastic dynamics of a compartmentalized biochemical system can be effectively studied using moment equations. With this technique, the time evolution of a compartment population is summarized using a finite number of ordinary differential equations, which can be analyzed very efficiently. However, the derivation of moment equations by hand can become time-consuming for systems comprising multiple reactants and interactions. Here we present Compartor, a toolbox that automatically generates the moment equations associated with a user-defined compartmentalized system. Through the moment equation method, Compartor renders the analysis of stochastic population models accessible to a broader scientific community.

Cells respond to external signals and stresses by activating transcription factors (TF), which induce gene expression changes. Prior work suggests that signal-specific gene expression changes are partly achieved because different gene promoters exhibit distinct induction dynamics in response to the same TF input signal. Here, using high-throughput quantitative single-cell measurements and a novel statistical method, we systematically analyzed transcriptional responses to a large number of dynamic TF inputs. In particular, we quantified the scaling behavior among different transcriptional features extracted from the measured trajectories such as the gene activation delay or duration of promoter activity. Surprisingly, we found that even the same gene promoter can exhibit qualitatively distinct induction and scaling behaviors when exposed to different dynamic TF contexts. While it was previously known that promoters fall into distinct classes, here we show that the same promoter can switch between different classes depending on context. Thus, promoters can adopt context-dependent "manifestations". Our analysis suggests that the full complexity of signal processing by genetic circuits may be significantly underestimated when studied in only specific contexts.

A grand challenge in bottom–up synthetic biology is the designand construction of synthetic multicellular systems usingnonliving molecular components. Abstracting key features ofcompartmentalisation, reaction and diffusion, and communica-tion provides the blueprint for assembling synthetic multiscalesystems with emergent properties. The diverse range of chem-istries for building encapsulated reactions in micron-sized com-partments offers combinatorial flexibility and modularity inbuilding synthetic multicellular systems with molecular-levelcontrol. Here, we focus onrecent advancesin the emergingareaof bottom–up approaches to create biologically inspired multi-cellular systems. Specifically, we consider how intercellularcommunication and feedback loops can be integrated intopopulations of synthetic cells and summarise recent de-velopments for the 2D/3D spatial localisation of microcompart-ments. Although building bottom–upmulticellular systems isstillin its infancy, progress in this field offers tractable models to un-derstand the minimal requirements for generating multiscalesystems from the molecular level for fundamental research andinnovative technological applications.

Compartmentalization of biochemical processes underlies all biological systems, from the organelle to the tissue scale. Theoretical models to study the interplay between noisy reaction dynamics and compartmentalization are sparse, and typically very challenging to analyze computationally. Recent studies have made progress toward addressing this problem in the context of specific biological systems, but a general and sufficiently effective approach remains lacking. In this work, we propose a mathematical framework based on counting processes that allows us to study dynamic compartment populations with arbitrary interactions and internal biochemistry. We derive an efficient description of the dynamics in terms of differential equations which capture the statistics of the population. We demonstrate the relevance of our approach by analyzing models inspired by different biological processes, including subcellular compartmentalization and tissue homeostasis.

During development, cells need to make decisions about their fate in order to ensure that the correct numbers and types of cells are established at the correct time and place in the embryo. Such cell fate decisions are often classified as deterministic or stochastic. However, although these terms are clearly defined in a mathematical sense, they are sometimes used ambiguously in biological contexts. Here, we provide some suggestions on how to clarify the definitions and usage of the terms stochastic and deterministic in biological experiments. We discuss the frameworks within which such clear definitions make sense and highlight when certain ambiguity prevails. As an example, we examine how these terms are used in studies of neuronal cell fate decisions and point out areas in which definitions and interpretations have changed and matured over time. We hope that this Review will provide some clarification and inspire discussion on the use of terminology in relation to fate decisions.

Many systems in biology and other sciences employ collaborative, collective communication strategies for improved efficiency and adaptive benefit. One such paradigm of particular interest is the community estimation of a dynamic signal, when, for example, an epithelial tissue of cells must decide whether to react to a given dynamic external concentration of stress-signaling molecules. At the level of dynamic cellular communication, however, it remains unknown what effect, if any, arises from communication beyond the mean field level. What are the limits and benefits to communication across a network of neighbor interactions? What is the role of Poissonian versus super-Poissonian dynamics in such a setting? How does the particular topology of connections impact the collective estimation and that of the individual participating cells? In this article we construct a robust and general framework of signal estimation over continuous-time Markov chains in order to address and answer these questions. Our results show that in the case of Possonian estimators, the communication solely enhances convergence speed of the mean squared error (MSE) of the estimators to their steady-state values while leaving these values unchanged. However, in the super-Poissonian regime, the MSE of estimators significantly decreases by increasing the number of neighbors. Surprisingly, in this case, the clustering coefficient of an estimator does not enhance its MSE while still reducing the total MSE of the population.

Expression of proteins inside cells is noisy, causing variability in protein concentration among identical cells. A central problem in cellular control is how cells cope with this inherent noise. Compartmentalization of proteins through phase separation has been suggested as a potential mechanism to reduce noise, but systematic studies to support this idea have been missing. In this study, we used a physical model that links noise in protein concentration to theory of phase separation to show that liquid droplets can effectively reduce noise. We provide experimental support for noise reduction by phase separation using engineered proteins that form liquid-like compartments in mammalian cells. Thus, phase separation can play an important role in biological signal processing and control.

Living cells encode and transmit information in the temporal dynamics of signaling molecules. Gaining a quantitative understanding of how intracellular networks process dynamic signals requires measures that capture the interdependence between complete time trajectories of network components. Mutual information provides such a measure but its calculation in the context of stochastic reaction networks is associated with computational challenges. Here we propose a method to calculate the mutual information between complete time-continuous paths of two molecular species that interact with each other through chemical reactions. We demonstrate our approach using three simple case studies.

Cells can utilize chemical communication to exchange information and coordinate their behavior in the presence of noise. Communication can reduce noise to shape a collective response, or amplify noise to generate distinct phenotypic subpopulations. Here we discuss a moment-based approach to study how cell-cell communication affects noise in biochemical networks that arises from both intrinsic and extrinsic sources. We derive a system of approximate differential equations that captures lower-order moments of a population of cells, which communicate by secreting and sensing a diffusing molecule. Since the number of obtained equations grows combinatorially with number of considered cells, we employ a previously proposed model reduction technique, which exploits symmetries in the underlying moment dynamics. Importantly, the number of equations obtained in this way is independent of the number of considered cells such that the method scales to arbitrary population sizes. Based on this approach, we study how cell-cell communication affects population variability in several biochemical networks. Moreover, we analyze the accuracy and computational efficiency of the moment-based approximation by comparing it with moments obtained from stochastic simulations.

Our results offer the potential for straightforward, additive-free, and molecularly simple routes to building complex bioreactors based on in vitro transcription-translation systems and lipid vesicles.

The variability in transcription factor concentration among cells is an important developmental determinant, yet how variability is controlled remains poorly understood. Studies of variability have focused predominantly on monitoring mRNA production noise. Little information exists about transcription factor protein variability, as this requires the use of quantitative methods with single-molecule sensitivity. Using Fluorescence Correlation Spectroscopy (FCS), we have characterized the concentration and variability of 14 endogenously tagged TFs in live Drosophila imaginal discs. For the Hox TF Antennapedia, we investigated whether protein variability results from random stochastic events or is developmentally regulated. We found that Antennapedia transitioned from low concentration/high variability early, to high concentration/low variability later, in development. FCS and temporally resolved genetic studies uncovered that Antennapedia itself is necessary and sufficient to drive a developmental regulatory switch from auto-activation to auto-repression, thereby reducing variability. This switch is controlled by progressive changes in relative concentrations of preferentially activating and repressing Antennapedia isoforms, which bind chromatin with different affinities. Mathematical modeling demonstrated that the experimentally supported auto-regulatory circuit can explain the increase of Antennapedia concentration and suppression of variability over time.

Stochastic simulations of biochemical networks are of vital importance for understanding complex dynamics in cells and tissues. However, existing methods to perform such simulations are associated with computational difficulties and addressing those remains a daunting challenge to the present. Here we introduce the selected-node stochastic simulation algorithm (snSSA), which allows us to exclusively simulate an arbitrary, selected subset of molecular species of a possibly large and complex reaction network. The algorithm is based on an analytical elimination of chemical species, thereby avoiding explicit simulation of the associated chemical events. These species are instead described continuously in terms of statistical moments derived from a stochastic filtering equation, resulting in a substantial speedup when compared to Gillespie's stochastic simulation algorithm (SSA). Moreover, we show that statistics obtained via snSSA profit from a variance reduction, which can significantly lower the number of Monte Carlo samples needed to achieve a certain performance. We demonstrate the algorithm using several biological case studies for which the simulation time could be reduced by orders of magnitude.

Transcription is often stochastic. This is seemingly incompatible with the importance of gene expression during development. Here we show that during zebrafish embryogenesis, transcription activation is stochastic due to (1) genes acquiring transcriptional competence at different times in different cells, (2) differences in cell cycle stage between cells, and (3) the stochastic nature of transcription. Initially, stochastic transcription causes large cell-to-cell differences in transcript levels. However, variability is reduced by lengthening cell cycles and the accumulation of transcription events in each cell. Temporal averaging might provide a general context in which to understand how embryos deal with stochastic transcription.

With the advance of experimental techniques such as time-lapse fluorescence microscopy, the availability of single-cell trajectory data has vastly increased, and so has the demand for computational methods suitable for parameter inference with this type of data. Most of currently available methods treat single-cell trajectories independently, ignoring the mother-daughter relationships and the information provided by the population structure. However, this information is essential if a process of interest happens at cell division, or if it evolves slowly compared to the duration of the cell cycle.

Determining the sensitivity of certain system states or outputs to variations in parameters facilitates our understanding of the inner working of that system and is an essential design tool for the de novo construction of robust systems. In cell biology, the output of interest is often the response of a certain reaction network to some input (e.g., stressors or nutrients) and one aims to quantify the sensitivity of this response in the presence of parameter heterogeneity. We argue that for such applications, parametric sensitivities in their standard form do not paint a complete picture of a system's robustness since one assumes that all cells in the population have the same parameters and are perturbed in the same way. Here, we consider stochasticreaction networks in which the parameters are randomly distributed over the population and propose a new sensitivity index that captures the robustness of system outputs upon changes in the characteristics of the parameter distribution, rather than the parameters themselves. Subsequently, we make use of Girsanov's likelihood ratio method to construct a Monte Carlo estimator of this sensitivity index. However, it turns out that this estimator has an exceedingly large variance. To overcome this problem, we propose a novel estimation algorithm that makes use of a marginalization of the path distribution of stochasticreaction networks and leads to Rao-Blackwellized estimators with reduced variance.

The invention of the Kalman filter is a crowning achievement of filtering theory-one that has revolutionized technology in countless ways. By dealing effectively with noise, the Kalman filter has enabled various applications in positioning, navigation, control, and telecommunications. In the emerging field of synthetic biology, noise and context dependency are among the key challenges facing the successful implementation of reliable, complex, and scalable synthetic circuits. Although substantial further advancement in the field may very well rely on effectively addressing these issues, a principled protocol to deal with noise-as provided by the Kalman filter-remains completely missing. Here we develop an optimal filtering theory that is suitable for noisy biochemical networks. We show how the resulting filters can be implemented at the molecular level and provide various simulations related to estimation, system identification, and noise cancellation problems. We demonstrate our approach in vitro using DNA strand displacement cascades as well as in vivo using flow cytometry measurements of a light-inducible circuit in Escherichia coli.

The dynamics of stochastic reaction networks within cells are inevitably modulated by factors considered extrinsic to the network such as, for instance, the fluctuations in ribosome copy numbers for a gene regulatory network. While several recent studies demonstrate the importance of accounting for such extrinsic components, the resulting models are typically hard to analyze. In this work we develop a general mathematical framework that allows to uncouple the network from its dynamic environment by incorporating only the environment's effect onto the network into a new model. More technically, we show how such fluctuating extrinsic components (e.g., chemical species) can be marginalized in order to obtain this decoupled model. We derive its corresponding process- and master equations and show how stochastic simulations can be performed. Using several case studies, we demonstrate the significance of the approach.

Mathematical methods combined with measurements of single-cell dynamics provide a means to reconstruct intracellular processes that are only partly or indirectly accessible experimentally. To obtain reliable reconstructions, the pooling of measurements from several cells of a clonal population is mandatory. However, cell-to-cell variability originating from diverse sources poses computational challenges for such process reconstruction. We introduce a scalable Bayesian inference framework that properly accounts for population heterogeneity. The method allows inference of inaccessible molecular states and kinetic parameters; computation of Bayes factors for model selection; and dissection of intrinsic, extrinsic and technical noise. We show how additional single-cell readouts such as morphological features can be included in the analysis. We use the method to reconstruct the expression dynamics of a gene under an inducible promoter in yeast from time-lapse microscopy data.

Recent computational studies indicate that the molecular noise of a cellular process may be a rich source of information about process dynamics and parameters. However, accessing this source requires stochastic models that are usually difficult to analyze. Therefore, parameter estimation for stochastic systems using distribution measurements, as provided for instance by flow cytometry, currently remains limited to very small and simple systems. Here we propose a new method that makes use of low-order moments of the measured distribution and thereby keeps the essential parts of the provided information, while still staying applicable to systems of realistic size. We demonstrate how cell-to-cell variability can be incorporated into the analysis obviating the need for the ubiquitous assumption that the measurements stem from a homogeneous cell population. We demonstrate the method for a simple example of gene expression using synthetic data generated by stochastic simulation. Subsequently, we use time-lapsed flow cytometry data for the osmo-stress induced transcriptional response in budding yeast to calibrate a stochastic model, which is then used as a basis for predictions. Our results show that measurements of the mean and the variance can be enough to determine the model parameters, even if the measured distributions are not well-characterized by low-order moments only--e.g., if they are bimodal.