What cannot be seen correctly in 2D visualizations of single-cell 'omics data?

A common strategy for exploring single-cell 'omics data is visualizing 2D nonlinear projections that aim to preserve high-dimensional data properties such as neighborhoods. Alternatively, mathematical theory and other computational tools can directly describe data geometry, while also showing that neighborhoods and other properties cannot be well-preserved in any 2D projection.

Epigenetic factor competition reshapes the EMT landscape.

The emergence of and transitions between distinct phenotypes in isogenic cells can be attributed to the intricate interplay of epigenetic marks, external signals, and gene-regulatory elements. These elements include chromatin remodelers, histone modifiers, transcription factors, and regulatory RNAs. Mathematical models known as gene-regulatory networks (GRNs) are an increasingly important tool to unravel the workings of such complex networks. In such models, epigenetic factors are usually proposed to act on the chromatin regions directly involved in the expression of relevant genes. However, it has been well-established that these factors operate globally and compete with each other for targets genome-wide. Therefore, a perturbation of the activity of a regulator can redistribute epigenetic marks across the genome and modulate the levels of competing regulators. In this paper, we propose a conceptual and mathematical modeling framework that incorporates both local and global competition effects between antagonistic epigenetic regulators, in addition to local transcription factors, and show the counterintuitive consequences of such interactions. We apply our approach to recent experimental findings on the epithelial-mesenchymal transition (EMT). We show that it can explain the puzzling experimental data, as well as provide verifiable predictions.

A synthetic distributed genetic multi-bit counter.

A design for genetically encoded counters is proposed via repressor-based circuits. An N-bit counter reads sequences of input pulses and displays the total number of pulses, modulo 2 N . The design is based on distributed computation with specialized cell types allocated to specific tasks. This allows scalability and bypasses constraints on the maximal number of circuit genes per cell due to toxicity or failures due to resource limitations. The design starts with a single-bit counter. The N-bit counter is then obtained by interconnecting (using diffusible chemicals) a set of N single-bit counters and connector modules. An optimization framework is used to determine appropriate gate parameters and to compute bounds on admissible pulse widths and relaxation (inter-pulse) times, as well as to guide the construction of novel gates. This work can be viewed as a step toward obtaining circuits that are capable of finite automaton computation in analogy to digital central processing units.

Perfect adaptation of CD8+ T cell responses to constant antigen input over a wide range of affinities is overcome by costimulation.

Reduced T cell responses by contrast antigen stimulation can be rescued by signals from costimulatory receptors.

Maximizing average throughput in oscillatory biochemical synthesis systems: an optimal control approach.

A dynamical system entrains to a periodic input if its state converges globally to an attractor with the same period. In particular, for a constant input, the state converges to a unique equilibrium point for any initial condition. We consider the problem of maximizing a weighted average of the system's output along the periodic attractor. The gain of entrainment is the benefit achieved by using a non-constant periodic input relative to a constant input with the same time average. Such a problem amounts to optimal allocation of resources in a periodic manner. We formulate this problem as a periodic optimal control problem, which can be analysed by means of the Pontryagin maximum principle or solved numerically via powerful software packages. We then apply our framework to a class of nonlinear occupancy models that appear frequently in biological synthesis systems and other applications. We show that, perhaps surprisingly, constant inputs are optimal for various architectures. This suggests that the presence of non-constant periodic signals, which frequently appear in biological occupancy systems, is a signature of an underlying time-varying objective functional being optimized.

Derivation of stationary distributions of biochemical reaction networks via structure transformation.

Long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models and stationary distributions in stochastic models. Unlike deterministic steady states, stationary distributions capturing inherent fluctuations of reactions are extremely difficult to derive analytically due to the curse of dimensionality. Here, we develop a method to derive analytic stationary distributions from deterministic steady states by transforming BRNs to have a special dynamic property, called complex balancing. Specifically, we merge nodes and edges of BRNs to match in- and out-flows of each node. This allows us to derive the stationary distributions of a large class of BRNs, including autophosphorylation networks of EGFR, PAK1, and Aurora B kinase and a genetic toggle switch. This reveals the unique properties of their stochastic dynamics such as robustness, sensitivity, and multi-modality. Importantly, we provide a user-friendly computational package, CASTANET, that automatically derives symbolic expressions of the stationary distributions of BRNs to understand their long-term stochasticity.

Universal features of epidemic models under social distancing guidelines.

Social distancing as a form of nonpharmaceutical intervention has been enacted in many countries as a form of mitigating the spread of COVID-19. There has been a large interest in mathematical modeling to aid in the prediction of both the total infected population and virus-related deaths, as well as to aid government agencies in decision making. As the virus continues to spread, there are both economic and sociological incentives to minimize time spent with strict distancing mandates enforced, and/or to adopt periodically relaxed distancing protocols, which allow for scheduled economic activity. The main objective of this study is to reduce the disease burden in a population, here measured as the peak of the infected population, while simultaneously minimizing the length of time the population is socially distanced, utilizing both a single period of social distancing as well as periodic relaxation. We derive a linear relationship among the optimal start time and duration of a single interval of social distancing from an approximation of the classic epidemic SIR model. Furthermore, we see a sharp phase transition region in start times for a single pulse of distancing, where the peak of the infected population changes rapidly; notably, this transition occurs well before one would intuitively expect. By numerical investigation of more sophisticated epidemiological models designed specifically to describe the COVID-19 pandemic, we see that all share remarkably similar dynamic characteristics when contact rates are subject to periodic or one-shot changes, and hence lead us to conclude that these features are universal in epidemic models. On the other hand, the nonlinearity of epidemic models leads to non-monotone behavior of the peak of infected population under periodic relaxation of social distancing policies. This observation led us to hypothesize that an additional single interval social distancing at a proper time can significantly decrease the infected peak of periodic policies, and we verified this improvement numerically. While synchronous quarantine and social distancing mandates across populations effectively minimize the spread of an epidemic over the world, relaxation decisions should not be enacted at the same time for different populations.

A novel COVID-19 epidemiological model with explicit susceptible and asymptomatic isolation compartments reveals unexpected consequences of timing social distancing.

Motivated by the current COVID-19 epidemic, this work introduces an epidemiological model in which separate compartments are used for susceptible and asymptomatic "socially distant" populations. Distancing directives are represented by rates of flow into these compartments, as well as by a reduction in contacts that lessens disease transmission. The dynamical behavior of this system is analyzed, under various different rate control strategies, and the sensitivity of the basic reproduction number to various parameters is studied. One of the striking features of this model is the existence of a critical implementation delay (CID) in issuing distancing mandates: while a delay of about two weeks does not have an appreciable effect on the peak number of infections, issuing mandates even slightly after this critical time results in a far greater incidence of infection. Thus, there is a nontrivial but tight "window of opportunity" for commencing social distancing in order to meet the capacity of healthcare resources. However, if one wants to also delay the timing of peak infections - so as to take advantage of potential new therapies and vaccines - action must be taken much faster than the CID. Different relaxation strategies are also simulated, with surprising results. Periodic relaxation policies suggest a schedule which may significantly inhibit peak infective load, but that this schedule is very sensitive to parameter values and the schedule's frequency. Furthermore, we considered the impact of steadily reducing social distancing measures over time. We find that a too-sudden reopening of society may negate the progress achieved under initial distancing guidelines, but the negative effects can be mitigated if the relaxation strategy is carefully designed.

A mathematical model exhibiting the effect of DNA methylation on the stability boundary in cell-fate networks.

Cell-fate networks are traditionally studied within the framework of gene regulatory networks. This paradigm considers only interactions of genes through expressed transcription factors and does not incorporate chromatin modification processes. This paper introduces a mathematical model that seamlessly combines gene regulatory networks and DNA methylation (DNAm), with the goal of quantitatively characterizing the contribution of epigenetic regulation to gene silencing. The 'Basin of Attraction percentage' is introduced as a metric to quantify gene silencing abilities. As a case study, a computational and theoretical analysis is carried out for a model of the pluripotent stem cell circuit as well as a simplified self-activating gene model. The results confirm that the methodology quantitatively captures the key role that DNAm plays in enhancing the stability of the silenced gene state.

Short-Term Circulating Tumor Cell Dynamics in Mouse Xenograft Models and Implications for Liquid Biopsy.

MOTIVATION: Circulating tumor cells (CTCs) are widely studied using liquid biopsy methods that analyze fractionally-small peripheral blood (PB) samples. However, little is known about natural fluctuations in CTC numbers that may occur over short timescales in vivo, and how these may affect detection and enumeration of rare CTCs from small blood samples. METHODS: We recently developed an optical instrument called "diffuse in vivo flow cytometry" (DiFC) that uniquely allows continuous, non-invasive counting of rare, green fluorescent protein expressing CTCs in large blood vessels in mice. Here, we used DiFC to study short-term changes in CTC numbers in multiple myeloma and Lewis lung carcinoma xenograft models. We analyzed CTC detections in over 100 h of DiFC data, and considered intervals corresponding to approximately 1%, 5%, 10%, and 20% of the PB volume. In addition, we analyzed changes in CTC numbers over 24 h (diurnal) periods. RESULTS: For rare CTCs (fewer than 1 CTC per ml of blood), the use of short DiFC intervals (corresponding to small PB samples) frequently resulted in no detections. For more abundant CTCs, CTC numbers frequently varied by an order of magnitude or more over the time-scales considered. This variance in CTC detections far exceeded that expected by Poisson statistics or by instrument variability. Rather, the data were consistent with significant changes in mean numbers of CTCs on the timescales of minutes and hours. CONCLUSIONS: The observed temporal changes can be explained by known properties of CTCs, namely, the continuous shedding of CTCs from tumors and the short half-life of CTCs in blood. It follows that the number of cells in a blood sample are strongly impacted by the timing of the draw. The issue is likely to be compounded for multicellular CTC clusters or specific CTC subtypes, which are even more rare than single CTCs. However, we show that enumeration can in principle be improved by averaging multiple samples, analysis of larger volumes, or development of methods for enumeration of CTCs directly in vivo.

Integrating transcriptomics and bulk time course data into a mathematical framework to describe and predict therapeutic resistance in cancer.

A significant challenge in the field of biomedicine is the development of methods to integrate the multitude of dispersed data sets into comprehensive frameworks to be used to generate optimal clinical decisions. Recent technological advances in single cell analysis allow for high-dimensional molecular characterization of cells and populations, but to date, few mathematical models have attempted to integrate measurements from the single cell scale with other types of longitudinal data. Here, we present a framework that actionizes static outputs from a machine learning model and leverages these as measurements of state variables in a dynamic model of treatment response. We apply this framework to breast cancer cells to integrate single cell transcriptomic data with longitudinal bulk cell population (bulk time course) data. We demonstrate that the explicit inclusion of the phenotypic composition estimate, derived from single cell RNA-sequencing data (scRNA-seq), improves accuracy in the prediction of new treatments with a concordance correlation coefficient (CCC) of 0.92 compared to a prediction accuracy of CCC = 0.64 when fitting on longitudinal bulk cell population data alone. To our knowledge, this is the first work that explicitly integrates single cell clonally-resolved transcriptome datasets with bulk time-course data to jointly calibrate a mathematical model of drug resistance dynamics. We anticipate this approach to be a first step that demonstrates the feasibility of incorporating multiple data types into mathematical models to develop optimized treatment regimens from data.

Mathematical Details on a Cancer Resistance Model.

One of the most important factors limiting the success of chemotherapy in cancer treatment is the phenomenon of drug resistance. We have recently introduced a framework for quantifying the effects of induced and non-induced resistance to cancer chemotherapy (Greene et al., 2018a, 2019). In this work, we expound on the details relating to an optimal control problem outlined in Greene et al. (2018a). The control structure is precisely characterized as a concatenation of bang-bang and path-constrained arcs via the Pontryagin Maximum Principle and differential Lie algebraic techniques. A structural identifiability analysis is also presented, demonstrating that patient-specific parameters may be measured and thus utilized in the design of optimal therapies prior to the commencement of therapy. For completeness, a detailed analysis of existence results is also included.

Delicate Balances in Cancer Chemotherapy: Modeling Immune Recruitment and Emergence of Systemic Drug Resistance.

Metronomic chemotherapy can drastically enhance immunogenic tumor cell death. However, the mechanisms responsible are still incompletely understood. Here, we develop a mathematical model to elucidate the underlying complex interactions between tumor growth, immune system activation, and therapy-mediated immunogenic cell death. Our model is conceptually simple, yet it provides a surprisingly excellent fit to empirical data obtained from a GL261 SCID mouse glioma model treated with cyclophosphamide on a metronomic schedule. The model includes terms representing immune recruitment as well as the emergence of drug resistance during prolonged metronomic treatments. Strikingly, a single fixed set of parameters, adjusted neither for individuals nor for drug schedule, recapitulates experimental data across various drug regimens remarkably well, including treatments administered at intervals ranging from 6 to 12 days. Additionally, the model predicts peak immune activation times, rediscovering experimental data that had not been used in parameter fitting or in model construction. Notably, the validated model suggests that immunostimulatory and immunosuppressive intermediates are responsible for the observed phenomena of resistance and immune cell recruitment, and thus for variation of responses with respect to different schedules of drug administration.

Distributed Implementation of Boolean Functions by Transcriptional Synthetic Circuits.

Starting in the early 2000s, sophisticated technologies have been developed for the rational construction of synthetic genetic networks that implement specified logical functionalities. Despite impressive progress, however, the scaling necessary in order to achieve greater computational power has been hampered by many constraints, including repressor toxicity and the lack of large sets of mutually orthogonal repressors. As a consequence, a typical circuit contains no more than roughly seven repressor-based gates per cell. A possible way around this scalability problem is to distribute the computation among multiple cell types, each of which implements a small subcircuit, which communicate among themselves using diffusible small molecules (DSMs). Examples of DSMs are those employed by quorum sensing systems in bacteria. This paper focuses on systematic ways to implement this distributed approach, in the context of the evaluation of arbitrary Boolean functions. The unique characteristics of genetic circuits and the properties of DSMs require the development of new Boolean synthesis methods, distinct from those classically used in electronic circuit design. In this work, we propose a fast algorithm to synthesize distributed realizations for any Boolean function, under constraints on the number of gates per cell and the number of orthogonal DSMs. The method is based on an exact synthesis algorithm to find the minimal circuit per cell, which in turn allows us to build an extensive database of Boolean functions up to a given number of inputs. For concreteness, we will specifically focus on circuits of up to 4 inputs, which might represent, for example, two chemical inducers and two light inputs at different frequencies. Our method shows that, with a constraint of no more than seven gates per cell, the use of a single DSM increases the total number of realizable circuits by at least 7.58-fold compared to centralized computation. Moreover, when allowing two DSM's, one can realize 99.995% of all possible 4-input Boolean functions, still with at most 7 gates per cell. The methodology introduced here can be readily adapted to complement recent genetic circuit design automation software. A toolbox that uses the proposed algorithm was created and made available at https://github.com/sontaglab/DBC/.

A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks.

Complex molecular biological processes such as transcription and translation, signal transduction, post-translational modification cascades, and metabolic pathways can be described in principle by biochemical reactions that explicitly take into account the sophisticated network of chemical interactions regulating cell life. The ability to deduce the possible qualitative behaviors of such networks from a set of reactions is a central objective and an ongoing challenge in the field of systems biology. Unfortunately, the construction of complete mathematical models is often hindered by a pervasive problem: despite the wealth of qualitative graphical knowledge about network interactions, the form of the governing nonlinearities and/or the values of kinetic constants are hard to uncover experimentally. The kinetics can also change with environmental variations. This work addresses the following question: given a set of reactions and without assuming a particular form for the kinetics, what can we say about the asymptotic behavior of the network? Specifically, it introduces a class of networks that are "structurally (mono) attractive" meaning that they are incapable of exhibiting multiple steady states, oscillation, or chaos by virtue of their reaction graphs. These networks are characterized by the existence of a universal energy-like function called a Robust Lyapunov function (RLF). To find such functions, a finite set of rank-one linear systems is introduced, which form the extremals of a linear convex cone. The problem is then reduced to that of finding a common Lyapunov function for this set of extremals. Based on this characterization, a computational package, Lyapunov-Enabled Analysis of Reaction Networks (LEARN), is provided that constructs such functions or rules out their existence. An extensive study of biochemical networks demonstrates that LEARN offers a new unified framework. Basic motifs, three-body binding, and genetic networks are studied first. The work then focuses on cellular signalling networks including various post-translational modification cascades, phosphotransfer and phosphorelay networks, T-cell kinetic proofreading, and ERK signalling. The Ribosome Flow Model is also studied.