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Abstract Single-cell omics technologies can measure millions of cells for up to thousands of biomolecular features, enabling data-driven studies of complex biological networks. However, these high-throughput experimental techniques often cannot track individual cells over time, thus complicating the understanding of dynamics such as time trajectories of cell states. These “dynamical phenotypes” are key to understanding biological phenomena such as differentiation fates. We show by mathematical analysis that, in spite of high dimensionality and lack of individual cell traces, three time-points of single-cell omics data are theoretically necessary and sufficient to uniquely determine the network interaction matrix and associated dynamics. Moreover, we show through numerical simulations that an interaction matrix can be accurately determined with three or more time-points even in the presence of sampling and measurement noise typical of single-cell omics. Our results can guide the design of single-cell omics time-course experiments, and provide a tool for data-driven phase-space analysis.

Hong, Kim and colleagues develop a method for analytically deriving the stationary distributions of stochastic biochemical reaction networks using network structure transformation. They provide this method in a user-friendly computational package, CASTANET, available open-source.

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.

Summary: 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 2N. 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.

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.

Exact analytical and closed-form solutions to the transient diffusion in bi-layer composites with external mass transfer resistance are reported. Expressions for the concentrations and the mass permeated are derived in both the Laplace and time domains through the use of the Laplace transform Inversion Theorem. The lead and lag times, which are often of importance in the characterization of membranes and arise from the analysis of the asymptotic behavior of the mass permeated through the bi-layer composite, were also derived. The presented solutions are also compared to previously derived limiting cases of the diffusion in a bi-layer with an impermeable wall and constant concentrations at the upstream and downstream boundaries. Analysis of the time lag shows that this membrane property is independent of the direction of flow. Finally, an outline is provided of how these transient solutions in response to a step function increase in concentration can be used to derive more complex input conditions. The importance of adequately handling boundary layer effects has a wide array of applications such as the study of bi-layers undergoing phenomena of heat convection, gas film resistance, and absorption/desorption.

MotivationCirculating 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.MethodsWe 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.ResultsFor 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.ConclusionsThe 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.

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.

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.

It was recently reported that acute influenza infection of the lung promoted distal melanoma growth in the dermis of mice. Melanoma-specific CD8+ T cells were shunted to the lung in the presence of the infection, where they expressed high levels of inflammation-induced cell-activation blocker PD-1, and became incapable of migrating back to the tumor site. At the same time, co-infection virus-specific CD8+ T cells remained functional while the infection was cleared. It was also unexpectedly found that PD-1 blockade immunotherapy reversed this effect. Here, we proceed to ground the experimental observations in a mechanistic immunobiochemical model that incorporates T cell pathways that control PD-1 expression. A core component of our model is a kinetic motif, which we call a PD-1 Double Incoherent Feed-Forward Loop (DIFFL), and which reflects known interactions between IRF4, Blimp-1, and Bcl-6. The different activity levels of the PD-1 DIFFL components, as a function of the cognate antigen levels and the given inflammation context, manifest themselves in phenotypically distinct outcomes. Collectively, the model allowed us to put forward a few working hypotheses as follows: (i) the melanoma-specific CD8+ T cells re-circulating with the blood flow enter the lung where they express high levels of inflammation-induced cell-activation blocker PD-1 in the presence of infection; (ii) when PD-1 receptors interact with abundant PD-L1, constitutively expressed in the lung, T cells loose motility; (iii) at the same time, virus-specific cells adapt to strong stimulation by their cognate antigen by lowering the transiently-elevated expression of PD-1, remaining functional and mobile in the inflamed lung, while the infection is cleared. The role that T cell receptor (TCR) activation and feedback loops play in the underlying processes are also highlighted and discussed. We hope that the results reported in our study could potentially contribute to the advancement of immunological approaches to cancer treatment and, as well, to a better understanding of a broader complexity of fundamental interactions between pathogens and tumors.

The aim of this paper is to study the qualitative behaviour predicted by a mathematical model for the initial stage of T-cell activation. The state variables in the model are the concentrations of phosphorylation states of the T-cell receptor (TCR) complex and the phosphatase SHP-1 in the cell. It is shown that these quantities cannot approach zero and that the model possesses more than one positive steady state for certain values of the parameters. It can also exhibit damped oscillations. It is proved that the chemical concentration which represents the degree of activation of the cell, that of the maximally phosphorylated form of the TCR complex, is, in general, a non-monotone function of the activating signal. In particular, there are cases where there is a value of the dissociation constant of the ligand from the receptor which produces a maximal activation of the T cell. This suggests that mechanisms taking place in the first few minutes after activation and included in the model studied in this paper suffice to explain the optimal dissociation time seen in experiments. In this way, the results of certain simulations in the literature have been confirmed rigorously and some important features which had not previously been seen have been discovered.