Lattice structures that parameterize regulatory network dynamics.

We consider two types of models of regulatory network dynamics: Boolean maps and systems of switching ordinary differential equations. Our goal is to construct all models in each category that are compatible with the directed signed graph that describe the network interactions. This leads to consideration of lattice of monotone Boolean functions (MBF), poset of non-degenerate MBFs, and a lattice of chains in these sets. We describe explicit inductive construction of these posets where the induction is on the number of inputs in MBF. Our results allow enumeration of potential dynamic behavior of the network for both model types, subject to practical limitation imposed by the size of the lattice of MBFs described by the Dedekind number.

Quantifying robustness of the gap gene network.

Early development of Drosophila melanogaster (fruit fly) facilitated by the gap gene network has been shown to be incredibly robust, and the same patterns emerge even when the process is seriously disrupted. We investigate this robustness using a previously developed computational framework called DSGRN (Dynamic Signatures Generated by Regulatory Networks). Our mathematical innovations include the conceptual extension of this established modeling technique to enable modeling of spatially monotone environmental effects, as well as the development of a collection of graph theoretic robustness scores for network models. This allows us to rank order the robustness of network models of cellular systems where each cell contains the same genetic network topology but operates under a parameter regime that changes continuously from cell to cell. We demonstrate the power of this method by comparing the robustness of two previously introduced network models of gap gene expression along the anterior-posterior axis of the fruit fly embryo, both to each other and to a random sample of networks with same number of nodes and edges. We observe that there is a substantial difference in robustness scores between the two models. Our biological insight is that random network topologies are in general capable of reproducing complex patterns of expression, but that using measures of robustness to rank order networks permits a large reduction in hypothesis space for highly conserved systems such as developmental networks.

A Nonlinear Delay Model for Metabolic Oscillations in Yeast Cells.

We introduce two time-delay models of metabolic oscillations in yeast cells. Our model tests a hypothesis that the oscillations occur as multiple pathways share a limited resource which we equate to the number of available ribosomes. We initially explore a single-protein model with a constraint equation governing the total resource available to the cell. The model is then extended to include three proteins that share a resource pool. Three approaches are considered at constant delay to numerically detect oscillations. First, we use a spectral element method to approximate the system as a discrete map and evaluate the stability of the linearized system about its equilibria by examining its eigenvalues. For the second method, we plot amplitudes of the simulation trajectories in 2D projections of the parameter space. We use a history function that is consistent with published experimental results to obtain metabolic oscillations. Finally, the spectral element method is used to convert the system to a boundary value problem whose solutions correspond to approximate periodic solutions of the system. Our results show that certain combinations of total resource available and the time delay, lead to oscillations. We observe that an oscillation region in the parameter space is between regions admitting steady states that correspond to zero and constant production. Similar behavior is found with the three-protein model where all proteins require the same production time. However, a shift in the protein production rates peaks occurs for low available resource suggesting that our model captures the shared resource pool dynamics.

Ribosome Abundance Control in Prokaryotes.

Cell growth is an essential phenotype of any unicellular organism and it crucially depends on precise control of protein synthesis. We construct a model of the feedback mechanisms that regulate abundance of ribosomes in E. coli, a prototypical prokaryotic organism. Since ribosomes are needed to produce more ribosomes, the model includes a positive feedback loop central to the control of cell growth. Our analysis of the model shows that there can be only two coexisting equilibrium states across all 23 parameters. This precludes the existence of hysteresis, suggesting that the ribosome abundance changes continuously with parameters. These states are related by a transcritical bifurcation, and we provide an analytic formula for parameters that admit either state.

Assessing biological network dynamics: comparing numerical simulations with analytical decomposition of parameter space.

Mathematical modeling of the emergent dynamics of gene regulatory networks (GRN) faces a double challenge of (a) dependence of model dynamics on parameters, and (b) lack of reliable experimentally determined parameters. In this paper we compare two complementary approaches for describing GRN dynamics across unknown parameters: (1) parameter sampling and resulting ensemble statistics used by RACIPE (RAndom CIrcuit PErturbation), and (2) use of rigorous analysis of combinatorial approximation of the ODE models by DSGRN (Dynamic Signatures Generated by Regulatory Networks). We find a very good agreement between RACIPE simulation and DSGRN predictions for four different 2- and 3-node networks typically observed in cellular decision making. This observation is remarkable since the DSGRN approach assumes that the Hill coefficients of the models are very high while RACIPE assumes the values in the range 1-6. Thus DSGRN parameter domains, explicitly defined by inequalities between systems parameters, are highly predictive of ODE model dynamics within a biologically reasonable range of parameters.

Resource allocation accounts for the large variability of rate-yield phenotypes across bacterial strains.

Different strains of a microorganism growing in the same environment display a wide variety of growth rates and growth yields. We developed a coarse-grained model to test the hypothesis that different resource allocation strategies, corresponding to different compositions of the proteome, can account for the observed rate-yield variability. The model predictions were verified by means of a database of hundreds of published rate-yield and uptake-secretion phenotypes of Escherichia coli strains grown in standard laboratory conditions. We found a very good quantitative agreement between the range of predicted and observed growth rates, growth yields, and glucose uptake and acetate secretion rates. These results support the hypothesis that resource allocation is a major explanatory factor of the observed variability of growth rates and growth yields across different bacterial strains. An interesting prediction of our model, supported by the experimental data, is that high growth rates are not necessarily accompanied by low growth yields. The resource allocation strategies enabling high-rate, high-yield growth of E. coli lead to a higher saturation of enzymes and ribosomes, and thus to a more efficient utilization of proteomic resources. Our model thus contributes to a fundamental understanding of the quantitative relationship between rate and yield in E. coli and other microorganisms. It may also be useful for the rapid screening of strains in metabolic engineering and synthetic biology.

Robustness and reproducibility of simple and complex synthetic logic circuit designs using a DBTL loop.

Computational tools addressing various components of design-build-test-learn (DBTL) loops for the construction of synthetic genetic networks exist but do not generally cover the entire DBTL loop. This manuscript introduces an end-to-end sequence of tools that together form a DBTL loop called Design Assemble Round Trip (DART). DART provides rational selection and refinement of genetic parts to construct and test a circuit. Computational support for experimental process, metadata management, standardized data collection and reproducible data analysis is provided via the previously published Round Trip (RT) test-learn loop. The primary focus of this work is on the Design Assemble (DA) part of the tool chain, which improves on previous techniques by screening up to thousands of network topologies for robust performance using a novel robustness score derived from dynamical behavior based on circuit topology only. In addition, novel experimental support software is introduced for the assembly of genetic circuits. A complete design-through-analysis sequence is presented using several OR and NOR circuit designs, with and without structural redundancy, that are implemented in budding yeast. The execution of DART tested the predictions of the design tools, specifically with regard to robust and reproducible performance under different experimental conditions. The data analysis depended on a novel application of machine learning techniques to segment bimodal flow cytometry distributions. Evidence is presented that, in some cases, a more complex build may impart more robustness and reproducibility across experimental conditions. Graphical Abstract.

Experimental guidance for discovering genetic networks through hypothesis reduction on time series.

Large programs of dynamic gene expression, like cell cyles and circadian rhythms, are controlled by a relatively small "core" network of transcription factors and post-translational modifiers, working in concerted mutual regulation. Recent work suggests that system-independent, quantitative features of the dynamics of gene expression can be used to identify core regulators. We introduce an approach of iterative network hypothesis reduction from time-series data in which increasingly complex features of the dynamic expression of individual, pairs, and entire collections of genes are used to infer functional network models that can produce the observed transcriptional program. The culmination of our work is a computational pipeline, Iterative Network Hypothesis Reduction from Temporal Dynamics (Inherent dynamics pipeline), that provides a priority listing of targets for genetic perturbation to experimentally infer network structure. We demonstrate the capability of this integrated computational pipeline on synthetic and yeast cell-cycle data.

Positive feedback drives a secondary nonlinear product burst during a biphasic DNA amplification reaction.

Isothermal DNA amplification reactions are used in a broad variety of applications, from diagnostic assays to DNA circuits, with greater speed and less complexity than established PCR technologies. We recently reported a unique, high gain, biphasic isothermal DNA amplification reaction, called the Ultrasensitive DNA Amplification Reaction (UDAR). Here we present a detailed analysis of the UDAR reaction pathways that initiates with a first phase followed by a nonlinear product burst, which is caused by an autocatalytic secondary reaction. The experimental reaction output was reproduced using an ordinary differential equation model based on detailed reaction mechanisms. This model provides insight on the relative importance of each reaction mechanism during both phases, which could aid in the design of product output during DNA amplification reactions.

Modeling Transport Regulation in Gene Regulatory Networks.

A gene regulatory network summarizes the interactions between a set of genes and regulatory gene products. These interactions include transcriptional regulation, protein activity regulation, and regulation of the transport of proteins between cellular compartments. DSGRN is a network modeling approach that builds on traditions of discrete-time Boolean models and continuous-time switching system models. When all interactions are transcriptional, DSGRN uses a combinatorial approximation to describe the entire range of dynamics that is compatible with network structure. Here we present an extension of the DGSRN approach to transport regulation across a boundary between compartments, such as a cellular membrane. We illustrate our approach by searching a model of the p53-Mdm2 network for the potential to admit two experimentally observed distinct stable periodic cycles.

Environment Constrains Fitness Advantages of Division of Labor in Microbial Consortia Engineered for Metabolite Push or Pull Interactions.

Fitness benefits from division of labor are well documented in microbial consortia, but the dependency of the benefits on environmental context is poorly understood. Two synthetic Escherichia coli consortia were built to test the relationships between exchanged organic acid, local environment, and opportunity costs of different metabolic strategies. Opportunity costs quantify benefits not realized due to selecting one phenotype over another. The consortia catabolized glucose and exchanged either acetic or lactic acid to create producer-consumer food webs. The organic acids had different inhibitory properties and different opportunity costs associated with their positions in central metabolism. The exchanged metabolites modulated different consortial dynamics. The acetic acid-exchanging (AAE) consortium had a "push" interaction motif where acetic acid was secreted faster by the producer than the consumer imported it, while the lactic acid-exchanging (LAE) consortium had a "pull" interaction motif where the consumer imported lactic acid at a comparable rate to its production. The LAE consortium outperformed wild-type (WT) batch cultures under the environmental context of weakly buffered conditions, achieving a 55% increase in biomass titer, a 51% increase in biomass per proton yield, an 86% increase in substrate conversion, and the complete elimination of by-product accumulation all relative to the WT. However, the LAE consortium had the trade-off of a 42% lower specific growth rate. The AAE consortium did not outperform the WT in any considered performance metric. Performance advantages of the LAE consortium were sensitive to environment; increasing the medium buffering capacity negated the performance advantages compared to WT. IMPORTANCE Most naturally occurring microorganisms persist in consortia where metabolic interactions are common and often essential to ecosystem function. This study uses synthetic ecology to test how different cellular interaction motifs influence performance properties of consortia. Environmental context ultimately controlled the division of labor performance as shifts from weakly buffered to highly buffered conditions negated the benefits of the strategy. Understanding the limits of division of labor advances our understanding of natural community functioning, which is central to nutrient cycling and provides design rules for assembling consortia used in applied bioprocessing.

Genetic networks encode secrets of their past.

Research shows that gene duplication followed by either repurposing or removal of duplicated genes is an important contributor to evolution of gene and protein interaction networks. We aim to identify which characteristics of a network can arise through this process, and which must have been produced in a different way. To model the network evolution, we postulate vertex duplication and edge deletion as evolutionary operations on graphs. Using the novel concept of an ancestrally distinguished subgraph, we show how features of present-day networks require certain features of their ancestors. In particular, ancestrally distinguished subgraphs cannot be introduced by vertex duplication. Additionally, if vertex duplication and edge deletion are the only evolutionary mechanisms, then a graph's ancestrally distinguished subgraphs must be contained in all of the graph's ancestors. We analyze two experimentally derived genetic networks and show that our results accurately predict lack of large ancestrally distinguished subgraphs, despite this feature being statistically improbable in associated random networks. This observation is consistent with the hypothesis that these networks evolved primarily via vertex duplication. The tools we provide open the door for analyzing ancestral networks using current networks. Our results apply to edge-labeled (e.g. signed) graphs which are either undirected or directed.

Operon dynamics with state dependent transcription and/or translation delays.

Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we study Goodwin models of both inducible and repressible operons with state-dependent delays. The paper provides justification and derivation of the model, detailed analysis of the appropriate setting of the corresponding dynamical system, and extensive numerical analysis of its dynamics. Comparison with constant delay models shows significant differences in dynamics that include existence of stable periodic orbits in inducible systems and multistability in repressible systems. A combination of parameter space exploration, numerics, analysis of steady state linearization and bifurcation theory indicates the likely presence of Shilnikov-type homoclinic bifurcations in the repressible operon model.

Trade-offs among transcription elongation rate, number, and duration of ubiquitous pauses on highly transcribed bacterial genes.

In this paper, we study the limitations imposed on the transcription process by the presence of short ubiquitous pauses and crowding. These effects are especially pronounced in highly transcribed genes such as ribosomal genes (rrn) in fast growing bacteria. Our model indicates that the quantity and duration of pauses reported for protein-coding genes is incompatible with the average elongation rate observed in rrn genes. When maximal elongation rate is high, pause-induced traffic jams occur, increasing promoter occlusion, thereby lowering the initiation rate. This lowers average transcription rate and increases average transcription time. Increasing maximal elongation rate in the model is insufficient to match the experimentally observed average elongation rate in rrn genes. This suggests that there may be rrn-specific modifications to RNAP, which then experience fewer pauses, or pauses of shorter duration than those in protein-coding genes. We identify model parameter triples (maximal elongation rate, mean pause duration time, number of pauses) which are compatible with experimentally observed elongation rates. Average transcription time and average transcription rate are the model outputs investigated as proxies for cell fitness. These fitness functions are optimized for different parameter choices, opening up a possibility of differential control of these aspects of the elongation process, with potential evolutionary consequences. As an example, a gene's average transcription time may be crucial to fitness when the surrounding medium is prone to abrupt changes. This paper demonstrates that a functional relationship among the model parameters can be estimated using a standard statistical analysis, and this functional relationship describes the various trade-offs that must be made in order for the gene to control the elongation process and achieve a desired average transcription time. It also demonstrates the robustness of the system when a range of maximal elongation rates can be balanced with transcriptional pause data in order to maintain a desired fitness.

Rational design of complex phenotype via network models.

We demonstrate a modeling and computational framework that allows for rapid screening of thousands of potential network designs for particular dynamic behavior. To illustrate this capability we consider the problem of hysteresis, a prerequisite for construction of robust bistable switches and hence a cornerstone for construction of more complex synthetic circuits. We evaluate and rank most three node networks according to their ability to robustly exhibit hysteresis where robustness is measured with respect to parameters over multiple dynamic phenotypes. Focusing on the highest ranked networks, we demonstrate how additional robustness and design constraints can be applied. We compare our results to more traditional methods based on specific parameterization of ordinary differential equation models and demonstrate a strong qualitative match at a small fraction of the computational cost.

Limitation by a shared mutualist promotes coexistence of multiple competing partners.

Although mutualisms are often studied as simple pairwise interactions, they typically involve complex networks of interacting species. How multiple mutualistic partners that provide the same service and compete for resources are maintained in mutualistic networks is an open question. We use a model bacterial community in which multiple 'partner strains' of Escherichia coli compete for a carbon source and exchange resources with a 'shared mutualist' strain of Salmonella enterica. In laboratory experiments, competing E. coli strains readily coexist in the presence of S. enterica, despite differences in their competitive abilities. We use ecological modeling to demonstrate that a shared mutualist can create temporary resource niche partitioning by limiting growth rates, even if yield is set by a resource external to a mutualism. This mechanism can extend to maintain multiple competing partner species. Our results improve our understanding of complex mutualistic communities and aid efforts to design stable microbial communities.

An intrinsic oscillator drives the blood stage cycle of the malaria parasite Plasmodium falciparum.

The blood stage of the infection of the malaria parasite Plasmodium falciparum exhibits a 48-hour developmental cycle that culminates in the synchronous release of parasites from red blood cells, which triggers 48-hour fever cycles in the host. This cycle could be driven extrinsically by host circadian processes or by a parasite-intrinsic oscillator. To distinguish between these hypotheses, we examine the P. falciparum cycle in an in vitro culture system and show that the parasite has molecular signatures associated with circadian and cell cycle oscillators. Each of the four strains examined has a different period, which indicates strain-intrinsic period control. Finally, we demonstrate that parasites have low cell-to-cell variance in cycle period, on par with a circadian oscillator. We conclude that an intrinsic oscillator maintains Plasmodium's rhythmic life cycle.

Multistability in the epithelial-mesenchymal transition network.

BACKGROUND: The transitions between epithelial (E) and mesenchymal (M) cell phenotypes are essential in many biological processes like tissue development and cancer metastasis. Previous studies, both modeling and experimental, suggested that in addition to E and M states, the network responsible for these phenotypes exhibits intermediate phenotypes between E and M states. The number and importance of such states is subject to intense discussion in the epithelial-mesenchymal transition (EMT) community. RESULTS: Previous modeling efforts used traditional bifurcation analysis to explore the number of the steady states that correspond to E, M and intermediate states by varying one or two parameters at a time. Since the system has dozens of parameters that are largely unknown, it remains a challenging problem to fully describe the potential set of states and their relationship across all parameters. We use the computational tool DSGRN (Dynamic Signatures Generated by Regulatory Networks) to explore the intermediate states of an EMT model network by computing summaries of the dynamics across all of parameter space. We find that the only attractors in the system are equilibria, that E and M states dominate across parameter space, but that bistability and multistability are common. Even at extreme levels of some of the known inducers of the transition, there is a certain proportion of the parameter space at which an E or an M state co-exists with other stable steady states. CONCLUSIONS: Our results suggest that the multistability is broadly present in the EMT network across parameters and thus response of cells to signals may strongly depend on the particular cell line and genetic background.

Multi-parameter exploration of dynamics of regulatory networks.

Over the last twenty years advances in systems biology have changed our views on microbial communities and promise to revolutionize treatment of human diseases. In almost all scientific breakthroughs since time of Newton, mathematical modeling has played a prominent role. Regulatory networks emerged as preferred descriptors of how abundances of molecular species depend on each other. However, the central question on how cellular phenotypes emerge from dynamics of these network remains elusive. The principal reason is that differential equation models in the field of biology (while so successful in areas of physics and physical chemistry), do not arise from first principles, and these models suffer from lack of proper parameterization. In response to these challenges, discrete time models based on Boolean networks have been developed. In this review, we discuss an emerging modeling paradigm that combines ideas from differential equations and Boolean models, and has been developed independently within dynamical systems and computer science communities. The result is an approach that can associate a range of potential dynamical behaviors to a network, arrange the descriptors of the dynamics in a searchable database, and allows for multi-parameter exploration of the dynamics akin to bifurcation theory. Since this approach is computationally accessible for moderately sized networks, it allows, perhaps for the first time, to rationally compare different network topologies based on their dynamics.

Using extremal events to characterize noisy time series.

Experimental time series provide an informative window into the underlying dynamical system, and the timing of the extrema of a time series (or its derivative) contains information about its structure. However, the time series often contain significant measurement errors. We describe a method for characterizing a time series for any assumed level of measurement error [Formula: see text] by a sequence of intervals, each of which is guaranteed to contain an extremum for any function that [Formula: see text]-approximates the time series. Based on the merge tree of a continuous function, we define a new object called the normalized branch decomposition, which allows us to compute intervals for any level [Formula: see text]. We show that there is a well-defined total order on these intervals for a single time series, and that it is naturally extended to a partial order across a collection of time series comprising a dataset. We use the order of the extracted intervals in two applications. First, the partial order describing a single dataset can be used to pattern match against switching model output (Cummins et al. in SIAM J Appl Dyn Syst 17(2):1589-1616, 2018), which allows the rejection of a network model. Second, the comparison between graph distances of the partial orders of different datasets can be used to quantify similarity between biological replicates.