Core signalling motif displaying multistability through multi-state enzymes

Compartmental exchange regulates steady states and stochastic switching of a phosphorylation network

The phosphoregulation of proteins with multiple phosphorylation sites is governed by biochemical reaction networks that can exhibit multistable behavior. However, the behavior of such networks is typically studied in a single reaction volume, while cells are spatially organized into compartments that can exchange proteins. In this work, we use stochastic simulations to study the impact of compartmentalization on a two-site phosphorylation network. We characterize steady states and fluctuation-driven transitions between them as a function of the rate of protein exchange between two compartments. Surprisingly, the average time spent in a state before stochastically switching to another depends nonmonotonically on the protein exchange rate, with the most frequent switching occurring at intermediate exchange rates. At sufficiently small exchange rates, the state of the system and mean switching time are controlled largely by fluctuations in the balance of enzymes in each compartment. This leads to negatively correlated states in the compartments. For large exchange rates, the two compartments behave as a single effective compartment. However, when the compartmental volumes are unequal, the behavior differs from a single compartment with the same total volume. These results demonstrate that exchange of proteins between distinct compartments can regulate the emergent behavior of a common signaling motif.

Turing Instabilities are Not Enough to Ensure Pattern Formation

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Systematic analysis and optimization of early warning signals for critical transitions using distribution data

Abrupt shifts between alternative regimes occur in complex systems, from cell regulation to brain functions to ecosystems. Several model-free early warning signals (EWS) have been proposed to detect impending transitions, but failure or poor performance in some systems have called for better investigation of their generic applicability. Notably, there are still ongoing debates whether such signals can be successfully extracted from data in particular from biological experiments. In this work, we systematically investigate properties and performance of dynamical EWS in different deteriorating conditions, and we propose an optimized combination to trigger warnings as early as possible, eventually verified on experimental data from microbiological populations. Our results explain discrepancies observed in the literature between warning signs extracted from simulated models and from real data, provide guidance for EWS selection based on desired systems and suggest an optimized composite indicator to alert for impending critical transitions using distribution data.

Multisite Enzymes as a Mechanism for Bistability in Reaction Networks

Here, we focus on a common class of enzymes that have multiple substrate binding sites (multisite enzymes) and analyze their capacity to generate bistable dynamics in the reaction networks that they are embedded in. These networks include both substrate-product-substrate cycles and substrate-to-product conversion with subsequent product consumption. Using mathematical techniques, we show that the inherent binding and catalysis reactions arising from multiple substrate-enzyme complexes create a potential for bistable dynamics in such reaction networks. We construct a generic model of an enzyme with n-substrate binding sites and derive an analytical solution for the steady-state concentration of all enzyme-substrate complexes. By studying these expressions, we obtain a mechanistic understanding of bistability, derive parameter combinations that guarantee bistability, and show how changing specific enzyme kinetic parameters and enzyme levels can lead to bistability in reaction networks involving multisite enzymes. Thus, the presented findings provide a biochemical and mathematical basis for predicting and engineering bistability in multisite enzymes.

A numerical approach for detecting switch-like bistability in mass action chemical reaction networks with conservation laws

BACKGROUND

Theoretical analysis of signaling pathways can provide a substantial amount of insight into their function. One particular area of research considers signaling pathways capable of assuming two or more stable states given the same amount of signaling ligand. This phenomenon of bistability can give rise to switch-like behavior, a mechanism that governs cellular decision making. Investigation of whether or not a signaling pathway can confer bistability and switch-like behavior, without knowledge of specific kinetic rate constant values, is a mathematically challenging problem. Recently a technique based on optimization has been introduced, which is capable of finding example parameter values that confer switch-like behavior for a given pathway. Although this approach has made it possible to analyze moderately sized pathways, it is limited to reaction networks that presume a uniterminal structure. It is this limited structure we address by developing a general technique that applies to any mass action reaction network with conservation laws.

RESULTS

In this paper we developed a generalized method for detecting switch-like bistable behavior in any mass action reaction network with conservation laws. The method involves (1) construction of a constrained optimization problem using the determinant of the Jacobian of the underlying rate equations, (2) minimization of the objective function to search for conditions resulting in a zero eigenvalue, (3) computation of a confidence level that describes if the global minimum has been found and (4) evaluation of optimization values, using either numerical continuation or directly simulating the ODE system, to verify that a bistability region exists. The generalized method has been tested on three motifs known to be capable of bistability.

CONCLUSIONS

We have developed a variation of an optimization-based method for the discovery of bistability, which is not limited to uniterminal chemical reaction networks. Successful completion of the method provides an S-shaped bifurcation diagram, which indicates that the network acts as a bistable switch for the given optimization parameters.

Maximization of information transmission influences selection of native phosphorelay architectures

Phosphorelays are signal transduction circuits that sense environmental changes and adjust cellular metabolism. Five different circuit architectures account for 99% of all phosphorelay operons annotated in over 9,000 fully sequenced genomes. Here we asked what biological design principles, if any, could explain selection among those architectures in nature. We began by studying kinetically well characterized phosphorelays (Spo0 of Bacillus subtilis and Sln1 of Saccharomyces cerevisiae). We find that natural circuit architecture maximizes information transmission in both cases. We use mathematical models to compare information transmission among the architectures for a realistic range of concentration and parameter values. Mapping experimentally determined phosphorelay protein concentrations onto that range reveals that the native architecture maximizes information transmission in sixteen out of seventeen analyzed phosphorelays. These results suggest that maximization of information transmission is important in the selection of native phosphorelay architectures, parameter values and protein concentrations.

Topological signatures in regulatory network enable phenotypic heterogeneity in small cell lung cancer

Phenotypic (non-genetic) heterogeneity has significant implications for the development and evolution of organs, organisms, and populations. Recent observations in multiple cancers have unraveled the role of phenotypic heterogeneity in driving metastasis and therapy recalcitrance. However, the origins of such phenotypic heterogeneity are poorly understood in most cancers. Here, we investigate a regulatory network underlying phenotypic heterogeneity in small cell lung cancer, a devastating disease with no molecular targeted therapy. Discrete and continuous dynamical simulations of this network reveal its multistable behavior that can explain co-existence of four experimentally observed phenotypes. Analysis of the network topology uncovers that multistability emerges from two teams of players that mutually inhibit each other, but members of a team activate one another, forming a 'toggle switch' between the two teams. Deciphering these topological signatures in cancer-related regulatory networks can unravel their 'latent' design principles and offer a rational approach to characterize phenotypic heterogeneity in a tumor.

Interrogating metabolism as an electron flow system

Metabolism is generally considered as a neatly organised system of modular pathways, shaped by evolution under selection for optimal cellular growth. This view falls short of explaining and predicting a number of key observations about the structure and dynamics of metabolism. We highlight these limitations of a pathway-centric view on metabolism and summarise studies suggesting how these could be overcome by viewing metabolism as a thermodynamically and kinetically constrained, dynamical flow system. Such a systems-level, first-principles based view of metabolism can open up new avenues of metabolic engineering and cures for metabolic diseases and allow better insights to a myriad of physiological processes that are ultimately linked to metabolism. Towards further developing this view, we call for a closer interaction among physical and biological disciplines and an increased use of electrochemical and biophysical approaches to interrogate cellular metabolism together with the microenvironment in which it exists.

In Silico Evolution of Signaling Networks Using Rule-Based Models: Bistable Response Dynamics

One of the ultimate goals in biology is to understand the design principles of biological systems. Such principles, if they exist, can help us better understand complex, natural biological systems and guide the engineering of de novo ones. Toward deciphering design principles, in silico evolution of biological systems with proper abstraction is a promising approach. Here, we demonstrate the application of in silico evolution combined with rule-based modeling for exploring design principles of cellular signaling networks. This application is based on a computational platform, called BioJazz, which allows in silico evolution of signaling networks with unbounded complexity. We provide a detailed introduction to BioJazz architecture and implementation and describe how it can be used to evolve and/or design signaling networks with defined dynamics. For the latter, we evolve signaling networks with switch-like response dynamics and demonstrate how BioJazz can result in new biological insights into network structures that can endow bistable response dynamics. This example also demonstrated both the power of BioJazz in evolving and designing signaling networks and its limitations at the current stage of development.

Enzyme sequestration by the substrate: An analysis in the deterministic and stochastic domains

This paper is concerned with the potential multistability of protein concentrations in the cell. That is, situations where one, or a family of, proteins may sit at one of two or more different steady state concentrations in otherwise identical cells, and in spite of them being in the same environment. For models of multisite protein phosphorylation for example, in the presence of excess substrate, it has been shown that the achievable number of stable steady states can increase linearly with the number of phosphosites available. In this paper, we analyse the consequences of adding enzyme docking to these and similar models, with the resultant sequestration of phosphatase and kinase by the fully unphosphorylated and by the fully phosphorylated substrates respectively. In the large molecule numbers limit, where deterministic analysis is applicable, we prove that there are always values for these rates of sequestration which, when exceeded, limit the extent of multistability. For the models considered here, these numbers are much smaller than the affinity of the enzymes to the substrate when it is in a modifiable state. As substrate enzyme-sequestration is increased, we further prove that the number of steady states will inevitably be reduced to one. For smaller molecule numbers a stochastic analysis is more appropriate, where multistability in the large molecule numbers limit can manifest itself as multimodality of the probability distribution; the system spending periods of time in the vicinity of one mode before jumping to another. Here, we find that substrate enzyme sequestration can induce bimodality even in systems where only a single steady state can exist at large numbers. To facilitate this analysis, we develop a weakly chained diagonally dominant M-matrix formulation of the Chemical Master Equation, allowing greater insights in the way particular mechanisms, like enzyme sequestration, can shape probability distributions and therefore exhibit different behaviour across different regimes.