Direct Lyapunov exponent analysis enables parametric study of transient signalling governing cell behaviour

Computational models aid in the quantitative understanding of cell signalling networks. One important goal is to ascertain how multiple network components work together to govern cellular responses, that is, to determine cell 'signal-response' relationships. Several methods exist to study steady-state signals in the context of differential equation-based models. However, many biological networks influence cell behaviour through time-varying signals operating during a transient activated state that ultimately returns to a basal steady-state. A computational approach adapted from dynamical systems analysis to discern how diverse transient signals relate to alternative cell fates is described. Direct finite-time Lyapunov exponents (DLEs) are employed to identify phase-space domains of high sensitivity to initial conditions. These domains delineate regions exhibiting qualitatively different transient activities that would be indistinguishable using steady-state analysis but which correspond to different outcomes. These methods are applied to a physicochemical model of molecular interactions among caspase-3, caspase-8 and X-linked inhibitor of apoptosis--proteins whose transient activation determines cell death against survival fates. DLE analysis enabled identification of a separatrix that quantitatively characterises network behaviour by defining initial conditions leading to apoptotic cell death. It is anticipated that DLE analysis will facilitate theoretical investigation of phenotypic outcomes in larger models of signalling networks.     

Role of network branching in eliciting differential short-term signaling responses in the hypersensitive epidermal growth factor receptor mutants implicated in lung cancer

We study the effects of EGFR inhibition in wild-type and mutant cell lines upon tyrosine kinase inhibitor TKI treatment through a systems level deterministic and spatially homogeneous model to help characterize the hypersensitive response of the cancer cell lines harboring constitutively active mutant kinases to inhibitor treatment. By introducing a molecularly resolved branched network systems model (the molecular resolution is introduced for EGFR reactions and interactions in order to distinguish differences in activation between wild-type and mutants), we are able to quantify differences in (1) short-term signaling in downstream ERK and Akt activation, (2) the changes in the cellular inhibition EC50 associated with receptor phosphorylation (i.e., 50% inhibition of receptor phosphorylation in the cellular context), and (3) EC50 for the inhibition of activated downstream markers ERK-(p) and Akt-(p), where (p) denotes phosphorylated, upon treatment with the inhibitors in cell lines carrying both wild-type and mutant forms of the receptor. Using the branched signaling model, we illustrate a possible mechanism for preferential Akt activation in the cell lines harboring the oncogenic mutants of EGFR implicated in non-small-cell lung cancer and the enhanced efficacy of the inhibitor erlotinib especially in ablating the cellular Akt-(p) response. Using a simple phenomenological model to describe the effect of Akt activation on cellular decisions, we discuss how this preferential Akt activation is conducive to cellular oncogene addiction and how its disruption can lead to dramatic apoptotic response and hence remarkable inhibitor efficacies. We also identify key network nodes of our branched signaling model through sensitivity analysis as those rendering the network hypersensitive to enhanced ERK-(p) and Akt-(p); intriguingly, the identified nodes have a strong correlation with species implicated in oncogenic transformations in human cancers as well as in drug resistance mechanisms identified for the inhibitors in non-small-cell lung cancer therapy.     

Estimating the Stochastic Bifurcation Structure of Cellular Networks

High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.     

Cell signaling as a cognitive process

Cellular identity as defined through morphology and function emerges from intracellular signaling networks that communicate between cells. Based on recursive interactions within and among these intracellular networks, dynamical solutions in terms of biochemical behavior are generated that can differ from those in isolated cells. In this way, cellular heterogeneity in tissues can be established, implying that cell identity is not intrinsically predetermined by the genetic code but is rather dynamically maintained in a cognitive manner. We address how to experimentally measure the flow of information in intracellular biochemical networks and demonstrate that even simple causality motifs can give rise to rich, context‐dependent dynamic behavior. The concept how intercellular communication can result in novel dynamical solutions is applied to provide a contextual perspective on cell differentiation and tumorigenesis.     

Compensatory effects in the PI3K/PTEN/AKT signaling network following receptor tyrosine kinase inhibition

Overcoming de novo and acquired resistance to anticancer drugs that target signaling networks is a formidable challenge for drug design and effective cancer therapy. Understanding the mechanisms by which this resistance arises may offer a route to addressing the insensitivity of signaling networks to drug intervention and restore the efficacy of anticancer therapy. Extending our recent work identifying PTEN as a key regulator of Herceptin sensitivity, we present an integrated theoretical and experimental approach to study the compensatory mechanisms within the PI3K/PTEN/AKT signaling network that afford resistance to receptor tyrosine kinase (RTK) inhibition by anti-HER2 monoclonal antibodies. In a computational model representing the dynamics of the signaling network, we define a single control parameter that encapsulates the balance of activities of the enzymes involved in the PI3K/PTEN/AKT cycle. By varying this control parameter we are able to demonstrate both distinct dynamic regimes of behavior of the signaling network and the transitions between those regimes. We demonstrate resistance, sensitivity, and suppression of RTK signals by the signaling network. Through model analysis we link the sensitivity-to-resistance transition to specific compensatory mechanisms within the signaling network. We study this transition in detail theoretically by variation of activities of PTEN, PI3K, AKT enzymes, and use the results to inform experiments that perturb the signaling network using combinatorial inhibition of RTK, PTEN, and PI3K enzymes in human ovarian carcinoma cell lines. We find good alignment between theoretical predictions and experimental results. We discuss the application of the results to the challenges of hypersensitivity of the signaling network to RTK signals, suppression of drug resistance, and efficacy of drug combinations in anticancer therapy.     

Multisite phosphorylation and network dynamics of cyclin-dependent kinase signaling in the eukaryotic cell cycle

Multisite phosphorylation of regulatory proteins has been proposed to underlie ultrasensitive responses required to generate nontrivial dynamics in complex biological signaling networks. We used a random search strategy to analyze the role of multisite phosphorylation of key proteins regulating cyclin-dependent kinase (CDK) activity in a model of the eukaryotic cell cycle. We show that multisite phosphorylation of either CDK, CDC25, wee1, or CDK-activating kinase is sufficient to generate dynamical behaviors including bistability and limit cycles. Moreover, combining multiple feedback loops based on multisite phosphorylation do not destabilize the cell cycle network by inducing complex behavior, but rather increase the overall robustness of the network. In this model we find that bistability is the major dynamical behavior of the CDK signaling network, and that negative feedback converts bistability into limit cycle behavior. We also compare the dynamical behavior of several simplified models of CDK regulation to the fully detailed model. In summary, our findings suggest that multisite phosphorylation of proteins is a critical biological mechanism in generating the essential dynamics and ensuring robust behavior of the cell cycle.     

Using cell fate attractors to uncover transcriptional regulation of HL60 neutrophil differentiation

Background  

The process of cellular differentiation is governed by complex dynamical biomolecular networks consisting of a multitude of genes and their products acting in concert to determine a particular cell fate. Thus, a systems level view is necessary for understanding how a cell coordinates this process and for developing effective therapeutic strategies to treat diseases, such as cancer, in which differentiation plays a significant role. Theoretical considerations and recent experimental evidence support the view that cell fates are high dimensional attractor states of the underlying molecular networks. The temporal behavior of the network states progressing toward different cell fate attractors has the potential to elucidate the underlying molecular mechanisms governing differentiation.     

The signaling petri net-based simulator: a non-parametric strategy for characterizing the dynamics of cell-specific signaling networks

Reconstructing cellular signaling networks and understanding how they work are major endeavors in cell biology. The scale and complexity of these networks, however, render their analysis using experimental biology approaches alone very challenging. As a result, computational methods have been developed and combined with experimental biology approaches, producing powerful tools for the analysis of these networks. These computational methods mostly fall on either end of a spectrum of model parameterization. On one end is a class of structural network analysis methods; these typically use the network connectivity alone to generate hypotheses about global properties. On the other end is a class of dynamic network analysis methods; these use, in addition to the connectivity, kinetic parameters of the biochemical reactions to predict the network's dynamic behavior. These predictions provide detailed insights into the properties that determine aspects of the network's structure and behavior. However, the difficulty of obtaining numerical values of kinetic parameters is widely recognized to limit the applicability of this latter class of methods. Several researchers have observed that the connectivity of a network alone can provide significant insights into its dynamics. Motivated by this fundamental observation, we present the signaling Petri net, a non-parametric model of cellular signaling networks, and the signaling Petri net-based simulator, a Petri net execution strategy for characterizing the dynamics of signal flow through a signaling network using token distribution and sampling. The result is a very fast method, which can analyze large-scale networks, and provide insights into the trends of molecules' activity-levels in response to an external stimulus, based solely on the network's connectivity. We have implemented the signaling Petri net-based simulator in the PathwayOracle toolkit, which is publicly available at http://bioinfo.cs.rice.edu/pathwayoracle. Using this method, we studied a MAPK1,2 and AKT signaling network downstream from EGFR in two breast tumor cell lines. We analyzed, both experimentally and computationally, the activity level of several molecules in response to a targeted manipulation of TSC2 and mTOR-Raptor. The results from our method agreed with experimental results in greater than 90% of the cases considered, and in those where they did not agree, our approach provided valuable insights into discrepancies between known network connectivities and experimental observations.     

Ingeneue: a versatile tool for reconstituting genetic networks,with examples from the segment polarity network

Here we describe a software tool for synthesizing molecular genetic data into models of genetic networks. Our software program Ingeneue, written in Java, lets the user quickly turn a map of a genetic network into a dynamical model consisting of a set of ordinary differential equations. We developed Ingeneue as part of an ongoing effort to explore the design and evolvability of genetic networks. Ingeneue has three principal advantages over other available mathematical software: it automates instantiation of the same network model in each cell in a 2-D sheet of cells; it constructs model equations from pre-made building blocks corresponding to common biochemical processes; and it automates searches through parameter space, sensitivity analyses, and other common tasks. Here we discuss the structure of the software and some of the issues we have dealt with. We conclude with some examples of results we have achieved with Ingeneue for the Drosophila segment polarity network.     

Phase Transitions in the Multi-cellular Regulatory Behavior of Pancreatic Islet Excitability

The pancreatic islets of Langerhans are multicellular micro-organs integral to maintaining glucose homeostasis through secretion of the hormone insulin. β-cells within the islet exist as a highly coupled electrical network which coordinates electrical activity and insulin release at high glucose, but leads to global suppression at basal glucose. Despite its importance, how network dynamics generate this emergent binary on/off behavior remains to be elucidated. Previous work has suggested that a small threshold of quiescent cells is able to suppress the entire network. By modeling the islet as a Boolean network, we predicted a phase-transition between globally active and inactive states would emerge near this threshold number of cells, indicative of critical behavior. This was tested using islets with an inducible-expression mutation which renders defined numbers of cells electrically inactive, together with pharmacological modulation of electrical activity. This was combined with real-time imaging of intracellular free-calcium activity [Ca²⁺]_i and measurement of physiological parameters in mice. As the number of inexcitable cells was increased beyond ∼15%, a phase-transition in islet activity occurred, switching from globally active wild-type behavior to global quiescence. This phase-transition was also seen in insulin secretion and blood glucose, indicating physiological impact. This behavior was reproduced in a multicellular dynamical model suggesting critical behavior in the islet may obey general properties of coupled heterogeneous networks. This study represents the first detailed explanation for how the islet facilitates inhibitory activity in spite of a heterogeneous cell population, as well as the role this plays in diabetes and its reversal. We further explain how islets utilize this critical behavior to leverage cellular heterogeneity and coordinate a robust insulin response with high dynamic range. These findings also give new insight into emergent multicellular dynamics in general which are applicable to many coupled physiological systems, specifically where inhibitory dynamics result from coupled networks.     

Impossibility of multiple steady states in a family of active membrane transport models

To study the dynamical behavior of active membrane transport models, Vieira and Bisch proposed a complex chemical network (model 3) with two cycles. One cycle involves monomers as pump units while the other cycle uses dimers. In their work, the stoichiometric network analysis was used to study the stability of steady states and the bifurcation analysis was done through numerical methods. They concluded that the possibility of multiple steady states in the model 3 could not be discarded. Here, a zero eigenvalue analysis is applied to prove the impossibility of multiple positive steady states in the model 3. (A positive steady state is one for which all species have positive concentrations.) Moreover, the result is generalized to its family networks. Received: 6 April 1998 / Revised version: 16 October 1998 / Accepted: 28 October 1998     

Chaotic gene regulatory networks can be robust against mutations and noise

Robustness to mutations and noise has been shown to evolve through stabilizing selection for optimal phenotypes in model gene regulatory networks. The ability to evolve robust mutants is known to depend on the network architecture. How do the dynamical properties and state-space structures of networks with high and low robustness differ? Does selection operate on the global dynamical behavior of the networks? What kind of state-space structures are favored by selection? We provide damage propagation analysis and an extensive statistical analysis of state spaces of these model networks to show that the change in their dynamical properties due to stabilizing selection for optimal phenotypes is minor. Most notably, the networks that are most robust to both mutations and noise are highly chaotic. Certain properties of chaotic networks, such as being able to produce large attractor basins, can be useful for maintaining a stable gene-expression pattern. Our findings indicate that conventional measures of stability, such as damage propagation, do not provide much information about robustness to mutations or noise in model gene regulatory networks.     

Modeling the topological organization of cellular processes

The cell as a dynamical system presents the characteristics of having a dynamical structure. That is, the exact phase space of the system cannot be fixed before the evolution and integrative cell models must state the evolution of the structure jointly with the evolution of the cell state. This kind of dynamical systems is very challenging to model and simulate. New programming concepts must be developed to ease their modeling and simulation. In this context, the goal of the MGS project is to develop an experimental programming language dedicated to the simulation of this kind of systems. MGS proposes a unified view on several computational mechanisms (CHAM, Lindenmayer systems, Paun systems, cellular automata) enabling the specification of spatially localized computations on heterogeneous entities. The evolution of a dynamical structure is handled through the concept of transformation which relies on the topological organization of the system components. An example based on the modeling of spatially distributed biochemical networks is used to illustrate how these notions can be used to model the spatial and temporal organization of intracellular processes.     

A robust model to describe the differentiation of T-helper cells

There is a wealth of information regarding the differentiation of T-helper cells. Nevertheless, there is no general agreement on the topology and dynamical properties of the molecular network controlling the differentiation of these cells. This paper presents a continuous dynamical system to model the signaling network that controls the differentiation process of T-helper cells. The model is able to represent the differentiation from the precursor Th0 cell to any of the four effectors types (Th1, Th2, Th17, and Treg), as well as the phenotype of single null mutants. We present the first sensitivity analysis of the equations defining the Th model, showing that the qualitative dynamical behavior of the model is very robust against changes in three out of four tested parameters. The robustness of the model is in agreement with our claim that the qualitative behavior of the system is to a large extent independent of the methodological framework used for modeling.     

Learning and evolution in bacterial taxis: an operational amplifier circuit modeling the computational dynamics of the prokaryotic 'two component system' protein network

Adaptive behavior in unicellular organisms (i.e., bacteria) depends on highly organized networks of proteins governing purposefully the myriad of molecular processes occurring within the cellular system. For instance, bacteria are able to explore the environment within which they develop by utilizing the motility of their flagellar system as well as a sophisticated biochemical navigation system that samples the environmental conditions surrounding the cell, searching for nutrients or moving away from toxic substances or dangerous physical conditions. In this paper we discuss how proteins of the intervening signal transduction network could be modeled as artificial neurons, simulating the dynamical aspects of the bacterial taxis. The model is based on the assumption that, in some important aspects, proteins can be considered as processing elements or McCulloch-Pitts artificial neurons that transfer and process information from the bacterium's membrane surface to the flagellar motor. This simulation of bacterial taxis has been carried out on a hardware realization of a McCulloch-Pitts artificial neuron using an operational amplifier. Based on the behavior of the operational amplifier we produce a model of the interaction between CheY and FliM, elements of the prokaryotic two component system controlling chemotaxis, as well as a simulation of learning and evolution processes in bacterial taxis. On the one side, our simulation results indicate that, computationally, these protein 'switches' are similar to McCulloch-Pitts artificial neurons, suggesting a bridge between evolution and learning in dynamical systems at cellular and molecular levels and the evolutive hardware approach. On the other side, important protein 'tactilizing' properties are not tapped by the model, and this suggests further complexity steps to explore in the approach to biological molecular computing.     

Influence of parameter values on the oscillation sensitivities of two p53–Mdm2 models

Biomolecular networks that present oscillatory behavior are ubiquitous in nature. While some design principles for robust oscillations have been identified, it is not well understood how these oscillations are affected when the kinetic parameters are constantly changing or are not precisely known, as often occurs in cellular environments. Many models of diverse complexity level, for systems such as circadian rhythms, cell cycle or the p53 network, have been proposed. Here we assess the influence of hundreds of different parameter sets on the sensitivities of two configurations of a well-known oscillatory system, the p53 core network. We show that, for both models and all parameter sets, the parameter related to the p53 positive feedback, i.e. self-promotion, is the only one that presents sizeable sensitivities on extrema, periods and delay. Moreover, varying the parameter set values to change the dynamical characteristics of the response is more restricted in the simple model, whereas the complex model shows greater tunability. These results highlight the importance of the presence of specific network patterns, in addition to the role of parameter values, when we want to characterize oscillatory biochemical systems.

Electronic supplementary material

The online version of this article (doi:10.1007/s11693-015-9173-y) contains supplementary material, which is available to authorized users.     

A domino effect in drug action: from metabolic assault towards parasite differentiation

Awareness is growing that drug target validation should involve systems analysis of cellular networks. There is less appreciation, though, that the composition of networks may change in response to drugs. If the response is homeostatic (e.g. through upregulation of the target protein), this may neutralize the inhibitory effect. In this scenario the effect on cell growth and survival would be less than anticipated based on affinity of the drug for its target. Glycolysis is the sole free-energy source for the deadly parasite Trypanosoma brucei and is therefore a possible target pathway for anti-trypanosomal drugs. Plasma-membrane glucose transport exerts high control over trypanosome glycolysis and hence the transporter is a promising drug target. Here we show that at high inhibitor concentrations, inhibition of trypanosome glucose transport causes cell death. Most interestingly, sublethal concentrations initiate a domino effect in which network adaptations enhance inhibition. This happens via (i) metabolic control exerted by the target protein, (ii) decreases in mRNAs encoding the target protein and other proteins in the same pathway, and (iii) partial differentiation of the cells leading to (low) expression of immunogenic insect-stage coat proteins. We discuss how these 'anti-homeostatic' responses together may facilitate killing of parasites at an acceptable drug dosage.     

Integration of cellular signals in chattering environments

Cells are constantly exposed to fluctuating environmental conditions. External signals are sensed, processed and integrated by cellular signal transduction networks, which translate input signals into specific cellular responses by means of biochemical reactions. These networks have a complex nature, and we are still far from having a complete characterization of the process through which they integrate information, specially given the noisy environment in which that information is embedded. Guided by the many instances of constructive influences of noise that have been reported in the physical sciences in the last decades, here we explore how multiple signals are integrated in an eukaryotic cell in the presence of background noise, or chatter. To that end, we use a Boolean model of a typical human signal transduction network. Despite its complexity, we find that the network is able to display simple patterns of signal integration. Furthermore, our computational analysis shows that these integration patterns depend on the levels of fluctuating background activity carried by other cell inputs. Taken together, our results indicate that signal integration is sensitive to environmental fluctuations, and that this background noise effectively determines the information integration capabilities of the cell.     

Dynamical approaches to modeling developmental gene regulatory networks

The network of interacting regulatory signals within a cell comprises one of the most complex and powerful computational systems in biology. Gene regulatory networks (GRNs) play a key role in transforming the information encoded in a genome into morphological form. To achieve this feat, GRNs must respond to and integrate environmental signals with their internal dynamics in a robust and coordinated fashion. The highly dynamic nature of this process lends itself to interpretation and analysis in the language of dynamical models. Modeling provides a means of systematically untangling the complicated structure of GRNs, a framework within which to simulate the behavior of reconstructed systems and, in some cases, suites of analytic tools for exploring that behavior and its implications. This review provides a general background to the idea of treating a regulatory network as a dynamical system, and describes a variety of different approaches that have been taken to the dynamical modeling of GRNs. Birth Defects Research (Part C) 87:131–142, 2009. © 2009 Wiley‐Liss, Inc.