Mathematical and computational analysis of adaptation via feedback inhibition in signal transduction pathways

Negative feedback systems for modelling NF-κB transcription factor oscillatory activity

Low-dimensional negative feedback systems (NFSs) were developed within a signal flow model to describe the oscillatory activities of NF-κB caused by interactions with its inhibitor IκBα. The NFSs were established as 3rd- and 4th-order linear systems containing unperturbed and perturbed negative feedback (NF) loops with constant or time-varying NF strengths and a feed-forward loop. NF-related analytical solutions to the NFSs representing the time courses of NF-κB and IκBα were determined and their exact mathematical relationship was found. The NFS's parameters were determined to fit the experimental time courses of NF-κB in TNF-α-stimulated embryonic fibroblasts, rela-/- embryonic fibroblasts reconstituted with RelA, C9L cells, GFP-p65 knock-in embryonic fibroblasts and embryogenic fibroblasts lacking Iκβ and IκBε, LPS-stimulated IC-21 macrophages treated or not with DCPA, and anti-IgM-stimulated DT40 B-lymphocytes. The unperturbed and perturbed NFSs describing the above biosystems generated isochronous and non-isochronous solutions, depending on a constant or time-varying NF strength, respectively. The oscillation period of the NF-coupled solutions, the phase difference between them and the time delays in the appearance of cytoplasmic IκBα after stimulation of NF-κB were determined. A significant divergence between the IκBα solutions to the NFSs and the IκBα experimental courses led to a rejection of the NF coupling between NF-κB and IκBα in the above biosystems. It was shown that neither the linearity nor the low dimensionality of the NFSs altered the NF relationship and the divergence between the IκBα solutions to the NFS and IκBα experimental time courses. Although the NF relationship between IκBα and NF-κB was not confirmed in all the experimental data analyzed, delayed negative feedback was found in some cases.

Design Principles for Biological Adaptation: A Systems and Control-Theoretic Treatment

Establishing a mapping between (from and to) the functionality of interest and the underlying network structure (design principles) remains a crucial step toward understanding and design of bio-systems. Perfect adaptation is one such crucial functionality that enables every living organism to regulate its essential activities in the presence of external disturbances. Previous approaches to deducing the design principles for adaptation have either relied on computationally burdensome brute-force methods or rule-based design strategies detecting only a subset of all possible adaptive network structures. This chapter outlines a scalable and generalizable method inspired by systems theory that unravels an exhaustive set of adaptation-capable structures. We first use the well-known performance parameters to characterize perfect adaptation. These performance parameters are then mapped back to a few parameters (poles, zeros, gain) characteristic of the underlying dynamical system constituted by the rate equations. Therefore, the performance parameters evaluated for the scenario of perfect adaptation can be expressed as a set of precise mathematical conditions involving the system parameters. Finally, we use algebraic graph theory to translate these abstract mathematical conditions to certain structural requirements for adaptation. The proposed algorithm does not assume any particular dynamics and is applicable to networks of any size. Moreover, the results offer a significant advancement in the realm of understanding and designing complex biochemical networks.

Evolutionary Stability of Small Molecular Regulatory Networks That Exhibit Near-Perfect Adaptation

Large-scale protein regulatory networks, such as signal transduction systems, contain small-scale modules ('motifs') that carry out specific dynamical functions. Systematic characterization of the properties of small network motifs is therefore of great interest to molecular systems biologists. We simulate a generic model of three-node motifs in search of near-perfect adaptation, the property that a system responds transiently to a change in an environmental signal and then returns near-perfectly to its pre-signal state (even in the continued presence of the signal). Using an evolutionary algorithm, we search the parameter space of these generic motifs for network topologies that score well on a pre-defined measure of near-perfect adaptation. We find many high-scoring parameter sets across a variety of three-node topologies. Of all possibilities, the highest scoring topologies contain incoherent feed-forward loops (IFFLs), and these topologies are evolutionarily stable in the sense that, under 'macro-mutations' that alter the topology of a network, the IFFL motif is consistently maintained. Topologies that rely on negative feedback loops with buffering (NFLBs) are also high-scoring; however, they are not evolutionarily stable in the sense that, under macro-mutations, they tend to evolve an IFFL motif and may-or may not-lose the NFLB motif.

On biological networks capable of robust adaptation in the presence of uncertainties: A linear systems-theoretic approach

Biological adaptation, the tendency of every living organism to regulate its essential activities in environmental fluctuations, is a well-studied functionality in systems and synthetic biology. In this work, we present a generic methodology inspired by systems theory to discover the design principles for robust adaptation, perfect and imperfect, in two different contexts: (1) in the presence of deterministic external and parametric disturbances and (2) in a stochastic setting. In all the cases, firstly, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as design requirements for the underlying networks. Thus, contrary to the existing approaches, the proposed methodologies provide an exhaustive set of admissible network structures without resorting to computationally burdensome brute-force techniques. Further, the proposed frameworks do not assume prior knowledge about the particular rate kinetics, thereby validating the conclusions for a large class of biological networks. In the deterministic setting, we show that unlike the incoherent feed-forward network structures (IFFLP or opposer modules), the modules containing negative feedback with buffer action (NFBLB or balancer modules) are robust to parametric fluctuations when a specific part of the network is assumed to remain unaffected. To this end, we propose a sufficient condition for imperfect adaptation and show that adding negative feedback in an IFFLP topology improves the robustness concerning parametric fluctuations. Further, we propose a stricter set of necessary conditions for imperfect adaptation. Turning to the stochastic scenario, we adopt a Wiener-Kolmogorov filter strategy to tune the parameters of a given network structure towards minimum output variance. We show that both NFBLB and IFFLP can be used as a reduced-order W-K filter. Further, we define the notion of nearest neighboring motifs to compare the output variances across different network structures. We argue that the NFBLB achieves adaptation at the cost of a variance higher than its nearest neighboring motifs whereas the IFFLP topology produces locally minimum variance while compared with its nearest neighboring motifs. We present numerical simulations to support the theoretical results. Overall, our results present a generic, systematic, and robust framework for advancing the understanding of complex biological networks.

Discovering adaptation-capable biological network structures using control-theoretic approaches

Constructing biological networks capable of performing specific biological functionalities has been of sustained interest in synthetic biology. Adaptation is one such ubiquitous functional property, which enables every living organism to sense a change in its surroundings and return to its operating condition prior to the disturbance. In this paper, we present a generic systems theory-driven method for designing adaptive protein networks. First, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as ‘design requirements’ for the underlying networks. We go on to prove that a protein network with different input–output nodes (proteins) needs to be at least of third-order in order to provide adaptation. Next, we show that the necessary design principles obtained for a three-node network in adaptation consist of negative feedback or a feed-forward realization. We argue that presence of a particular class of negative feedback or feed-forward realization is necessary for a network of any size to provide adaptation. Further, we claim that the necessary structural conditions derived in this work are the strictest among the ones hitherto existed in the literature. Finally, we prove that the capability of producing adaptation is retained for the admissible motifs even when the output node is connected with a downstream system in a feedback fashion. This result explains how complex biological networks achieve robustness while keeping the core motifs unchanged in the context of a particular functionality. We corroborate our theoretical results with detailed and thorough numerical simulations. Overall, our results present a generic, systematic and robust framework for designing various kinds of biological networks.

Biological systems display a remarkable diversity of functionalities, many of which can be conceived as the response of a large network composed of small interconnecting modules. Unravelling the connection pattern, i.e. design principles, behind important biological functionalities is one of the most challenging problems in systems biology. One such phenomenon is perfect adaptation, which merits special attention owing to its universal presence ranging from chemotaxis in bacterial cells to calcium homeostasis in mammalian cells. The present work focuses on finding the design principles for perfect adaptation in the presence of a stair-case type disturbance. To this end, the current work proposes a systems-theoretic approach to deduce precise mathematical (hence structural) conditions that comply with the key performance parameters for adaptation. The approach is agnostic to the particularities of the reaction kinetics, underlining the dominant role of the topological structure on the response of the network. Notably, the design principles obtained in this work serve as the most strict necessary structural conditions for a network of any size to provide perfect adaptation.

Molecular switch architecture determines response properties of signaling pathways

Many intracellular signaling pathways are composed of molecular switches, proteins that transition between two states-on and off Typically, signaling is initiated when an external stimulus activates its cognate receptor that, in turn, causes downstream switches to transition from off to on using one of the following mechanisms: activation, in which the transition rate from the off state to the on state increases; derepression, in which the transition rate from the on state to the off state decreases; and concerted, in which activation and derepression operate simultaneously. We use mathematical modeling to compare these signaling mechanisms in terms of their dose-response curves, response times, and abilities to process upstream fluctuations. Our analysis elucidates several operating principles for molecular switches. First, activation increases the sensitivity of the pathway, whereas derepression decreases sensitivity. Second, activation generates response times that decrease with signal strength, whereas derepression causes response times to increase with signal strength. These opposing features allow the concerted mechanism to not only show dose-response alignment, but also to decouple the response time from stimulus strength. However, these potentially beneficial properties come at the expense of increased susceptibility to upstream fluctuations. We demonstrate that these operating principles also hold when the models are extended to include additional features, such as receptor removal, kinetic proofreading, and cascades of switches. In total, we show how the architecture of molecular switches govern their response properties. We also discuss the biological implications of our findings.

VEGAS as a Platform for Facile Directed Evolution in Mammalian Cells

Directed evolution, artificial selection toward designed objectives, is routinely used to develop new molecular tools and therapeutics. Successful directed molecular evolution campaigns repeatedly test diverse sequences with a designed selective pressure. Unicellular organisms and their viral pathogens are exceptional for this purpose and have been used for decades. However, many desirable targets of directed evolution perform poorly or unnaturally in unicellular backgrounds. Here, we present a system for facile directed evolution in mammalian cells. Using the RNA alphavirus Sindbis as a vector for heredity and diversity, we achieved 24-h selection cycles surpassing 10^-3 mutations per base. Selection is achieved through genetically actuated sequences internal to the host cell, thus the system's name: viral evolution of genetically actuating sequences, or "VEGAS." Using VEGAS, we evolve transcription factors, GPCRs, and allosteric nanobodies toward functional signaling endpoints each in less than 1 weeks' time.

Importance of Feedback and Feedforward Loops to Adaptive Immune Response Modeling

The human adaptive immune system is a very complex network of different types of cells, cytokines, and signaling molecules. This complex network makes it difficult to understand the system level regulations. To properly explain the immune system, it is necessary to explicitly investigate the presence of different feedback and feedforward loops (FFLs) and their crosstalks. Considering that these loops increase the complexity of the system, the mathematical modeling has been proved to be an important tool to explain such complex biological systems. This review focuses on these regulatory loops and discusses their importance on systems modeling of the immune system.

Core signalling motif displaying multistability through multi-state enzymes

Bistability, and more generally multistability, is a key system dynamics feature enabling decision-making and memory in cells. Deciphering the molecular determinants of multistability is thus crucial for a better understanding of cellular pathways and their (re)engineering in synthetic biology. Here, we show that a key motif found predominantly in eukaryotic signalling systems, namely a futile signalling cycle, can display bistability when featuring a two-state kinase. We provide necessary and sufficient mathematical conditions on the kinetic parameters of this motif that guarantee the existence of multiple steady states. These conditions foster the intuition that bistability arises as a consequence of competition between the two states of the kinase. Extending from this result, we find that increasing the number of kinase states linearly translates into an increase in the number of steady states in the system. These findings reveal, to our knowledge, a new mechanism for the generation of bistability and multistability in cellular signalling systems. Further the futile cycle featuring a two-state kinase is among the smallest bistable signalling motifs. We show that multi-state kinases and the described competition-based motif are part of several natural signalling systems and thereby could enable them to implement complex information processing through multistability. These results indicate that multi-state kinases in signalling systems are readily exploited by natural evolution and could equally be used by synthetic approaches for the generation of multistable information processing systems at the cellular level.

Oscillatory stimuli differentiate adapting circuit topologies

Adapting pathways consist of negative feedback loops (NFLs) or incoherent feedforward loops (IFFLs), which we show can be differentiated using oscillatory stimulation: NFLs but not IFFLs generically show ‘refractory period stabilization’ or ‘period skipping’. Using these signatures and genetic rewiring we identified the circuit dominating cell cycle timing in yeast. In C. elegans AWA neurons we uncovered a Ca2+-NFL, diffcult to find by other means, especially in wild-type, intact animals. (70 words)

Nonequilibrium nature of adaptation in bacterial chemotaxis: A fluctuation-dissipation theorem approach

Adaptation is a crucial biological function possessed by many sensory systems. In this paper, we show that the fluctuation-dissipation theorem (FDT) serves as an ideal mathematical tool to study adaptation. With the aid of the nonequilibrium FDT developed by Seifert and Speck [Europhys. Lett. 89, 10007 (2010)EULEEJ0295-507510.1209/0295-5075/89/10007], we demonstrate the nonequilibrium nature of adaptation in bacterial chemotaxis. We further show that nonequilibrium is a necessary condition for adaptation even beyond the linear response regime using the spectral theory of generator matrices. In particular, the results of this paper are irrelevant to the specific functional forms of the model parameters. This suggests that the nonequilibrium nature of adaptation is a topological property, rather than a geometric property, of the underlying biochemical reaction network.

A shadow detector for photosynthesis efficiency

Plants tolerate large variations in the intensity of the light environment by controlling the efficiency of solar to chemical energy conversion. To do this, plants have a mechanism to detect the intensity, duration, and change in light as they experience moving shadows, flickering light, and cloud cover. Sugars are the primary products of CO₂ fixation, a metabolic pathway that is rate limited by this solar energy conversion. We propose that sugar is a signal encoding information about the intensity, duration and change in the light environment. We previously showed that the Arabidopsis heterotrimeric G protein complex including its receptor-like Regulator of G signaling protein, AtRGS1, detects both the concentration and the exposure time of sugars (Fu et al., 2014. Cell 156: 1084-1095). This unique property, designated dose-duration reciprocity, is a behavior that emerges from the system architecture / system motif. Here, we show that another property of the signaling system is to detect large changes in light while at the same time, filtering types of fluctuation in light that do not affect photosynthesis efficiency. When AtRGS1 is genetically ablated, photosynthesis efficiency is reduced in a changing- but not a constant-light environment. Mathematical modeling revealed that information about changes in the light environment is encoded in the amount of free AtRGS1 that becomes compartmentalized following stimulation. We propose that this property determines when to adjust photosynthetic efficiency in an environment where light intensity changes abruptly caused by moving shadows on top of a background of light changing gradually from sun rise to sun set and fluctuating light such as that caused by fluttering leaves.

Computational modeling of cytokine signaling in microglia

Neuroinflammation due to glial activation has been linked to many CNS diseases. We developed a computational model of a microglial cytokine interaction network to study the regulatory mechanisms of microglia-mediated neuroinflammation. We established a literature-based cytokine network, including TNFα, TGFβ, and IL-10, and fitted a mathematical model to published data from LPS-treated microglia. The addition of a previously unreported TGFβ autoregulation loop to our model was required to account for experimental data. Global sensitivity analysis revealed that TGFβ- and IL-10-mediated inhibition of TNFα was critical for regulating network behavior. We assessed the sensitivity of the LPS-induced TNFα response profile to the initial TGFβ and IL-10 levels. The analysis showed two relatively shifted TNFα response profiles within separate domains of initial condition space. Further analysis revealed that TNFα exhibited adaptation to sustained LPS stimulation. We simulated the effects of functionally inhibiting TGFβ and IL-10 on TNFα adaptation. Our analysis showed that TGFβ and IL-10 knockouts (TGFβ KO and IL-10 KO) exert divergent effects on adaptation. TFGβ KO attenuated TNFα adaptation whereas IL-10 KO enhanced TNFα adaptation. We experimentally tested the hypothesis that IL-10 KO enhances TNFα adaptation in murine macrophages and found supporting evidence. These opposing effects could be explained by differential kinetics of negative feedback. Inhibition of IL-10 reduced early negative feedback that results in enhanced TNFα-mediated TGFβ expression. We propose that differential kinetics in parallel negative feedback loops constitute a novel mechanism underlying the complex and non-intuitive pro- versus anti-inflammatory effects of individual cytokine perturbations.

Ultrasensitive Negative Feedback Control: A Natural Approach for the Design of Synthetic Controllers

Many of the most important potential applications of Synthetic Biology will require the ability to design and implement high performance feedback control systems that can accurately regulate the dynamics of multiple molecular species within the cell. Here, we argue that the use of design strategies based on combining ultrasensitive response dynamics with negative feedback represents a natural approach to this problem that fully exploits the strongly nonlinear nature of cellular information processing. We propose that such feedback mechanisms can explain the adaptive responses observed in one of the most widely studied biomolecular feedback systems—the yeast osmoregulatory response network. Based on our analysis of such system, we identify strong links with a well-known branch of mathematical systems theory from the field of Control Engineering, known as Sliding Mode Control. These insights allow us to develop design guidelines that can inform the construction of feedback controllers for synthetic biological systems.

Adaptation through proportion

Adaptation is a ubiquitous feature in biological sensory and signaling networks. It has been suggested that adaptive systems may follow certain simple design principles across diverse organisms, cells and pathways. One class of networks that can achieve adaptation utilizes an incoherent feedforward control, in which two parallel signaling branches exert opposite but proportional effects on the output at steady state. In this paper, we generalize this adaptation mechanism by establishing a steady-state proportionality relationship among a subset of nodes in a network. Adaptation can be achieved by using any two nodes in the sub-network to respectively regulate the output node positively and negatively. We focus on enzyme networks and first identify basic regulation motifs consisting of two and three nodes that can be used to build small networks with proportional relationships. Larger proportional networks can then be constructed modularly similar to LEGOs. Our method provides a general framework to construct and analyze a class of proportional and/or adaptation networks with arbitrary size, flexibility and versatile functional features.

Nonequilibrium Enhances Adaptation Efficiency of Stochastic Biochemical Systems

Adaptation is a crucial biological function possessed by many sensory systems. Early work has shown that some influential equilibrium models can achieve accurate adaptation. However, recent studies indicate that there are close relationships between adaptation and nonequilibrium. In this paper, we provide an explanation of these two seemingly contradictory results based on Markov models with relatively simple networks. We show that as the nonequilibrium driving becomes stronger, the system under consideration will undergo a phase transition along a fixed direction: from non-adaptation to simple adaptation then to oscillatory adaptation, while the transition in the opposite direction is forbidden. This indicates that although adaptation may be observed in equilibrium systems, it tends to occur in systems far away from equilibrium. In addition, we find that nonequilibrium will improve the performance of adaptation by enhancing the adaptation efficiency. All these results provide a deeper insight into the connection between adaptation and nonequilibrium. Finally, we use a more complicated network model of bacterial chemotaxis to validate the main results of this paper.

Enzyme Sequestration as a Tuning Point in Controlling Response Dynamics of Signalling Networks

Signalling networks result from combinatorial interactions among many enzymes and scaffolding proteins. These complex systems generate response dynamics that are often essential for correct decision-making in cells. Uncovering biochemical design principles that underpin such response dynamics is a prerequisite to understand evolved signalling networks and to design synthetic ones. Here, we use in silico evolution to explore the possible biochemical design space for signalling networks displaying ultrasensitive and adaptive response dynamics. By running evolutionary simulations mimicking different biochemical scenarios, we find that enzyme sequestration emerges as a key mechanism for enabling such dynamics. Inspired by these findings, and to test the role of sequestration, we design a generic, minimalist model of a signalling cycle, featuring two enzymes and a single scaffolding protein. We show that this simple system is capable of displaying both ultrasensitive and adaptive response dynamics. Furthermore, we find that tuning the concentration or kinetics of the sequestering protein can shift system dynamics between these two response types. These empirical results suggest that enzyme sequestration through scaffolding proteins is exploited by evolution to generate diverse response dynamics in signalling networks and could provide an engineering point in synthetic biology applications.

Finding Robust Adaptation Gene Regulatory Networks Using Multi-Objective Genetic Algorithm

Robust adaptation plays a key role in gene regulatory networks, and it is thought to be an important attribute for the organic or cells to survive in fluctuating conditions. In this paper, a simplified three-node enzyme network is modeled by the Michaelis-Menten rate equations for all possible topologies, and a family of topologies and the corresponding parameter sets of the network with satisfactory adaptation are obtained using the multi-objective genetic algorithm. The proposed approach improves the computation efficiency significantly as compared to the time consuming exhaustive searching method. This approach provides a systemic way for searching the feasible topologies and the corresponding parameter sets to make the gene regulatory networks have robust adaptation. The proposed methodology, owing to its universality and simplicity, can be used to address more complex issues in biological networks.

Spatio-temporal dynamics of a cell signal pathway with negative feedbacks: the MAPK/ERK pathway

We studied the spatio-temporal dynamics of a cell signal cascade with negative feedback that quantitatively emulates the regulative process that occurs in the Mitogen Activated Protein Kinase/Extracellular Regulated Kinase (MAPK/ERK) pathway. The model consists of a set of six coupled reaction-diffusion equations that describes the dynamics of the six-module pathway. In the basic module the active form of the protein transmits the signal to the next pathway’s module. As suggested by experiments, the model considers that the fifth module's kinase down-regulates the first and third modules. The feedback parameter is defined as, μ(r)( j)= k(kin)5/k(kin)(j), (j = 1, 3). We analysed the pathway's dynamics for μ(r)( j) = 0.10, 1.0, and 10 in the kinetic regimes: i) saturation of both kinases and phosphatases, ii) saturation of the phosphatases and iii) saturation of the kinases. For a regulated pathway the Total Activated Protein Profiles (TAPPs) as a function of time develop a maximum during the transient stage in the three kinetic regimes. These maxima become higher and their positions shift to longer times downstream. This scenario also applies to the TAPP's regulatory kinase that sums up its inhibitory action to that of the phosphatases leading to a maximum. Nevertheless, when μ(r)(j)= 1.0 , the TAPPs develop two maxima, with the second maximum being almost imperceptible. These results are in qualitative agreement with experimental data obtained from NIH 3T3 mouse fibroblasts. In addition, analyses of the stationary states as a function of position indicate that in the kinetic regime i) which is of physiological interest, signal transduction occurs with a relatively large propagation length for the three values of the regulative parameter. However, for μ(r)(j)= 0.10 , the sixth module concentration profile is transmitted with approximately 45% of its full value. The results obtained for μ(r)(j) = 10 , indicate that the first five concentration profiles are small with a short propagation length; nonetheless, the last concentration profile, c6, attains more than 90% of its full value with a relatively large propagation length as an indication of signal transduction. Signal transduction also occurred favourably in the kinetic regimes ii) and iii), but the signal was longer-ranged in the regime ii).

Aptamer-Enabled Manipulation of the Hsp70 Chaperone System Suggests a Novel Strategy for Targeted Ubiquitination

The Hsp70 chaperone system plays an important role in protein quality control by assisting in the folding and clearance of misfolded proteins. However, the mechanism by which it chooses between folding and degradation pathways is not fully understood. In this study, we used an RNA aptamer for Hsp70 to perturb the function of Hsp70 in cell-free systems. We found that the aptamer inhibited both Hsp70-mediated folding and Hsp70-CHIP-mediated ubiquitination/degradation of a misfolded protein substrate. Based on these results, we explored a novel strategy for targeted protein ubiquitination, using an engineered bifunctional aptamer to tether a protein substrate to Hsp70. We demonstrated that increased Hsp70-CHIP-mediated ubiquitination of the tethered protein substrate can be specifically induced by this bifunctional aptamer. This strategy may be useful in selective degradation of disease-causing proteins for therapeutic purposes. In addition, these studies provide insight into the mechanism of Hsp70-mediated protein triage.

BioJazz: in silico evolution of cellular networks with unbounded complexity using rule-based modeling

Systems biologists aim to decipher the structure and dynamics of signaling and regulatory networks underpinning cellular responses; synthetic biologists can use this insight to alter existing networks or engineer de novo ones. Both tasks will benefit from an understanding of which structural and dynamic features of networks can emerge from evolutionary processes, through which intermediary steps these arise, and whether they embody general design principles. As natural evolution at the level of network dynamics is difficult to study, in silico evolution of network models can provide important insights. However, current tools used for in silico evolution of network dynamics are limited to ad hoc computer simulations and models. Here we introduce BioJazz, an extendable, user-friendly tool for simulating the evolution of dynamic biochemical networks. Unlike previous tools for in silico evolution, BioJazz allows for the evolution of cellular networks with unbounded complexity by combining rule-based modeling with an encoding of networks that is akin to a genome. We show that BioJazz can be used to implement biologically realistic selective pressures and allows exploration of the space of network architectures and dynamics that implement prescribed physiological functions. BioJazz is provided as an open-source tool to facilitate its further development and use. Source code and user manuals are available at: http://oss-lab.github.io/biojazz and http://osslab.lifesci.warwick.ac.uk/BioJazz.aspx.

Ptch1 and Gli regulate Shh signalling dynamics via multiple mechanisms

In the vertebrate neural tube, the morphogen Sonic Hedgehog (Shh) establishes a characteristic pattern of gene expression. Here we quantify the Shh gradient in the developing mouse neural tube and show that while the amplitude of the gradient increases over time, the activity of the pathway transcriptional effectors, Gli proteins, initially increases but later decreases. Computational analysis of the pathway suggests three mechanisms that could contribute to this adaptation: transcriptional upregulation of the inhibitory receptor Ptch1, transcriptional downregulation of Gli and the differential stability of active and inactive Gli isoforms. Consistent with this, Gli2 protein expression is downregulated during neural tube patterning and adaptation continues when the pathway is stimulated downstream of Ptch1. Moreover, the Shh-induced upregulation of Gli2 transcription prevents Gli activity levels from adapting in a different cell type, NIH3T3 fibroblasts, despite the upregulation of Ptch1. Multiple mechanisms therefore contribute to the intracellular dynamics of Shh signalling, resulting in different signalling dynamics in different cell types.

Arrhythmogenic transient dynamics in cardiac myocytes

Cardiac action potential alternans and early afterdepolarizations (EADs) are linked to cardiac arrhythmias. Periodic action potentials (period 1) in healthy conditions bifurcate to other states such as period 2 or chaos when alternans or EADs occur in pathological conditions. The mechanisms of alternans and EADs have been extensively studied under steady-state conditions, but lethal arrhythmias often occur during the transition between steady states. Why arrhythmias tend to develop during the transition is unclear. We used low-dimensional mathematical models to analyze dynamical mechanisms of transient alternans and EADs. We show that depending on the route from one state to another, action potential alternans and EADs may occur during the transition between two periodic steady states. The route taken depends on the time course of external perturbations or intrinsic signaling, such as β-adrenergic stimulation, which regulate cardiac calcium and potassium currents with differential kinetics.

An allosteric model of the inositol trisphosphate receptor with nonequilibrium binding

The inositol trisphosphate receptor (IPR) is a crucial ion channel that regulates the Ca(2+) influx from the endoplasmic reticulum (ER) to the cytoplasm. A thorough study of the IPR channel contributes to a better understanding of calcium oscillations and waves. It has long been observed that the IPR channel is a typical biological system which performs adaptation. However, recent advances on the physical essence of adaptation show that adaptation systems with a negative feedback mechanism, such as the IPR channel, must break detailed balance and always operate out of equilibrium with energy dissipation. Almost all previous IPR models are equilibrium models assuming detailed balance and thus violate the dissipative nature of adaptation. In this article, we constructed a nonequilibrium allosteric model of single IPR channels based on the patch-clamp experimental data obtained from the IPR in the outer membranes of isolated nuclei of the Xenopus oocyte. It turns out that our model reproduces the patch-clamp experimental data reasonably well and produces both the correct steady-state and dynamic properties of the channel. Particularly, our model successfully describes the complicated bimodal [Ca(2+)] dependence of the mean open duration at high [IP3], a steady-state behavior which fails to be correctly described in previous IPR models. Finally, we used the patch-clamp experimental data to validate that the IPR channel indeed breaks detailed balance and thus is a nonequilibrium system which consumes energy.

Overshoot in biological systems modelled by Markov chains: a non-equilibrium dynamic phenomenon

A number of biological systems can be modelled by Markov chains. Recently, there has been an increasing concern about when biological systems modelled by Markov chains will perform a dynamic phenomenon called overshoot. In this study, the authors found that the steady-state behaviour of the system will have a great effect on the occurrence of overshoot. They showed that overshoot in general cannot occur in systems that will finally approach an equilibrium steady state. They further classified overshoot into two types, named as simple overshoot and oscillating overshoot. They showed that except for extreme cases, oscillating overshoot will occur if the system is far from equilibrium. All these results clearly show that overshoot is a non-equilibrium dynamic phenomenon with energy consumption. In addition, the main result in this study is validated with real experimental data.

Multiplexing oscillatory biochemical signals

In recent years it has been increasingly recognized that biochemical signals are not necessarily constant in time and that the temporal dynamics of a signal can be the information carrier. Moreover, it is now well established that the protein signaling network of living cells has a bow-tie structure and that components are often shared between different signaling pathways. Here we show by mathematical modeling that living cells can multiplex a constant and an oscillatory signal: they can transmit these two signals simultaneously through a common signaling pathway, and yet respond to them specifically and reliably. We find that information transmission is reduced not only by noise arising from the intrinsic stochasticity of biochemical reactions, but also by crosstalk between the different channels. Yet, under biologically relevant conditions more than 2 bits of information can be transmitted per channel, even when the two signals are transmitted simultaneously. These observations suggest that oscillatory signals are ideal for multiplexing signals.

A network characteristic that correlates environmental and genetic robustness

As scientific advances in perturbing biological systems and technological advances in data acquisition allow the large-scale quantitative analysis of biological function, the robustness of organisms to both transient environmental stresses and inter-generational genetic changes is a fundamental impediment to the identifiability of mathematical models of these functions. An approach to overcoming this impediment is to reduce the space of possible models to take into account both types of robustness. However, the relationship between the two is still controversial. This work uncovers a network characteristic, transient responsiveness, for a specific function that correlates environmental imperturbability and genetic robustness. We test this characteristic extensively for dynamic networks of ordinary differential equations ranging up to 30 interacting nodes and find that there is a power-law relating environmental imperturbability and genetic robustness that tends to linearity as the number of nodes increases. Using our methods, we refine the classification of known 3-node motifs in terms of their environmental and genetic robustness. We demonstrate our approach by applying it to the chemotaxis signaling network. In particular, we investigate plausible models for the role of CheV protein in biochemical adaptation via a phosphorylation pathway, testing modifications that could improve the robustness of the system to environmental and/or genetic perturbation.

Advances in the ways that living systems can be perturbed in order to study how they function and sharp reductions in the cost of computer resources have allowed the collection of large amounts of data. The aim of biological system modeling is to analyze this data in order to pin down the precise interactions of molecules that underlie the observed functions. This is made difficult due to two features of biological systems: (1) Living things do not show an appreciable loss of function across large ranges of environmental factors. (2) Their function is inherited from parent to child more or less unchanged in spite of random mutations in genetic sequences. We find that these two features are more correlated in a specific subset of networks and show how to use this observation to find networks in which these two features appear together. Working within this smaller space of networks may make it easier to find suitable underlying models from data.

Tunable signal processing through a kinase control cycle: the IKK signaling node

The transcription factor NFκB, a key component of the immune system, shows intricate stimulus-specific temporal dynamics. Those dynamics are thought to play a role in controlling the physiological response to cytokines and pathogens. Biochemical evidence suggests that the NFκB inducing kinase, IKK, a signaling hub onto which many signaling pathways converge, is regulated via a regulatory cycle comprising a poised, an active, and an inactive state. We hypothesize that it operates as a modulator of signal dynamics, actively reshaping the signals generated at the receptor proximal level. Here we show that a regulatory cycle can function in at least three dynamical regimes, tunable by regulating a single kinetic parameter. In particular, the simplest three-state regulatory cycle can generate signals with two well-defined phases, each with distinct coding capabilities in terms of the information they can carry about the stimulus. We also demonstrate that such a kinase cycle can function as a signal categorizer classifying diverse incoming signals into outputs with a limited set of temporal activity profiles. Finally, we discuss the extension of the results to other regulatory motifs that could be understood in terms of the regimes of the three-state cycle.

The cost of sensitive response and accurate adaptation in networks with an incoherent type-1 feed-forward loop

The incoherent type-1 feed-forward loop (I1-FFL) is ubiquitous in biological regulatory circuits. Although much is known about the functions of the I1-FFL motif, the energy cost incurred in the network and how it affects the performance of the network have not been investigated. Here, we study a generic I1-FFL enzymatic reaction network modelled after the GEF-GAP-Ras pathway responsible for chemosensory adaptation in eukaryotic cells. Our analysis shows that the I1-FFL network always operates out of equilibrium. Continuous energy dissipation is necessary to drive an internal phosphorylation-dephosphorylation cycle that is crucial in achieving strong short-time response and accurate long-time adaptation. In particular, we show quantitatively that the energy dissipated in the I1-FFL network is used (i) to increase the system's initial response to the input signals; (ii) to enhance the adaptation accuracy at steady state; and (iii) to expand the range of such accurate adaptation. Moreover, we find that the energy dissipation rate, the catalytic speed and the maximum adaptation accuracy in the I1-FFL network satisfy the same energy-speed-accuracy relationship as in the negative-feedback-loop (NFL) networks. Because the I1-FFL and NFL are the only two basic network motifs that enable accurate adaptation, our results suggest that a universal cost-performance trade-off principle may underlie all cellular adaptation processes independent of the detailed biochemical circuit architecture.

The Neurospora photoreceptor VIVID exerts negative and positive control on light sensing to achieve adaptation

Light adaptation in Neurospora is mediated by the photoreceptor VIVID, which exerts both a negative and positive effect on light sensing. These apparently paradoxical roles of VIVID are explained by the dynamics of a network motif that utilizes futile cycling.

The fungus Neurospora detects relative changes in light intensity by adapting to the ambient light level and remaining responsive to increases in light intensity.

Both the downregulation of the acute light response and maintained responsiveness are mediated by the photoreceptor VIVID (VVD).

Data-based mathematical modeling shows that this paradoxical function of VVD can be realized by a futile-cycle network motif that turns feedback inhibition into sensory adaptation.

The light response in Neurospora is mediated by the photoreceptor and circadian transcription factor White Collar Complex (WCC). The expression rate of the WCC target genes adapts in daylight and remains refractory to moonlight, despite the extraordinary light sensitivity of the WCC. To explain this photoadaptation, feedback inhibition by the WCC interaction partner VIVID (VVD) has been invoked. Here we show through data-driven mathematical modeling that VVD allows Neurospora to detect relative changes in light intensity. To achieve this behavior, VVD acts as an inhibitor of WCC-driven gene expression and, at the same time, as a positive regulator that maintains the responsiveness of the photosystem. Our data indicate that this paradoxical function is realized by a futile cycle that involves the light-induced sequestration of active WCC by VVD and the replenishment of the activatable WCC pool through the decay of the photoactivated state. Our quantitative study uncovers a novel network motif for achieving sensory adaptation and defines a core input module of the circadian clock in Neurospora.

Principles of adaptive sorting revealed by in silico evolution

Many biological networks have to filter out useful information from a vast excess of spurious interactions. In this Letter, we use computational evolution to predict design features of networks processing ligand categorization. The important problem of early immune response is considered as a case study. Rounds of evolution with different constraints uncover elaborations of the same network motif we name "adaptive sorting." Corresponding network substructures can be identified in current models of immune recognition. Our work draws a deep analogy between immune recognition and biochemical adaptation.

Common dynamical features of sensory adaptation in photoreceptors and olfactory sensory neurons

Sensory systems adapt, i.e., they adjust their sensitivity to external stimuli according to the ambient level. In this paper we show that single cell electrophysiological responses of vertebrate olfactory receptors and of photoreceptors to different input protocols exhibit several common features related to adaptation, and that these features can be used to investigate the dynamical structure of the feedback regulation responsible for the adaptation. In particular, we point out that two different forms of adaptation can be observed, in response to steps and to pairs of pulses. These two forms of adaptation appear to be in a dynamical trade-off: the more adaptation to a step is close to perfect, the slower is the recovery in adaptation to pulse pairs and viceversa. Neither of the two forms is explained by the dynamical models currently used to describe adaptation, such as the integral feedback model.

Optimal design of stimulus experiments for robust discrimination of biochemical reaction networks

MOTIVATION

Biochemical reaction networks in the form of coupled ordinary differential equations (ODEs) provide a powerful modeling tool for understanding the dynamics of biochemical processes. During the early phase of modeling, scientists have to deal with a large pool of competing nonlinear models. At this point, discrimination experiments can be designed and conducted to obtain optimal data for selecting the most plausible model. Since biological ODE models have widely distributed parameters due to, e.g. biologic variability or experimental variations, model responses become distributed. Therefore, a robust optimal experimental design (OED) for model discrimination can be used to discriminate models based on their response probability distribution functions (PDFs).

RESULTS

In this work, we present an optimal control-based methodology for designing optimal stimulus experiments aimed at robust model discrimination. For estimating the time-varying model response PDF, which results from the nonlinear propagation of the parameter PDF under the ODE dynamics, we suggest using the sigma-point approach. Using the model overlap (expected likelihood) as a robust discrimination criterion to measure dissimilarities between expected model response PDFs, we benchmark the proposed nonlinear design approach against linearization with respect to prediction accuracy and design quality for two nonlinear biological reaction networks. As shown, the sigma-point outperforms the linearization approach in the case of widely distributed parameter sets and/or existing multiple steady states. Since the sigma-point approach scales linearly with the number of model parameter, it can be applied to large systems for robust experimental planning.

AVAILABILITY

An implementation of the method in MATLAB/AMPL is available at http://www.uni-magdeburg.de/ivt/svt/person/rf/roed.html.

CONTACT

flassig@mpi-magdeburg.mpg.de

SUPPLEMENTARY INFORMATION

Supplementary data are are available at Bioinformatics online.

Adaptive temperature compensation in circadian oscillations

A temperature independent period and temperature entrainment are two defining features of circadian oscillators. A default model of distributed temperature compensation satisfies these basic facts yet is not easily reconciled with other properties of circadian clocks, such as many mutants with altered but temperature compensated periods. The default model also suggests that the shape of the circadian limit cycle and the associated phase response curves (PRC) will vary since the average concentrations of clock proteins change with temperature. We propose an alternative class of models where the twin properties of a fixed period and entrainment are structural and arise from an underlying adaptive system that buffers temperature changes. These models are distinguished by a PRC whose shape is temperature independent and orbits whose extrema are temperature independent. They are readily evolved by local, hill climbing, optimization of gene networks for a common quality measure of biological clocks, phase anticipation. Interestingly a standard realization of the Goodwin model for temperature compensation displays properties of adaptive rather than distributed temperature compensation.

An investigation of spatial signal transduction in cellular networks

Background

Spatial signal transduction plays a vital role in many intracellular processes such as eukaryotic chemotaxis, polarity generation and cell division. Furthermore it is being increasingly realized that the spatial dimension to signalling may play an important role in other apparently purely temporal signal transduction processes. It is increasingly being recognized that a conceptual basis for studying spatial signal transduction in signalling networks is necessary.

Results

In this work we examine spatial signal transduction in a series of standard motifs/networks. These networks include coherent and incoherent feedforward, positive and negative feedback, cyclic motifs, monostable switches, bistable switches and negative feedback oscillators. In all these cases, the driving signal has spatial variation. For each network we consider two cases, one where all elements are essentially non-diffusible, and the other where one of the network elements may be highly diffusible. A careful analysis of steady state signal transduction provides many insights into the behaviour of all these modules. While in the non-diffusible case for the most part, spatial signalling reflects the temporal signalling behaviour, in the diffusible cases, we see significant differences between spatial and temporal signalling characteristics. Our results demonstrate that the presence of diffusible elements in the networks provides important constraints and capabilities for signalling.

Conclusions

Our results provide a systematic basis for understanding spatial signalling in networks and the role of diffusible elements therein. This provides many insights into the signal transduction capabilities and constraints in such networks and suggests ways in which cellular signalling and information processing is organized to conform to or bypass those constraints. It also provides a framework for starting to understand the organization and regulation of spatial signal transduction in individual processes.

A dynamical feedback model for adaptation in the olfactory transduction pathway

Olfactory transduction exhibits two distinct types of adaptation, which we denote multipulse and step adaptation. In terms of measured transduction current, multipulse adaptation appears as a decrease in the amplitude of the second of two consecutive responses when the olfactory neuron is stimulated with two brief pulses. Step adaptation occurs in response to a sustained steplike stimulation and is characterized by a return to a steady-state current amplitude close to the prestimulus value, after a transient peak. In this article, we formulate a dynamical model of the olfactory transduction pathway, which includes the kinetics of the CNG channels, the concentration of Ca ions flowing through them, and the Ca-complexes responsible for the regulation. Based on this model, a common dynamical explanation for the two types of adaptation is suggested. We show that both forms of adaptation can be well described using different time constants for the kinetics of Ca ions (faster) and the kinetics of the feedback mechanisms (slower). The model is validated on experimental data collected in voltage-clamp conditions using different techniques and animal species.

Mathematical modeling reveals the functional implications of the different nuclear shuttling rates of Erk1 and Erk2

The mitogen-activated protein kinase (MAPK) family of proteins is involved in regulating cellular fates such as proliferation, differentiation and apoptosis. In particular, the dynamics of the Erk/Mek system, which has become the canonical example for MAPK signaling systems, have attracted considerable attention. Erk is encoded by two genes, Erk1 and Erk2, that until recently had been considered equivalent as they differ only subtly at the sequence level. However, these proteins exhibit radically different trafficking between cytoplasm and nucleus and this fact may have functional implications. Here we use spatially resolved data on Erk1/2 to develop and analyze spatio-temporal models of these cascades, and we discuss how sensitivity analysis can be used to discriminate between mechanisms. Our models elucidate some of the factors governing the interplay between signaling processes and the Erk1/2 localization in different cellular compartments, including competition between Erk1 and Erk2. Our approach is applicable to a wide range of signaling systems, such as activation cascades, where translocation of molecules occurs. Our study provides a first model of Erk1 and Erk2 activation and their nuclear shuttling dynamics, revealing a role in the regulation of the efficiency of nuclear signaling.

Phospholemman is a negative feed-forward regulator of Ca2+ in β-adrenergic signaling, accelerating β-adrenergic inotropy

Sympathetic stimulation enhances cardiac contractility by stimulating β-adrenergic signaling and protein kinase A (PKA). Recently, phospholemman (PLM) has emerged as an important PKA substrate capable of regulating cytosolic Ca(2+) transients. However, it remains unclear how PLM contributes to β-adrenergic inotropy. Here we developed a computational model to clarify PLM's role in the β-adrenergic signaling response. Simulating Na(+) and sarcoplasmic reticulum (SR) Ca(2+) clamps, we identify an effect of PLM phosphorylation on SR unloading as the key mechanism by which PLM confers cytosolic Ca(2+) adaptation to long-term β-adrenergic receptor (β-AR) stimulation. Moreover, we show that phospholamban (PLB) opposes and overtakes these actions on SR load, forming a negative feed-forward loop in the β-adrenergic signaling cascade. This network motif dominates the negative feedback conferred by β-AR desensitization and accelerates β-AR-induced inotropy. Model analysis therefore unmasks key actions of PLM phosphorylation during β-adrenergic signaling, indicating that PLM is a critical component of the fight-or-flight response.

Sensitivity control through attenuation of signal transfer efficiency by negative regulation of cellular signalling

Sensitivity is one of the hallmarks of biological and pharmacological responses. However, the principle of controlling sensitivity remains unclear. Here we theoretically analyse a simple biochemical reaction and find that the signal transfer efficiency of the transient peak amplitude attenuates depending on the strength of negative regulation. We experimentally find that many signalling pathways in various cell lines, including the Akt and ERK pathways, can be approximated by simple biochemical reactions and that the same property of the attenuation of signal transfer efficiency was observed for such pathways. Because of this property, a downstream molecule should show higher sensitivity to an activator and lower sensitivity to an inhibitor than an upstream molecule. Indeed, we experimentally verify that S6, which lies downstream of Akt, shows lower sensitivity to an epidermal growth factor receptor inhibitor than Akt. Thus, cells can control downstream sensitivity through the attenuation of signal transfer efficiency by changing the expression level of negative regulators.

A model for how signal duration can determine distinct outcomes of gene transcription programs

The reason why IL-6 induces a pro-inflammatory response, while IL-10 induces an anti-inflammatory response, despite both cytokines activating the same transcription factor, STAT3, is not well understood. It is known that IL-6 induces a transient STAT3 signal and that IL-10 induces a sustained STAT3 signal due to the STAT3-induced inhibitor SOCS3's ability to bind to the IL-6R and not the IL-10R. We sought to develop a general transcriptional network that is capable of translating sustained signals into one response, while translating transient signals into a second response. The general structure of such a network is that the transcription factor STAT3 can induce both an inflammatory response and an anti-inflammatory response by inducing two different genes. The anti-inflammatory gene can bind to and inhibit the inflammatory gene's production and the inflammatory gene can bind to its own promoter and induce its own transcription in the absence of the signal. One prediction that can be made from such a network is that in SOCS3-/- mice, where IL-6 induces a sustained STAT3 signal, that IL-6 would act as an anti-inflammatory cytokine, which has indeed been observed experimentally in the literature.

Excitable behavior can explain the "ping-pong" mode of communication between cells using the same chemoattractant

Here we elucidate a paradox: how a single chemoattractant-receptor system in two individuals is used for communication despite the seeming inevitability of self-excitation. In the filamentous fungus Neurospora crassa, genetically identical cells that produce the same chemoattractant fuse via the homing of individual cell protrusions toward each other. This is achieved via a recently described "ping-pong" pulsatile communication. Using a generic activator-inhibitor model of excitable behavior, we demonstrate that the pulse exchange can be fully understood in terms of two excitable systems locked into a stable oscillatory pattern of mutual excitation. The most puzzling properties of this communication are the sudden onset of oscillations with final amplitude, and the absence of seemingly inevitable self-excitation. We show that these properties result directly from both the excitability threshold and refractory period characteristic of excitable systems. Our model suggests possible molecular mechanisms for the ping-pong communication.

Incoherent feedforward control governs adaptation of activated ras in a eukaryotic chemotaxis pathway

Adaptation in signaling systems, during which the output returns to a fixed baseline after a change in the input, often involves negative feedback loops and plays a crucial role in eukaryotic chemotaxis. We determined the dynamical response to a uniform change in chemoattractant concentration of a eukaryotic chemotaxis pathway immediately downstream from G protein-coupled receptors. The response of an activated Ras showed near-perfect adaptation, leading us to attempt to fit the results using mathematical models for the two possible simple network topologies that can provide perfect adaptation. Only the incoherent feedforward network accurately described the experimental results. This analysis revealed that adaptation in this Ras pathway is achieved through the proportional activation of upstream components and not through negative feedback loops. Furthermore, these results are consistent with a local excitation, global inhibition mechanism for gradient sensing, possibly with a Ras guanosine triphosphatase-activating protein acting as a global inhibitor.

Predictive models for cellular signaling networks

This chapter provides an introduction to the formulation and analysis of differential-equation-based models for biological regulatory networks. In the first part, we discuss basic reaction types and the use of mass action kinetics and of simplifying approximations in the development of models for biological signaling. In the second part we introduce phase plane and linear stability analysis to evaluate the time evolution and identify the long-term attractors of dynamic systems. We then discuss the use of bifurcation diagrams to evaluate the parameter dependency of qualitative network behaviors (i.e., the emergence of oscillations or switches), and we give measures for the sensitivity and robustness of the signaling output.

A generic model for open signaling cascades with forward activation

In this work, cellular signal transduction in an open cascade with forward activation was studied. By proposing a generic model which captures the common features of major existing models in the literature, it is showed how signaling profile changes during the propagation along the cascade. In particular, a typical OFF-ON-OFF switch-like transient behavior with prolonged temporary ON state is revealed, where OFF and ON represent the states of low level and high level concentrations, respectively. Analytically this phenomenon is closely related to uniform convergence of the active protein concentration of downstream cycles in the finite time range and its failure in the entire time domain. Consequently a classification of open signaling cascade which can sustain OFF-ON-OFF behavior in the far downstream cycles is accessible. Relevant biological issues, such as delayed activation of downstream reaction cycles, signal amplification and prolonged signal duration, to the generic model is discussed.

Adaptation as a genome-wide autoregulatory principle in the stress response of yeast

The gene expression response of yeast to various types of stresses/perturbations shows a common functional and dynamical pattern for the vast majority of genes, characterised by a quick transient peak (affecting primarily short genes) followed by a return to the pre-stimulus level. Kinetically, this process of adaptation following the transient excursion can be modelled using a genome-wide autoregulatory mechanism by means of which yeast aims at maintaining a preferential concentration in its mRNA levels. The resulting feedback system explains well the different time constants observable in the transient response, while being in agreement with all the known experimental dynamical features. For example, it suggests that a very rapid transient can be induced also by a slowly varying concentration of the gene products.