Characterizing multistationarity regimes in biochemical reaction networks

Switch like responses appear as common strategies in the regulation of cellular systems. Here we present a method to characterize bistable regimes in biochemical reaction networks that can be of use to both direct and reverse engineering of biological switches. In the design of a synthetic biological switch, it is important to study the capability for bistability of the underlying biochemical network structure. Chemical Reaction Network Theory (CRNT) may help at this level to decide whether a given network has the capacity for multiple positive equilibria, based on their structural properties. However, in order to build a working switch, we also need to ensure that the bistability property is robust, by studying the conditions leading to the existence of two different steady states. In the reverse engineering of biological switches, knowledge collected about the bistable regimes of the underlying potential model structures can contribute at the model identification stage to a drastic reduction of the feasible region in the parameter space of search. In this work, we make use and extend previous results of the CRNT, aiming not only to discriminate whether a biochemical reaction network can exhibit multiple steady states, but also to determine the regions within the whole space of parameters capable of producing multistationarity. To that purpose we present and justify a condition on the parameters of biochemical networks for the appearance of multistationarity, and propose an efficient and reliable computational method to check its satisfaction through the parameter space.     

Multistationarity in mass action networks with applications to ERK activation

Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).     

Chemical Reaction Systems with Toric Steady States

Mass-action chemical reaction systems are frequently used in computational biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parameterized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to have toric steady states. Furthermore, we analyze the capacity of such a system to exhibit positive steady states and multistationarity. Examples of systems with toric steady states include weakly-reversible zero-deficiency chemical reaction systems. An important application of our work concerns the networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism.     

Model identification of signal transduction networks from data using a state regulator problem

Advances in molecular biology provide an opportunity to develop detailed models of biological processes that can be used to obtain an integrated understanding of the system. However, development of useful models from the available knowledge of the system and experimental observations still remains a daunting task. In this work, a model identification strategy for complex biological networks is proposed. The approach includes a state regulator problem (SRP) that provides estimates of all the component concentrations and the reaction rates of the network using the available measurements. The full set of the estimates is utilised for model parameter identification for the network of known topology. An a priori model complexity test that indicates the feasibility of performance of the proposed algorithm is developed. Fisher information matrix (FIM) theory is used to address model identifiability issues. Two signalling pathway case studies, the caspase function in apoptosis and the MAP kinase cascade system, are considered. The MAP kinase cascade, with measurements restricted to protein complex concentrations, fails the a priori test and the SRP estimates are poor as expected. The apoptosis network structure used in this work has moderate complexity and is suitable for application of the proposed tools. Using a measurement set of seven protein concentrations, accurate estimates for all unknowns are obtained. Furthermore, the effects of measurement sampling frequency and quality of information in the measurement set on the performance of the identified model are described.     

Multistationarity in Sequential Distributed Multisite Phosphorylation Networks

Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control, or nuclear signal integration. In this contribution, networks describing the phosphorylation and dephosphorylation of a protein at n sites in a sequential distributive mechanism are considered. Multistationarity (i.e., the existence of at least two positive steady state solutions of the associated polynomial dynamical system) has been analyzed and established in several contributions. It is, for example, known that there exist values for the rate constants where multistationarity occurs. However, nothing else is known about these rate constants. Here, we present a sign condition that is necessary and sufficient for multistationarity in n-site sequential, distributive phosphorylation. We express this sign condition in terms of linear systems, and show that solutions of these systems define rate constants where multistationarity is possible. We then present, for n≥2, a collection of feasible linear systems, and hence give a new and independent proof that multistationarity is possible for n≥2. Moreover, our results allow to explicitly obtain values for the rate constants where multistationarity is possible. Hence, we believe that, for the first time, a systematic exploration of the region in parameter space where multistationarity occurs has become possible. One consequence of our work is that, for any pair of steady states, the ratio of the steady state concentrations of kinase-substrate complexes equals that of phosphatase-substrate complexes.     

STDP in a bistable synapse model based on CaMKII and associated signaling pathways

The calcium/calmodulin-dependent protein kinase II (CaMKII) plays a key role in the induction of long-term postsynaptic modifications following calcium entry. Experiments suggest that these long-term synaptic changes are all-or-none switch-like events between discrete states. The biochemical network involving CaMKII and its regulating protein signaling cascade has been hypothesized to durably maintain the evoked synaptic state in the form of a bistable switch. However, it is still unclear whether experimental LTP/LTD protocols lead to corresponding transitions between the two states in realistic models of such a network. We present a detailed biochemical model of the CaMKII autophosphorylation and the protein signaling cascade governing the CaMKII dephosphorylation. As previously shown, two stable states of the CaMKII phosphorylation level exist at resting intracellular calcium concentration, and high calcium transients can switch the system from the weakly phosphorylated (DOWN) to the highly phosphorylated (UP) state of the CaMKII (similar to a LTP event). We show here that increased CaMKII dephosphorylation activity at intermediate Ca²⁺ concentrations can lead to switching from the UP to the DOWN state (similar to a LTD event). This can be achieved if protein phosphatase activity promoting CaMKII dephosphorylation activates at lower Ca²⁺ levels than kinase activity. Finally, it is shown that the CaMKII system can qualitatively reproduce results of plasticity outcomes in response to spike-timing dependent plasticity (STDP) and presynaptic stimulation protocols. This shows that the CaMKII protein network can account for both induction, through LTP/LTD-like transitions, and storage, due to its bistability, of synaptic changes.     

The fission yeast septation initiation network (SIN) kinase, Sid2, is required for SIN asymmetry and regulates the SIN scaffold, Cdc11

The Schizosaccharomyces pombe septation initiation network (SIN) is an Spg1-GTPase-mediated protein kinase cascade that triggers actomyosin ring constriction, septation, and cell division. The SIN is assembled at the spindle pole body (SPB) on the scaffold proteins Cdc11 and Sid4, with Cdc11 binding directly to SIN signaling components. Proficient SIN activity requires the asymmetric distribution of its signaling components to one of the two SPBs during anaphase, and Cdc11 hyperphosphorylation correlates with proficient SIN activity. In this paper, we show that the last protein kinase in the signaling cascade, Sid2, feeds back to phosphorylate Cdc11 during mitosis. The characterization of Cdc11 phosphomutants provides evidence that Sid2-mediated Cdc11 phosphorylation promotes the association of the SIN kinase, Cdc7, with the SPB and maximum SIN signaling during anaphase. We also show that Sid2 is crucial for the establishment of SIN asymmetry, indicating a positive-feedback loop is an important element of the SIN.     

MAPK cascade possesses decoupled controllability of signal amplification and duration

The three important characteristics of the output signal of mitogen activated protein kinase (MAPK) cascade are time delay between stimulus and response, amplitude gain, and duration of the output signal. In this study, we performed a sensitivity analysis on the computational model of epidermal growth factor receptor (EGFR) activated MAPK cascade developed by Schoeberl and co-workers (1) to identify the sensitive steps of the pathway affecting these characteristics. We show that the signaling network is sensitive in a decoupled manner, which provides the ability to control its output amplitude and duration one at a time. Signal duration is found sensitive only to the phosphatase reactions at the MEK level. In contrast, signal amplitude is found most sensitive to the phosphatase reactions at the ERK level. Time delay is found to be a robust characteristic of the system.     

MAP kinases function downstream of HSP90 and upstream of mitochondria in TMV resistance gene N-mediated hypersensitive cell death

Although the involvement of heat shock protein 90 (HSP90), mitogen-activated protein kinase (MAPK) cascades and organelle dysfunction in plant hypersensitive cell death has been suggested, the mutual relationship among them has not been elucidated. Here, we show the molecular network of HSP90, the wound-induced protein kinase (WIPK)/salicylic acid-induced protein kinase (SIPK)-mediated MAPK cascade and mitochondrial dysfunction in tobacco mosaic virus (TMV) resistance gene N-dependent cell death. p50, the Avr component for N, NtMEK2(DD), a constitutively active form of a MAPK kinase of WIPK/SIPK, and a mammalian pro-apoptotic factor Bax were used for cell death induction. Suppression of HSP90 and treatment with geldanamycin, a specific inhibitor of HSP90, compromised p50- but not NtMEK2(DD)- or Bax-mediated cell death accompanying the reduction of NtMEK2, WIPK and SIPK activation. In WIPK/SIPK-double knockdown plants, p50- and NtMEK2(DD)- but not Bax-mediated cell death was suppressed. All three types of cell death induced mitochondrial dysfunction, but they were similarly suppressed by Bcl-xL, which is a mammalian anti-apoptotic factor, and prevents mitochondrial dysfunction in plants as it does in animals in the cell death signal pathway. Taken together with the expression profile of hypersensitive reaction marker genes, it was indicated that the MAPK cascade functions downstream of HSP90 and transduces the cell death signal to mitochondria for N gene-dependent cell death. Furthermore, we found that WIPK and SIPK are functionally redundant in cell death signaling using WIPK/SIPK single or double knockdown plants.     

In transduced cells, the US3 protein kinase of herpes simplex virus 1 precludes activation and induction of apoptosis by transfected procaspase 3

The US3 protein kinase of herpes simplex virus 1 blocks apoptosis induced by replication-incompetent virus mutants, proapoptotic members of the Bcl-2 family of proteins, and by a variety of other agents that act at the premitochondrial level in the proapoptotic cascade. To define the role of US3 in blocking apoptosis at the postmitochondrial level, we investigated the US3 protein kinase in transduced cells that were either transfected with a plasmid encoding procaspase 3 or superinfected with a proapoptotic mutant virus lacking the gene encoding the infected cell protein no. 4. (i) We show that US3 blocks the proteolytic cleavage that generates active caspase 3 from the transfected zymogen procaspase 3, concomitant with inhibition of apoptosis. (ii) Studies based on detection of fluorescence emitted upon cleavage of a synthetic caspase 3 substrate showed that expression of the US3 kinase and appearance of the cleaved substrate were mutually exclusive. (iii) An affinity-purified glutathione S-transferase (GST)-US3 fusion protein, but not the inactive GST-US3(K220N) protein, phosphorylated procaspase 3 in vitro. The studies published earlier on the effect of US3 on the upstream regulatory proteins and current studies suggest that the US3 protein kinase may act on several proteins in the proapoptotic cascade to enable the virus to complete its replication.     

Enhancement of Tunability of MAPK Cascade Due to Coexistence of Processive and Distributive Phosphorylation Mechanisms

The processive phosphorylation mechanism becomes important when there is macromolecular crowding in the cytoplasm. Integrating the processive phosphorylation mechanism with the traditional distributive one, we propose a mixed dual-site phosphorylation (MDP) mechanism in a single-layer phosphorylation cycle. Further, we build a degree model by applying the MDP mechanism to a three-layer mitogen-activated protein kinase (MAPK) cascade. By bifurcation analysis, our study suggests that the crowded-environment-induced pseudoprocessive mechanism can qualitatively change the response of this biological network. By adjusting the degree of processivity in our model, we find that the MAPK cascade is able to switch between the ultrasensitivity, bistability, and oscillatory dynamical states. Sensitivity analysis shows that the theoretical results remain unchanged within a reasonably chosen variation of parameter perturbation. By scaling the reaction rates and also introducing new connections into the kinetic scheme, we further construct a proportion model of the MAPK cascade to validate our findings. Finally, it is illustrated that the spatial propagation of the activated MAPK signal can be improved (or attenuated) by increasing the degree of processivity of kinase (or phosphatase). Our research implies that the MDP mechanism makes the MAPK cascade become a flexible signal module, and the coexistence of processive and distributive phosphorylation mechanisms enhances the tunability of the MAPK cascade.     

Enhancement of Tunability of MAPK Cascade Due to Coexistence of Processive and Distributive Phosphorylation Mechanisms

The processive phosphorylation mechanism becomes important when there is macromolecular crowding in the cytoplasm. Integrating the processive phosphorylation mechanism with the traditional distributive one, we propose a mixed dual-site phosphorylation (MDP) mechanism in a single-layer phosphorylation cycle. Further, we build a degree model by applying the MDP mechanism to a three-layer mitogen-activated protein kinase (MAPK) cascade. By bifurcation analysis, our study suggests that the crowded-environment-induced pseudoprocessive mechanism can qualitatively change the response of this biological network. By adjusting the degree of processivity in our model, we find that the MAPK cascade is able to switch between the ultrasensitivity, bistability, and oscillatory dynamical states. Sensitivity analysis shows that the theoretical results remain unchanged within a reasonably chosen variation of parameter perturbation. By scaling the reaction rates and also introducing new connections into the kinetic scheme, we further construct a proportion model of the MAPK cascade to validate our findings. Finally, it is illustrated that the spatial propagation of the activated MAPK signal can be improved (or attenuated) by increasing the degree of processivity of kinase (or phosphatase). Our research implies that the MDP mechanism makes the MAPK cascade become a flexible signal module, and the coexistence of processive and distributive phosphorylation mechanisms enhances the tunability of the MAPK cascade.     

The modularity of biological regulatory networks

A useful approach to complex regulatory networks consists of modeling their elements and interactions by Boolean equations. In this context, feedback circuits (i.e. circular sequences of interactions) have been shown to play key dynamical roles: whereas positive circuits are able to generate multistationarity, negative circuits may generate oscillatory behavior. In this paper, we principally focus on the case of gene networks. These are represented by fully connected Boolean networks where each element interacts with all elements including itself. Flexibility in network design is introduced by the use of Boolean parameters, one associated with each interaction or group of interactions affecting a given element. Within this formalism, a feedback circuit will generate its typical dynamical behavior (i.e. multistationarity or oscillations) only for appropriate values of some of the logical parameters. Whenever it does, we say that the circuit is 'functional'. More interestingly, this formalism allows the computation of the constraints on the logical parameters to have any feedback circuit functional in a network. Using this methodology, we found that the fraction of the total number of consistent combinations of parameter values that make a circuit functional decreases geometrically with the circuit length. From a biological point of view, this suggests that regulatory networks could be decomposed into small and relatively independent feedback circuits or 'regulatory modules'.     

Emergence of bimodal cell population responses from the interplay between analog single-cell signaling and protein expression noise

ABSTRACT: BACKGROUND: Cell-to-cell variability in protein expression can be large, and its propagation through signaling networks affects biological outcomes. Here, we apply deterministic and probabilistic models and biochemical measurements to study how network topologies and cell-to-cell protein abundance variations interact to shape signaling responses. RESULTS: We observe bimodal distributions of extracellular signal-regulated kinase (ERK) responses to epidermal growth factor (EGF) stimulation, which are generally thought to indicate bistable or ultrasensitive signaling behavior in single cells. Surprisingly, we find that a simple MAPK/ERK-cascade model with negative feedback that displays graded, analog ERK responses at a single cell level can explain the experimentally observed bimodality at the cell population level. Model analysis suggests that a conversion of graded input--output responses in single cells to digital responses at the population level is caused by a broad distribution of ERK pathway activation thresholds brought about by cell-to-cell variability in protein expression. CONCLUSIONS: Our results show that bimodal signaling response distributions do not necessarily imply digital (ultrasensitive or bistable) single cell signaling, and the interplay between protein expression noise and network topologies can bring about digital population responses from analog single cell dose responses. Thus, cells can retain the benefits of robustness arising from negative feedback, while simultaneously generating population-level on/off responses that are thought to be critical for regulating cell fate decisions.     

Molecular cloning and mRNA expression analysis of a novel rice (Oryzasativa L.) MAPK kinase kinase,OsEDR1, an ortholog of Arabidopsis AtEDR1, reveal its role in defense/stress signalling pathways and development

Mitogen-activated protein kinase (MAPK) cascade(s) is important for plant defense/stress responses. Though MAPKs have been identified and characterized in rice (Oryza sativa L.), a monocot cereal crop research model, the first upstream component of the kinase cascade, namely MAPK kinase kinase (MAPKKK) has not yet been identified. Here we report the cloning of a novel rice gene encoding a MAPKKK, OsEDR1, designated based on its homology with the Arabidopsis MAPKKK, AtEDR1. OsEDR1, a single copy gene in the genome of rice, encodes a predicted protein with molecular mass of 113046.13 and a pI of 9.03. Using our established two-week-old rice seedling in vitro model system, we show that OsEDR1 has a constitutive expression in seedling leaves and is further up-regulated within 15 min upon wounding by cut, treatment with the global signals jasmonic acid (JA), salicylic acid (SA), ethylene (ethephon, ET), abscisic acid, and hydrogen peroxide. In addition, protein phosphatase inhibitors, fungal elicitor chitosan, drought, high salt and sugar, and heavy metals also dramatically induce its expression. Moreover, OsEDR1 expression was altered by co-application of JA, SA, and ET, and required de novo synthesized protein factor(s) in its transient regulation. Furthermore, using an in vivo system we also show that OsEDR1 responds to changes in temperature and environmental pollutants-ozone and sulfur dioxide. Finally, OsEDR1 expression varied significantly in vegetative and reproductive tissues. These results suggest a role for OsEDR1 in defense/stress signalling pathways and development.     

Optimal phosphorylation step number of intracellular signal-transduction pathway

Eukaryotic cells use signal-transduction network to respond in specific ways to external signals from their environment. Several signal-transduction pathway are composed of multi-step chemical reactions. We here theoretically study what determines the number of kinase phosphorylation steps composing of the intracellular signal-transduction cascade. We examine a simple mathematical model for the association and phosphorylation process of kinases in the signal-transduction cascade. We focus on the speed of signal transduction as the criterion for determining the optimal response. The present model first reveals that the initial expression level of kinase in each step of the cascade must be the same in the optimal response under the constraint of the constant total kinase concentration. The second conclusion is that the optimal step number of kinase cascade is primarily determined by the ratio of the target concentration of the final phosphorylated kinase in the cascade to that of input signal molecule, C/S. A multi-step cascade is optimum if the amplification of the final product concentration C relative to the input signal S is sufficiently large; whereas, a single-step reaction is optimum if C/S is small. This suggests that multi-step phosphorylation would have evolved to accelerate the speed of transduction of weak signals.     

Competing docking interactions can bring about bistability in the MAPK cascade

Mitogen-activated protein kinases are crucial regulators of various cell fate decisions including proliferation, differentiation, and apoptosis. Depending on the cellular context, the Raf-Mek-Erk mitogen-activated protein kinase cascade responds to extracellular stimuli in an all-or-none manner, most likely due to bistable behavior. Here, we describe a previously unrecognized positive-feedback mechanism that emerges from experimentally observed sequestration effects in the core Raf-Mek-Erk cascade. Unphosphorylated/monophosphorylated Erk sequesters Mek into Raf-inaccessible complexes upon weak stimulation, and thereby inhibits cascade activation. Mek, once phosphorylated by Raf, triggers Erk phosphorylation, which in turn induces dissociation of Raf-inaccessible Mek-Erk heterodimers, and thus further amplifies Mek phosphorylation. We show that this positive circuit can bring about bistability for parameter values measured experimentally in living cells. Previous studies revealed that bistability can also arise from enzyme depletion effects in the Erk double (de)phosphorylation cycle. We demonstrate that the feedback mechanism proposed in this article synergizes with such enzyme depletion effects to bring about a much larger bistable range than either mechanism alone. Our results show that stable docking interactions and competition effects, which are common in protein kinase cascades, can result in sequestration-based feedback, and thus can have profound effects on the qualitative behavior of signaling pathways.     

Spatiotemporal dynamics of the Calvin cycle: multistationarity and symmetry breaking instabilities

The possibility of controlling the Calvin cycle has paramount implications for increasing the production of biomass. Multistationarity, as a dynamical feature of systems, is the first obvious candidate whose control could find biotechnological applications. Here we set out to resolve the debate on the multistationarity of the Calvin cycle. Unlike the existing simulation-based studies, our approach is based on a sound mathematical framework, chemical reaction network theory and algebraic geometry, which results in provable results for the investigated model of the Calvin cycle in which we embed a hierarchy of realistic kinetic laws. Our theoretical findings demonstrate that there is a possibility for multistationarity resulting from two sources, homogeneous and inhomogeneous instabilities, which partially settle the debate on multistability of the Calvin cycle. In addition, our tractable analytical treatment of the bifurcation parameters can be employed in the design of validation experiments.