Examining Sodium and Potassium Channel Conductances Involved in Hyperexcitability of Chemotherapy-Induced Peripheral Neuropathy: A Mathematical and Cell Culture-Based Study.

Chemotherapy-induced peripheral neuropathy (CIPN) is a prevalent, painful side effect which arises due to a number of chemotherapy agents. CIPN can have a prolonged effect on quality of life. Chemotherapy treatment is often reduced or stopped altogether because of the severe pain. Currently, there are no FDA-approved treatments for CIPN partially due to its complex pathogenesis in multiple pathways involving a variety of channels, specifically, voltage-gated ion channels. One aspect of neuropathic pain in vitro is hyperexcitability in dorsal root ganglia (DRG) peripheral sensory neurons. Our study employs bifurcation theory to investigate the role of voltage-gated ion channels in inducing hyperexcitability as a consequence of spontaneous firing due to the common chemotherapy agent paclitaxel. Our mathematical investigation of a reductionist DRG neuron model comprised of sodium channel Nav1.7, sodium channel Nav1.8, delayed rectifier potassium channel, A-type transient potassium channel, and a leak channel suggests that Nav1.8 and delayed rectifier potassium channel conductances are critical for hyperexcitability of small DRG neurons. Introducing paclitaxel into the model, our bifurcation analysis predicts that hyperexcitability is highest for a medium dose of paclitaxel, which is supported by multi-electrode array (MEA) recordings. Furthermore, our findings using MEA reveal that Nav1.8 blocker A-803467 and delayed rectifier potassium enhancer L-alpha-phosphatidyl-D-myo-inositol 4,5-diphosphate, dioctanoyl (PIP2) can reduce paclitaxel-induced hyperexcitability of DRG neurons. Our approach can be readily extended and used to investigate various other contributors of hyperexcitability in CIPN.

Computational analysis of a 9D model for a small DRG neuron.

Small dorsal root ganglion (DRG) neurons are primary nociceptors which are responsible for sensing pain. Elucidation of their dynamics is essential for understanding and controlling pain. To this end, we present a numerical bifurcation analysis of a small DRG neuron model in this paper. The model is of Hodgkin-Huxley type and has 9 state variables. It consists of a Nav1.7 and a Nav1.8 sodium channel, a leak channel, a delayed rectifier potassium, and an A-type transient potassium channel. The dynamics of this model strongly depend on the maximal conductances of the voltage-gated ion channels and the external current, which can be adjusted experimentally. We show that the neuron dynamics are most sensitive to the Nav1.8 channel maximal conductance ([Formula: see text]). Numerical bifurcation analysis shows that depending on [Formula: see text] and the external current, different parameter regions can be identified with stable steady states, periodic firing of action potentials, mixed-mode oscillations (MMOs), and bistability between stable steady states and stable periodic firing of action potentials. We illustrate and discuss the transitions between these different regimes. We further analyze the behavior of MMOs. As the external current is decreased, we find that MMOs appear after a cyclic limit point. Within this region, bifurcation analysis shows a sequence of isolated periodic solution branches with one large action potential and a number of small amplitude peaks per period. For decreasing external current, the number of small amplitude peaks is increasing and the distance between the large amplitude action potentials is growing, finally tending to infinity and thereby leading to a stable steady state. A closer inspection reveals more complex concatenated MMOs in between these periodic MMO branches, forming Farey sequences. Lastly, we also find small solution windows with aperiodic oscillations which seem to be chaotic. The dynamical patterns found here-as consequences of bifurcation points regulated by different parameters-have potential translational significance as repetitive firing of action potentials imply pain of some form and intensity; manipulating these patterns by regulating the different parameters could aid in investigating pain dynamics.

Trade-off and flexibility in the dynamic regulation of the cullin-RING ubiquitin ligase repertoire.

Cullin-RING ubiquitin ligases (CRLs) catalyze the ubiquitylation of substrates many of which are degraded by the 26S proteasome. Their modular architecture enables recognition of numerous substrates via exchangeable substrate receptors that competitively bind to a cullin scaffold with high affinity. Due to the plasticity of these interactions there is ongoing uncertainty how cells maintain a flexible CRL repertoire in view of changing substrate loads. Based on a series of in vivo and in vitro studies, different groups proposed that the exchange of substrate receptors is mediated by a protein exchange factor named Cand1. Here, we have performed mathematical modeling to provide a quantitative underpinning of this hypothesis. First we show that the exchange activity of Cand1 necessarily leads to a trade-off between high ligase activity and fast receptor exchange. Supported by measurements we argue that this trade-off yields an optimal Cand1 concentration in cells where the time scale for substrate degradation becomes minimal. In a second step we show through simulations that (i) substrates bias the CRL repertoire leading to preferential assembly of ligases for which substrates are available and (ii) differences in binding affinities or substrate receptor abundances create a temporal hierarchy for the degradation of substrates. Finally, we compare the Cand1-mediated exchange cycle with an alternative architecture lacking Cand1 which indicates superiority of a system with exchange factor if substrate receptors bind substrates and the cullin scaffold in a random order. Together, our results provide general constraints for the operating regimes of molecular exchange systems and suggest that Cand1 endows the CRL network with the properties of an "on demand" system allowing cells to dynamically adjust their CRL repertoire to fluctuating substrate abundances.

N-Site phosphorylation systems with 2n-1 steady states.

Multisite protein phosphorylation plays a prominent role in intracellular processes like signal transduction, cell-cycle control and nuclear signal integration. Many proteins are phosphorylated in a sequential and distributive way at more than one phosphorylation site. Mathematical models of n-site sequential distributive phosphorylation are therefore studied frequently. In particular, in Wang and Sontag (J Math Biol 57:29­52, 2008), it is shown that models of n-site sequential distributive phosphorylation admit at most 2n - 1 steady states.Wang and Sontag furthermore conjecture that for odd n, there are at most n and that, for even n, there are at most n + 1 steady states. This, however, is not true: building on earlier work in Holstein et al. (BullMath Biol 75(11):2028­2058, 2013), we present a scalar determining equation for multistationarity which will lead to parameter values where a 3-site system has 5 steady states and parameter values where a 4-site system has 7 steady states. Our results therefore are counterexamples to the conjecture of Wang and Sontag.We furthermore study the inherent geometric properties of multistationarity in n-site sequential distributive phosphorylation: the complete vector of steady state ratios is determined by the steady state ratios of free enzymes and unphosphorylated protein and there exists a linear relationship between steady state ratios of phosphorylated protein.

Nested autoinhibitory feedbacks alter the resistance of homeostatic adaptive biochemical networks.

Negative feedback control is a ubiquitous feature of biochemical systems, as is time delay between a signal and its response. Negative feedback in conjunction with time delay can lead to oscillations. In a cellular context, it might be beneficial to mitigate oscillatory behaviour to avoid recurring stress situations. This can be achieved by increasing the distance between the parameters of the system and certain thresholds, beyond which oscillations occur. This distance has been termed resistance. Here, we prove that in a generic three-dimensional negative feedback system the resistance of the system is modified by nested autoinhibitory feedbacks. Our system features negative feedbacks through both input-inhibition as well as output-activation, a signalling component with mass conservation and perfect adaptation. We show that these features render the system applicable to biological data, exemplified by the high osmolarity glycerol system in yeast and the mammalian p53 system. Output-activation is better supported by data than input-inhibition and also shows distinguished properties with respect to the system's stimulus. Our general approach might be useful in designing synthetic systems in which oscillations can be tuned by synthetic autoinhibitory feedbacks.

Model-based derivation, analysis and control of unstable microaerobic steady-states--considering Rhodospirillum rubrum as an example.

Microaerobic (oxygen-limited) conditions are critical for inducing many important microbial processes in industrial or environmental applications. At very low oxygen concentrations, however, the process performance often suffers from technical limitations. Available dissolved oxygen measurement techniques are not sensitive enough and thus control techniques, that can reliable handle these conditions, are lacking. Recently, we proposed a microaerobic process control strategy, which overcomes these restrictions and allows to assess different degrees of oxygen limitation in bioreactor batch cultivations. Here, we focus on the design of a control strategy for the automation of oxygen-limited continuous cultures using the microaerobic formation of photosynthetic membranes (PM) in Rhodospirillum rubrum as model phenomenon. We draw upon R. rubrum since the considered phenomenon depends on the optimal availability of mixed-carbon sources, hence on boundary conditions which make the process performance challenging. Empirically assessing these specific microaerobic conditions is scarcely practicable as such a process reacts highly sensitive to changes in the substrate composition and the oxygen availability in the culture broth. Therefore, we propose a model-based process control strategy which allows to stabilize steady-states of cultures grown under these conditions. As designing the appropriate strategy requires a detailed knowledge of the system behavior, we begin by deriving and validating an unstructured process model. This model is used to optimize the experimental conditions, and identify properties of the system which are critical for process performance. The derived model facilitates the good process performance via the proposed optimal control strategy. In summary the presented model-based control strategy allows to access and maintain microaerobic steady-states of interest and to precisely and efficiently transfer the culture from one stable microaerobic steady-state into another. Therefore, the presented approach is a valuable tool to study regulatory mechanisms of microaerobic phenomena in response to oxygen limitation alone. Biotechnol. Bioeng. 2014;111: 734-747. © 2013 Wiley Periodicals, Inc.

Continuous influenza virus production in cell culture shows a periodic accumulation of defective interfering particles.

Influenza viruses are a major public health burden during seasonal epidemics and a continuous threat due to their potential to cause pandemics. Annual vaccination provides the best protection against the contagious respiratory illness caused by influenza viruses. However, the current production capacities for influenza vaccines are insufficient to meet the increasing demands. We explored the possibility to establish a continuous production process for influenza viruses using the duck-derived suspension cell line AGE1.CR. A two-stage bioreactor setup was designed in which cells were cultivated in a first stirred tank reactor where an almost constant cell concentration was maintained. Cells were then constantly fed to a second bioreactor where virus infection and replication took place. Using this two-stage reactor system, it was possible to continuously produce influenza viruses. Surprisingly, virus titers showed a periodic increase and decrease during the run-time of 17 days. These titer fluctuations were caused by the presence of defective interfering particles (DIPs), which we detected by PCR. Mathematical modeling confirmed this observation showing that constant virus titers can only emerge in the absence of DIPs. Even with very low amounts of DIPs in the seed virus and very low rates for de novo DIP generation, defective viruses rapidly accumulate and, therefore, represent a serious challenge for continuous vaccine production. Yet, the continuous replication of influenza virus using a two-stage bioreactor setup is a novel tool to study aspects of viral evolution and the impact of DIPs.

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.

Structural analysis of a core model for carbohydrate uptake in Escherichia coli.

The analysis of biochemical reaction schemes is often limited by uncertainties in the kinetic parameters. Here, we apply Metabolic Control Analysis to explore structurally determined properties of a feed-forward loop in carbohydrate uptake and glycolysis in Escherichia coli. This part of the metabolic network is a starting point for a sensory system and therefore an important element for the understanding of the coordination between metabolism and signal transduction. By analytical calculations for various model structures of increasing complexity, we explore for which kinetic reaction rates and in which parameter ranges the resulting core models offer the decaying degree of phosphorylation that is observed in experiments. In particular, the consequences of allowing reversibility will be described and the role of the feed-forward enzymatic activation will be discussed. Finally, it is shown that the proposed model can be used as a generic basis for an extended and more realistic version of the network.

An extended model for the repression of photosynthesis genes by the AppA/PpsR system in Rhodobacter sphaeroides.

Purple bacteria derive energy from aerobic respiration or photosynthesis depending on the availability of oxygen and light. Under aerobic conditions, photosynthesis genes are specifically repressed by the PpsR protein. In Rhodobacter sphaeroides, the repressive action of PpsR is antagonized by the blue-light and redox-sensitive flavoprotein AppA, which sequesters PpsR under anaerobic conditions into transcriptionally inactive complexes. However, under semi-aerobic conditions, blue-light excitation of AppA causes the AppA-PpsR complexes to dissociate, again leading to a repression of photosynthesis genes. We have recently developed a simple mathematical model suggesting that this phenotype arises from the formation of a maximum in the response curve of reduced PpsR at intermediate oxygen concentrations. However, this model focused mainly on the oxygen-dependent interactions whereas light regulation was only implemented in a simplified manner. In the present study, we incorporate a more detailed mechanism for the light-dependent interaction between AppA and PpsR, which now allows for a direct comparison with experiments. Specifically, we take into account that, upon blue-light excitation, AppA undergoes a conformational change, creating a long-lived signalling state causing the dissociation of the AppA-PpsR complexes. The predictions of the extended model are found to be in good agreement with experimental results on the light-dependent repression of photosynthesis genes under semi-aerobic conditions. We also identify the potential kinetic and stoichiometric constraints that the interplay between light and redox regulation imposes on the functionality of the AppA/PpsR system, especially with respect to a possible bistable response.

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).

Modeling the light- and redox-dependent interaction of PpsR/AppA in Rhodobacter sphaeroides.

Facultative photosynthetic bacteria switch their energy generation mechanism from respiration to photosynthesis depending on oxygen tension and light. Part of this transition is mediated by the aerobic transcriptional repressor PpsR. In Rhodobacter sphaeroides, the repressive action of PpsR is antagonized by the redox- and blue-light-sensitive flavoprotein AppA which results in a unique phenotype: the repression of photosynthesis genes at intermediate oxygen levels and high light intensity, which is believed to reduce the risk of photooxidative stress. To analyze the underlying mechanism we developed a simple mathematical model based on the AppA-dependent reduction of a disulfide bond in PpsR and the light-sensitive complex formation between the reduced forms of AppA and PpsR. A steady-state analysis shows that high light repression can indeed occur at intermediate oxygen levels if PpsR is reduced on a faster timescale than AppA and if the electron transfer from AppA to PpsR is effectively irreversible. The model further predicts that if AppA copy numbers exceed those of PpsR by at least a factor of two, the transition from aerobic to anaerobic growth mode can occur via a bistable regime. We provide necessary conditions for the emergence of bistability and discuss possible experimental verifications.

A thermodynamically consistent explicit competitive adsorption isotherm model based on second-order single component behaviour.

A competitive adsorption isotherm model is derived for binary mixtures of components characterized by single component isotherms which are second-order truncations of higher order equilibrium models suggested by multi-layer theory and statistical thermodynamics. The competitive isotherms are determined using the ideal adsorbed solution (IAS) theory which, in case of complex single component isotherms, does not generate explicit expressions to calculated equilibrium loadings and causes time consuming iterations in simulations of adsorption processes. The explicit model derived in this work is based on an analysis of the roots of a cubic polynomial resulting from the set of IAS equations. The suggested thermodynamically consistent and widely applicable competitive isotherm model can be recommended as a flexible tool for efficient simulations of fixed-bed adsorber dynamics.

Size-exclusion chromatography as a linear transfer system: purification of human influenza virus as an example.

Preparative size-exclusion chromatography suffers from low selectivity and productivity. Empirical optimization of operating conditions constitutes a laborious task due to many parameters. Here, a modeling framework based on linear systems theory is presented for predicting the influence of volume overloading. Impulse-responses characterizing system behavior are derived from experimental data by maximum entropy deconvolution. Theoretical derivations are validated experimentally by study of a model system and chromatography of human influenza virus. By application of the theory it is demonstrated how group separation operations can be optimized with respect to yield, purity, productivity and dilution of the product.

Multistationarity in the activation of a MAPK: parametrizing the relevant region in parameter space.

Mathematical models of biochemical reaction networks in the form of ordinary differential equations can exhibit all sorts of complex dynamical behaviour. It is for example known, that even a single layer of a MAPK cascade can exhibit bistability (i.e. there exist multiple (positive) steady state solutions). It is almost a common-place that bistability or some other form of multistationarity are observed in many biochemical reaction networks, especially if the focus is on signal transduction or cell cycle regulation. However, multistationarity is only exhibited if the parameter vector is located in an appropriate region of parameter space. To find these regions, for example by using numerical tools like bifurcation analysis, is a non-trivial task as it amounts to searching the whole parameter space. In this paper we show that for a model of a single layer of a MAPK cascade it is possible to derive analytical descriptions of these regions, if mass action kinetics are used. Moreover, our results give rise to a straightforward explanation for the 'robust yet fragile' behaviour in the activation of a MAPK.

Subnetwork analysis reveals dynamic features of complex (bio)chemical networks.

In analyzing and mathematical modeling of complex (bio)chemical reaction networks, formal methods that connect network structure and dynamic behavior are needed because often, quantitative knowledge of the networks is very limited. This applies to many important processes in cell biology. Chemical reaction network theory allows for the classification of the potential network behavior-for instance, with respect to the existence of multiple steady states-but is computationally limited to small systems. Here, we show that by analyzing subnetworks termed elementary flux modes, the applicability of the theory can be extended to more complex networks. For an example network inspired by cell cycle control in budding yeast, the approach allows for model discrimination, identification of key mechanisms for multistationarity, and robustness analysis. The presented methods will be helpful in modeling and analyzing other complex reaction networks.

Coexistence in the chemostat as a result of metabolic by-products.

Classical chemostat models assume that competition is purely exploitative and mediated via a common, limiting and single resource. However, in laboratory experiments with pathogens related to the genetic disease Cystic Fibrosis, species specific properties of production, inhibition and consumption of a metabolic by-product, acetate, were found. These assumptions were implemented into a mathematical chemostat model which consists of four nonlinear ordinary differential equations describing two species competing for one limiting nutrient in an open system. We derive classical chemostat results and find that our basic model supports the competitive exclusion principle, the bistability of the system as well as stable coexistence. The analytical results are illustrated by numerical simulations performed with experimentally measured parameter values. As a variant of our basic model, mimicking testing of antibiotics for therapeutic treatments in mixed cultures instead of pure ones, we consider the introduction of a lethal inhibitor, which cannot be eliminated by one of the species and is selective for the stronger competitor. We discuss our theoretical results in relation to our experimental model system and find that simulations coincide with the qualitative behavior of the experimental result in the case where the metabolic by-product serves as a second carbon source for one of the species, but not the producer.

Comment on "Identification of low order manifolds: validating the algorithm of Maas and Pope" [Chaos 9, 108-123 (1999)].

It is claimed by Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] that the projection algorithm of Maas and Pope [Combust. Flame 88, 239-264 (1992)] identifies the slow invariant manifold of a system of ordinary differential equations with time-scale separation. A transformation to Fenichel normal form serves as a tool to prove this statement. Furthermore, Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] conjectured that away from a slow manifold, the criterion of Maas and Pope will never be fulfilled. We present two examples that refute the assertions of Rhodes, Morari, and Wiggins. In the first example, the algorithm of Maas and Pope leads to a manifold that is not invariant but close to a slow invariant manifold. The claim of Rhodes, Morari, and Wiggins that the Maas and Pope projection algorithm is invariant under a coordinate transformation to Fenichel normal form is shown to be not correct in this case. In the second example, the projection algorithm of Maas and Pope leads to a manifold that lies in a region where no slow manifold exists at all. This rejects the conjecture of Rhodes, Morari, and Wiggins mentioned above.

Mathematical model of influenza A virus production in large-scale microcarrier culture.

A mathematical model that describes the replication of influenza A virus in animal cells in large-scale microcarrier culture is presented. The virus is produced in a two-step process, which begins with the growth of adherent Madin-Darby canine kidney (MDCK) cells. After several washing steps serum-free virus maintenance medium is added, and the cells are infected with equine influenza virus (A/Equi 2 (H3N8), Newmarket 1/93). A time-delayed model is considered that has three state variables: the number of uninfected cells, infected cells, and free virus particles. It is assumed that uninfected cells adsorb the virus added at the time of infection. The infection rate is proportional to the number of uninfected cells and free virions. Depending on multiplicity of infection (MOI), not necessarily all cells are infected by this first step leading to the production of free virions. Newly produced viruses can infect the remaining uninfected cells in a chain reaction. To follow the time course of virus replication, infected cells were stained with fluorescent antibodies. Quantitation of influenza viruses by a hemagglutination assay (HA) enabled the estimation of the total number of new virions produced, which is relevant for the production of inactivated influenza vaccines. It takes about 4-6 h before visibly infected cells can be identified on the microcarriers followed by a strong increase in HA titers after 15-16 h in the medium. Maximum virus yield Vmax was about 1x10(10) virions/mL (2.4 log HA units/100 microL), which corresponds to a burst size ratio of about 18,755 virus particles produced per cell. The model tracks the time course of uninfected and infected cells as well as virus production. It suggests that small variations (<10%) in initial values and specific rates do not have a significant influence on Vmax. The main parameters relevant for the optimization of virus antigen yields are specific virus replication rate and specific cell death rate due to infection. Simulation studies indicate that a mathematical model that neglects the delay between virus infection and the release of new virions gives similar results with respect to overall virus dynamics compared with a time delayed model.

Reduction of chemical reaction networks using quasi-integrals.

We present a numerical method to identify possible candidates for quasi-stationary manifolds in complex reaction networks governed by systems of ordinary differential equations. Inspired by singular perturbation theory, we examine the ratios of certain components of the reaction rate vector. Those ratios that rapidly approach a nearly constant value define a slow manifold for the original flow in terms of quasi-integrals, that is, functions that are nearly constant along the trajectories. The dimensionality of the original system is thus effectively reduced without reliance on a priori knowledge of the different time scales in the system. We also demonstrate the relation of our approach to singular perturbation theory which, in its simplest form, is just the well-known quasi-steady-state approximation. In two case studies, we apply our method to oscillatory chemical systems: the 6-dimensional hemin-hydrogen peroxide-sulfite pH oscillator and a 10-dimensional mechanistic model for the peroxidase-oxidase (PO) reaction system. We conjecture that the presented method is especially suited for a straightforward reduction of higher dimensional dynamical systems where analytical methods fail to identify the different time scales associated with the slow invariant manifolds present in the system.