Derivation of a Quantitative Kinetic Model for a Visual Pigment from Observations of Early Receptor Potential

A “complete” and quantitative kinetic model for the states and transitions of the barnacle visual pigment in situ has been constructed from intracellular recordings of the early receptor potential responses to long light pulses. The model involves two stable and four thermolabile states and 10 photochemical, thermal, and metabolic transitions among them. The existence of each state and transition is demonstrated by qualitative examination of the response resulting from a carefully chosen experimental paradigm (combination of intensity, duration, and wavelength of adaptation and stimulation). Quantitative examination of the same responses determines all of the model transition rates, but only puts constraints on the state dipole moments. The latter are determined, and the former refined, by quantitative comparison of the predictions of the complete model with the responses to a set of paradigms chosen to involve as many states and transitions as possible. The fact that good fits can be obtained to these responses without further modification of the model supports its completeness.     

Candidate states of Helicobacter pylori's genome-scale metabolic network upon application of "loop law" thermodynamic constraints

Constraint-based modeling has proven to be a useful tool in the analysis of biochemical networks. To date, most studies in this field have focused on the use of linear constraints, resulting from mass balance and capacity constraints, which lead to the definition of convex solution spaces. One additional constraint arising out of thermodynamics is known as the "loop law" for reaction fluxes, which states that the net flux around a closed biochemical loop must be zero because no net thermodynamic driving force exists. The imposition of the loop-law can lead to nonconvex solution spaces making the analysis of the consequences of its imposition challenging. A four-step approach is developed here to apply the loop-law to study metabolic network properties: 1), determine linear equality constraints that are necessary (but not necessarily sufficient) for thermodynamic feasibility; 2), tighten V(max) and V(min) constraints to enclose the remaining nonconvex space; 3), uniformly sample the convex space that encloses the nonconvex space using standard Monte Carlo techniques; and 4), eliminate from the resulting set all solutions that violate the loop-law, leaving a subset of steady-state solutions. This subset of solutions represents a uniform random sample of the space that is defined by the additional imposition of the loop-law. This approach is used to evaluate the effect of imposing the loop-law on predicted candidate states of the genome-scale metabolic network of Helicobacter pylori.     

Oscillations in an epidemiological model based on asynchronous probabilistic cellular automaton

Consider a contagious disease affecting a host population composed of two groups with distinct habits. At each time step, each individual of this population can be in one of two states: susceptible (S) or infective (I). Here, a SIS epidemic model based on cellular automaton (CA) is proposed to study the disease spreading in such a population. In this model, the state transitions are described by probabilistic rules and each group has its own schedule to update the states of its individuals. We also propose a set of difference equations (DE) to analyze this population dynamics and we show how these two approaches (CA and DE) can be equivalent. We noticed that oscillations can be found in the composition of the group with more active social life, but not in the composition of the other group.     

On the admittance of membranes associated with channel conduction. Application to channels at non-equilibrium steady state

The small-signal admittance of membranes associated with channel conduction is derived for a general channel model. A general channel model is represented by a set of chemical reactions with each species of the reactions representing a channel state. The membrane admittance is shown to be related to the phenomenological relaxation matrix of the reactions. If the kinetic reactions are at a non-equilibrium steady state, the relaxation matrix may have complex eigenvalues and the equivalent circuit of the membrane admittance may contain RLC or RLC-like branches. For equilibrium kinetic systems, on the other hand, the equivalent circuit contains only RL or RC branches. Thus, the membrane admittance of equilibrium channels is quite different from that of non-equilibrium channels. In particular, we show that the low frequency feature in the admittance of squid axons as observed by Fishman, Poussart, Moore &; Siebenga (1977) can be obtained easily from a non-equilibrium cycling steady-state model.     

Mitoenergetic Dysfunction Triggers a Rapid Compensatory Increase in Steady-State Glucose Flux

ATP can be produced in the cytosol by glycolytic conversion of glucose (GLC) into pyruvate. The latter can be metabolized into lactate, which is released by the cell, or taken up by mitochondria to fuel ATP production by the tricarboxylic acid cycle and oxidative phosphorylation (OXPHOS) system. Altering the balance between glycolytic and mitochondrial ATP generation is crucial for cell survival during mitoenergetic dysfunction, which is observed in a large variety of human disorders including cancer. To gain insight into the kinetic properties of this adaptive mechanism we determined here how acute (30 min) inhibition of OXPHOS affected cytosolic GLC homeostasis. GLC dynamics were analyzed in single living C2C12 myoblasts expressing the fluorescent biosensor FLII¹²Pglu-700μδ6 (FLII). Following in situ FLII calibration, the kinetic properties of GLC uptake (V₁) and GLC consumption (V₂) were determined independently and used to construct a minimal mathematical model of cytosolic GLC dynamics. After validating the model, it was applied to quantitatively predict V₁ and V₂ at steady-state (i.e., when V₁ = V₂ = V_steady-state) in the absence and presence of OXPHOS inhibitors. Integrating model predictions with experimental data on lactate production, cell volume, and O₂ consumption revealed that glycolysis and mitochondria equally contribute to cellular ATP production in control myoblasts. Inhibition of OXPHOS induced a twofold increase in V_steady-state and glycolytic ATP production flux. Both in the absence and presence of OXPHOS inhibitors, GLC was consumed at near maximal rates, meaning that GLC consumption is rate-limiting under steady-state conditions. Taken together, we demonstrate here that OXPHOS inhibition increases steady-state GLC uptake and consumption in C2C12 myoblasts. This activation fully compensates for the reduction in mitochondrial ATP production, thereby maintaining the balance between cellular ATP supply and demand.     

Describing behavioral states using a system model of the primate brain.

A system model of the primate neocortex is presented, based mainly on the neuroanatomy of the rhesus macaque monkey and consisting of a set of processing modules arranged as a perception-action hierarchy. These modules correspond to regions of the neocortex and their connectivity to that of the neocortex. A computational approach based on predicate logic is explained, and the results of a computer implementation of the model are reported, which demonstrate social behaviors involving affiliation and social conflict. The behavioral states of primates involved in these behaviors can be represented by the states of the system model, which have a logical representation and a diagrammatic form. It is shown how the behavioral states in goal-directed behaviors can be represented and also their short term moment-to-moment development in time. It is then shown how the state of social interaction among two or more primates can be represented, using their individual behavioral states, with interindividual action and perception. The causal dynamics of behavioral states is explained and also a control mechanism, namely, the use of confirmation signals, which stabilizes behavioral states and their dynamics. Stabilized behavioral states are seen as corresponding to coherent activations of the system, resulting from successful selection of module activations and intermodule communication with confirmation.     

Global Kinetic Analysis of Mammalian E3 Reveals pH-dependent NAD+/NADH Regulation,Physiological Kinetic Reversibility,and Catalytic Optimum

Mammalian E3 is an essential mitochondrial enzyme responsible for catalyzing the terminal reaction in the oxidative catabolism of several metabolites. E3 is a key regulator of metabolic fuel selection as a component of the pyruvate dehydrogenase complex (PDHc). E3 regulates PDHc activity by altering the affinity of pyruvate dehydrogenase kinase, an inhibitor of the enzyme complex, through changes in reduction and acetylation state of lipoamide moieties set by the NAD⁺/NADH ratio. Thus, an accurate kinetic model of E3 is needed to predict overall mammalian PDHc activity. Here, we have combined numerous literature data sets and new equilibrium spectroscopic experiments with a multitude of independently collected forward and reverse steady-state kinetic assays using pig heart E3. The latter kinetic assays demonstrate a pH-dependent transition of NAD⁺ activation to inhibition, shown here, to our knowledge, for the first time in a single consistent data set. Experimental data were analyzed to yield a thermodynamically constrained four-redox-state model of E3 that simulates pH-dependent activation/inhibition and active site redox states for various conditions. The developed model was used to determine substrate/product conditions that give maximal E3 rates and show that, due to non-Michaelis-Menten behavior, the maximal flux is different compared with the classically defined k_cat.     

Exploring the Mechanism of General Anesthesia: Kinetic Analysis of GABAA Receptor Electrophysiology

A kinetic model of the effect of agonist and anesthetics on ligand-gated ion channels, developed in earlier work, is further refined and used to predict traces observed in fast-perfusion electrophysiological studies on recombinant GABA_A receptors under a wide range of agonist and/or anesthetic concentrations. The model incorporates only three conformational states (resting, open, and desensitized) but allows for the modulation of the conformational free energy landscape connecting these states resulting from adsorption of agonist and/or anesthetic to the bilayer in which the protein is embedded. The model is shown to reproduce the diverse and complex features of experimental traces remarkably well, including both anesthetic-induced and agonist-induced traces, as well as the modulation of agonist-induced traces by anesthetic, either coapplied or continuously present. The solutions to the kinetic equations, which give the time-dependence of each of the nine protein states (three ligation states for each of the three conformations), describe the flow of probability among these states and thus reveal the kinetic underpinnings of the traces. Many of the parameters in the model, such as the desorption rate constants of anesthetic and agonist, are directly related to model-independent experimental measurements and thus can serve as a definitive test of its validity.     

Flux module decomposition for parameter estimation in a multiple-feedback loop model of biochemical networks

Computer simulation is an important technique to capture the dynamics of biochemical networks. Since few quantitative values are measured in vivo, the values for unmeasured parameters should be estimated so that the simulation agrees with the experimental data. Considering the sparsity and error rates of experimentally measured data, the first thing is not to find a numerically exact and global solution but to explore a variety of the plausible parameter solutions. To find many plausible parameter solutions without any biases, we developed the two-phase search (TPS) method. However, calculation complexity makes it hard for TPS to optimize a large-scale dynamic model. In this study divide-and-conquer methods are used to solve this problem. The flux module decomposition (FMD) is first proposed that separates a complex, large-scale dynamic model into multiple flux modules without deteriorating its basic control architectures. FMD is combined with TPS, named FMD-TPS, to find many plausible parameter solutions for a dynamic model. To demonstrate the feasibility of FMD-TPS, it is applied to the E. coli ammonia assimilation system that consists of multiple-feedback loops. The variability of the solutions is verified by measuring the space distribution of the parameter solution vectors and by defining the binary vectors checking the consistency with biological behaviors. Compared with non-decomposition methods, FMD-TPS efficiently explored a variety of plausible parameter solutions that reproduce the dynamic behaviors in vivo.     

Kinetic model for DBT desulphurization by resting whole cells of Pseudomonas putida CECT5279

Bio-desulphurization kinetics of dibenzothiophene (DBT) using Pseudomonas putida CECT 5279, a genetically modified micro-organism (GMO), is studied. A kinetic model describing the 4S route of DBT desulphurization is proposed. Bio-desulphurization experiments have been carried out using resting whole cells of P. putida CECT 5279 obtained at different growth times as biocatalysts. The kinetic equations proposed for each reaction have been previously checked by studying each reaction of the 4S route individually, employing different substrates in different experiments. Finally, simple Michaelis–Menten kinetic equations for the three first reactions catalyzed by two mono-oxygenases (DszC and DszA) and a kinetic equation taking into account competitive inhibition due to product for the final reaction catalyzed by a desulfinase (DszB) have been adopted. DBT has been desulphurized using cells obtained at different growth times (5, 10, 23, 30 and 45 h). The overall kinetic model proposed involving the four reactions of the 4S route was fitted to all the experimental data yielding a set of kinetic parameters able to describe the system evolution. Cell age has influence on the rates of all the reactions: reactions (1), (2) and (3) present maximum rates for cell grown during 30 h, while reaction (4) shows a maximum rate for cells with around 10 h of growth time. However, affinities of each substrate and the inhibition constant of the last reaction are not influenced by the time of growth.     

A [P]NAD-based method to identify and quantitate long residence time enoyl-acyl carrier protein reductase inhibitors

The classical methods for quantifying drug–target residence time (t_R) use loss or regain of enzyme activity in progress curve kinetic assays. However, such methods become imprecise at very long residence times, mitigating the use of alternative strategies. Using the NAD(P)H-dependent FabI enoyl-acyl carrier protein (enoyl-ACP) reductase as a model system, we developed a Penefsky column-based method for direct measurement of t_R, where the off-rate of the drug was determined with radiolabeled [adenylate-³²P]NAD(P⁺) cofactor. In total, 23 FabI inhibitors were analyzed, and a mathematical model was used to estimate limits to the t_R values of each inhibitor based on percentage drug–target complex recovery following gel filtration. In general, this method showed good agreement with the classical steady-state kinetic methods for compounds with t_R values of 10 to 100 min. In addition, we were able to identify seven long t_R inhibitors (100–1500 min) and to accurately determine their t_R values. The method was then used to measure t_R as a function of temperature, an analysis not previously possible using the standard kinetic approach due to decreased NAD(P)H stability at elevated temperatures. In general, a 4-fold difference in t_R was observed when the temperature was increased from 25 to 37 °C.     

Phage and bacteria support mutual diversity in a narrowing staircase of coexistence

The competitive exclusion principle states that phage diversity M should not exceed bacterial diversity N. By analyzing the steady-state solutions of multistrain equations, we find a new constraint: the diversity N of bacteria living on the same resources is constrained to be M or M+1 in terms of the diversity of their phage predators. We quantify how the parameter space of coexistence exponentially decreases with diversity. For diversity to grow, an open or evolving ecosystem needs to climb a narrowing ‘diversity staircase'' by alternatingly adding new bacteria and phages. The unfolding coevolutionary arms race will typically favor high growth rate, but a phage that infects two bacterial strains differently can occasionally eliminate the fastest growing bacteria. This context-dependent fitness allows abrupt resetting of the ‘Red-Queen''s race'' and constrains the local diversity.     

Ergodic Sets as Cell Phenotype of Budding Yeast Cell Cycle

It has been suggested that irreducible sets of states in Probabilistic Boolean Networks correspond to cellular phenotype. In this study, we identify such sets of states for each phase of the budding yeast cell cycle. We find that these “ergodic sets” underly the cyclin activity levels during each phase of the cell cycle. Our results compare to the observations made in several laboratory experiments as well as the results of differential equation models. Dynamical studies of this model: (i) indicate that under stochastic external signals the continuous oscillating waves of cyclin activity and the opposing waves of CKIs emerge from the logic of a Boolean-based regulatory network without the need for specific biochemical/kinetic parameters; (ii) suggest that the yeast cell cycle network is robust to the varying behavior of cell size (e.g., cell division under nitrogen deprived conditions); (iii) suggest the irreversibility of the Start signal is a function of logic of the G1 regulon, and changing the structure of the regulatory network can render start reversible.     

Kinetic behavior of mixed populations of activated sludge

The kinetic behavior of heterogeneous microbial populations of sewage origin was studied in a single-stage isothermal continuous flow completely mixed aeration tank. A series of experiments were carried out at various dilution rates using glucose as the growth limiting substrate. The steady-state behavior of the system was observed at each dilution rate and the results were found to fit fairly well with the steady-state equation bayed on the Monod model with an endogenous respiration term included, i.e., μ = μ_mS/(K_s + S) ? K_d. The growth kinetics of cells harvested at steady state for each dilution rate were studied using batch experiments. The multiple response data of the system as functions of time were used to estimate the parameter values in the above kinetic model. It was found that values of the growth parameters changed significantly and systematically with cell population. For example, values of μ_m were high at high dilution rates and low at low dilution rates. It was also found that only those batch growth parameters from cells obtained at fairly high dilution rates are comparable with those estimated by the results of steady-state operations. The results of this investigation suggest that (1) different cell populations pre dominated at different steady-state dilution rates, with high dilution rates resulting in predominantly fast-growing organisms and low dilution rates resulting in predominantly slow-growing cells, and (2) risk exists in any randomly picked batch experiment to predict the steady-state behavior of the system when heterogeneous microbial populations must be used.     

Allosteric kinetics of pyruvate kinase of Saccharomyces carlsbergensis

The allosteric model of Monod et al. (1965) has been used to analyse the steadystate kinetics of pyruvate kinase from Saccharomyces carlsbergensis. The dissociation constants for the substrate phosphoenolpyruvate, the inhibitor ATP as well as the activator fructose-1, 6-diphosphate from the R and T state were calculated using a series of computer programs. On the basis of a crucial relation (derived in the Appendix), which correlates the Hill coefficient and the half-saturating concentration of substrate saturation curves with the parameters of the model of Monod et al., it is possible to differentiate between exclusive and non-exclusive ligand binding. On the other hand, this relation makes it possible to fit the experimental data to an extended model assuming only partially concerted transitions in each enzyme molecule.The physical data of yeast pyruvate kinase point to a tetrameric structure, whereas the steady-state kinetics favour a trimeric one. This discrepancy in the number of protomers can be overcome by the use of an extended model, which permits the occurrence of hybrid states R_tT_n?t. The introduction of one symmetrical hybrid state R₂T₂ into the model explains the kinetic data of yeast pyruvate kinase on the basis of four, probably identical, protomers. The equilibrium constants between the states are given.In the Appendix the derivation of the equation describing the occurrence of hybrid states is reported.     

A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

Adsorption with biodegradation for decolorization of reactive black 5 by Funalia trogii 200800 on a fly ash-chitosan medium in a fluidized bed bioreactor-kinetic model and reactor performance

A non-steady-state mathematical model system for the kinetics of adsorption and biodegradation of reactive black 5 (RB5) by Funalia trogii (F. trogii) ATCC 200800 biofilm on fly ash-chitosan bead in the fluidized bed process was derived. The mechanisms in the model system included adsorption by fly ash-chitosan beads, biodegradation by F. trogii cells and mass transport diffusion. Batch kinetic tests were independently performed to determine surface diffusivity of RB5, adsorption parameters for RB5 and biokinetic parameters of F. trogii ATCC 200800. A column test was conducted using a continuous-flow fluidized bed reactor with a recycling pump to approximate a completely-mixed flow reactor for model verification. The experimental results indicated that F. trogii biofilm bioregenerated the fly ash-chitosan beads after attached F. trogii has grown significantly. The removal efficiency of RB5 was about 95 % when RB5 concentration in the effluent was approximately 0.34 mg/L at a steady-state condition. The concentration of suspended F. trogii cells reached up to about 1.74 mg/L while the thickness of attached F. trogii cells was estimated to be 80 μm at a steady-state condition by model prediction. The comparisons of experimental data and model prediction show that the model system for adsorption and biodegradation of RB5 can predict the experimental results well. The approaches of experiments and mathematical modeling in this study can be applied to design a full-scale fluidized bed process to treat reactive dye in textile wastewater.     

RankAggreg, an R package for weighted rank aggregation

Background

Determining the parameters of a mathematical model from quantitative measurements is the main bottleneck of modelling biological systems. Parameter values can be estimated from steady-state data or from dynamic data. The nature of suitable data for these two types of estimation is rather different. For instance, estimations of parameter values in pathway models, such as kinetic orders, rate constants, flux control coefficients or elasticities, from steady-state data are generally based on experiments that measure how a biochemical system responds to small perturbations around the steady state. In contrast, parameter estimation from dynamic data requires time series measurements for all dependent variables. Almost no literature has so far discussed the combined use of both steady-state and transient data for estimating parameter values of biochemical systems.

Results

In this study we introduce a constrained optimization method for estimating parameter values of biochemical pathway models using steady-state information and transient measurements. The constraints are derived from the flux connectivity relationships of the system at the steady state. Two case studies demonstrate the estimation results with and without flux connectivity constraints. The unconstrained optimal estimates from dynamic data may fit the experiments well, but they do not necessarily maintain the connectivity relationships. As a consequence, individual fluxes may be misrepresented, which may cause problems in later extrapolations. By contrast, the constrained estimation accounting for flux connectivity information reduces this misrepresentation and thereby yields improved model parameters.

Conclusion

The method combines transient metabolic profiles and steady-state information and leads to the formulation of an inverse parameter estimation task as a constrained optimization problem. Parameter estimation and model selection are simultaneously carried out on the constrained optimization problem and yield realistic model parameters that are more likely to hold up in extrapolations with the model.     

Effect of atoms evaporated from the anode on the structure of the vacuum arc space charge sheath

The structure of the anode space charge sheath of a vacuum arc is studied with allowance for the dependence of the negative anode fall on the ratio of the directed electron velocity v ₀ to the electron thermal velocity v _T for different values of the flux density of atoms evaporated from the anode. Poisson’s equation for the sheath potential is solved taking into account the electron space charge, fast cathode ions, and slow ions produced due to the ionization of atoms evaporated from the anode. The kinetic equation for atoms and slow anode ions is solved with allowance for ionization in the collision integral. Analytic solutions for the velocity distribution functions of atoms and slow ions and the density of slow ions are obtained. It is shown that the flux of slow ions substantially affects the spatial distribution of the electric field E(z) in the sheath. As the flux density increases, the nonmonotonic dependence E(z) transforms into a monotonic one and the sheath narrows. For a given flux of evaporated atoms Πa, the increase in the ratio of the directed electron velocity to the electron thermal velocity leads again to a nonmonotonic dependence E(z). As z increases, the electric field first increases, passes through the maximum, decreases, passes through the minimum E _min, and then again increases toward the anode. There is a limiting value of the ratio (v ₀/v _T )* at which E _min(z) vanishes. At v ₀/v _T > (v ₀/V _T )*, the condition for the existence of a steady-state sheath is violated and the profiles of the field and potential in the sheath become oscillating. The dependence of (v ₀/v _T )* on the flux density of evaporated atoms Π _a is obtained. It is shown that the domain of existence of steady-state solutions in the sheath broadens with increasing Π _a .