Validity of quasi-steady-state and transfer-function representations for input-output relation in a Michaelis-Menten reaction

In relation to the input-output characteristics of enzymatic reactions in the cellular metabolism and biochemical reactors, the validity of the quasi-steady-state and transfer-function representations of reaction velocity has been examined for a basic Michaelis-Menten reaction employing computer simulation, that is, numerical integration of the rate equation. The well-known S-v relationship (relationship between substrate concentration and reaction velocity)derived on the quasi-steady-state assumption is found to be in general a good approximation to the actual velocity throughout the temporal progress of the reaction. The validity of the approximation depends on a ratio of the Michaelis constant to the total enzyme concentration in the reaction system rather than on the individual rate constants. A transfer-function representation is derived on assuming an exponential change in the reaction velocity for the indicial response to the substrate influx rate. The representation has a wider valid region with a decrease in influx rate than with an increase in the influx rate. The validity is most dependent on a ratio of total enzyme concentration to the steady-state concentration of the substrate. The analysis of the linear sensitivity of the reaction velocity to rate constants reveals that the characteristics of these valid representations in systems analysis change according to the phase of the reaction.     

A transfer-function representation for regulatory responses of a controlled metabolic pathway

A transfer-function representation for the response of a controlled metabolic pathway to the changes in influx and efflux rates of metabolites is formulated to describe analytically and approximately the regulatory behavior of the pathway around a steady state. The pathway model analysed is an open and homogeneous system which consists of two consecutive enzymatic reactions catalyzed by an allosteric enzyme of Monod-Wyman-Changeux (MWC) dimeric model and a Michaelis-Menten-type enzyme, respectively, and undergoes the feedback inhibition by the end product. The rate equation for the system (a system of ordinary differential equations) is linearized about a steady state, so that the responses of the reaction rates to the changes in influx rate of the substrate and efflux rate of the end product are expressed in a form of transfer function. The formulation leads to the transfer function for the response of production rate of the end product to the change in its efflux rate to clarify the regulatory response of feedback mechanism in controlled metabolic pathways. The relationship among the chemical species in the system at steady stete also supports a reasonable assumption that the regulatory mechanisms in metabolic pathways are to control the production of end product against the change in its demand from the cellular environments.     

Evolution of allosteric models for hemoglobin

We compare various allosteric models that have been proposed to explain cooperative oxygen binding to hemoglobin, including the two-state allosteric model of Monod, Wyman, and Changeux (MWC), the Cooperon model of Brunori, the model of Szabo and Karplus (SK) based on the stereochemical mechanism of Perutz, the generalization of the SK model by Lee and Karplus (SKL), and the Tertiary Two-State (TTS) model of Henry, Bettati, Hofrichter and Eaton. The preponderance of experimental evidence favors the TTS model which postulates an equilibrium between high (r)- and low (t)-affinity tertiary conformations that are present in both the T and R quaternary structures. Cooperative oxygenation in this model arises from the shift of T to R, as in MWC, but with a significant population of both r and t conformations in the liganded T and in the unliganded R quaternary structures. The TTS model may be considered a combination of the SK and SKL models, and these models provide a framework for a structural interpretation of the TTS parameters. The most compelling evidence in favor of the TTS model is the nanosecond - millisecond carbon monoxide (CO) rebinding kinetics in photodissociation experiments on hemoglobin encapsulated in silica gels. The polymeric network of the gel prevents any tertiary or quaternary conformational changes on the sub-second time scale, thereby permitting the subunit conformations prior to CO photodissociation to be determined from their ligand rebinding kinetics. These experiments show that a large fraction of liganded subunits in the T quaternary structure have the same functional conformation as liganded subunits in the R quaternary structure, an experimental finding inconsistent with the MWC, Cooperon, SK, and SKL models, but readily explained by the TTS model as rebinding to r subunits in T. We propose an additional experiment to test another key prediction of the TTS model, namely that a fraction of subunits in the unliganded R quaternary structure has the same functional conformation (t) as unliganded subunits in the T quaternary structure.     

Alternative time representation in dopamine models

Dopaminergic neuron activity has been modeled during learning and appetitive behavior, most commonly using the temporal-difference (TD) algorithm. However, a proper representation of elapsed time and of the exact task is usually required for the model to work. Most models use timing elements such as delay-line representations of time that are not biologically realistic for intervals in the range of seconds. The interval-timing literature provides several alternatives. One of them is that timing could emerge from general network dynamics, instead of coming from a dedicated circuit. Here, we present a general rate-based learning model based on long short-term memory (LSTM) networks that learns a time representation when needed. Using a naïve network learning its environment in conjunction with TD, we reproduce dopamine activity in appetitive trace conditioning with a constant CS-US interval, including probe trials with unexpected delays. The proposed model learns a representation of the environment dynamics in an adaptive biologically plausible framework, without recourse to delay lines or other special-purpose circuits. Instead, the model predicts that the task-dependent representation of time is learned by experience, is encoded in ramp-like changes in single-neuron activity distributed across small neural networks, and reflects a temporal integration mechanism resulting from the inherent dynamics of recurrent loops within the network. The model also reproduces the known finding that trace conditioning is more difficult than delay conditioning and that the learned representation of the task can be highly dependent on the types of trials experienced during training. Finally, it suggests that the phasic dopaminergic signal could facilitate learning in the cortex.     

Remark on the input-output relation of formalized neurons in case of randomized stimuli

A formalized neuron receiving stimuli at random is considered. Formulae are derived for the probability distribution and expected value of the intervals between successive firings.     

Connectionistic models of boolean category representation

Several distinct connectionistic/neural representations capable of computing arbitrary Boolean functions are described and discussed in terms of possible tradeoffs between time, space, and expressive clarity. It is suggested that the ability of a threshold logic unit (TLU) to represent prototypical groupings has significant advantages for representing real world categories. Upper and lower bounds on the number of nodes needed for Boolean completeness are demonstrated. The necessary number of nodes is shown to increase exponentially with the number of input features, the exact rate of increase depending on the representation scheme. In addition, in non-recurrent networks, connection weights are shown to increase exponentially with a linear reduction in the number of nodes below approximately 2^d. This result suggests that optimum memory efficiency may require unacceptable learning time. Finally, two possible extensions to deal with non-Boolean values are considered.     

Computational method for muscle-path representation in musculoskeletal models

 This paper presents a new and efficient method to calculate the line-of-action of a muscle as it wraps over bones and other tissues on its way from origin to insertion. The muscle is assumed to be a one-dimensional, massless, taut string, and the surfaces of bones that the muscle may wrap around are approximated by cross-sectional boundaries obtained by slicing geometrical models of bones. Each cross-sectional boundary is approximated by a series of connected line segments. Thus, the muscle path to be calculated is piecewise linear with vertices being the contact points on the cross-sectional boundaries of the bones. Any level of geometric accuracy can be obtained by increasing the number of cross sections and the number of line segments in each cross section. The algorithm is computationally efficient even for large numbers of cross sections. Received: 9 July 2001 / Accepted in revised form: 11 March 2002     

The matrix representation of pharmacokinetic models.

An equation is developed from the matrix of rate constants which describes the behaviour of linear pharmacokinetic models for any initial condition as a function of time. This general matrix equation is then used to derive analogous expressions for drug distribution after a period of infusion, at the steady state, or during a multiple constant-dosage regimen. Matrix expressions are also derived for areas under drug concentration curves for any compartment after single doses or during multiple dosing. General matrix equations are shown to yield loading dosage schedules to achieve plateau concentrations throughout any open system.It is suggested that matrix methods have advantages over previously used mathematical techniques in pharmacokinetics in the simplicity of the algebraic expressions, and their ease of manipulation. An algebraic example of an open two-compartment model is worked to indicate the applicability of the general expressions.     

The relation between input-output transformation and gastrointestinal nematode infections on dairy farms

Efficiency analysis is used for assessing links between technical efficiency (TE) of livestock farms and animal diseases. However, previous studies often do not make the link with the allocation of inputs and mainly present average effects that ignore the often huge differences among farms. In this paper, we studied the relationship between exposure to gastrointestinal (GI) nematode infections, the TE and the input allocation on dairy farms. Although the traditional cost allocative efficiency (CAE) indicator adequately measures how a given input allocation differs from the cost-minimising input allocation, they do not represent the unique input allocation of farms. Similar CAE scores may be obtained for farms with different input allocations. Therefore, we propose an adjusted allocative efficiency index (AAEI) to measure the unique input allocation of farms. Combining this AAEI with the TE score allows determining the unique input-output position of each farm. The method is illustrated by estimating efficiency scores using data envelopment analysis (DEA) on a sample of 152 dairy farms in Flanders for which both accountancy and parasitic monitoring data were available. Three groups of farms with a different input-output position can be distinguished based on cluster analysis: (1) technically inefficient farms, with a relatively low use of concentrates per 100 l milk and a high exposure to infection, (2) farms with an intermediate TE, relatively high use of concentrates per 100 l milk and a low exposure to infection, (3) farms with the highest TE, relatively low roughage use per 100 l milk and a relatively high exposure to infection. Correlation analysis indicates for each group how the level of exposure to GI nematodes is associated or not with improved economic performance. The results suggest that improving both the economic performance and exposure to infection seems only of interest for highly TE farms. The findings indicate that current farm recommendations regarding GI nematode infections could be improved by also accounting for the allocation of inputs on the farm.     

Adipyl crosslinked bovine hemoglobins as new models of allosteric systems

As indicated by peptide analyses and mass spectrometry estimations, intramolecular crosslink with bis(3,5-dibromosalicyl)adipate of bovine hemoglobin results in the formation of two main components covalently bridged across the beta-cleft. In one component the crosslink joins the beta(1)V1-beta(2)K81 residues (XL-Peak-1), in the other the bridge is between the beta(1)K81-beta(2)K81 residues (XL-Peak-2). Both components are tetrameric with a mass near MW = 67 kDa as estimated by gel filtration, and a hydrodynamic radius near 3. 20 nm, estimated by dynamic light scattering. They have very low oxygen affinity with Pm near 100 mmHg (XL-Peak-1) and near 70 mmHg (XL-Peak-2) respectively at 37 degrees C, at neutral pH. The Bohr effect is almost absent in XL-Peak-1, while in XL-Peak-2 it is very near normal. Both systems show oxygen binding cooperativity with an index near n = 2.0. Flash photolysis kinetics of the recombination with CO could be resolved into a fast and a slow component. The amplitude of the fast rates were not concentration-dependent. The stopped-flow kinetics were autoaccelerating, consistent with their ligand-binding cooperativity. All rates were very similar to those of normal hemoglobin, suggesting that the oxy- rather than the deoxy-forms of the systems were affected by the crosslink. Proteins 2000;39:166-169.     

Alternative to Ritt's pseudodivision for finding the input-output equations of multi-output models

Differential algebra approaches to structural identifiability analysis of a dynamic system model in many instances heavily depend upon Ritt’s pseudodivision at an early step in analysis. The pseudodivision algorithm is used to find the characteristic set, of which a subset, the input-output equations, is used for identifiability analysis. A simpler algorithm is proposed for this step, using Gröbner Bases, along with a proof of the method that includes a reduced upper bound on derivative requirements. Efficacy of the new algorithm is illustrated with several biosystem model examples.     

Intrinsic dynamics of enzymes in the unbound state and relation to allosteric regulation

In recent years, there has been a surge in the number of studies exploring the relationship between proteins' equilibrium dynamics and structural changes involved in function. An emerging concept, supported by both theory and experiments, is that under native state conditions proteins have an intrinsic ability to sample conformations that meet functional requirements. A typical example is the ability of enzymes to sample open and closed forms, irrespective of substrate, succeeded by the stabilization of one form (usually closed) upon substrate binding. This ability is structure-encoded, and plays a key role in facilitating allosteric regulation, which suggests complementing the sequence-encodes-structure paradigm of protein science by structure-encodes-dynamics-encodes-function. The emerging connection implies an evolutionary role in selecting/conserving structures based on their ability to achieve functional dynamics, and in turn, selecting sequences that fold into such 'apt' structures.     

A Volterra representation for some neuron models

A Volterra-like polynomial representation is derived and its convergence discussed for two neuronal models in which subthreshold inputs are integrated either without loss (integrate and fire) or with a decay which follows an exponential time course (leaky integrator). This polynomial representation provides a kind of nonlinear transfer function for the nonlinear encoding process. Standard formulae are used to derive explicitely the output for various inputs as in linear system theory. Moreover, the nonlinear transfer function associated with cascades or networks of neurons can be also obtained. Finally, extensions and implications of these results are discussed.     

Multi-way metamodelling facilitates insight into the complex input-output maps of nonlinear dynamic models

Background

The physical periphery of a biological cell is mainly described by signaling pathways which are triggered by transmembrane proteins and receptors that are sentinels to control the whole gene regulatory network of a cell. However, our current knowledge about the gene regulatory mechanisms that are governed by extracellular signals is severely limited.

Results

The purpose of this paper is three fold. First, we infer a gene regulatory network from a large-scale B-cell lymphoma expression data set using the C3NET algorithm. Second, we provide a functional and structural analysis of the largest connected component of this network, revealing that this network component corresponds to the peripheral region of a cell. Third, we analyze the hierarchical organization of network components of the whole inferred B-cell gene regulatory network by introducing a new approach which exploits the variability within the data as well as the inferential characteristics of C3NET. As a result, we find a functional bisection of the network corresponding to different cellular components.

Conclusions

Overall, our study allows to highlight the peripheral gene regulatory network of B-cells and shows that it is centered around hub transmembrane proteins located at the physical periphery of the cell. In addition, we identify a variety of novel pathological transmembrane proteins such as ion channel complexes and signaling receptors in B-cell lymphoma.     

Linear network representation of multistate models of transport.

By introducing external driving forces in rate-theory models of transport we show how the Eyring rate equations can be transformed into Ohm's law with potentials that obey Kirchhoff's second law. From such a formalism the state diagram of a multioccupancy multicomponent system can be directly converted into linear network with resistors connecting nodal (branch) points and with capacitances connecting each nodal point with a reference point. The external forces appear as emf or current generators in the network. This theory allows the algebraic methods of linear network theory to be used in solving the flux equations for multistate models and is particularly useful for making proper simplifying approximation in models of complex membrane structure. Some general properties of linear network representation are also deduced. It is shown, for instance, that Maxwell's reciprocity relationships of linear networks lead directly to Onsager's relationships in the near equilibrium region. Finally, as an example of the procedure, the equivalent circuit method is used to solve the equations for a few transport models.