A metabolic control analysis of kinetic controls in ATP free energy metabolism in contracting skeletal muscle

A system analysis ofATP free energy metabolism in skeletal muscle was made using theprinciples of metabolic control theory. We developed a network model ofATP free energy metabolism in muscle consisting of actomyosin ATPase,sarcoplasmic reticulum (SR) Ca²⁺-ATPase, and mitochondria.These components were sufficient to capture the major aspects of theregulation of the cytosolic ATP-to-ADP concentration ratio (ATP/ADP) inmuscle contraction and had inherent homeostatic properties regulatingthis free energy potential. As input for the analysis, we used ATPmetabolic flux and the cytosolic ATP/ADP at steady state at sixcontraction frequencies between 0 and 2 Hz measured in human forearmflexor muscle by ³¹P-NMR spectroscopy. We used themathematical formalism of metabolic control theory to analyze thedistribution of fractional kinetic control of ATPase flux and theATP/ADP in the network at steady state among the components over thisexperimental range and an extrapolated range of stimulation frequencies(up to 10 Hz). The control analysis showed that the contractileactomyosin ATPase has dominant kinetic control of ATP flux in forearmflexor muscle over the 0- to 1.6-Hz range of contraction frequenciesthat resulted in steady states, as determined by ³¹P-NMR.However, flux control begins to shift toward mitochondria at >1 Hz.This inversion of flux control from ATP demand to ATP supply controlhierarchy progressed as the contraction frequency increased past 2 Hzand was nearly complete at 10 Hz. The functional significance of thisresult is that, at steady state, ATP free energy consumption cannotoutstrip the ATP free energy supply. Therefore, this reduced,three-component muscle ATPase system is inherently homeostatic.

     

Integration of temporal analysis and control analysis of metabolic systems.

A theory is developed that integrates approaches to the analysis of pathway transient response and metabolic control analysis. A Temporal Control Coefficient is defined that is a measure of the system's transient response to modulation of enzyme activity or concentration. The approach allows for the analysis of the establishment of a steady state from rest, of the system's 'agility' of response to minor perturbations of a pre-existing steady state and of the macroscopic transition between steady states. In the last-mentioned case it is shown that, like the transient time itself, the control of transient response retains the property of independence from the mechanism of the transition. In consequence, the Temporal Control Coefficient can be defined in terms of the control properties of the initial and final states alone without reference to the mechanism of transition. A summation property is shown to apply to the Temporal Control Coefficients in each case. Connectivity relationships between elasticities and Temporal Control Coefficients are also established.     

Mathematical analysis of multienzyme systems. II. Steady state and transient control.

The control theory of steady states, previously presented for linear enzymatic systems (Heinrich and Rapoport, 1974) is extended to nonlinear systems. On the basis of three theorems a new procedure for the calculation of the control strength and of the control matrix is developed. The theory is applied to the extended model of glycolysis of erythrocytes, which includes also ATP-consuming processes. Also in this model the glycolytic flux is mainly controlled by the hexokinase-phosphofructokinase-system. The control strengths of the pyruvate kinase and of the enzymes of the 2.3 P2G-bypass are negligibly small. The control strength of the ATPase is negative, i.e. an activation of this enzyme leads to a decrease of the flux. For transition states of multienzyme systems definitions are given for the mean time required for the transition of the metabolites and for the "transient control" of enzymes. Enzymes with a pronounced influence on the transition time are called time-limiting enzymes. Enzymes which excert strong control on the time-dependent processes may have little influence under steady state conditions and vice versa. The transition times of ATP have been calculated for transient states of glycolysis.     

Systems analysis of the tricarboxylic acid cycle in Dictyostelium discoideum. II. Control analysis.

A steady-state computer model of the tricarboxylic cycle in Dictyostelium discoideum was analyzed using metabolic control theory. The steady state had variations of less than 0.04% over the last half of the simulation for both metabolite concentrations and fluxes. Metabolite and flux control coefficients were determined by varying enzymatic activities within 2% of their initial values and simulating the responses of metabolite concentrations and fluxes to these changes. Under these conditions, summation properties were met for most metabolite and all flux control coefficients. Maximum flux control coefficients were found for succinate dehydrogenase (0.35), malic enzyme (0.24), and malate dehydrogenase (-0.18). Comparable control was found for the reaction supplying pyruvate (0.14) and for the sum of the input amino acids (0.43), which serve as an energy source for D. discoideum. The time-dependent processes by which a new steady state was established were examined after increasing malic enzyme or malate dehydrogenase activities. This provided a method for an analysis of the mechanisms by which the observed control coefficients were generated. In addition, the effects of increasing the stimuli within 5-20% of the original enzyme activity were examined. Under these conditions, more typical of experimental stimuli and measurable responses, the metabolic model failed to return to steady state, and thus summation properties were not met. Whether "true" steady states ever occur or whether metabolic control theory can be applied in vivo is discussed.     

A theory on auto-oscillation and contraction in striated muscle

It is widely accepted that muscle cells take either force-generating or relaxing state in an all-or-none fashion through the so-called excitation–contraction coupling. On the other hand, the membrane-less contractile apparatus takes the third state, i.e., the auto-oscillation (SPOC) state, at the activation level that is intermediate between full activation and relaxation. Here, to explain the dynamics of all three states of muscle, we construct a novel theoretical model based on the balance of forces not only parallel but also perpendicular to the long axis of myofibrils, taking into account the experimental fact that the spacing of myofilament lattice changes with sarcomere length and upon contraction. This theory presents a phase diagram composed of several states of the contractile apparatus and explains the dynamic behavior of SPOC, e.g., periodical changes in sarcomere length with the saw-tooth waveform. The appropriate selection of the constant of the molecular friction due to the cross-bridge formation can explain the difference in the SPOC periods observed under various activating conditions and in different muscle types, i.e., skeletal and cardiac. The theory also predicts the existence of a weak oscillation state at the boundary between SPOC and relaxation regions in the phase diagram. Thus, the present theory comprehensively explains the characteristics of auto-oscillation and contraction in the contractile system of striated muscle.     

A generalized theory of the transition time for sequential enzyme reactions.

In a sequence of coupled enzyme reactions the steady-state production of product is preceded by a lag period or transition time during which the intermediates of the sequence are accumulating. Provided that a steady state is eventually reached, the magnitude of this lag may be calculated, even when the differentiation equations describing the process have no analytical solution. The calculation may be made for simple systems in which the enzymes obey Michaelis-Menten kinetics or for more complex pathways in which intermediates act as modifiers of the enzymes. The transition time associated with each intermediate in the sequence is given by the ratio of the appropriate steady-state intermediate concentration to the steady-state flux. The theory is also applicable to the transition between steady states produced by flux changes. Application of the theory to coupled enzyme assays allows a definition of the minimum requirements for successful operation of the assay. The theory can be extended to deal with sequences in which the enzyme concentration exceeds substrate concentration.     

Chemostat-cultivated Escherichia coli at high dilution rate: multiple steady states and drift

The representation of metabolic network reaction kinetics in a scaled, polynomial form can allow for the prediction of multiple steady states. The polynomial formalism is used to study chemostat-cultured Escherichia coli which has been observed to exhibit two multiple steady states under ammonium ion-limited growth conditions: a high cell density-low ammonium ion concentration steady state and a low cell density-high ammonium ion concentration steady state. Additionally, the low-cell-density steady state has been observed to drift to the high-cell-density steady state. Inspection of the steady-state rate expressions for the ammonium ion transport/assimilation network (in polynomial form) suggests that at low ammonium ion concentrations, two steady states are possible. One corresponds to heavy use of the glutamine synthetase-glutamate synthase (GLNS-GS) branch and the second to heavy use of the glutamate dehydrogenase (GDH) branch. Realization of the predicted intracellular steady states is also found to be dependent on the parameters of the transport process. Moreover, the two steady states differ in where their energy intensity lies. To explain the drift, GLNS, which is inducible under low ammonium ion concentrations, is suggested to be a "memory element." A chemostat-based model is developed to illustrate that perturbations in dilution rate can lead to drift between the two steady states provided that the disturbance in dilution rate is sufficiently large and/or long in duration.     

The steady state kinetics of some biological systems: III. Thermodynamic aspects

Some thermodynamic aspects of steady systems are considered. The time rates of changes, “flux”, of various thermodynamic quantities are formulated. In particular the free energy flux in the steady state, the difference between the free energy flux in the steady and time dependent states and the change in free energy flux upon transition between steady states are discussed. Equations are derived which exhibit the formal similarities and differences between the free energy flux and the conventional free energy change. The temperature dependence of the steady state rate is examined and conditions for “mastery” by a single step discussed. A brief discussion of the role ofrate in the coupling of exergonic and endergonic reactions is given.     

Stability of steady states in nucleic acid poly (A-T) synthesis and the stirred flow reactor.

The template directed synthesis of poly[d(A-T)] from the nucleoside triphosphates in the presence of DNA polymerase I is carried out continuously in a stirred flow reactor for the first time. The initial objective is to test the kinetic stability of the established steady states at various flow rates. Graphical analysis predicts instable steady states for certain high flow rates. As a consequence of instabilities multiple steady states and steady-state hysteresis may occur. Steady-state hysteresis has now been found experimentally. For a different enzyme fraction of low exonuclease activity we found the steady-state absorbance at 260 nm to be almost invariant with flow rate at high enzyme concentrations even if the flow rate was increased by a large factor. We call this phenomenon kinetic buffering. Relaxation of a large flow perturbation approaches the steady state in a sigmoidal fashion. Concentration oscillations at 260 nm occurred in one experiment using an enzyme fraction of low exonuclease activity after perturbing the steady state by monomer (dATP). Advantages of the stirred flow reactor method over serial transfer are discussed.     

Kinetic model for isometric contraction in smooth muscle on the basis of myosin phosphorylation hypothesis.

A kinetic model was proposed to simulate an isometric contraction curve in smooth muscle on the basis of the myosin phosphorylation hypothesis. The Ca2+-calmodulin-dependent activation of myosin light-chain kinase and the phosphorylation-dephosphorylation reaction of myosin were mathematically treated. Solving the kinetic equations at a steady state, we could calculate the relationship between the Ca2+ concentration and the myosin phosphorylation. Assuming that two-head-phosphorylated myosin has an actin-activated Mg2+-ATPase activity and that this state corresponds to an active state, we computed the time courses of the myosin phosphorylation and the active state for various Ca2+ transients. The time course of the active state was converted into that of isometric tension by use of Sandow's model composed of a contractile element and a series elastic component. The model could simulate not only the isometric contraction curves for any given Ca2+ transient but also the following experimental results: the calmodulin-dependent shift of the Ca2+ sensitivity of isometric tension observed in skinned muscle fibers, the disagreement between the Ca2+ sensitivity of myosin phosphorylation and that of isometric tension at a steady state, and the disagreement between the time course of myosin phosphorylation and that of isometric tension development.     

The dual effect of internal tetramethylammonium ions on the open states of the sodium channel in the squid giant axon.

Voltage-clamp recordings of INa in squid axons dialysed with Cs or TMA, and bathed in low Na choline seawater, showed that, except close to threshold, the initial peak of fast-inactivating current was invariably decreased by TMA, whereas the non-inactivating current in the steady state was simultaneously increased. The results suggest that although TMA does not act directly on the movements of the voltage sensors that activate the sodium system, it blocks single-channel conductance in a voltage-dependent fashion in both the open states of the Na channel, while it has an entirely different type of action by increasing the probability of late openings in the steady state. Another difference between the two open states was that the sodium permeability coefficient had a Q10 of 1.8 in the initial open state, whereas in the steady state the effect of temperature was much smaller or even negative.     

An analysis of extended and exponentially-fed-batch cultures.

Under certain conditions it is shown that an extended culture is equivalent to an exponentially-fed-batch culture, that an exponentially-fed-batch culture (and an extended culture) can be maintained at a steady state and that an exponentially-fed-batch culture may be mimicked by a continuous-flow culture with a constant dilution rate. Operational conditions required to maintain steady states are specified.     

Mathematical model of the primary CD8 T cell immune response: stability analysis of a nonlinear age-structured system

The primary CD8 T cell immune response, due to a first encounter with a pathogen, happens in two phases: an expansion phase, with a fast increase of T cell count, followed by a contraction phase. This contraction phase is followed by the generation of memory cells. These latter are specific of the antigen and will allow a faster and stronger response when encountering the antigen for the second time. We propose a nonlinear mathematical model describing the T CD8 immune response to a primary infection, based on three nonlinear ordinary differential equations and one nonlinear age-structured partial differential equation, describing the evolution of CD8 T cell count and pathogen amount. We discuss in particular the roles and relevance of feedback controls that regulate the response. First we reduce our system to a system with a nonlinear differential equation with a distributed delay. We study the existence of two steady states, and we analyze the asymptotic stability of these steady states. Second we study the system with a discrete delay, and analyze global asymptotic stability of steady states. Finally, we show some simulations that we can obtain from the model and confront them to experimental data.     

Time scale and dimension analysis of a budding yeast cell cycle model

Background  

The progress through the eukaryotic cell division cycle is driven by an underlying molecular regulatory network. Cell cycle progression can be considered as a series of irreversible transitions from one steady state to another in the correct order. Although this view has been put forward some time ago, it has not been quantitatively proven yet. Bifurcation analysis of a model for the budding yeast cell cycle has identified only two different steady states (one for G1 and one for mitosis) using cell mass as a bifurcation parameter. By analyzing the same model, using different methods of dynamical systems theory, we provide evidence for transitions among several different steady states during the budding yeast cell cycle.     

Equilibrium points for nonlinear compartmental models.

Equilibrium points for nonlinear autonomous compartmental models with constant input are discussed. Upper and lower bounds for the steady states are derived. Theorems guaranteeing existence and uniqueness of equilibrium points for a large collection of system are proved. New information relating to mean residence times is developed. Asymptotic results and a section on stability are included. A recursive process is discussed that generates iterates that converge to steady states for certain types of models. An interesting range of models are included as examples. An attempt is made to provide general qualitative theory for such nonlinear compartmental systems.     

Identification of all steady states in large networks by logical analysis

The goal of generalized logical analysis is to model complex biological systems, especially so-called regulatory systems, such as genetic networks. This theory is mainly characterized by its capacity to find all the steady states of a given system and the functional positive and negative circuits, which generate multistationarity and a cycle in the state sequence graph, respectively. So far, this has been achieved by exhaustive enumeration, which severely limits the size of the systems that can be analysed. In this paper, we introduce a mathematical function, called image function, which allows the calculation of the value of the logical parameter associated with a logical variable depending on the state of the system. Thus the state table of the system is represented analytically. We then show how all steady states can be derived as solutions to a system of steady-state equations. Constraint programming, a recent method for solving constraint satisfaction problems, is applied for that purpose. To illustrate the potential of our approach, we present results from computer experiments carried out on very large randomly-generated systems (graphs) with hundreds, or even thousands, of interacting components, and show that these systems can be solved using moderate computing time. Moreover, we illustrate the approach through two published applications, one of which concerns the computation times of all steady states for a large genetic network.     

A biochemically structured model for Saccharomyces cerevisiae.

A biochemically structured model for the aerobic growth of Saccharomyces cerevisiae on glucose and ethanol is presented. The model focuses on the pyruvate and acetaldehyde branch points where overflow metabolism occurs when the growth changes from oxidative to oxido-reductive. The model is designed to describe the onset of aerobic alcoholic fermentation during steady-state as well as under dynamical conditions, by triggering an increase in the glycolytic flux using a key signalling component which is assumed to be closely related to acetaldehyde. An investigation of the modelled process dynamics in a continuous cultivation revealed multiple steady states in a region of dilution rates around the transition between oxidative and oxido-reductive growth. A bifurcation analysis using the two external variables, the dilution rate, D, and the inlet concentration of glucose, S(f), as parameters, showed that a fold bifurcation occurs close to the critical dilution rate resulting in multiple steady-states. The region of dilution rates within which multiple steady states may occur depends strongly on the substrate feed concentration. Consequently a single steady state may prevail at low feed concentrations, whereas multiple steady states may occur over a relatively wide range of dilution rates at higher feed concentrations.     

Equilibria and stability of a class of positive feedback loops

Positive feedback loops are common regulatory elements in metabolic and protein signalling pathways. The length of such feedback loops determines stability and sensitivity to network perturbations. Here we provide a mathematical analysis of arbitrary length positive feedback loops with protein production and degradation. These loops serve as an abstraction of typical regulation patterns in protein signalling pathways. We first perform a steady state analysis and, independently of the chain length, identify exactly two steady states that represent either biological activity or inactivity. We thereby provide two formulas for the steady state protein concentrations as a function of feedback length, strength of feedback, as well as protein production and degradation rates. Using a control theory approach, analysing the frequency response of the linearisation of the system and exploiting the Small Gain Theorem, we provide conditions for local stability for both steady states. Our results demonstrate that, under some parameter relationships, once a biological meaningful on steady state arises, it is stable, while the off steady state, where all proteins are inactive, becomes unstable. We apply our results to a three-tier feedback of caspase activation in apoptosis and demonstrate how an intermediary protein in such a loop may be used as a signal amplifier within the cascade. Our results provide a rigorous mathematical analysis of positive feedback chains of arbitrary length, thereby relating pathway structure and stability.     

The sliding filament model of muscle contraction. II. The energetic and dynamical predictions of a quantum mechanical transducer model

A theoretical model of a molecular energy transducing unit designed for the production of mechanical work is constructed and its consequences examined and compared with the experimentally determined myothermal and dynamic properties of vertebrate striated muscle. The model rests on a number of independent assumptions which include: the almost instantaneous generation of mechanical force by the occurrence of a radiationless transition between vibronic states of the transducer (crossbridge) at a point of potential energy surface crossing; transmission of this force to the load via the active sites on the thin filament by means of non-bonding repulsive forces, no energy being required for detachment; “detachment” consists of a second radiationless transition at a lower energy point than the first force generating transition, the energy difference appearing largely as work. The method of force generation completely avoids problems such as the “force-rate dilemma” which occur repeatedly in any discussion where state populations are near-Boltzmann and also leads without further arbitrary assumptions to such concepts as “attached but non force-producing states” and strongly position dependent “attachment” and “detachment” rate constants since these can only be appreciable near potential energy surface crossings. The kinetics and energetics of a transducer of this type operating cyclically and converting ATP → ADP + P_i are considered and shown to lead to length-tension and energetic behaviour very similar to that exhibited by vertebrate striated muscle, both for contraction and stretching. The existence of a limiting tension for stretching is predicted by the model as is the decrease of the rate of enthalpy release rate below the isometric value. At the limiting tension the rate of enthalpy release by the transducers is virtually zero, as observed. However, the stretching only inhibits the ATP hydrolysis, the cyclic synthesis from ADP and work being impossible with this model. The response to rapid length step changes automatically contains the asymmetry observed experimentally (with respect to lengthening and shortening) and arbitrary assumptions over and above those giving adequate explanation of the steady-state properties are not required. The asymmetry arises mainly as a consequence of the non-bonded pushing action of the crossbridges. This same assumption predicts the occurrence of an asymmetric thermoelastic ratio for active muscle with respect to stretching and contraction. The quantitative aspects of the model are satisfactory as it simultaneously reconciles the numerical magnitudes of macroscopic quantities such as isometric tension, maximum contraction velocity, limiting tension sustainable on stretching, isometric heat rate and resting heat rate with molecular parameters such as the filament and crossbridge periodicities, molecular vibrational relaxation rates, recurrence times for the radiationless transitions occurring, etc. This is achieved without any parameter optimization and only a very much smaller number of unknown parameters than the number of observed results accounted for. Many of the entities occurring in the model cycle (vibronic states of crossbridges, ATP, etc.) appear to be in one-to-one correspondence with many of the kinetic entities postulated to account for the biochemical kinetic results obtained for the actomyosin ATPase system in vitro. Finally, the rigor state has to be viewed in a different way from the conventional one; on the basis that the present model states which are part of the contraction cycle but sparsely populated during the latter (and hence are of chemical kinetic but not dynamical importance) are heavily populated during the rigor state. The mechanical properties of the rigor state would then be determined by these molecular states which would be very short-lived during the contraction cycle. If this is correct the rigor state could yield much more information about inaccessible parts of the contraction cycle than is presently supposed. The model leads one to expect a rather different response to quick length step changes in the rigor state from that of the active state, in contrast to current interpretations in terms of a large number of attached crossbridges, unable to detach due to the absence of ATP.     

pH-dependent changes of 2,3-bisphosphoglycerate.

A systematic study of the pH dependent changes in the range 6.6--7.4 of 2,3 bisphosphoglycerate (2,3-DPG) was performed in the presence and absence of glucose during transitional and steady states. The results indicate that 2,3-DPGase breaks down 2,3-DPG nealy independent of pH at a rate of 480 mu moles 2,3-DPG/1 cells.h. The 2,3-DPG mutase is practically completely inhibited below pH 6.9. The 2,3-DPG level in the presence of glucose reaches a pH dependent steady state after about 18 h. The share of the 2,3-DPG bypass in the steady state decreases from 24% at pH 7.4 to 12% at pH 7.0. The formation of pyruvate corresponds to the beadkdown of 2,3-DPG after consumption of an unknown reducing substance.