CHEMICALLY-CONTROLLED REACTION KINETICS ON FRACTALS: APPLICATION TO HYDROGEN EXCHANGE IN LYSOZYME

Fractals ◽  
1995 ◽  
Vol 03 (02) ◽  
pp. 251-267 ◽  
Author(s):  
T. GREGORY DEWEY
Keyword(s):  
Steady State ◽  
Rate Constants ◽  
Isotope Exchange ◽  
Time Course ◽  
Hydrogen Exchange ◽  
Chemical Processes ◽  
General Mechanism ◽  
Structure Information ◽  
Diffusion Controlled ◽  
Encounter Complex

The effect of time-dependent diffusional rate constants on chemically-controlled reactions in solution is considered. A general mechanism is examined that consists of a two step process. First the reactants diffuse together to form an “encounter complex.” This is followed by the collapse of the complex to the final product. The first step is diffusion controlled and the second step is chemically controlled. For reactions in restricted geometries or on fractals the rate constants associated with the diffusive process will scale with time as t−h where h is a constant between 0 and 1. The chemical processes are assumed to have time-independent rate constants. For reactions in which the encounter complex achieves a steady state, the differential equations governing the time course of the reaction can be solved exactly. At short times, the concentration of the reactants decays exponentially, reflecting the time constant of the chemical processes. At longer times, the decreasing diffusive rate constants result in the process being diffusion controlled. A stretched exponential of the form, exp{−kt1−h}, is observed. Approximate solutions for the pre-steady state behavior of the system are also determined using a Liouville transformation and corresponding asymptotic expansions. The short time regime shows power law decays of reactants. These decays will depend both on the dimensionality of the system as well as on the value of the rate constants associated with individual steps in the mechanism. Conditions can exist where a transformation to logarithmic oscillations will occur. Using this theoretical foundation a model is developed to analyze the kinetics of hydrogen isotope exchange kinetics in proteins. The exchange reaction is assume to occur in the boundary volume of the protein. Using the predicted fractal dimension of this boundary volume, scaling exponents are calculated and used as an unadjusted parameter. Highly accurate fits to the experimental data are achieved and activation energies are obtained that reflect the energetics of isotope exchange. This approach allows chemical kinetic behavior to be predicted from X-ray structure information.

Towards a complete account of diastereotopic hydrogen isotope exchange of carbon acids. III. Base-catalyzed exchange of sulfoxides. Evidence for exchange by inversion

1985 ◽  
Vol 63 (8) ◽  
pp. 2100-2109 ◽  
Author(s):  
Nick Henry Werstiuk ◽  
Sujit Banerjee
Keyword(s):  
Steady State ◽  
Rate Constants ◽  
Hydrogen Isotope ◽  
Isotope Exchange ◽  
Hydrogen Isotope Exchange ◽  
Upper Limit ◽  
Steady State Assumption ◽  
Carbon Acids ◽  
State Assumption ◽  
Complete Account

The rate constants for deuteroxide and hydroxide catalyzed H → D and D → H exchange of benzodihydrothiophene oxide (1a) and several of its deuterated analogs 1b, 1c, and 1d at 30.00 ± 0.05 °C are reported along with the rate constants for D → H exchange of deuterated benzyl methyl sulfoxides 2b and 2c. Application of the steady-state assumption to schemes involving equilibrating pyramidal anions yield equations which are used to fit experimentally determined (kf/ks)H → D and (kf/ks)D → H ratios. The analysis supports our view that exchange of the diastereotopic protons (deuterons) occurs by inversion. The inversion component contributes significantly to exchange of the "slow" proton (deuteron) of a diastereotopic pair and this accounts for the observation that (kf/ks)D → H < (kf/ks)H → D. This study establishes an upper limit of 6 kcal for the barrier to inversion of the carbanions derivable from 1 and 2, if pyramidal anions are formed.

Effect of Calcium ion on the interaction between Thrombin and Heparin. Thermal Denaturation

1977 ◽  
Vol 38 (03) ◽  
pp. 0677-0684 ◽  
Author(s):  
Raymund Machovich ◽  
Péter Arányi
Keyword(s):  
Calcium Ions ◽  
Rate Constants ◽  
Thermal Denaturation ◽  
Time Course ◽  
Complex Analysis ◽  
Heat Inactivation ◽  
Second Phase ◽  
Denaturation Kinetics ◽  
Enzyme Protection ◽  
Enzyme Denaturation

SummaryHeat inactivation of thrombin at 54° C followed first order kinetics with a rate constant of 1.0 min−1 approximately. Addition of heparin resulted in protection against thermal denaturation and, at the same time, rendered denaturation kinetics more complex. Analysis of the biphasic curve of heat inactivation in the presence of heparin revealed that the rate constants of the second phase changed systematically with heparin concentrations. Namely, at 4.5 × 10−6M, 9 × 10−6M, 1.8 × 10−5M and 3.6 × 10−5M heparin concentrations, the rate constants were 0.27 min−1, 0.17 min−1, 0.11 min−1 and 0.06 min−1, respectively.Sulfate as well as phosphate ions displayed also enzyme protection against heat inactivation, however, the same effect was obtained already at a heparin concentration, lower by three orders of magnitude.The kinetics of enzyme denaturation was not affected by calcium ions, whereas in the presence of heparin the inactivation rate of thrombin changed, i. e. calcium ions abolished the biphasic character of time course of thermal denaturation.Thus, the data suggest that calcium ions contribute to the effect of heparin on thrombin.

Inward rectification in rat nucleus accumbens neurons

1989 ◽  
Vol 62 (6) ◽  
pp. 1280-1286 ◽  
Author(s):  
N. Uchimura ◽  
E. Cherubini ◽  
R. A. North
Keyword(s):  
Steady State ◽  
Nucleus Accumbens ◽  
Time Course ◽  
Potassium Conductance ◽  
Resting Potential ◽  
Inward Rectification ◽  
Resting Membrane Potential ◽  
Potential Range ◽  
Inward Current ◽  
Rat Nucleus Accumbens

1. Intracellular recordings were made from neurons in slices cut from the rat nucleus accumbens septi. Membrane currents were measured with a single-electrode voltage-clamp amplifier in the potential range -50 to -140 mV. 2. In control conditions (2.5 mM potassium), the resting membrane potential of the neurons was -83.4 +/- 1.1 (SE) mV (n = 157). Steady state membrane conductance was voltage dependent, being 34.8 +/- 1.7 nS (n = 25) at -100 mV and 8.0 +/- 0.7 nS (n = 25) at -60 mV. 3. Barium (1 microM) markedly reduced the inward rectification and caused a small inward current (40.6 +/- 8.7 pA, n = 8) at the resting potential. These effects became larger with higher barium concentrations, and, in 100 microM barium, the current-voltage relation was straight. 4. The block of the inward current by barium (at -130 mV) occurred with an exponential time course; the time constant was approximately 1 s at 1 microM barium and less than 90 ms with 100 microM. Strontium had effects similar to those of barium, but 1000-fold higher concentrations were required. Cesium chloride (2 mM) and rubidium chloride (2 mM) also blocked the inward rectification; their action reached steady state within 50 ms. 5. It is concluded that the nucleus accumbens neurons have a potassium conductance with many features of a typical inward rectifier and that this contributes to the potassium conductance at the resting potential.

scholarly journals Reaction mechanism of the magnesium ion-dependent adenosine triphosphatase of frog muscle myosin and subfragment 1

1978 ◽  
Vol 171 (1) ◽  
pp. 165-175 ◽  
Author(s):  
M A Ferenczi ◽  
E Homsher ◽  
R M Simmons ◽  
D R Trentham
Keyword(s):  
Steady State ◽  
Rate Constants ◽  
Myosin Atpase ◽  
Forward Rate ◽  
Magnesium Ion ◽  
Skeletal Muscle Myosin ◽  
General Terms ◽  
Predominant Component ◽  
Main Components ◽  
Muscle Myosin

The Mg2+-dependent ATPase (adenosine 5′-triphosphatase) mechanism of myosin and subfragment 1 prepared from frog leg muscle was investigated by transient kinetic technique. The results show that in general terms the mechanism is similar to that of the rabbit skeletal-muscle myosin ATPase. During subfragment-1 ATPase activity at 0-5 degrees C pH 7.0 and I0.15, the predominant component of the steady-state intermediate is a subfragment-1-products complex (E.ADP.Pi). Binary subfragment-1-ATP (E.ATP) and subfragment-1-ADP (E.ADP) complexes are the other main components of the steady-state intermediate, the relative concentrations of the three components E.ATP, E.ADP.Pi and E.ADP being 5.5:92.5:2.0 respectively. The frog myosin ATPase mechanism is distinguished from that of the rabbit at 0-5 degrees C by the low steady-state concentrations of E.ATP and E.ADP relative to that of E.ADP.Pi and can be described by: E + ATP k' + 1 in equilibrium k' − 1 E.ATP k' + 2 in equilibrium k' − 2 E.ADP.Pi k' + 3 in equilibrium k' − 3 E.ADP + Pi k' + 4 in equilibrium k' − 4 E + ADP. In the above conditions successive forward rate constants have values: k' + 1, 1.1 × 10(5)M-1.S-1; k' + 2 greater than 5s-1; k' + 3, 0.011 s-1; k' + 4, 0.5 s-1; k'-1 is probably less than 0.006s-1. The observed second-order rate constants of the association of actin to subfragment 1 and of ATP-induced dissociation of the actin-subfragment-1 complex are 5.5 × 10(4) M-1.S-1 and 7.4 × 10(5) M-1.S-1 respectively at 2-5 degrees C and pH 7.0. The physiological implications of these results are discussed.

scholarly journals Determination of the steady-state turnover rates of the metabolically active pools of phosphatidylinositol 4-phosphate and phosphatidylinositol 4,5-bisphosphate in human erythrocytes

1989 ◽  
Vol 259 (3) ◽  
pp. 893-896 ◽  
Author(s):  
C E King ◽  
P T Hawkins ◽  
L R Stephens ◽  
R H Michell
Keyword(s):  
Steady State ◽  
Rate Constants ◽  
Human Erythrocytes ◽  
Turnover Rates ◽  
Metabolic Fluxes ◽  
Metabolic Pool ◽  
Phosphatidylinositol 4 ◽  
Kinetics Of ◽  
Phosphate Groups

When intact human erythrocytes are incubated at metabolic steady state in a chloride-free medium containing [32P]Pi, there is rapid labelling of the gamma-phosphate of ATP, followed by a slower labelling of the monoester phosphate groups of phosphatidylinositol 4-phosphate (PtdIns4P) and phosphatidylinositol 4,5-bisphosphate [PtdIns(4,5)P2] [King, Stephens, Hawkins, Guy & Michell (1987) Biochem. J. 244, 209-217]. We have analysed the early kinetics of the labelling of these phosphate groups, in order to determine: (a) the steady-state rates of the interconversions of phosphatidylinositol, PtdIns4P and PtdIns(4,5)P2; and (b) the fractions of the total cellular complement of PtdIns4P and PtdIns(4,5)P2 that participate in this steady-state turnover. The experimental data most closely fit a pattern of PtdIns4P and PtdIns(4,5)P2 turnover in which one-quarter of the total cellular complement of each lipid is in the metabolic pool that participates in rapid metabolic turnover, with rate constants of 0.028 min-1 for the interconversion of PtdIns and PtdIns4P, and of 0.010 min-1 for the PtdIns4P/PtdIns(4,5)P2 cycle. These rate constants represent metabolic fluxes of approx. 2.1 nmol of lipid/h per ml of packed erythrocytes between PtdIns and PtdIns4P and of approx. 5.7 nmol/h per ml of cells between PtdIns4P and PtdIns(4,5)P2.

scholarly journals Zero Eigenvalue Analysis for the Determination of Multiple Steady States in Reaction Networks

1998 ◽  
Vol 53 (3-4) ◽  
pp. 171-177
Author(s):  
Hsing-Ya Li
Keyword(s):  
Steady State ◽  
Rate Constants ◽  
Reaction Network ◽  
Steady States ◽  
Eigenvalue Analysis ◽  
Zero Eigenvalue ◽  
Positive Steady State ◽  
Positive Steady States ◽  
Positive Rate ◽  
System Of Inequalities

Abstract A chemical reaction network can admit multiple positive steady states if and only if there exists a positive steady state having a zero eigenvalue with its eigenvector in the stoichiometric subspace. A zero eigenvalue analysis is proposed which provides a necessary and sufficient condition to determine the possibility of the existence of such a steady state. The condition forms a system of inequalities and equations. If a set of solutions for the system is found, then the network under study is able to admit multiple positive steady states for some positive rate constants. Otherwise, the network can exhibit at most one steady state, no matter what positive rate constants the system might have. The construction of a zero-eigenvalue positive steady state and a set of positive rate constants is also presented. The analysis is demonstrated by two examples.

Faster adjustment of O2 uptake to the energy requirement of exercise in the trained state

1978 ◽  
Vol 44 (6) ◽  
pp. 877-881 ◽  
Author(s):  
R. C. Hickson ◽  
H. A. Bomze ◽  
J. O. Hollozy
Keyword(s):  
Steady State ◽  
Exercise Training ◽  
Training Program ◽  
Endurance Exercise ◽  
Time Course ◽  
Energy Requirement ◽  
Constant Load ◽  
O2 Uptake ◽  
Endurance Exercise Training

The purpose of this study was to determine the effect of endurance exercise training on the time course of the increase in VO2 toward steady state in response to submaximal constant load work. Seven men participated in a strenuous program of endurance exercise for 40 min/day, 6 days/wk for 10 wk. Their average VO2max increased from 3.29 liters before training to 4.53 liters at the end of the training program. VO2 was measured continuously on a breath-by-breath basis at work rates requiring 40%, 50%, 60%, or 70% of VO2max before training. After training the subjects were retested both at the same absolute and the same relative work rates. The increases in VO2 toward steady state occurred more rapidly in the trained than in the untrained state both at the same absolute and at the same relative work rates. The finding that O2 uptake rises to meet O2 demand more rapidly in the trained than in the untrained state provides evidence that the working muscles become less hypoxic at the onset of exercise of the same intensity after training.

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