Ion induced deformation of soft tissue

In this paper the effects of changing the ion concentration in and around a sample of soft tissue are investigated. The triphasic theory developed by Laiet al. (1990,Biomechanics of Diarthrodial Joints, Vol. 1, Berlin, Springer-Verlag) is reduced to two coupled partial differential equations involving fluid ion concentration and tissue solid deformation. These equations are given in general form for Cartesian, cylindrical and spherical geometries. After solving the two equations quantities such as fluid velocity, fluid pressure, chemical potentials and chemical expansion stress may be easily calculated. In the Cartesian geometry comparison is made with the experimental and theoretical work of Myerset al. (1984,ASME J. biomech. Engng,106, 151–158). This dealt with changing the ion concentration of a salt shower on a strip of bovine articular cartilage. Results were obtained in both free swelling and isometric tension states, using an empirical formula to acount for ion induced deformation. The present theory predicts lower ion concentrations inside the tissue than this earlier work. A spherical sample of tissue subjected to a change in salt bath ion concentration is also considered. Numerical results are obtained for both hypertonic and hypotonic bathing solutions. Of particular interest is the finding that tissue may contract internally before reaching a final swollen equilibrium state or swell internally before finally contracting. By considering the relative magnitude, and also variation throughout the time course of terms in the governing equations, an even simpler system is deduced. As well as being linear the concentration equation in the new system is uncoupled. Results obtained from the linear system compare well with those from the spherical section. Thus, biological swelling situations may be modelled by a simple system of equations with the possibility, of approximate analytic solutions in certain cases.     

Regulation of differentiation in a population of cells interacting through a common pool

We consider a model of a suspension of a cell population in a well-mixed medium. There are two chemical substances, say A and H, reacting in each cell of the population and the substance H can only diffuse from the inside of cell to the medium or vice versa across the cell membrane. The medium is well mixed that the concentration of H is kept uniform over the medium. Cells interact indirectly with each other through the medium. The differential equations governing the dynamics of the suspension are analyzed using standard techniques for differential equations. It is shown that the cell population divides into several groups in respect of the chemical concentrations as time elapses. It is also shown how the fraction of the number of cells belonging to each subgroup to the total number of cells is regulated. The results may be used to explain the mechanism for differentiation of multi-cellular organisms.     

A thermodynamical model of cell distributions in the slug of cellular slime mold

A theoretical model is proposed for the formation of cell distribution patterns in the slug stage of the cellular slime moldDictyostelium discoideum. The equilibrium distribution of two types of cells, prestalk and prespore, is obtained by minimizing the free energy, which is defined in terms of differential chemotaxis, differential cell adhesion and randomness of cell movement. Resulting distributions show various segregation patterns of cell types. The condition for cell sorting is obtained from stability analysis of the set of diffusion equations governing the evolution of cell type distribution and the concentration of chemoattractant. The intensities of differential chemotaxis and random cell movement are quantitatively evaluated from experimental data to show that two cell types can sort themselves completely by these forces.     

Kinetic analysis of the michaelis-menten mechanism in which the substrate and the product are unstable

• 1.1. A kinetic analysis of the Michaelis-Menten mechanism for the case in which both the substrate and the product are unstable, either spontaneously or as the result of the addition of a reagent, has been made.
• 2.2. The explicit time course equations of the immediate product and the species into which it subsequently is transformed have been derived under the conditions of rapid equilibrium and limiting substrate concentration.
• 3.3. The validity of these equations has been checked using numerical simulations.
• 4.4. The kinetic data analysis which we suggest is based on the time progress curves of the product or, in the case in which the product accumulation cannot be monitored experimentally, on the time progress curve of the species into which the immediate product is transformed.
• 5.5. This analysis allows the determination of the rate and the equilibrium constants if adequate experimental results are available.
• 6.6. We have chosen a numerical example, with which we illustrate the procedure of the kinetic data analysis by simulating some curves with assumed experimental errors.

     

Deriving the rate equations for product inhibition patterns in bisubstrate enzyme reactions

In this work, the full rate equations for 17 completely reversible bisubstrate enzyme kinetic mechanisms, with two substrates in the forward and two in the reverse direction, have been presented; among these are rapid equilibrium, steady-state, and mixed steady-state and rapid equilibrium mechanisms. From each rate equation eight product inhibition equations were derived, four for the forward and four for the reverse direction. All the corresponding product inhibition equations were derived in full; thus a total of 17 × 8 = 136 equations, were presented. From these equations a list of product inhibition patterns were constructed and presented in a tabular form, both for the primary plots (intercept effects) and the secondary plots (slope effects).

The purpose of this work is to help investigators in practical work, especially biologists working with enzymes, to choose quickly an appropriate product inhibition pattern for the identification of the kinetic mechanism. The practical application of above product inhibition analysis was illustrated with three examples of yeast alcohol dehydrogenase-catalyzed reactions.     

Perspectives of taste reception

Summary Ion: solute cotransporters frequency are incapable of achieving equilibrium between the solute accumulation and the transmembrane difference of the electrochemical potential of the ion. The presence of uncoupled flows of ion and solutes (leaks) is often advanced as an explanation. Here an alternative is discussed. The net accumulation of solute may be so slow that equilibrium can never be attained at finite times (e.g., several hours). Cotransporters may exhibit strong product inhibition, and the net influx of solute approaches zero far from equilibrium. The inherent slowness of net transport under these conditions is termed catalytic inefficiency. The likelihood that galactoside: H⁺ cotransport inEscherichia coli, hexose: H⁺ cotransport inChlorella vulgaris, andd-glucose: Na⁺ cotransport in brush-border membranes exhibit catalytic inefficiency is examined. The existence of strong product inhibition complicates the determination of the stoichiometry of cotransport and the characterization of chemically modified or mutant cotransporters.     

Deriving the rate equations for product inhibition patterns in bisubstrate enzyme reactions

In this work, the full rate equations for 17 completely reversible bisubstrate enzyme kinetic mechanisms, with two substrates in the forward and two in the reverse direction, have been presented; among these are rapid equilibrium, steady-state, and mixed steady-state and rapid equilibrium mechanisms. From each rate equation eight product inhibition equations were derived, four for the forward and four for the reverse direction. All the corresponding product inhibition equations were derived in full; thus a total of 17 x 8 = 136 equations, were presented. From these equations a list of product inhibition patterns were constructed and presented in a tabular form, both for the primary plots (intercept effects) and the secondary plots (slope effects). The purpose of this work is to help investigators in practical work, especially biologists working with enzymes, to choose quickly an appropriate product inhibition pattern for the identification of the kinetic mechanism. The practical application of above product inhibition analysis was illustrated with three examples of yeast alcohol dehydrogenase-catalyzed reactions.     

Oxygen-absorption rate-controlled feeding of substrate into aerobic microbial cultures

A system of automatic control of substrate inflow into an aerated culture of microorganisms which depend on oxygen-absorption rate (OAR) has been devised and tested. As the control variable, dissolved oxygen concentration (DOC), which shows the equilibrium between OAR and oxygen-uptake rate in the microbial culture, was chosen. If the equilibrium is disturbed by changes in OAR, then the oxygen-uptake rate is changed by substrate limitation. The DOC is measured by means of a Clark-type polarographic electrode, and the signal is used to actuate the substrate inflow valve or pump. The oxygen-uptake rate changes of microorganisms, after the addition or exhaustion of substrate in the medium, are so rapid that they can be used for this type of control. Fundamental equations were derived and graphical solutions for the control system parameters were suggested for the steady-state relations between DOC, oxygen-uptake rate, specific growth rate, substrate concentration, K_La, and concentration of microorganisms. The system is stable in the entire range of the uptake rates up to nearly the maximum attainable in unlimited substrate conditions, and can be operated in batch feed or continuous flow modifications. It was experimentally tested with the yeast Saccharomyces cerevisiae. The complete utilization of aeration-system capacity of the fermentor was achieved with high yeast yield and no alcohol formation. The quality of the product was excellent.     

Externally driven countercurrent multiplication in a mathematical model of the urinary concentrating mechanism of the renal inner medulla

Substitution of measured permeabilities into mathematical models of the concentrating mechanism of the renal inner medulla yields less than the known urine osmolalities. To gain a better understanding of the mechanism we analyse a model in which a force of unspecified origin [expressed as fraction, ɛ, of entering descending thin limb (DTL) concentration] drives fluid from DTL to interstitial vascular space (CORE), thus concentrating the solution in DTL. When flow in the DTL reverses at the hairpin bend of the loop of Henle, the high solute permeability of ascending thin limb (ATL) permits solute to diffuse into the CORE thus permitting ɛ to be multiplied many-fold. Behavior of the model is described by two non-linear differential equations. In the limit for infinite salt permeability of ATL the two equations reduce to a single equation that is formally identical with that for the Hargitay and Kuhn multiplier, which assumes fluid transport directly from DTL to ATL (Z. Electrochem. Angew. Phys. Chem. 55, 539, 1951). Solutions of the equations describing the model with parameters taken from perfused thin limbs show that urine osmolalities of the order of 5000 mosm L⁻¹ can be generated by forces of the order of 20 mosm L⁻¹. It seems probable that mammals including desert rodents use some variant of this basic mechanism for inner medullary concentration.     

Membrane transport models with fast and slow reactions: General analytical solution for a single relaxation

Summary Membrane transport models are usually expressed on the basis of chemical kinetics. The states of a transporter are related by rate constants, and the time-dependent changes of these states are given by linear differential equations of first order. To calculate the time-dependent transport equation, it is necessary to solve a system of differential equations which does not have a general analytical solution if there are more than five states. Since transport measurements in a complex system rarely provide all the time constants because some of them are too rapid, it is more appropriate to obtain approximate analytical solutions, assuming that there are fast and slow reaction steps. The states of the fast steps are related by equilibrium constants, thus permitting the elimination of their differential equations and leaving only those for the slow steps. With a system having only two slow steps, a single differential equation is obtained and the state equations have a single relaxation. Initial conditions for the slow reactions are determined after the perturbation which redistribute the states related by fast reactions. Current and zero-trans uptake equations are calculated. Curve fitting programs can be used to implement the general procedure and obtain the model parameters.     

Equilibrium position, kinetics, and reactor concepts for the adipyl-7-ADCA-hydrolysis process

One of the building blocks of cephalosporin antibiotics is 7-amino-deacetoxycephalosporanic acid (7-ADCA). It is currently produced from penicillin G using an elaborate chemical ring-expansion step followed by an enzyme-catalyzed hydrolysis. However, 7-ADCA-like components can also be produced by direct fermentation. This is of scientific and economic interest because the elaborate ring-expansion step is performed within the microorganism. In this article, the hydrolysis of the fermentation product adipyl-7-ADCA is studied. Adipyl-7-ADCA can be hydrolyzed in an equilibrium reaction to adipic acid and 7-ADCA using glutaryl-acylase. The equilibrium reaction yield is described as a function of pH, temperature, and initial adipyl-7-ADCA concentration. Reaction rate equations were derived for adipyl-7-ADCA-hydrolysis using three (pH-independent) reaction rate constants and the apparent equilibrium constant. The reaction rate constants were calculated from experimental data. Based on the equilibrium position and reaction rate equations the hydrolysis reaction was optimized and standard reactor configurations were evaluated. It was found that equilibrium yields are high at high pH, high temperature and low-initial adipyl-7-ADCA concentration. The course of the reaction could be described well as a function of pH (7-9), temperature (20-40 degrees C) and concentration using the reaction rate equations. It was shown that a series of CSTR's is the best alternative for the process.     

The sedimentation equilibrium of heterogeneously associating systems and mixtures of non-interacting solutes: analysis without determination of molecular weight averages

Sedimentation equilibrium is first considered of a system in which a ligand of any size binds to an acceptor at p sites, the experimental result, obtained with either interference or absorption optics, being a distribution of total solute concentration as a function of radial distance. Theory illustrated by a numerical example, is presented which shows that this distribution may be analysed to give the activity of the unbound ligand as a function of total weight concentration. It is shown that this information may be used together with conservation of mass equations written in terms of the initial mixing composition to evaluate the equilibrium constant(s) relevant to the system. Correlation with composition evaluation by use of absorption optics (when possible) is also discussed. The procedure does not involve solution of simultaneous equations which are sums of exponentials nor differentiation of experimental results to obtain apparent weight-average molecular weights. It is general in that it leads to the evaluation of the activity of the species characterized by the smallest M(l-$\text{v}$π) product and, accordingly, is shown to be useful in the analysis of non-interacting as well as of interacting systems.     

Kinetic Formulations for the Reduction of Ketomalonate by Lactate Dehydrogenase

Abstract

Initial rate kinetic studies of lactate dehydrogenase with ketomalonate and NADH as substrates suggest that this enzymatic system is adapted to a rapid equilibrium ordered bi-bi ternary complex mechanism. The application of the reaction product inhibition method reveals the existence of the enzyme-NADH-hydroxy-malonate and enzyme-NAD⁺-ketomalonate abortive complexes. This kinetic behaviour is confirmed by the differential inhibition induced by several alternate products on the pyruvate-lactate dehydrogenase-NADH and ketomalonate-lactate dehydrogenase-NADH systems.     

A differential game theoretical analysis of mechanistic models for territoriality

In this paper, elements of differential game theory are used to analyze a spatially explicit home range model for interacting wolf packs when movement behavior is uncertain. The model consists of a system of partial differential equations whose parameters reflect the movement behavior of individuals within each pack and whose steady-state solutions describe the patterns of space-use associated to each pack. By controlling the behavioral parameters in a spatially-dynamic fashion, packs adjust their patterns of movement so as to find a Nash-optimal balance between spreading their territory and avoiding conflict with hostile neighbors. On the mathematical side, we show that solving a nonzero-sum differential game corresponds to finding a non-invasible function-valued trait. From the ecological standpoint, when movement behavior is uncertain, the resulting evolutionarily stable equilibrium gives rise to a buffer-zone, or a no-wolf’s land where deer are known to find refuge.     

Mathematical theory of periodic relapsing catatonia

Two first-order, non-linear differential equations are presented to describe the interaction of the thyroid and pituitary glands and to explain some of the phenomena of periodic relapsing catatonia. Topographical study of these equations shows a mechanism in which either random fluctuations of hormone levels or system stability may occur. The variations in hormone concentrations are assumed to be caused by system malfunction, while stability is shown to result from the administration of exogenous thyroid which, if supplied at an optimum rate, suppresses the output of both glands. The system, under treatment, shows a zero level of thyrotropin and a thyroid concentration which is larger than the equilibrium level in the untreated system. The theory is consistent with what is known of the course and proper treament of periodic relapsing catatonia. Read before the American Chemical Society, Chicago, Ill., September 11, 1953.     

Periodic metabolic systems: Oscillations in multiple-loop negative feedback biochemical control networks

Summary For a general multiple loop feedback inhibition system in which the end product can inhibit any or all of the intermediate reactions it is shown that biologically significant behaviour is always confined to a bounded region of reaction space containing a unique equilibrium. By explicit construction of a Liapunov function for the general n dimensional differential equation it is shown that some values of reaction parameters cause the concentration vector to approach the equilibrium asymptotically for all physically realizable initial conditions. As the parameter values change, periodic solutions can appear within the bounded region. Some information about these periodic solutions can be obtained from the Hopf bifurcation theorem. Alternatively, if specific parameter values are known a numerical method can be used to find periodic solutions and determine their stability by locating a zero of the displacement map. The single loop Goodwin oscillator is analysed in detail. The methods are then used to treat an oscillator with two feedback loops and it is found that oscillations are possible even if both Hill coefficients are equal to one.     

Dynamical machinery of a biochemical clock

Closed positive feedback loops of catalytic reactions between macromolecules, or hypercycles, provide a kinetic mechanism whereby each Species serves to catalyze selfreproduction of its successor in the loop. Hypercycles of five members or more evolve into limit cycles characteristic of a biochemical clock. Computer study of the coupled non-linear differential equations which describe these systems shows that the periodT _n of then-species limit cycle is given byT _n=nτ_n, where τ_n is an elemental repeat period reflecting translational time invariance. Analytic solutions of the equations are developed so that the time evolution of elementaryn-hypercycles can be traced in dynamical detail. It is shown that the magnitude of τ_n is, to good approximation, a linear function ofn. For a givenn, τ_n is a very sensitive function of the relative concentration a given member of the loop has at the time its predecessor dominates the state of the hypercycle. These concentrations decrease with increasingn. Aroundn=15 they become so small that elementary hypercycles become unstable against disruptive concentration fluctuations. Species concentrations for more realistic hypercycles tend not to be as small, so that the present estimate of a maximum number of components is a lower bound.     

Kinetic characterization of all steps of the interaction between acetylcholinesterase and eserine

The three-step carbamylenzyme mechanism of the action of eserine on acetylcholinesterase (acetylcholine acetylhydrolase, EC 3.1.1.7) has been known for a long time, but its complete kinetic characterization has never been done. Some of our investigations indicated that the determination of missing kinetic parameters should include the inspection of the enzyme-eserine interaction in a very wide range of eserine concentrations. Therefore, the activity of acetylcholinesterase as a function of time in the presence of low concentrations of eserine comparable to the enzyme concentration was followed. The reaction mechanism was analysed by fitting numerically integrated differential equations that describe the time dependences of all reactants and reaction intermediates to these data. Additionally, the progress curve measurements at higher eserine concentrations were carried out on a stopped-flow apparatus. The corresponding progress curve equations were derived and the kinetic parameters evaluated by non-linear regression treatment. The complex analysis confirmed the three-step mechanism. The values of the constants showed that the very high affinity of eserine for binding into the active centre of the enzyme is not so much a consequence of the fast initial complex formation but rather a consequence of its slow dissociation. The subsequent covalent bonding of eserine is also slow, but faster than the dissociation of the initial complex. In this manner, the decarbamoylation is the only process responsible for the reactivation of acetylcholinesterase after removal of eserine.     

An analysis of a stochastic model for bacteriophage systems

In this article, we analyze a system modeling bacteriophage treatments for infections in a noisy context. In the small noise regime, we show that after a reasonable amount of time the system is close to a bacteria free equilibrium (which is a relevant biologic information) with high probability. Mathematically speaking, our study hinges on concentration techniques for delayed stochastic differential equations.