The switch mechanism in periodic catatonia and manic-depressive disorder.

Both in periodic catatonia and in manic-depressive disorders sudden switches occur in behavior, in the autonomic nervous system and in the catecholamine metabolism during the transition from interval or depression into catatonia or mania. Both the manic and the catatonic attacks seem to be superimposed on the basic depressive or schizophrenic illnesses. The attacks can be counteracted or suppressed by psychotropic drugs such as alpha-methyldopa, disulfiram, reserpine, haloperidol or chloropromazine which interfere with the catecholamine metabolism or their receptor sites. The involvement of the catecholamines may however be secondary to primary defects in the thyroid, the hypothalamus or the limbic system. The strict periodicity in periodic catatonia points to an accumulation of some active metabolite which may be produced centrally during the interval. At a certain level it may trigger the switch-mechanism and then be reduced during the catatonic phase. In periodic catatonia both the basic schizophrenic disease as well as the periodic manifestations are compensated by thyroxine-thyroid treatment.     

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.     

Characterization of chaotic dynamics in the human menstrual cycle

Background  

The human menstrual cycle is known to exhibit a significant amount of unexplained variability. This variation is typically dismissed as random fluctuations in an otherwise periodic and predictable system. Given the many delayed nonlinear feedbacks in the multiple levels of the reproductive endocrine system, however, the menstrual cycle can properly be construed as the output of a nonlinear dynamical system, and such a system has the possibility of being in a chaotic trajectory. We hypothesize that this is in fact the case and that it accounts for the observed variability.     

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.     

Endocrine changes in male stumptailed macaques (Macaca arctoides) as a response to odor stimulation with vaginal secretions

In mammalian species, social chemosignals are important in modulating endocrine reproductive functions. In nonhuman primates, previous studies have described a high frequency of mounting behavior by females in the follicular and periovulatory phases of the menstrual cycle. Stumptailed macaque females do not signal receptivity by means of sexual swellings, as do others macaques, therefore providing a good model in which to study chemical signaling of reproductive status. We exposed anesthetized stumptailed males to vaginal secretions of either late follicular or menses phase or to saline solution to determine the endocrine changes promoting male sexual behavior. In males exposed to follicular secretions, plasma testosterone concentrations were sustained up to 120 min after exposure. Such an effect was not observed in animals exposed to menses or saline odor sources. A luteinizing hormone surge, occurring 30 minutes after exposure to late follicular phase secretion swabs, preceded this sustained testosterone effect. The fact that late follicular scents induce sustained testosterone concentrations provides support to the idea that stumptailed males draw information concerning female reproductive status from the female's vaginal odor.     

Computer simulation of sustained oscillations in peroxidase-oxidase reaction

A system of differential equations of second order exhibiting transitional behaviour and sustained oscillations has been obtained for a complete scheme of the peroxidase-oxidase reaction. The concentrations of hydrogen peroxide and of hydrogen donor radicals are slow variables of the system. The most essential reactions responsible for oscillations have been selected. Analysis of the system in phase plane and in parameter space has been carried out. The dependence of oscillation period and amplitude on the parameter values has been investigated.     

On periodicities in metabolizing systems

Following a previous study by A. Weinberg, the author investigates periodical diffusion phenomena produced in a spherical cell by a simple coupled set of chemical reactions. The general solution even for a spherical cell does not possess spherical symmetry. It is found that periodic oscillations are possible with a frequency spectrum determined by a set of “eigenvalues”. However, these oscillations are all damped even if the system of coupled reactions which is responsible for them has non-damped solutions. Therefore, although very complex and highly asymmetrical configurations of concentrations may be thus produced in the cell, none of those configurations, except some possible centrally symmetric ones, is lasting.     

Identification of possible two-reactant sources of oscillations in the Calvin photosynthesis cycle and ancillary pathways

A systematic search for possible sources of experimentally observed oscillations in the photosynthetic reaction system has been performed by application of recent theoretical results characterizing the transient-state rate behaviour of metabolic reactions involving two independent concentration variables. All subsystems involving two independent reactants in metabolically fundamental parts of the Calvin cycle and the ancillary pathways of starch and sucrose synthesis have been examined in order to decide on basis of their kinetic and stoichiometric structure whether or not they may trigger oscillations. The results show that no less than 20 possible oscillators can be identified in the examined reaction system, only three of which have been previously considered as potential sources of experimentally observed oscillations. This illustrates the superiority of the method now applied over those previously used to identify possible two-reactant sources of metabolic oscillations and indicates that there should be no difficulty in complex metabolic pathways to point to a multitude of interactions that may trigger an oscillatory rate behaviour of the system.     

Allosteric oscillatory enzymes: influence of the number of protomers on metabolic periodicities.

The study of a concerted allosteric model for an enzyme activated by the reaction product shows that this system can generate sustained metabolic oscillations regardless of the number of protomers constituting the enzyme. The analysis extends the results previously obtained in a dimeric model for the phosphofructokinase reaction which produces glycolytic periodicities. When the substrate and product concentrations evolve on comparable time scales, the amplitude of oscillations significantly drops as the number of enzyme subunits evolves from 2 to 8. The width of the domain of substrate injection rates which produce oscillations and the periodic variation in enzyme activity also depend on the number of protomers and on the time scale structure of the system. Theoretical predictions are compared with the experiments on glycolytic oscillations in yeast and muscle, and with the structural characteristics of phosphofructokinase. The results are also discussed in relation with the mechanism of cyclic AMP oscillations in the slime mold Dictyostelium discoideum.     

Cyclic changes of plasma spermine concentrations in women

Based on previous studies which suggest that blood polyamines fluctuate during the menstrual cycle, the present study was set to determine whether plasma concentrations of the polyamine spermine show menstrual cycle-associated changes and if so, how these changes relate to phasic variations in other female hormones. Blood samples were collected from a group of 9 healthy women of various ages at 5 defined periods during their menstrual cycle including 1 woman on oral contraceptives. Spermine concentrations were determined in plasma acid extracts by reversed-phase high performance liquid chromatography method. Plasma estradiol, LH and FSH were measured by microparticle enzyme immunoassay using an automatic analyzer. Spermine concentrations, 104.4 +/- 12.2 nmol/ml at 1-3 day of the cycle, were increased transiently with a peak (263.8 +/- 22.1 nmol/ml) at 8-10 day and declined to 85.4 +/- 29.8 nmol/ml by 21-23 day of the cycle. The peak spermine concentrations coincided with the first increase in plasma estrogen levels. The individual variations in the temporal profile of spermine concentrations were of similar magnitude as individual differences in other female hormones. We conclude that: a) Plasma spermine concentrations undergo distinct cyclic alterations during the menstrual cycle with peak concentrations coinciding with the first estradiol increase, and b) Peak plasma spermine concentrations occur during the follicular phase, just prior to ovulation, during the period of rapid endometrial growth.     

A theoretical treatment of damped oscillations in the transient state kinetics of single-enzyme reactions

An extension of the available kinetic theory for reactions in the transient state is presented which establishes that single-enzyme reactions may exhibit damped oscillations under the conditions of standard kinetic experiments performed by stopped-flow techniques. Such oscillations may occur for reasonable magnitudes of rate constants in the enzymic reaction mechanism and at physiological concentrations of enzyme and substrate. In the simplest reaction systems, the oscillations will be strongly damped and lead to progress curves resembling those of a reaction governed by standard exponential transients; statistical regression methods may then have to be applied for their detection and characterization. The observation that single-enzyme reactions may exhibit oscillatory behaviour points to a previously unrecognized possible source of the damped oscillations observed in metabolic systems such as the pathways of glycolysis or photosynthesis.     

Whole-cell modeling framework in which biochemical dynamics impact aspects of cellular geometry

A mathematical framework for modeling biological cells from a physicochemical perspective is described. Cells modeled within this framework consist of at least two regions, including a cytosolic volume encapsulated by a membrane surface. The cytosol is viewed as a well-stirred chemical reactor capable of changing volume while the membrane is assumed to be an oriented 2-D surface capable of changing surface area. Two physical properties of the cell, namely volume and surface area, are determined by (and determine) the reaction dynamics generated from a set of chemical reactions designed to be occurring in the cell. This framework allows the modeling of complex cellular behaviors, including self-replication. This capability is illustrated by constructing two self-replicating prototypical whole-cell models. One protocell was designed to be of minimal complexity; the other to incorporate a previously reported well-known mechanism of the eukaryotic cell cycle. In both cases, self-replicative behavior was achieved by seeking stable physically possible oscillations in concentrations and surface-to-volume ratio, and by synchronizing the period of such oscillations to the doubling of cytosolic volume and membrane surface area. Rather than being enforced externally or artificially, growth and division occur naturally as a consequence of the assumed chemical mechanism operating within the framework.     

A framework for whole-cell mathematical modeling

The default framework for modeling biochemical processes is that of a constant-volume reactor operating under steady-state conditions. This is satisfactory for many applications, but not for modeling growth and division of cells. In this study, a whole-cell modeling framework is developed that assumes expanding volumes and a cell-division cycle. A spherical newborn cell is designed to grow in volume during the growth phase of the cycle. After 80% of the cycle period, the cell begins to divide by constricting about its equator, ultimately affording two spherical cells with total volume equal to twice that of the original. The cell is partitioned into two regions or volumes, namely the cytoplasm (Vcyt) and membrane (Vmem), with molecular components present in each. Both volumes change during the cell cycle; Vcyt changes in response to osmotic pressure changes as nutrients enter the cell from the environment, while Vmem changes in response to this osmotic pressure effect such that membrane thickness remains invariant. The two volumes change at different rates; in most cases, this imposes periodic or oscillatory behavior on all components within the cell. Since the framework itself rather than a particular set of reactions and components is responsible for this behavior, it should be possible to model various biochemical processes within it, affording stable periodic solutions without requiring that the biochemical process itself generates oscillations as an inherent feature. Given that these processes naturally occur in growing and dividing cells, it is reasonable to conclude that the dynamics of component concentrations will be more realistic than when modeled within constant-volume and/or steady-state frameworks. This approach is illustrated using a symbolic whole cell model.     

Allosteric oscillatory enzymes : Influence of the number of protomers on metabolic periodicities

The study of a concerted allosteric model for an enzyme activated by the reaction product shows that this system can generate sustained metabolic oscillations regardless of the number of protomers constituting the enzyme. The analysis extends the results previously obtained in a dimeric model for the phosphofructokinase reaction which produces glycolytic periodicities. When the substrate and product concentrations evolve on comparable time scales, the amplitude of oscillations significantly drops as the number of enzyme subunits evolves from 2 to 8. The width of the domain of substrate injection rates which produce oscillations and the periodic variation in enzyme activity also depend on the number of protomers and on the time scale structure of the system. Theoretical predictions are compared with the experiments on glycolytic oscillations in yeast and muscle, and with the structural characteristics of phosphofructokinase. The results are also discussed in relation with the mechanism of cyclic AMP oscillations in the slime mold Dictyostelium discoideum.     

Constraints and missing reactions in the urea cycle.

The stoichiometric relations in a series of biochemical reactions are summarized by a stoichiometric number matrix (with a column for each reaction) and a conservation matrix (with a row for each constraint). These two matrices for a series or cycle of biochemical reactions are related because the columns of the stoichiometric number matrix are in the null space of the conservation matrix, and the rows of the transpose of the conservation matrix are in the null space of the transpose of the stoichiometric number matrix. The conservation matrix for a system of biochemical reactions is of interest because it shows the nature of the constraints in addition to the conservation of atoms and groups. Constraints beyond those for the conservation of atoms and groups indicate "missing reactions" that do not occur because the enzymes involved couple reactions that could occur and still conserve atoms and groups. The interpretation of conservation matrices and stoichiometric matrices for a reaction system is complicated by the fact that they are not unique. However, their row-reduced forms are unique, as are their dimensions, which represent the number of reactants and number of independent reactions. Two matrices that look different contain the same information if they have the same row-reduced form. The urea cycle, which involves five enzyme-catalyzed reactions, and its net reaction are discussed in terms of the linear constraints produced by enzyme catalysis. A procedure to obtain a set of conservation equations that will yield the correct net reaction is described.     

Plasma insulin and glucagon concentrations and biochemical variables in regularly menstruating females with ovulatory and anovulatory menstrual cycles.

Ovarian hormones are known to affect endocrine pancreas function. However, data concerning the effects of anovulatory menstrual cycles in regularly menstruating women on endocrine pancreas and blood metabolites are lacking. We examined plasma insulin, glucagon, glucose, lactate, urea and glycerol concentrations in reproductive-age, regularly menstruating females classified as ovulating or non-ovulating on the basis of basal body temperature measurements and plasma 17beta-estradiol and progesterone determinations. All measurements were performed twice--in the follicular and again in the luteal phases of the menstrual cycle. There were no differences in plasma lactate and glycerol concentrations between the two groups of subjects. Plasma insulin concentrations tended to be lower in non-ovulating than in ovulating women. In addition, plasma glucagon did not differ in the follicular (33.2 pmol/l) or luteal phase of the menstrual cycle in females with disturbed ovarian hormone secretion (34.1 pmol/l). In contrast, plasma glucagon concentrations in the luteal phase (32.8 pmol/l) were significantly higher than in the follicular phase (24.9 pmol/l) of the menstrual cycle in ovulating women. Plasma glucose concentrations in the follicular phase of the menstrual cycle in non-ovulating women (4.1 mmol/l) were slightly but significantly lower than in their ovulating counterparts (5.3 mmol/l). Furthermore, no correlations were noted between plasma glucose and insulin-to-glucagon molar ratio in non-ovulating subjects. Plasma urea concentrations in non-ovulating women were markedly lower than in ovulating women in both follicular and luteal phases of the menstrual cycle (4.1 and 3.9 mmol/l vs. 5.3 and 5.4 mmol/l in non-ovulating and ovulating women, respectively). In ovulating women, plasma urea levels in both cycle phases were significantly correlated with plasma glucagon concentrations, but no such correlation was found in non-ovulating women. In conclusion, anovulatory menstrual cycles in premenopausal females slightly altered pancreatic hormone plasma levels but markedly impaired their action on plasma glucose and urea concentrations.     

On the role of enzyme cooperativity in metabolic oscillations: analysis of the Hill coefficient in a model for glycolytic periodicities.

The role of enzyme cooperativity in the mechanism of metabolic oscillations is analyzed in a concerted allosteric model for the phosphofructokinase reaction. This model of a dimer enzyme activated by the reaction product accounts quantitatively for glycolytic periodicities observed in yeast and muscle. The Hill coefficient characteristic of enzyme-substrate interactions is determined in the model, both at the steady state and in the course of sustained oscillations. Positive cooperativity is a prerequisite for periodic behavior. A necessary condition for oscillation in a dimer K system is a Hill coefficient larger than 1.6 at the unstable stationary state. The analysis suggests that positive as well as negative effectors of phosphofructokinase inhibit glycolytic oscillations by inducing a decrease in enzyme cooperativity. The results are discussed with respect to glycolytic and other metabolic periodicities.     

A suggestion of a new approach to the theory of some biological periodicities

Previous studies of L. Danziger and G. Elmergreen (Bull. Math. Biophysics,16, 15–21, 1954;18, 1–13, 1956) of possible biochemical periodicities in organisms assumed non-linear biochemical interaction between different metabolites, because linear systems do not lead to undamped ocsillations. They treated homogeneous systems. Later N. Rashevsky generalized their results to a more realistic case where the non-homogeneity due to the histological structure is considered. (Some Medical Aspects of Mathematical Biology, Springfield, Illinois: Charles C. Thomas, Publisher, 1964;Bull. Math. Biophysics,29, 389–393, 1967.) As long as the histological structure remains constant, the existence of sustained periodicities requires the assumption of non-linearity of biochemical interactions. If, however, the secretions of an endocrine gland affect the histological structure of the target organ, notably as in the menstrual cycle, and if there is a feed-back, the equations become non-linear and may admit sustained periodic solutions even if the purely biochemical interactions are linear.     

Oscillating reactions in open systems

The probable existence of oscillating chemical reactions has been attracting some interest in recent years for their possible role in explaining certain biological phenomena. Perhaps the simplest model of oscillating reactions is that of Lotka (1910), which consists of a chain of autocatalytic reactions. Two “reactor systems” in which such a chain of reactions could take place are considered in this work and are called homogeneous and compartmental models, respectively. The differential equations governing the temporal behavior of the reacting species are solved on an analog computer, and the conditions under which sustained oscillations occur are obtained and discussed. Comparisons of the solution obtained in the two models are discussed.