Resonance effect in self-renewing tissues.

It was predicted previously that a depletion of cell population depends extremely on the period of periodic phase-specific killing of cells and at periods near the integer multiple of the mean cell-cycle time there is a considerable reduction in treatment-induced decrease of cell-population size (resonance effect). This paper is devoted to theoretical and experimental analysis of the resonance effect in self-renewing tissues. The main biological assumptions underlying the idea of the resonance effect are discussed and then reformulated in terms of a mathematical model. The model describes kinetics of cell populations under periodic treatment by high doses of a phase-specific cytotoxic agent with blocking effect. The dependence of cell survival on cell-cycle parameters and parameters of treatment is studied and the results are presented in a suitable form for experimental testing and practical usage. The obtained results give a theoretical basis for using the resonance effect in a self-renewing tissue control study and estimation of cell-cycle parameters of various cells including morphologically unrecognizable ones. The quantitative comparison of the theoretical predictions with experimental data obtained for mice spleen colony forming units (CFUs), cells of mice small intestine epithelium as well L1210 leukemic cells is carried out. The excellent agreement between theoretical and experimental results is demonstrated and the estimations of cycle parameters of CFUs, cryptogenic and transit cells of mice intestine epithelium are obtained. Biological consequences of the obtained results are discussed and the new hypothesis relative to self-renewing tissue proliferative control is put forward.

[Dynamics of proliferating cell populations during exposure to periodic phase-specific factors]. / Dinamika populiatsii proliferiruiushchikh kletok pri periodicheskom fazospetsificheskom vozdeistvii.

Dynamics of the numbers of population of proliferating cells under periodic phase-specific cytotoxic effect with and without blocking action was studied on the basis of a mathematical model. It has been shown that at real values of the model parameters after the beginning of the effects the population number exponentially depends of time. Dependence of the population number dynamics on integral parameters of the cell cycle and the exposure regime was studied. It has been shown that at certain periods a resonance decrease of the damage of the population cells must be observed. The values of the periods corresponding to the resonance decrease of the damage are determined mainly by the mean duration of the cell cycle and the time of blocking action, at small duration of the blocking action they are approximately multiple to the average time of the cell cycle. The theoretical predictions are proved experimentally in the experiments on determining the relationship between the damage of small intestine epithelium and mouse survival and the period of repeated periodic injections of S-phase-specific cytotoxic agent hydroxyurea. A distinct resonance increase of mouse survival and decreased damage of the epithelium were observed under the injections of oxyurea with the periods near to the mean and doubled mean time of the cell cycle of crypt enterocytes. The results obtained not only support the correctness of theoretical predictions, but make it possible to estimate the parameters of the stem cell cycle of mouse small intestine epithelium. They also show that this approach can be used for reducing aftereffects of chemotherapy by means of phase-specific agents.

[The resonance nature of the dependence of epithelial lesions of the small intestine in mice on the interval between injections of an S-phase-specific agent--hydroxyurea]. / Rezonansnyi kharakter zavisimosti porazheniia épiteliia tonkoi kishki myshei ot intervala mezhdu in''ektsiiami S-fazospetsificheskogo agenta--oksimocheviny.

The dependence of the injury of murine small intestinal epithelium on the interval between multiple regular injections of hydroxyurea (HU) was investigated. Mice were injected 8 times with HU (5 mg per injection) in different experimental groups of animals, and the interval between injections varied from 6 to 19 hours. With the intervals between the injections close to 8 or 16.5 hours the resonance decrease of the injury was observed whereas the intervals of 6, 12 and 19 hours corresponded to maximum injury.

[Significance of the intervals of hydroxyurea administration for epithelial damage to the small intestine in mice]. / Znachenie intervalov vvedeniia oksimocheviny dlia povrezhdeniia épiteliia tonkoi kishki myshei.

The states of the murine small intestinal epithelium 6, 30 and 78 h after the end of the multiple regular injections of hydroxyurea (HU) were analysed with the aid of the light and electron microscopy. The course of 6 regular injections of 5 mg/mouse HU was begun 24 h after the initiating gamma-irradiation in a dose 200 rad and the interval between injections was varied from 7 to 19 h for different experimental groups of mice. The analysis of the epithelial state revealed two minima of the tissue damage which correspond to the courses of HU injections with the intervals close 9 h and 16.5 h.

[Apoptosis and regeneration of enterocytes in experimental atrophy of the small intestine mucosa]. / Apoptoz i obnovlenie énterotsitov pri éksperimental'noi atrofii slizistoi obolochki tonkoi kishki.

Apoptosis, one of the types of cell deaths, participates in regulating the size of regenerated tissue. Severe atrophy of small intestine mucosa in mice was caused by the administration of hydroxyurea solution. The degree of atrophy correlated with a lowering mitotic activity and DNA synthesis in the epithelium of crypts. Apoptotic bodies were situated above the basal membrane, in crypt lumen or were phagocytized by adjacent epithelial cells. The development of atrophy, as well as the regeneration of mucosa can be predicted by the relation between mitosis and apoptosis.

[Resonance nature of the relation of mouse survival to the interval between administrations of the S-specific agent--hydroxyurea]. / Rezonansnyi kharakter zavisimosti vyzhivaemosti myshei ot intervala mezhdu vvedeniiami S-fazospetsificheskogo agenta--oksimocheviny.

The dependence of murine survival on the intervals between periodic hydroxyurea (HU) injections was studied. The single dose of HU comprised 250 mg per kg. Intervals between injections varied from 5 to 19 hours while their number changed from 6 to 9 in different experiments. A resonant increase in the survival was observed under HU administered every 8-9 or 16 1/2 hours.

[Relation between the toxic killing of mouse small intestine enterocytes and the interval between the administration of the S-specific agent hydroxyurea]. / Zavisimost' toksicheskoi gibeli énterotsitov tonkoi kishki myshei ot intervala mezhdu vvedeniiami S-fazovo-spetsificheskogo agenta-oksimocheviny.

The dependence of small intestinal lesions on the interval between repeated regular injections of hydroxyurea was studied. The doses of 1 g/kg and 0.2 g/kg were used. To activate the enterocyte proliferation the mice were pre-irradiated at 2 Gy. 3-4 hours after the last hydroxyurea injection the two distinct minimum points of epithelial lesions were observed at interinjection intervals of 9 and 16.5 hours. The result was dose-independent.

[Optimization of the exposure of normal and transformed cell populations to phase-specific agents]. / Optimizatsiia vozdeistviia na normal'nye i transformirovannye populiatsii kletok fazospetsificheskimi agentami.

Dynamic behavior of stem cells population of the "critical" tissue (normal population) and tumor cell population under periodic treatment with a phase-specific cytotoxic agent was considered. The results were used for optimization of anticancer chemotherapy. The schedules of treatment were found which provide a maximum rate of tumor-cell elimination for any given rate of the normal population size decrease. If the mean generation times of normal and tumor populations differ (which was stated for many tumors), usage of the optimal period markedly increases the selectivity of therapy, while application of other periods can result in selective elimination of the normal population. Problems concerned with practical realization of the proposed regimes are discussed.

[Relation of the survival of mouse hematopoietic stem cells to the interval between periodic injections of hydroxyurea]. / Zavisimost' vyzhivaniia stvolovykh krovetvornykh kletok mysh ot intervala mezhdu periodicheskimi vvedeniiami oksimocheviny.

The dependence was studied between the survival of proliferating hemopoietic stem cells (CFUs) on the interval between regular multiple injections of hydroxyurea (HU). Before HU injection mice were irradiated in a dose of 200 rad to trigger the proliferation of CFUs. The dependence obtained is resonance in character, with a maximum being attained during injections made with a 12-hour interval. Comparison with the mathematical model indicates that interval corresponding with the maximal survival should correlate with the mean time of the generation of proliferating CFUs.

[Influence of erythrocyte destruction kinetics on the dynamic behavior of the erythropoiesis system]. / Vliianie kinetiki raspada éritrotsitov na dinamicheskoe povedenie sistemy éritropoéza.

A simple mathematical model was used to analyse the influence of cell destruction kinetics on the dynamics of erythropoietic system. Two ways of erythrocyte destruction were compared: random destruction and destruction of cells at their maximum age. It has been shown that mostly used description of erythrocyte destruction as a random process results in a great stability increase (about three times in the critical feedback coefficient). Oscillations with large periods (20 divided by 200 days) may exist in the system.

Dynamic stability of steady states and static stabilization in unbranched metabolic pathways.

The paper is concerned with the conditions of dynamic (asymptotic) stability of steady states in unbranched metabolic pathways. The stationary flux in such pathways is generally determined by the concentration of the end product due to the effector action of this product on the reactions proceeding in its synthetic pathway. The delay in feedback circuits causes violation of dynamic stability at large static stabilization factors. A methods permitting analytic estimation of the critical stabilization factor is suggested. Sufficient and necessary conditions for asymptotic stability of the steady state in the general case of the pathway with a single feedback loop have been established. Mechanisms for maintenance of the steady state asymptotic stability at large static stabilization factors are studied. It has been shown that the range of dynamic stability can be widened greatly, if the pathway contains one or two reactions (but not more) of relatively small effective rate constants. Short strong negative feedback is also found to extend considerably the range of dynamic stability of the pathway. The feedback is more effective if it acts on the reaction with small effective rate constant.

[Regulation of erythrocyte energy metabolism. Dependence of glycolysis characteristics on donor individual parameters]. / Reguliatsiia énergeticheskogo metabolizma éritrotsita. Zavisimost' kharakteristiki glikoliza ot individual'nykh parametrov krovi.

Approximate analytical expression is obtained for the dependence of erythrocyte glycolytic rate on ATP concentration and donor individual parameters: activation of hexokinase and phosphofructokinase, some kinetic constants and total concentration of adenine nucleotides in the cell.

[Mathematical model for energy metabolism in erythrocytes. Independence of scaled glycolytic characteristics of individual features of the donors]. / Matematicheskaia model' energeticheskogo metabolizma eritrotsitov. Nezavicimost' normirovannoi kharakteristiki glikoliza ot individual'nykh osobennostei donorov.

A mathematical model for energy metabolism of erythrocytes includes the first three enzymes of the Embden-Meyerhof pathway as well as total ATPase. It was assumed that the ATPase is strongly inhibited by AMP. The model was used to calculate the dependence of the rate of ATP production and glucose consumption on ATP concentration, i. e. characteristics of glycolysis. The scaled characteristics were plotted in relative units, the normal physiological values of the rates of glucose consumption and ATP concentration being taken for 100%. It was shown that the scaled characteristics show a coincidence, the precision being 20%, when the values of the enzyme activities and other parameters of the model vary within 100%. This agrees with the data, where the scaled characteristics of glycolysis in different donors (in all donors) coincide with experimental precision, while the values of the rates and concentrations with respect to the absolute value varied within 100%.

[Interrelationship between the rate of ATP-consuming processes and ATP concentration in intact erythrocytes]. / Sviaz' mezhdu skorost'iu ATP-potrebliiaiushchikh protsessov i kontsentratsiei ATP v intaktnykh eritrotsitakh.

The behaviour of ATP concentrations in intact erythrocytes upon glucose addition to the erythrocyte suspension at a constant rate was studied. At different rates of glucose addition lower than the rate of glucose consumption at normalcy, the corresponding ATP concentrations are established, which are dependent on the rate of glucose influx. The 20% decrease of ATP concentration from the physiological value is observed at a 2-fold decrease of the flux. The results obtained made it possible to establish the dependence of the total rate of ATP consumption on ATP concentration in intact erythrocytes. This dependence is non-linear and shows a dramatic rise in the proximity of the physiological point.

[Nonlinear resistance in a diffuse trigger system. Biological barrier]. / Nelineinaia ustoichivost' v raspredelennoi triggernoi sisteme. Biologicheskii bar'er.

Mathematical model of antagonistic interrelations of biological species in which the migration is taken into account is considered. The ecological nish, occupied by one of the antagonistic species cannot be populated by the other if the initial quantity of the new species does not exceed the particular value. This is considered as the demonstration of the ecological barrier. The approach to the investigation of the nonlinear stability for spatially distributed systems can be applied to other biophysical problems.

[Mathematical model of the immune response. IV. Threshold character of the infectious process]. / Matematicheskaia model' immunnoi reaktsii. IV. Porogovyi kharakter infektsionnogo protsessa.

A model of immune reaction is described which takes into account the agressive behaviour of infectious microorganisms in the macroorganism. The model allows a description of threshold relationship between the infection process and the initial dose of the antigen. According to the model depending on the parameters and initial data the following regimes are possible: destruction of the antigen without a disease, its destruction after a disease, coexistence of the antigen with specific antibodies, recurrent course of the disease, unlimited multiplication of the antigen.

[Mathematical model of the immune reaction. III. Description of the infectious process in relation to changes in the number of B-lymphocytes]. / Matematicheskaia model' immunnoi reaktsii. III. Opisanie infektsionnogo protsessa s uchetom izmeneniia chislennosti B-limfotsitov.

A model of humoral immune reaction which takes into account explicitely the time lags in the immune response is considered. Possible regimes are described.