Description of Na, K-ATPase activation by monovalent cations using a simplified mathematical model

A simple mathematic model describing the activation Na,K-ATPase system by univalent cations is proposed. The constants for the enzyme activation values by each of the ions in the presence of a fixed concentration of the other ion have been calculated. The substitution of these values into the common equation describing the behaviour of the whole system according to the given model gives the curve of Na,K-ATPase activity change in dependence of Na/K ration at the same total concentration 150 mM. The experimental points correspond to the curve.     

A triangle lattice model that predicts transmembrane helix configuration using a polar jigsaw puzzle

We developed a method of predicting the tertiary structures of seven transmembrane helical proteins in triangle lattice models, assuming that the configuration of helices is stabilized by polar interactions. Triangle lattice models having 12 or 11 nearest neighbor pairs were used as general templates of a seven-helix system, then the orientation angles of all helices were varied at intervals of 15 degrees. The polar interaction energy for all possible positions of each helix was estimated using the calculated polar indices of transmembrane helices. An automated system was constructed and applied to bacteriorhodopsin, a typical membrane protein with seven transmembrane helices. The predicted optimal and actual structures were similar. The top 100 predicted helical configurations indicated that the helix-triangle, CFG, occurred at the highest frequency. In fact, this helix-triangle of bacteriorhodopsin forms an active proton-pumping site, suggesting that the present method can identify functionally important helices in membrane proteins. The possibility of studying the structure change of bacteriorhodopsin during the functional process by this method is discussed, and may serve to explain the experimental structures of photointermediate states.     

Dynamic state-dependent modelling predicts optimal usage patterns of responsive defences

Chemical defences against predation often involve responses to specific predation events where the prey expels fluids, such as haemolymph or gut contents, which are aversive to the predator. The common link is that each predation attempt that is averted results in an energetic cost and a reduction in the chemical defences of the prey, which might leave the prey vulnerable if the next predation attempt occurs soon afterwards. Since prey appear to be able to control the magnitude of their responses, we should expect them to trade-off the need to repel the current threat against the need to preserve defences against future threats and conserve energy for other essential activities. Here we use dynamic state-dependent models to predict optimal strategies of defence deployment in the juvenile stage of an animal that has to survive to maturation. We explore the importance of resource level, predator density, and the costs of making defences on the magnitude of the responses and optimal age and size at maturation. We predict the patterns of investment and the magnitude of the deployment of defences to potentially multiple attacks over the juvenile period, and show that responses should be smaller when the costs of defences and/or predation risk are higher. The model enables us to predict that animals in which defences benefit the adult stage will employ different strategies than those that do not use the same defences as adults, and thereby experience a smaller reduction in body size as a result of repeated attacks. We also explore the effect of the importance of adult size, and find that the sex and mating system of the prey should also affect defensive strategies. Our work provides the first predictive theory of the adaptive use of responsive defences across taxa.     

Artificial neural network model for the generation of muscle activation patterns for human locomotion.

Skilled locomotor behaviour requires information from various levels within the central nervous system (CNS). Mathematical models have permitted researchers to simulate various mechanisms in order to understand the organization of the locomotor control system. While it is difficult to adequately characterize the numerous inputs to the locomotor control system, an alternative strategy may be to use a kinematic movement plan to represent the complex inputs to the locomotor control system based on the possibility that the CNS may plan movements at a kinematic level. We propose the use of artificial neural network (ANN) models to represent the transformation of a kinematic plan into the necessary motor patterns. Essentially, kinematic representation of the actual limb movement was used as the input to an ANN model which generated the EMG activity of 8 muscles of the lower limb and trunk. Data from a wide variety of gait conditions was necessary to develop a robust model that could accommodate various environmental conditions encountered during everyday activity. A total of 120 walking strides representing normal walking and ten conditions where the normal gait was modified in terms of cadence, stride length, stance width or required foot clearance. The final network was assessed on its ability to predict the EMG activity on individual walking trials as well as its ability to represent the general activation pattern of a particular gait condition. The predicted EMG patterns closely matched those recorded experimentally, exhibiting the appropriate magnitude and temporal phasing required for each modification. Only 2 of the 96 muscle/gait conditions had RMS errors above 0.10, only 5 muscle/gait conditions exhibited correlations below 0.80 (most were above 0.90) and only 25 muscle/gait conditions deviated outside the normal range of muscle activity for more than 25% of the gait cycle. These results indicate the ability of single network ANNs to represent the transformation between a kinematic movement plan and the necessary muscle activations for normal steady state locomotion but they were also able to generate muscle activation patterns for conditions requiring changes in walking speed, foot placement and foot clearance. The abilities of this type of network have implications towards both the fundamental understanding of the control of locomotion and practical realizations of artificial control systems for use in rehabilitation medicine.     

Fermentation process diagnosis using a mathematical model

Summary Intriguing physiology of a solvent-producing strain ofClostridium acetobutylicum led to the synthesis of a mathematical model of the acetone-butanol fermentation process. The model presented is capable of describing the process dynamics and the culture behavior during a standard and a substandard acetone-butanol fermentation. In addition to the process kinetic parameters, the model includes the culture physiological parameters, such as the cellular membrane permeability and the number of membrane sites for active tansport of sugar. Computer process simulation studies for different culture conditions used the model, and quantitatively pointed out the importance of selected culture parameters that characterize the cell membrane behaviour and play an important role in the control of solvent synthesis by the cell. The theoretical predictions by the new model were confirmed by experimental determination of the cellular membrane permeability.     

Trunk muscle activation patterns during walking at different speeds.

Investigations of trunk muscle activation during gait are rare in the literature. As yet, the small body of literature on trunk muscle activation during gait does not include any systematic study on the influence of walking speed. Therefore, the aim of this study was to analyze trunk muscle activation patterns at different walking speeds. Fifteen healthy men were investigated during walking on a treadmill at speeds of 2, 3, 4, 5 and 6 km/h. Five trunk muscles were investigated using surface EMG (SEMG). Data were time normalized according to stride time and grand averaged SEMG curves were calculated. From these data stride characteristics were extracted: mean SEMG amplitude, minimum SEMG level and the variation coefficient (VC) over the stride period. With increasing walking speed, muscle activation patterns remained similar in terms of phase dependent activation during stride, but mean amplitudes increased generally. Phasic activation, indicated by VC, increased also, but remained almost unchanged for the back muscles (lumbar multifidus and erector spinae) between 4 and 6 km/h. During stride, minimum amplitude reached a minimum at 4 km/h for the back muscles, but for internal oblique muscle it decreased continuously from 2 to 6 km/h. Cumulative sidewise activation of all investigated muscles reached maximum amplitudes during the contralateral heel strike and propulsion phases. The observed changes argue for a speed dependent modulation of activation of trunk muscles within the investigated range of walking speeds prior to strictly maintaining certain activation characteristics for all walking speeds.     

On the optimal duration of memory of losing a conflict--a mathematical model approach

In male broad-horned flour beetles, Gnatocerus cornutus, losers of conflicts avoid fighting at subsequent encounters. The loser effect lasts for 4 days. It is considered that the memory of losing remains for 4 days. The duration of the memory is expected to affect the fitness, and the duration, 4 days, is expected to be optimal. We consider the fitness of a mutant in an homogeneous population to obtain the optimal duration. Here we carry out simulations using an individual-based model. The results suggest that the trade-off of getting mating chances and avoiding damage can cause the optimal duration of the memory, and that the decay in time of the female population is an important factor.     

Force-frequency relations in hypertrophic heart muscle: a mathematical model for excitation-contraction coupling.

Static and dynamic chrono-inotropic responses were recorded from both normal and hypertrophic rat auricular myocardium. The slope of the static force-frequency relation for hypertrophic hearts was steeper than that for control hearts. Computer experiments were designed to study the cellular mechanisms underlying the changes in the force-frequency response associated with heart hypertrophy, with the aid of a mathematical model for excitation-contraction coupling in rat heart. A set of equations was derived which permitted to study the effects on the chronoinotropic relations of both the geometrical dimensions of cardiomyocytes and the sarcoplasmic reticulum, and of the variation in activity of mechanisms for Ca movements through the sarcolemma and the sarcoreticular membrane. A comparison of data obtained from simulated and real experiments suggested that the features characteristic of force-frequency relations for hypertrophic heart are a result of an enhanced volume of intracellular Ca-stores rather than of the total volume of the cardiomyocyte.     

Neutron and X-ray effects on small intestine summarized by using a mathematical model or paradigm.

The responses of intestinal tissues to ionizing radiation can be described by comparing irradiated cell populations qualitatively or quantitatively with corresponding controls. This paper describes quantitative data obtained from resin-embedded sections of neutron-irradiated mouse small intestine at different times after treatment. Information is collected by counting cells or structures present per complete circumference. The data are assessed by using standard statistical tests, which show that early mitotic arrest precedes changes in goblet, absorptive, endocrine and stromal cells and a decrease in crypt numbers. The data can also produce ratios of irradiated: control figures for cells or structural elements. These ratios, along with tissue area measurements, can be used to summarize the structural damage as a composite graph and table, including a total figure, known as the Morphological Index. This is used to quantify the temporal response of the wall as a whole and to compare the effects of different qualities of radiation, here X-ray and cyclotron-produced neutron radiations. It is possible that such analysis can be used predictively along with other reference data to identify the treatment, dose and time required to produce observed tissue damage.     

A mathematical model for calculating the vector magnetic field of a single muscle fiber.

A mathematical model is described for calculating the volume-conducted magnetic field from active muscle fibers in an anisotropic bundle. With earlier models, the azimuthal magnetic field of a nerve bundle was calculated and the results were compared with the fields measured by toroidal pickup coils. The present model is capable of evaluating all three of the magnetic field components and is thus applicable for analyzing SQUID magnetometer recordings of fields from a muscle bundle. The component of the magnetic field parallel to the fiber axis is more than an order of magnitude smaller than either of the other two components. The amplitude of the magnetic signal is strongly dependent upon the anisotropy of the muscle bundle, the intracellular conductivity, the radius of the muscle fiber, the radius of the muscle bundle, and the location of the fiber in the muscle bundle. The peak-to-peak amplitude of the single-muscle-fiber action field increases linearly with increasing intracellular conductivity, as the square of the radius of the muscle fiber, and exponentially with the distance between the location of the fiber and the center of the bundle.     

A mathematical model and a computerized simulation of PCR using complex templates.

A mathematical model and a computer simulation were used to study PCR specificity. The model describes the occurrences of non-targeted PCR products formed through random primer-template interactions. The PCR simulation scans DNA sequence databases with primers pairs. According to the model prediction, PCR with complex templates should rarely yield non-targeted products under typical reaction conditions. This is surprising as such products are often amplified in real PCR under conditions optimized for stringency. The causes for this 'PCR paradox' were investigated by comparing the model predictions with simulation results. We found that deviations from randomness in sequences from real genomes could not explain the frequent occurrence of non-targeted products in real PCR. The most likely explanation to the 'PCR paradox' is a relatively high tolerance of PCR to mismatches. The model also predicts that mismatch tolerance has the strongest effect on the number of non-targeted products, followed by primer length, template size and product size limit. The model and the simulation can be utilized for PCR studies, primer design and probing DNA uniqueness and randomness.     

A mathematical model on the optimal timing of offspring desertion

We consider the offspring desertion as the optimal strategy for the deserter parent, analyzing a mathematical model for its expected reproductive success. It is shown that the optimality of the offspring desertion significantly depends on the offsprings' birth timing in the mating season, and on the other ecological parameters characterizing the innate nature of considered animals. Especially, the desertion is less likely to occur for the offsprings born in the later period of mating season. It is also implied that the offspring desertion after a partially biparental care would be observable only with a specific condition.     

New determinants of firing rates and patterns of vasopressinergic magnocellular neurons: predictions using a mathematical model of osmodetection

Arginine vasopressin (AVP), one of the most important hormones involved in hydromineral homeostasis, is secreted by hypothalamic magnocellular neurons (MCNs). Here, we implemented two critical parameters for MCN physiology into a Hodgkin-Huxley simulation of the MCN. By incorporating the mechanosensitive channel (MSC) responsible for osmodetection and the synaptic inputs whose frequencies are modulated by changes in ambient osmolality into our model, we were able to develop an improved model with increased physiological relevance and gain new insight into the determinants of the firing patterns of AVP magnocellular neurons. Our results with this MCN model predict that 1) a single MCN is able to display all the firing patterns experimentally observed: silent, irregular, phasic and continuous firing patterns; 2) under conditions of hyperosmolality, burst durations are regulated by the frequency-dependent fatigue of dynorphin secretion; and 3) the transitions between firing patterns are controlled by EPSP and IPSP frequencies (0, 2, 4, 8, 16, 32, 64 and 128 Hz). Moreover, this simulation predicts that EPSPs and IPSPs do not modify the spiking frequency (SF) of phasic firing patterns (0.0034 Hz/Hz [EPSP]; 0.0012 Hz/Hz [IPSP]). Rather, these afferents strongly regulate SF during irregular and continuous firing patterns (0.075 Hz/Hz [EPSP]; 0.027 Hz/Hz [IPSP]). The use of the realistic MCN model developed here allows for an improved understanding of the determinants driving the firing patterns and spiking frequencies of vasopressinergic magnocellular neurons.     

A biomechanical model to estimate corrective changes in muscle activation patterns for stroke patients

We have created a model to estimate the corrective changes in muscle activation patterns needed for a person who has had a stroke to walk with an improved gait-nearing that of an unimpaired person. Using this model, we examined how different functional electrical stimulation (FES) protocols would alter gait patterns. The approach is based on an electromyographically (EMG)-driven model to estimate joint moments. Different stimulation protocols were examined, which generated different corrective muscle activation patterns. These approaches grouped the muscles together into flexor and extensor groups (to simulate FES using surface electrodes) or left each muscle to vary independently (to simulate FES using intramuscular electrodes). In addition, we limited the maximal change in muscle activation (to reduce fatigue). We observed that with the two protocols (grouped and ungrouped muscles), the calculated corrective changes in muscle activation yielded improved joint moments nearly matching those of unimpaired subjects. The protocols yielded different muscle activation patterns, which could be selected based on practical condition. These calculated corrective muscle activation changes can be used in studying FES protocols, to determine the feasibility of gait retraining with FES for a given subject and to determine which protocols are most reasonable.     

Regulation of thrombocytopoiesis studied by a mathematical model

A mathematical model of thrombocytopoiesis is proposed which accounts for the recent data on its regulation. It is shown that the compensatory response of the system to a decrease in the level of thrombocytes in the blood is controlled by the total amount of thrombocytes and megakaryocytes. The proliferation intensity of megakaryocytes and the total number of thrombocytes reveal, respectively, a lineary and a logarithmical dependence on the total number of thrombocytes and megakaryocytes. The limits of the post-transfusion level of thrombocytes are defined, within which the thrombocytopoiesis is controlled only by the number of thrombocytes. The values of parameters characterizing the behaviour of the thrombocytopoiesis system are calculated.     

Numerical simulation of motility patterns of the small bowel. II. Comparative pharmacological validation of a mathematical model.

A model of a locus of the small bowel, described earlier by the authors (Miftakhov et al., 1999) was validated in a comparison of the results of numerical simulations of pharmacological compounds to their effects in biological studies. The actions of the following four classes of drugs were simulated, those: (i) acting on the sarcoplasmic reticulum, (ii) altering the permeability of L- and T-type Ca(2+)channels on the smooth muscle membrane, (iii) motilides, and (iv) benzodiazepines. The strong qualitative resemblance between the theoretical and experimental results supports the robustness of the model.     

A mathematical model for predicting relative muscle force with a perturbation analysis of selected muscle parameters

A mathematical model is presented that predicts relative muscle forces using a minimum of experimentally derived input data. Tests of this model against literature values for maximum muscle force of four cat hindlimb muscles show a maximum error of only 5%. A perturbation analysis using this model demonstrates its sensitivity and applicability, as well as the congruence between this model and previous theoretical discussions of muscle function.