Biological applications of the theory of birth-and-death processes

In this review, we discuss applications of the theory of birth-and-death processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birth-and-death processes, with some straightforward additions such as innovation, are a simple, natural and formal framework for modeling a vast variety of biological processes such as population dynamics, speciation, genome evolution, including growth of paralogous gene families and horizontal gene transfer and somatic evolution of cancers. We further describe how empirical data, e.g. distributions of paralogous gene family size, can be used to choose the model that best reflects the actual course of evolution among different versions of birth-death-and-innovation models. We conclude that birth-and-death processes, thanks to their mathematical transparency, flexibility and relevance to fundamental biological processes, are going to be an indispensable mathematical tool for the burgeoning field of systems biology.     

Changes of gas exchange parameters and of functional-biochemical properties of erythrocytes in dynamics of experimental anemia in rats

In experiments on male Wistar rats, in a specially constructed computerized installation, O₂ consumption by the animals in comparison with changes of hematological, biochemical, and rheological blood properties is studied after anemization—acute blood loss (12–15% of the total blood mass). An increase of the O₂ consumption by the organism and tissues by 18–28% has been revealed for the first 7 days after the blood loss, in spite of a pronounced decrease of hematocrit and of the amount of erythrocytes and hemoglobin in peripheral blood by 20–25% of the initial level. There was a 5–10-fold increase of the content of immature erythrocyte forms—reticulocytes and a progressive rise of cell acidic resistance, which is characteristic of young erythrocyte forms. An increase of O₂ consumption at a decrease of the blood oxygen capacity (a low hemoglobin level) seems to be due to the more efficient transport and yield of O₂ to tissues. At the 3rd and 7th day after the blood loss, activity of Na,K-ATPase has been found to increase by 60% and 20%, respectively. Analysis of the erythrocyte rheological properties has shown that the maximal firmness of aggregates (U_q) and the aggregation rate (1/T) decrease progressively beginning from 3 days after the blood loss; index of deformability (I_max) turned out to be elevated by 7–11%, probably due to an increase of the cell membrane elasticity. The conclusion is made that changes of erythrocyte rheological properties are interconnected with changes of the Na,K-ATPase activity and are directed at optimization of blood circulation in large vessels and the capillary network.     

Cortical potential imaging using L-curve and GCV method to choose the regularisation parameter

Background

The electroencephalography (EEG) is an attractive and a simple technique to measure the brain activity. It is attractive due its excellent temporal resolution and simple due to its non-invasiveness and sensor design. However, the spatial resolution of EEG is reduced due to the low conducting skull. In this paper, we compute the potential distribution over the closed surface covering the brain (cortex) from the EEG scalp potential. We compare two methods – L-curve and generalised cross validation (GCV) used to obtain the regularisation parameter and also investigate the feasibility in applying such techniques to N170 component of the visually evoked potential (VEP) data.

Methods

Using the image data set of the visible human man (VHM), a finite difference method (FDM) model of the head was constructed. The EEG dataset (256-channel) used was the N170 component of the VEP. A forward transfer matrix relating the cortical potential to the scalp potential was obtained. Using Tikhonov regularisation, the potential distribution over the cortex was obtained.

Results

The cortical potential distribution for three subjects was solved using both L-curve and GCV method. A total of 18 cortical potential distributions were obtained (3 subjects with three stimuli each – fearful face, neutral face, control objects).

Conclusions

The GCV method is a more robust method compared to L-curve to find the optimal regularisation parameter. Cortical potential imaging is a reliable method to obtain the potential distribution over cortex for VEP data.

     

Dynamics of hematologic parameters and of the erythrocyte deformability index at the juvenal period of rats and guinea pigs

In experiments on Wistar rats and guinea pigs, dynamics of blood hematological parameters and of erythrocyte deformability index (DI) was studied from birth to the 3-month age. In rats, significant changes were revealed in the picture of blood for the first month of postnatal ontogenesis. The number of erythrocytes and leukocytes sharply increased, the number of reticulocytes decreased, their ratio in the degree of maturity was changed. The mean erythrocyte volume and their acidic resistance also decreased. Prior to the 4-week age, DI increased, which was accompanied by an increase in the hemoglobin hydration and flattening of the erythrocyte shape. Then ID decreased and erythrocytes acquired the more spherical shape. After the critical period of 3-4 weeks the studied parameters reached plateau. In guinea pigs, at this period of ontogenesis the hematologic parameters had sufficiently stable values, except for a DI decrease from birth to 1.5 months, then it rose without reaching the initial values. Like in the case with rats, DI had a close correlation with the erythrocyte parameter of shape (O min).     

On the spread of epidemics in a closed heterogeneous population

Heterogeneity is an important property of any population experiencing a disease. Here we apply general methods of the theory of heterogeneous populations to the simplest mathematical models in epidemiology. In particular, an SIR (susceptible-infective-removed) model is formulated and analyzed when susceptibility to or infectivity of a particular disease is distributed. It is shown that a heterogeneous model can be reduced to a homogeneous model with a nonlinear transmission function, which is given in explicit form. The widely used power transmission function is deduced from the model with distributed susceptibility and infectivity with the initial gamma-distribution of the disease parameters. Therefore, a mechanistic derivation of the phenomenological model, which is believed to mimic reality with high accuracy, is provided. The equation for the final size of an epidemic for an arbitrary initial distribution of susceptibility is found. The implications of population heterogeneity are discussed, in particular, it is pointed out that usual moment-closure methods can lead to erroneous conclusions if applied for the study of the long-term behavior of the models.     

Two new species of Perichaena (Myxomycetes) from arid areas of Russia and Kazakhstan

Two new myxomycete species from dry steppe and desert communities of the Caspian Lowland (Russia) and central Kazakhstan are described and illustrated. They are placed tentatively within genus Perichaena, which does include species with a reduced capillitium and single-layered peridium. Both species were found repeatedly in moist chamber cultures; P. heterospinispora appeared on leaf litter and twigs, whereas P. polygonospora occurred on leaf litter and weathered dung of rodents. Both species have spore ornamentation that is unique for members of genera Licea and Perichaena. The spore ornamentation of the first species includes scattered large, pyramid-like spines 0.9-1.2 microm high that sometimes have enlarged ends. Among these spines the spore surface is covered by evenly and densely distributed warts that are visible only by SEM. The second species is characterized by angular spores with a coarse network of rounded ridges. The areas among these ridges bear scattered composite warts 0.3-0.5 microm high that sometimes coalesce to form clusters but more often are distributed evenly and densely and are visible only by SEM. The stability of the taxonomic characters of both species was confirmed by several collections from different regions obtained in 2 y. The morphology of the fructifications of the two myxomycetes was examined with both scanning electron and light microscopy, and micrographs of all relevant features are presented.     

Changes of physiological and biochemical characteristics of rat erythrocytes after blood loss

In experiments on Wistar male rats, changes are studied of erythrocyte hematological, biochemical (activities of transport ATPases), and rheological properties (capability for aggregation and deformability) 7 days after bloodletting of 12–15% of the total blood mass. It has been shown that alongside with an elevation of erythrocyte volume and of the number of immature cells—reticulocytes, there was a statistically significant increase of Na,K-ATPase and Ca-ATPase activities in the whole erythrocytes and in their membrane preparations—ghosts, the increment of activity in the case of Na,K-ATPase being essentially higher in the whole cells. This indicates the appearance of an enzyme activator inside the erythrocytes. There are also revealed a decrease of firmness of erythrocyte aggregates, a deceleration of spontaneous aggregation, and an increase of index of erythrocyte deformability. The conclusion is made that changes of erythrocyte rheological properties are interconnected with changes of the Na,K-ATPase activity and are aimed at optimization of blood circulation in large vessels and capillary network.     

Nonglutamate pore residues in ion selection and conduction in voltage-gated Ca(2+) channels

High-affinity, intrapore binding of Ca(2+) over competing ions is the essential feature in the ion selectivity mechanism of voltage-gated Ca(2+) channels. At the same time, several million Ca(2+) ions can travel each second through the pore of a single open Ca(2+) channel. How such high Ca(2+) flux is achieved in the face of tight Ca(2+) binding is a current area of inquiry, particularly from a structural point of view. The ion selectivity locus comprises four glutamate residues within the channel's pore. These glutamates make unequal contributions to Ca(2+) binding, underscoring a role for neighboring residues in pore function. By comparing two Ca(2+) channels (the L-type alpha(1C), and the non-L-type alpha(1A)) that differ in their pore properties but only differ at a single amino acid position near the selectivity locus, we have identified the amino-terminal neighbor of the glutamate residue in motif III as a determinant of pore function. This position is more important in the function of alpha(1C) channels than in alpha(1A) channels. For a systematic series of mutations at this pore position in alpha(1C), both unitary Ba(2+) conductance and Cd(2+) block of Ba(2+) current varied with residue volume. Pore mutations designed to make alpha(1C) more like alpha(1A) and vice versa revealed that relative selectivity for Ba(2+) over K(+) depended almost solely on pore sequence and not channel type. Analysis of thermodynamic mutant cycles indicates that the motif III neighbor normally interacts in a cooperative fashion with the locus, molding the functional behavior of the pore.     

Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics

Background:  

One of the mechanisms that ensure cancer robustness is tumor heterogeneity, and its effects on tumor cells dynamics have to be taken into account when studying cancer progression. There is no unifying theoretical framework in mathematical modeling of carcinogenesis that would account for parametric heterogeneity.