Mathematical biophysics of cell division

The new fundamental equation, based on the principle of maximum energy exchange, is applied to the case of elongation and constriction of a cell. A simple case is treated as an illustration, and it is shown that the elongation curves thus obtained are of the same character as in the old theory.     

Shear-layer detection in poststenotic flow by spectrum analysis of Doppler signals

Spectrum analysis of the Doppler signals was performed 0.5 tube diameters downstream from an axisymmetric constriction with an area reduction of 80 percent in steady flow at a jet Reynolds number of 2840. Both pulsed and continuous wave (CW) Doppler spectra showed significant reverse flow components in the separated flow. The pulsed Doppler spectra exhibited sudden changes when the sample volume crossed the shear layer between the center jet and the separated flow. A power spectrum equation was theoretically derived from continuity of flow to define the Doppler shift frequency for the shear layer velocity. The CW Doppler spectrum showed a minimum spectrum density at a frequency which equalled the shear layer Doppler shift frequency derived from the equation. The pulsed spectra exhibited the sudden changes at the same frequency as well.     

A mathematical analysis of elongation and constriction in cell division

An equation for the rate of elongation of a dividing egg is integrated and generalized. The rates of elongation and constriction of a number of eggs under various conditions are analyzed and compared with the theoretical predictions. The theory accounts rather well for a large body of data on elongation and constriction. The general shapes of the elongation and constriction curves are predicted and the orders of magnitude of the parameters are satisfactory. One of the parameters for the elongation curves is related theoretically to the parameter of the constriction curves, and the correct order of magnitude is obtained if one parameter is predicted from the other.     

Congenital amputation of the toes with replantation at the site of a constriction band in the mid-calf. Case report

We present a case of congenital amputation of the toes with replantation at the site of a constriction band in the mid-calf. The theories of the etiology of constriction bands are discussed, and the importance of this case in supporting the amniotic constriction band theory is emphasized.     

Tracheal stenosis: a flow dynamics study.

Patients referred for treatment of tracheal stenosis typically are asymptomatic until critical narrowing of the airway occurs, which then requires immediate intervention. To understand how tracheal stenosis affects local pressure drops and explore how a dramatic increase in pressure drop could possibly be detected at an early stage, a computational fluid dynamics (CFD) study was undertaken. We assessed flow patterns and pressure drops over tracheal stenoses artificially inserted into a realistic three-dimensional upper airway model derived from multislice computed tomography images obtained in healthy men. Solving the Navier-Stokes equations (with a Yang-shih k-epsilon turbulence model) for different degrees of tracheal constriction located approximately one tracheal diameter below the glottis, the simulated pressure drop over the stenosis (DeltaP) was seen to dramatically increase only when well over 70% of the tracheal lumen was obliterated. At 30 l/min, DeltaP increased from 7 Pa for a 50% stenosis to, respectively, 46 and 235 Pa for 80% and 90% stenosis. The pressure-flow relationship in the entire upper airway model (between mouth and end of trachea) in the flow range 0-60 l/min showed a power law relationship with best-fit flow exponent of 1.77 in the absence of stenosis. The exponent became 1.92 and 2.00 in the case of 60% and 85% constriction, respectively. The present simulations confirm that the overall pressure drop at rest is only affected in case of severe constriction, and the simulated flow dependence of pressure drop suggests a means of detecting stenosis at a precritical stage.     

On the stationary state of Kohonen's self-organizing sensory mapping

The stationary state of the self-organizing sensory mapping of Kohonen is investigated. For this purpose the equation for the stationary state is derived for the case of one-dimensional and two-dimensional mappings. The equation can be solved for special cases, including the general one-dimensional case, to yield an explicit expression for the local magnification factor of the map.     

Contributions to the mathematical biophysics of the cardiovascular system

The reflection of pressure waves in a fluid enclosed within a tube with an elastic wall is studied for the case of a localized change in diameter of the tube. The concept of impedance is introduced. The relation of the reflection characteristics of the parts of the tube at either side of the change is derived on the basis of the continuity of pressure and mass flow at the site of the change. This relations is used to derive the expression for the ratio of the pressure oscillations measured in front of, and behind, the constriction in terms of the constants of the system. As a result, a method is indicated to locate the coarctation from measurements of the pressures in front of, and behind it.     

A stochastic process determines the time at which cell division begins in Escherichia coli

The theoretical distributions of cell masses in exponential cultures of bacteria were derived for both total cells and cells having formed a constriction in preparation for division. The parameters used for this derivation include the mass doubling time, tau, the T-period, and 3 statistical parameters (h, sigma 1, sigma 2) which describe the variability of the cell cycle. The theoretical distributions were compared with observed distributions from E. coli B/rA growing in glucose minimal medium (45 min doubling time) to determine whether a stochastic process in the division pathway affects the time of initiation of constriction or the duration of the constriction process. The results indicate that the stochastic process determines the onset rather than the completion of constriction and that the timing of this process is coupled (6% variability, = sigma 1) to a given cell mass. The values obtained for the duration of the T-period, T = 9.3 min, and for a half-life parameter associated with the stochastic process, h = 4.3 min, agree with previously reported data.     

An unusual presentation of constriction band syndrome

Congenital constriction band syndrome is a rare entity with a wide spectrum of associated congenital anomalies. Review of the pathogenesis and an unusual case of constriction band syndrome in a newborn are presented. Surgical excision of the deformity and the band was performed within the first week of life. There were no vascular or neural structures within the excised tissue, and there were no other associated anomalies other than dextrocardia and an equinovarus deformity of the foot. The wound was closed primarily without the need for Z-plasties. This alternative method of treatment can be considered in such unusual locations of constriction band syndromes.     

On microelectrode ionophoresis

An equation for ionophoresis in large tip microelectrodes is derived from Nernst-Planck equations for the general case of a completely dissociated electrolyte. The relation between the release of ions and the applied electric current is mainly determined by two parameters: the transference number of the ions under consideration and the diffusional leak of the microelectrode. Also it is shown how the release of ions is affected by the concentration of the electrolyte within the electrode and that of the external solution. The equation describes the ionophoretic release of polyvalent spermine. In addition, new equations for tip potential and for tip resistance are derived.     

One-way blocks in cardiac tissue: A mechanism for propagation failure in purkinje fibres

The concept of a one-way block, arising from a region of depressed tissue, has remained central to theories for cardiac arrhythmias. We show that both the geometry of a depressed region and spatial heterogeneities in depression are key factors for inducing such a block. By using an asymptotic approximation, known as the eikonal equation, to model qualitatively the movement of a depolarization wave-front down a Purkinje fibre bundle, we show how a one-way block in conduction may result from asymmetric constriction in the width of a depressed bundle. We demonstrate that this theory is valid for biologically relevant parameters and simulate a one-way block by numerically solving the eikonal approximation. We consider the case of non-uniform depression, where the planar travelling wave speed is spatially dependent. Here, numerical simulations indicate that such a spatial dependency may, in itself, be sufficient to produce a one-way block.     

Molecular dynamics simulation of the cytosolic mouth in Kcv-type potassium channels

The functional effect of mutations near the intracellular mouth of the short viral Kcv potassium channel was studied by molecular dynamics simulations. As a model system we used the analogously mutated and truncated KirBac1.1, a channel with known crystal structure that shares genuine local sequence motifs with Kcv. By a novel simulated annealing methodology for structural averaging, information about the structure and dynamics of the intracellular mouth was extracted and complemented by Poisson-Boltzmann and 3D-RISM (reference interaction site model) integral equation theory for the determination of the K+ free energy surface. Besides the wild-type analogue of Kcv with its experimental reference activity (truncated KirBac1.1), two variants were studied: a deletion mutant where the N-terminus is further truncated by eight amino acids, showing inactivity in the Kcv reference system, and a point mutant where the kink-forming proline at position 13 is substituted by alanine, resulting in hyperactivity. The computations reveal that the change of activity is closely related to a hydrophilic intracellular constriction formed by the C-terminal residues of the monomers. Hyperactivity of the point mutant is correlated with both sterical and electrostatic factors, while inactivity of the deletion mutant is related to a loss of specific salt bridge patterns between the C- and N-terminus at the constriction and to the consequences for ion passage barriers, as revealed by integral equation theory. The cytosolic gate, however, is probably formed by the N-terminal segment up to the proline kink and not by the constriction. The results are compared with design principles found for other channels.     

Stochastic epidemics: major outbreaks and the duration of the endemic period

A study is made of a two-dimensional stochastic system that models the spread of an infectious disease in a population. An asymptotic expression is derived for the probability that a major outbreak of the disease will occur in case the number of infectives is small. For the case that a major outbreak has occurred, an asymptotic approximation is derived for the expected time that the disease is in the population. The analytical expressions are obtained by asymptotically solving Dirichlet problems based on the Fokker-Planck equation for the stochastic system. Results of numerical calculations for the analytical expressions are compared with simulation results.     

Penetration of external thermal perturbations into homeothermic organisms, part I (author's transl)

The general importance of the mean surface curvature for heat conduction problems is explained and a special symmetry with constant mean curvature on the isothermal surfaces is defined. The applicability for the body shapes of homeothermic organisms is demonstrated and the partial differential equation of heat conduction for this case is derived. The definition: heat release = real heat production + convective pseudoproduction eliminates the term of convective heat transfer through the blood stream and allows the reduction to a mere heat conduction problem. Formulas for the heat loss to the environment and for steady state temperature profiles are given. In case of sudden change of heat loss the partial differential equation is solved and a formula is derived, using dimensionless coordinates of time and distance. The mean surface curvature has strongest influence to the interior temperature field. The solution shows clearly the importance of thermal inertia of the homeothermic organism, for the external temperature wave penetrates into the body with a long phase displacement in time.     

Venus flytrap biomechanics: Forces in the Dionaea muscipula trap

Biomechanics of morphing structures in the Venus flytrap has attracted the attention of scientists during the last 140 years. The trap closes in a tenth of a second if a prey touches a trigger hair twice. The driving force of the closing process is most likely due to the elastic curvature energy stored and locked in the leaves, which is caused by a pressure differential between the upper and lower layers of the leaf. The trap strikes, holds and compresses the prey. We have developed new methods for measuring all these forces involved in the hunting cycle. We made precise calibration of the piezoelectric sensor and performed direct measurements of the average impact force of the trap closing using a high speed video camera for the determination of time constants. The new equation for the average impact force was derived. The impact average force between rims of two lobes in the Venus flytrap was found equal to 149 mN and the corresponding pressure between the rims was about 41 kPa. Direct measurements of the constriction force in the trap of Dionaea muscipula was performed during gelatin digestion. This force increases in the process of digestion from zero to 450 mN with maximal constriction pressure created by the lobes reaching to 9 kPa. The insects and different small prey have little chance to escape after the snap of the trap. The prey would need to overpower the “escaping” force which is very strong and can reach up to 4 N.     

Spike-timing-dependent plasticity for neurons with recurrent connections

The dynamics of the learning equation, which describes the evolution of the synaptic weights, is derived in the situation where the network contains recurrent connections. The derivation is carried out for the Poisson neuron model. The spiking-rates of the recurrently connected neurons and their cross-correlations are determined self- consistently as a function of the external synaptic inputs. The solution of the learning equation is illustrated by the analysis of the particular case in which there is no external synaptic input. The general learning equation and the fixed-point structure of its solutions is discussed.     

Nucleotide discrimination with DNA immobilized in the MspA nanopore

Nanopore sequencing has the potential to become a fast and low-cost DNA sequencing platform. An ionic current passing through a small pore would directly map the sequence of single stranded DNA (ssDNA) driven through the constriction. The pore protein, MspA, derived from Mycobacterium smegmatis, has a short and narrow channel constriction ideally suited for nanopore sequencing. To study MspA's ability to resolve nucleotides, we held ssDNA within the pore using a biotin-NeutrAvidin complex. We show that homopolymers of adenine, cytosine, thymine, and guanine in MspA exhibit much larger current differences than in α-hemolysin. Additionally, methylated cytosine is distinguishable from unmethylated cytosine. We establish that single nucleotide substitutions within homopolymer ssDNA can be detected when held in MspA's constriction. Using genomic single nucleotide polymorphisms, we demonstrate that single nucleotides within random DNA can be identified. Our results indicate that MspA has high signal-to-noise ratio and the single nucleotide sensitivity desired for nanopore sequencing devices.     

GLOBAL ATTRACTIVITY IN A N-SPECIES COMPETITION DIFFERENCE SYSTEM

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Computational fluid dynamic analysis of flow velocity waveform notching in umbilical arteries

Umbilical artery Doppler velocimetry waveform notching has long been associated with umbilical cord abnormalities, such as distortion, torsion, and/or compression (i.e., constriction). The physical mechanism by which the notching occurs has not been elucidated. Flow velocity waveforms (FVWs) from two-dimensional pulsatile flows in a constricted channel approximating a compressed umbilical cord are analyzed, leading to a clear relationship between the notching and the constriction. Two flows with an asymmetric, semi-elliptical constriction are computed using a stabilized finite-element method. In one case, the constriction blocks 75% of the flow passage, and in the other the constriction blocks 85%. Channel width and prescribed flow rates at the channel inflow are consistent with typical cord diameters and flow rates reported in the literature. Computational results indicate that waveform notching is caused by flow separation induced by the constriction, giving rise to a vortex (core) wave and associated eddies. Notching in FVWs based on centerline velocity (centerline FVW) is directly related to the passage of an eddy over the point of measurement on the centerline. Notching in FVWs based on maximum cross-sectional velocity (envelope FVW) is directly related to acceleration and deceleration of the fluid along the vortex wave. Results show that notching in envelope FVW is not present in flows with less than a 75% constriction. Furthermore, notching disappears as the vortex wave is attenuated at distances downstream of the constriction. In the flows with 75 and 85% constriction, notching of the envelope FVW disappears at ～3.8 and ～4.3 cm (respectively) downstream of the constriction. These results are of significant medical importance, given that envelope FVW is typically measured by commercial Doppler systems.     

Generalized transition state method and continuous diffusion in multi-dimensional systems with relation to ion transport in channels of biological membranes

An exact expression for the escape rate of a particle in a multi-dimensional system, with respect to an arbitrary reaction coordinate, is derived from first principles according to the transition state method, using a simple geometrical argument. It is shown how the mutual coupling of all degrees of freedom due to the interaction forces leads to the appearance of an effective mass and the potential of the mean force. The same relevant quantities dominate the effective one-particle Fokker-Planck equation, which is derived by a similar projection procedure from the multi-dimensional transport equation. In the limit of a large, position-dependent friction the respective effective Smoluchowski equation is obtained. It allows for the discussion of a diffusing particle which is subject to a temperature bath only through the coupled motion with the constituent lattice particles, or ligands in the case of a molecular ion channel. This treatment is of particular importance for the analysis of ion transport in membrane pores in which the ionic motion is mediated by internal ligand motion.