On the volume-flow mechanism of phloem transport

Summary A steady-state model of solution flow in a tubular semipermeable membrane is developed for an arbitrary distribution of solute sources and sinks along the translocation path. It is demonstrated that the volume-flow mechanism of phloem transport depends only on the two assumptions: 1. that the plasmalemma of the sieve tube is a differentially permeable membrane, and 2. that sugars are actively secreted into and absorbed from the lumen of the sieve tube. It is shown that in the absence of a pressure gradient, there is a negligible concentration gradient over most of the translocation path. However, in the presence of a pressure gradient a small concentration gradient develops as a result of the continually changing chemical potential of water along the direction of solution flow. For Poiseuille flow the concentration gradient is approximately proportional to the mean stream velocity.     

Transient magneto-peristaltic flow of couple stress biofluids: A magneto-hydro-dynamical study on digestive transport phenomena

Magnetic fields are increasingly being utilized in endoscopy and gastric transport control. In this regard, the present study investigates the influence of a transverse magnetic field in the transient peristaltic rheological transport. An electrically-conducting couple stress non-Newtonian model is employed to accurately simulate physiological fluids in peristaltic flow through a sinusoidally contracting channel of finite length. This model is designed for computing the intra-bolus oesophageal and intestinal pressures during the movement of food bolus in the digestive system under magneto-hydro-dynamic effects. Long wavelength and low Reynolds number approximations have been employed to reduce the governing equations from nonlinear to linear form, this being a valid approach for creeping flows which characterizes physiological dynamics. Analytical approximate solutions for axial velocity, transverse velocity, pressure gradient, local wall shear stress and volumetric flow rate are obtained for the non-dimensional conservation equations subject to appropriate boundary conditions. The effects of couple stress parameter and transverse magnetic field on the velocity profile, pressure distribution, local wall shear stress and the averaged flow rate are discussed with the aid of computational results. The comparative study of non-integral and integral number of waves propagating along the finite length channel is also presented. Magnetic field and non-Newtonian properties are found to strongly influence peristaltic transport.     

Peristaltic transport in a channel with a porous peripheral layer: model of a flow in gastrointestinal tract

Peristaltic transport in a two dimensional channel, filled with a porous medium in the peripheral region and a Newtonian fluid in the core region, is studied under the assumptions of long wavelength and low Reynolds number. The fluid flow is investigated in the waveframe of reference moving with the velocity of the peristaltic wave. Brinkman extended Darcy equation is utilized to model the flow in the porous layer. The interface is determined as a part of the solution using the conservation of mass in both the porous and fluid regions independently. A shear-stress jump boundary condition is used at the interface. The physical quantities of importance in peristaltic transport like pumping, trapping, reflux and axial velocity are discussed for various parameters of interest governing the flow like Darcy number, porosity, permeability, effective viscosity etc. It is observed that the peristalsis works as a pump against greater pressure in two-layered model with a porous medium compared with a viscous fluid in the peripheral layer. Increasing Darcy number Da decreases the pumping and increasing shear stress jump constant beta results in increasing the pumping. The limits on the time averaged flux Q for trapping in the core layer are obtained. The discussion on pumping, trapping and reflux may be helpful in understanding some of the fluid dynamic aspects of the transport of chyme in gastrointestinal tract.     

Evidence for solution flow in the phloem of willow

Sucrose specific mass transfer measurements were made in a translocating willow shoot (Salix viminalis L.) by a steady state labelling technique and the translocate sucrose specific activity, concentration and velocity monitored by analysis of the honeydew from two colonies of the willow aphid Tuberolachnus salignus Gmelin. The values of sucrose SMT obtained were related to the simultaneous measurements of translocate concentration and velocity and to the gradients of sucrose concentration within the stem transport path to determine if transport was a bulk flow or a diffusional analogue. Estimates of potassium ion concentration in the sieve tubes were made, using aphid honeydew, and related to the sucrose SMT measured simultaneously. Correlations were found between translocate concentration, velocity and SMT which suggested that solution flow was occurring rather than a process analogous to diffusion. Evidence was obtained that velocity of flow was a valid concept and that the measured velocity was being lowered by leakage of tracer from the sieve tubes. The analysis of potassium concentration suggested that if solution flow was occurring then potassium must be very exchangeable down the transport path. A good correlation was observed between the SMT of sucrose and the combined gradient of sucrose and potassium concentration, though this gradient was in the opposite direction to transport in some cases.Abbreviations SMT Sucrose specific mass transfer rate - SAR Specific activity ratio - OP Osmotic pressure     

Estimation of Osmotic Gradients in Soybean Sieve Tubes by Quantitative Autoradiography: Qualified Support for the MUnch Hypothesis

An attempt was made to evaluate Münch's hypothesis of osmotically generated pressure flow in soybean (Glycine max L.) sieve tubes from velocity measurements and calculations of pressure potentials and sieve tube resistances. Pressure potential was estimated from values for water potentials and osmotic potential. Leaf water potential measurements were made by isopiestic thermocouple psychrometry, while the water potential of the nutrient solution was made with a vapor pressure osmometer. Osmotic potential was measured by first bringing the sucrose pools in the entire plant to the same specific radioactivity by steady-state-labeling of the shoot with constant specific radioactivity ¹⁴CO₂ for 5 to 8 hours. Sucrose concentrations in sieve tubes were calculated from the disintegration rate per unit volume in sieve elements as measured by absolute quantitative microautoradiography of freeze-substituted, Eponembedded source (leaf) and sink (root) tissues.     

Peristaltic Transport of Carreau-Yasuda Fluid in a Curved Channel with Slip Effects

The wide occurrence of peristaltic pumping should not be surprising at all since it results physiologically from neuro-muscular properties of any tubular smooth muscle. Of special concern here is to predict the rheological effects on the peristaltic motion in a curved channel. Attention is focused to develop and simulate a nonlinear mathematical model for Carreau-Yasuda fluid. The progressive wave front of peristaltic flow is taken sinusoidal (expansion/contraction type). The governing problem is challenge since it has nonlinear differential equation and nonlinear boundary conditions even in the long wavelength and low Reynolds number regime. Numerical solutions for various flow quantities of interest are presented. Comparison for different flow situations is also made. Results of physical quantities are interpreted with particular emphasis to rheological characteristics.     

Peristaltic transport of a solid bolus

An analysis of the peristaltic propulsion of a solid spherical bolus enclosed in a contractile membrane is presented. The model is based on in vitro preparations of intestinal segments, and utilizes a simplified representation of the mechanical properties of the muscular coats of the wall. The sequence of deformed configurations of the membrane and the displacement of the bolus are obtained by numerical solution of the model equations. The analysis presented in this paper could be useful for other studies in biomechanics (e.g. uterine contraction and motion of red blood cells in narrow capillaries).     

A Mathematical Treatment of Munch's Pressure-Flow Hypothesis of Phloem Translocation

The steady state solutions of two mathematical models are used to evaluate Münch's pressure-flow hypothesis of phloem translocation. The models assume a continuous active loading and unloading of translocate but differ in the site of loading and unloading and the route of water to the sieve tube. The dimensions of the translocation system taken are the average observed values for sugar beet and are intended to simulate translocation from a mature source leaf to an expanding sink leaf. The volume flow rate of solution along the sieve tube, water flow rate into the sieve tube, hydrostatic pressure, and concentration of sucrose in the sieve tube are obtained from a numerical computer solution of the models. The mass transfer rate, velocity of translocation, and osmotic and hydrostatic pressures are consistent with empirical findings. Owing to the resistance to water flow offered by the lateral membranes, the hydrostatic pressure generated by the osmotic pressure can be considerably less than would be predicted by the solute concentration. These models suggest that translocation at observed rates and velocities can be driven by a water potential difference between the sieve tube and surrounding tissue and are consistent with the pressure-flow hypothesis of translocation.     

Peristaltic biofluids flow through vertical porous human vessels using third-grade non-Newtonian fluids model

In this paper, the heat and flow characteristic of third-grade non-Newtonian biofluids flow through a vertical porous human vessel due to peristaltic wall motion are studied. The third-grade model can describe shear thinning (or shear thickening) and normal stress differences, which is acceptable for biofluids modeling. In order to solve the governing equations, the assumption of long-wavelength approximation is utilized. This hypothesis emphasizes that the wavelength of the peristaltic wall motion is large in comparison with the radius of the human vessel, which is widely acceptable in biological investigations. The analytical perturbation method is employed to solve the governing equations. Consequently, analytical expressions for the velocity profile, shear stress, temperature field, and biofluid flow rate are obtained. In addition, the effects of the governing parameters such as the third-grade non-Newtonian parameter, Grashof Number, Eckert number, and porosity, on the results are examined.     

A mathematical model for the peristaltic flow of chyme movement in small intestine

A mathematical model based on viscoelastic fluid (fractional Oldroyd-B model) flow is considered for the peristaltic flow of chyme in small intestine, which is assumed to be in the form of an inclined cylindrical tube. The peristaltic flow of chyme is modeled more realistically by assuming that the peristaltic rush wave is a sinusoidal wave, which propagates along the tube. The governing equations are simplified by making the assumptions of long wavelength and low Reynolds number. Analytical approximate solutions of problem are obtained by using homotopy analysis method and convergence of the obtained series solution is properly checked. For the realistic values of the emerging parameters such as fractional parameters, relaxation time, retardation time, Reynolds number, Froude number and inclination of tube, the numerical results for the pressure difference and the frictional force across one wavelength are computed and discussed the roles played by these parameters during the peristaltic flow. On the basis of this study, it is found that the first fractional parameter, relaxation time and Froude number resist the movement of chyme, while, the second fractional parameter, retardation time, Reynolds number and inclination of tube favour the movement of chyme through the small intestine during pumping. It is further revealed that size of trapped bolus reduces with increasing the amplitude ratio whereas it is unaltered with other parameters.     

Fluid Dynamics Simulating Experiment of Pressure Gradient for Assimilate Translocation in Sieve Tubes

According to the similitude theory of fluid mechanics, a pressure-driven experiment in a sieve tube model has been performed. The relationships between the pressure gradient and the transport velocities at the various diameters of sieve pore have been obtained. The experimental data were compared with the theoretical prediction based on the poiseuille equation. The result shows that the pressure gradient required for translating the assimilate is directly proportional to the translating velocity, which agrees with Poiseuille equation. However, the resistance of the sieve plate to flow is much more than that estimated by the previous theory. Even though the translating velocity is a normal level and the diameter of the sieve tube is a typical size, the pressure gradient required for flow is about 1 bar/m. It will increase sharply with the decrease of the diameter of sieve pore. The ratios of the pressure gradients measured in the test to that predicted by Poiseuille equation are about 2–4 for the five kinds of diameters of Sieve pore. Additionally, a simulating test of the pores being partly plugged has been performed. The result shows that the pressure gradients measured are greatly beyond the range that may be kept in plants. This study will be helpful to assess Munch hypothesis.     

The mechanical advantage of local longitudinal shortening on peristaltic transport

Whereas bolus transport along the esophagus results from peristaltic contractions of the circular muscle layer, it has been suggested that local shortening of the longitudinal muscle layer concentrates circular muscle fibers in the region where the highest contractile pressures are required. Here we analyze the mechanical consequences of local longitudinal shortening (LLS) through a mathematical model based on lubrication theory. We find that local pressure and shear stress in the contraction zone are greatly reduced by the existence of LLS. In consequence, peak contractile pressure is reduced by nearly 2/3 at physiological LLS, and this reduction is greatest when peak in LLS is well aligned with peak contractile pressure. We conclude that a peristaltic wave of local longitudinal muscle contraction coordinated with the circular muscle contraction wave has both a great physiological advantage (concentrating circular muscle fibers), and a great mechanical advantage (reducing the level of contractile force required to transport the bolus), which combine to greatly reduce circular muscle tone during esophageal peristalsis.     

Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube

The effect of pulsatile flow on peristaltic transport in a circular cylindrical tube is analysed. The flow of a Newtonian viscous incompressible fluid in a flexible circular cylindrical tube on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The initial flow in the tube is induced by an arbitrary periodic pressure gradient. A perturbation solution with amplitude ratio (wave amplitude/tube radius) as a parameter is obtained when the frequency of the travelling wave and that of the imposed pressure gradient are equal. The interaction effects of periodic wall induced flow and periodic pressure imposed flow are visualized through the presence of substantially different components of steady and higher harmonic oscillating flow in the first order flow solution. Numerical results show a strong variation of steady state velocity profiles with boundary wave number and Reynolds number and a strong phase shift behaviour of the flow in the radial direction.     

Distorted phytochrome action spectra in green plants

An evaluation was made of the extent which a Münch-type pressure flow mechanism (i.e., osmotically-generated pressure flow) might contribute to phloem transport in soybean. Estimates of sucrose concentrations in source (leaf) and sink (root) sieve tubes were obtained by a negativestaining procedure. Water potential measurements of the leaf and of the nutrient solution allowed calculation of the turgor pressures in source and sink sieve tubes. The turgor difference between source and sink sieve tubes was compared to that required to drive translocation at the observed velocity between the source and sink, as measured by [¹⁴C] photosynthate movement. Sieve-tube conductivity was calculated from the sieve-tube dimensions, assuming an essentially unobstructed pathway. In three experiments, the sucrose concentration was consistently higher in source sieve tubes (an average of 11.5%) than in sink sieve tubes (an average of 5.3%). The ratio of these values (2.3:1) agreed reasonably well with an earlier ratio for source/sink sieve tube concentrations of 1.8:1, obtained by quantitative microautoradiography. The resulting calculated turgor difference (an average of 4.1 bars) was adequate to drive a pressure flow mechanism at the observed translocation velocities (calculated to require a turgor difference of 1.2 to 4.6 bars). No other force need be presumed to be involved.This work was presented in part at a joint U.S.-Australian Conference on Transport and Transfer Processes in Plants, Canberra, Australia, December 15–20, 1975; see Fisher (1976)     

A sterile, regulated, pulsation-dampened precision dosing system for biomedical technics

For sterile continuous metering of small and very small flow rates delivered by peristaltic pump, a gravimetric control system which permits a high accuracy of mass flow is described. This is achieved by a digital PI controller and a digital filter with which the very accurate signal from a high precision electronic scale is filtered. In addition proposals are made for solving the problem of refilling the reservoir and for compensating the pulsations caused by the peristaltic pump.     

A mathematical simulation of the ureter: effects of the model parameters on ureteral pressure/flow relations

Ureteral peristaltic mechanism facilitates urine transport from the kidney to the bladder. Numerical analysis of the peristaltic flow in the ureter aims to further our understanding of the reflux phenomenon and other ureteral abnormalities. Fluid-structure interaction (FSI) plays an important role in accuracy of this approach and the arbitrary Lagrangian-Eulerian (ALE) formulation is a strong method to analyze the coupled fluid-structure interaction between the compliant wall and the surrounding fluid. This formulation, however, was not used in previous studies of peristalsis in living organisms. In the present investigation, a numerical simulation is introduced and solved through ALE formulation to perform the ureteral flow and stress analysis. The incompressible Navier-Stokes equations are used as the governing equations for the fluid, and a linear elastic model is utilized for the compliant wall. The wall stimulation is modeled by nonlinear contact analysis using a rigid contact surface since an appropriate model for simulation of ureteral peristalsis needs to contain cell-to-cell wall stimulation. In contrast to previous studies, the wall displacements are not predetermined in the presented model of this finite-length compliant tube, neither the peristalsis needs to be periodic. Moreover, the temporal changes of ureteral wall intraluminal shear stress during peristalsis are included in our study. Iterative computing of two-way coupling is used to solve the governing equations. Two phases of nonperistaltic and peristaltic transport of urine in the ureter are discussed. Results are obtained following an analysis of the effects of the ureteral wall compliance, the pressure difference between the ureteral inlet and outlet, the maximum height of the contraction wave, the contraction wave velocity, and the number of contraction waves on the ureteral outlet flow. The results indicate that the proximal part of the ureter is prone to a higher shear stress during peristalsis compared with its middle and distal parts. It is also shown that the peristalsis is more efficient as the maximum height of the contraction wave increases. Finally, it is concluded that improper function of ureteropelvic junction results in the passage of part of urine back flow even in the case of slow start-up of the peristaltic contraction wave.     

Influence of Hall Current and Viscous Dissipation on Pressure Driven Flow of Pseudoplastic Fluid with Heat Generation: A Mathematical Study

In this paper, we study the influence of heat sink (or source) on the peristaltic motion of pseudoplastic fluid in the presence of Hall current, where channel walls are non-conducting in nature. Flow analysis has been carried out under the approximations of a low Reynolds number and long wavelength. Coupled equations are solved using shooting method for numerical solution for the axial velocity function, temperature and pressure gradient distributions. We analyze the influence of various interesting parameters on flow quantities. The present study can be considered as a mathematical presentation of the dynamics of physiological organs with stones.     

Synthesis of a hormonally active conjugate of alpha-MSH, ferritin, and fluorescein.

A flow assay device for the study of steady-state enzyme kinetics is described. The apparatus employs a peristaltic pump and has a fluorimeter as the monitoring device. An automatic data storage and handling system is used. Theoretical considerations are made on the determination of the apparent initial velocity and flow-induced distortion of the initial velocity versus initial substrate concentration profile. Several analytical expressions useful in calculating the magnitudes of errors and for designing experiments are given. The distortion of output evident in this and in a previously described apparatus of a similar kind is removed by use of Fourier deconvolution using the response of the instrument to a δ function input. Some data obtained with rabbit muscle glyceraldehyde-3-phosphate dehydrogenase, which displays nonhyperbolic kinetics, are discussed and used as an illustration of the application of the method.     

Forisome performance in artificial sieve tubes

In the legume phloem, sieve element occlusion (SEO) proteins assemble into Ca(2+)-dependent contractile bodies. These forisomes presumably control phloem transport by forming reversible sieve tube plugs. This function, however, has never been directly demonstrated, and appears questionable as forisomes were reported to be too small to plug sieve tubes, and failed to block flow efficiently in artificial microchannels. Moreover, plugs of SEO-related proteins in Arabidopsis sieve tubes do not affect phloem translocation. We improved existing procedures for forisome isolation and storage, and found that the degree of Ca(2+)-driven deformation that is possible in forisomes of Vicia faba, the standard object of earlier research, has been underestimated substantially. Forisomes deform particularly strongly under reducing conditions and high sugar concentrations, as typically found in sieve tubes. In contrast to our previous inference, Ca(2+)-inducible forisome swelling certainly seems sufficient to plug sieve tubes. This conclusion was supported by 3D-reconstructions of forisome plugs in Canavalia gladiata. For a direct test, we built microfluidics chips with artificial sieve tubes. Using fluorescent dyes to visualize flow, we demonstrated the complete blockage of these biomimetic microtubes by Ca(2+)-induced forisome plugs, and concluded by analogy that forisomes are capable of regulating phloem flow in vivo.     

Interaction of peristaltic motion with Poiseuille flow

The effect of Poiseuille flow on peristaltic transport has been investigated in a two-dimensional mathematical model of peristalsis for the case when the wall of the channel executes a sinusoidal motion of small amplitude. Closed-form solutions have been obtained for limiting values of Reynolds number and the Poiseuille flow parameter, while the method of Frobenius series solution has been used for the general case. It is found that the mean flow reversal is strongly dependent on the Poiseuille flow. The position of flow reversal may change drastically from the center of the channel to the boundaries. Numberical results are reported for various values of the physical parameters of interest.