Limitations of the whole cell patch clamp technique in the control of intracellular concentrations.

Recent experimental studies (Pusch and Neher, 1988) and theoretical studies (Oliva et al., 1988) have found that the pipette tip is a significant barrier to diffusion in the whole cell patch clamp configuration. In this paper, we extend the theoretical analysis of fluxes between the pipette and cell to include transmembrane fluxes. The general conclusions are: (a) within the pipette, ion fluxes are driven primarily by diffusion rather than voltage gradients. (b) At steady state there is a concentration difference between the bulk pipette and intracellular solution that is described by delta c = jRp/Dp, where delta c = 1 mM for a flux, j = 1 fmol/s, through a pipette of resistance, Rp = 1 M omega, filled with a solution of resistivity, p = 100 omega --cm, given a solute diffusion coefficient, D = 10(-5) cm2/s. (c) The time to steady state is always accelerated by membrane transport, regardless of the direction of transport. We apply our analysis to the measurement of transport by the Na/K pump and Na/Ca exchanger in cells from the ventricles of mammalian heart. We find that the binding curve for intracellular Na+ to the Na/K pump will appear significantly less steep and more linear if one does not correct for the concentration difference between intracellular and pipette Na+. Similar shifts in the binding curve for extracellular Na+ to the Na/Ca exchanger can occur due to depletion of intracellular Ca(+)+ when the exchanger is stimulated. Lastly, in Appendix we analyze the effects of mobile and fixed intracellular buffers on the movement of Ca(+)+ between the pipette and cell. Fixed buffers greatly slow the time for equilibration of pipette and intracellular Ca(+)+. Mobile buffers act like a shuttle system, as they carry Ca(+)+ from pipette to cell then diffuse back when they are empty. Vigorous transport by the Na/Ca exchanger depletes mobile buffered calcium, thus stimulating diffusion from the pipette to match the rate of Ca(+)+ transport. Moreover, we find that binding of Ca(+)+ to the exchanger can be affected by the mobile buffer.     

A stability analysis of the power-law steady state of marine size spectra

This paper investigates the stability of the power-law steady state often observed in marine ecosystems. Three dynamical systems are considered, describing the abundance of organisms as a function of body mass and time: a “jump-growth” equation, a first order approximation which is the widely used McKendrick–von Foerster equation, and a second order approximation which is the McKendrick–von Foerster equation with a diffusion term. All of these yield a power-law steady state. We derive, for the first time, the eigenvalue spectrum for the linearised evolution operator, under certain constraints on the parameters. This provides new knowledge of the stability properties of the power-law steady state. It is shown analytically that the steady state of the McKendrick–von Foerster equation without the diffusion term is always unstable. Furthermore, numerical plots show that eigenvalue spectra of the McKendrick–von Foerster equation with diffusion give a good approximation to those of the jump-growth equation. The steady state is more likely to be stable with a low preferred predator:prey mass ratio, a large diet breadth and a high feeding efficiency. The effects of demographic stochasticity are also investigated and it is concluded that these are likely to be small in real systems.     

Analysis of a nonlinear diffusion problem With Michaelis-Menten kinetics by an integral equation method

A mathematical model for oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is analyzed by means of an intergral equation method. It is shown that an integral equation formulation can be used to obtain a numerical solution associated with this boundary and initial value problem. Through an illustrative numerical calculation we are able to obtain an accurate solution for both the steady and transient problems. Finally, a comparison is made with the numerical solution of McElwain and the variational solution of Anderson and Arthurs for the steady state and Lin's result concerning the unsteady state.     

Ionic diffusion in voltage-clamped isolated cardiac myocytes. Implications for Na,K-pump studies.

The whole-cell voltage-clamp technique employing electrolyte-filled micro-pipette suction electrodes is widely used to investigate questions requiring an electrophysiological approach. With this technique, the ionic composition of the cytosol is assumed to be strongly influenced (as result of diffusion) by the ionic composition of the solution contained in the electrode. If this assumption is valid for isolated cardiac myocytes, the technique would be particularly powerful for studying the dependence of their Na,K-pump on the intracellular [Na+]. However, the relationship between the concentrations of ions in the solution filling the electrode and those in the cytosol has not been established. The relationship was investigated to determine in particular whether the [Na+] at the intracellular cation ligand binding sites for the Na-pump ([ Na+]ps) can be set and clamped by [Na+] in the pipette electrode ([ Na+]pip). If [Na+]pip can set and clamp [Na+]ps, this would provide a means for defining the dependence of the Na,K-pump on intracellular [Na+]. The relationship between [Na+]pip and [Na+]ps was analyzed using two approaches. First, a mathematical model of three-dimensional ionic diffusion within a whole-cell patch-clamped myocyte was developed and the effects of experimental parameters on mean [Na+]ps were investigated. When typical experimental values were simulated, the time course to achieve steady state mean [Na+]ps was found to be most sensitive to variations in electrode pore size, cell length and the Na+ pumping rate, but at steady state, mean [Na+]ps varies from [Na+]pip by 5% or less depending on pump rate. Second, to provide experimental support for the validity of the simulations, isolated ventricular myocytes were voltage-clamped and the reversal potential for the Na current was determined in order to estimate steady state intracellular [Na+]. The results of the mathematical and experimental analyses suggest that steady state [Na+]ps can be regulated by the [Na+] in suction pipette electrodes. These findings, while also having a broader significance, indicate for isolated cardiac myocytes that whole-cell suction micro-electrodes can provide a means to assess the dependence of the Na,K-pump on [Na+]ps.     

On a mathematical theory of the reaction of cells to X-ray irradiation

To describe the variation of concentration of certain substances in cells subjected to X-rays for a finite length of time, R. M. Sievert (1941) has given an equation of mixed differences involving time-lag and having discontinuous coefficients. In the present paper Laplace transform methods are applied to solve this equation in terms of a series valid for all values of time. In particular, the conditions given by J. Th. van der Werff (1943) that the concentration approaches a constant value are confirmed; otherwise steady oscillations in concentration may be expected, or oscillations appear whose amplitude steadily increases until the cell is destroyed or the assumed mathematical model otherwise fails to apply.     

Measurement and perturbation of morphogen lifetime: effects on gradient shape

Protein lifetime is of critical importance for most biological processes and plays a central role in cell signaling and embryonic development, where it impacts the absolute concentration of signaling molecules and, potentially, the shape of morphogen gradients. Early conceptual and mathematical models of gradient formation proposed that steady-state gradients are established by an equilibration between the lifetime of a morphogen and its rates of synthesis and diffusion, though whether gradients in fact reach steady state before being read out is a matter of controversy. In any case, this class of models predicts that protein lifetime is a key determinant of both the time to steady state and the spatial extent of a gradient. Using a method that employs repeated photoswitching of a fusion of the morphogen Bicoid (Bcd) and the photoconvertible fluorescent protein Dronpa, we measure and modify the lifetime of Dronpa-Bcd in living Drosophila embryos. We find that the lifetime of Bcd is dynamic, changing from 50 min before mitotic cycle 14 to 15 min during cellularization. Moreover, by measuring total quantities of Bcd over time, we find that the gradient does not reach steady state. Finally, using a nearly continuous low-level conversion to the dark state of Dronpa-Bcd to mimic the effect of increased degradation, we demonstrate that perturbation of protein lifetime changes the characteristic length of the gradient, providing direct support for a mechanism based on synthesis, diffusion, and degradation.     

Cyclic GMP diffusion coefficient in rod photoreceptor outer segments.

Cyclic GMP (cGMP) is the intracellular messenger that mediates phototransduction in retinal rods. As photoisomerizations of rhodopsin molecules are local events, the longitudinal diffusion of cGMP in the rod outer segment should be a contributing factor to the response of the cell to light. We have employed the truncated rod outer segment preparation from bullfrog (Rana catesbeiana) and tiger salamander (Ambystoma tigrinum) to measure the cGMP diffusion coefficient. In this preparation, the distal portion of a rod outer segment was drawn into a suction pipette for measuring membrane current, and the rest of the cell was then sheared off with a glass probe, allowing bath cGMP to diffuse into the outer segment and activate the cGMP-gated channels on the surface membrane. Addition and removal of bath cGMP were fast enough to produce effectively step changes in cGMP concentration at the open end of the outer segment. When cGMP hydrolysis is inhibited by isobutylmethylxanthine (IBMX), the equation for the diffusion of cGMP inside the truncated rod outer segment has a simple analytical solution, which we have used to analyze the rise and decay kinetics of the cGMP-elicited currents. From these measurements we have obtained a cGMP diffusion coefficient of approximately 70 x 10(-8) cm2 s-1 for bullfrog rods and approximately 60 x 10(-8) cm2 s-1 for tiger salamander rods. These values are six to seven times lower than the expected value in aqueous solution. The estimated diffusion coefficient is the same at high (20-1000 microM) and low (5-10 microM) concentrations of cGMP, suggesting no significant effect from buffering over these concentration ranges.     

On the dynamics of flying insects populations controlled by large scale information

We present a diffusion model adapted to an operational control of flying insects. The operational objective entails working with non-homogeneous regions and the corresponding non-uniform diffusion tensors. It the becomes necessary to study the problem of scale in order to justify the diffusion equation. We show that in typical cases, the spatial resolution must be coarser than 20 m and we have specified the appropriated time scales. We propose a method for computing the tensors at the given resolution, in relation to the landscape and the insects' species and a method for the numerical solution of the equations and present two applications.     

Determination of Brain Interstitial Concentrations by Microdialysis

Microdialysis is an extensively used technique for the study of solutes in brain interstitial space. The method is based on collection of substances by diffusion across a dialysis membrane positioned in the brain. The outflow concentration reflects the interstitial concentration of the substance of interest, but the relationship between these two entities is at present unclear. So far, most evaluations have been based solely on calibrations in saline. This procedure is misleading, because the ease by which molecules in saline diffuse into the probe is different from that of tissue. We describe here a mathematical analysis of mass transport into the dialysis probe in tissue based on diffusion equations in complex media. The main finding is that diffusion characteristics of a given substance have to be included in the formula. These include the tortuosity factor (lambda) and the extracellular volume fraction (alpha). We have substantiated this by studies in a well-defined complex medium (red blood cell suspensions) as well as in brain. We conclude that the traditional calculation procedure results in interstitial concentrations that are too low by a factor of lambda 2/alpha for a given compound.     

Directionality of stripes formed by anisotropic reaction-diffusion models

Turing mechanism explains the formation of striped patterns in a uniform field in which two substances interact locally and diffuse randomly. In a twin paper, to explain the directionality of stripes on fish skin in closely related species, we studied the effect of anisotropic diffusion of the two substances on the direction of stripes, in the cases in which both substances have high diffusivity in the same direction. In this paper, we study the direction of stripes in more general situations in which the diffusive direction may differ between the two substances. We derive a formula for the direction of stripes, based on a heuristic argument of unstable modes of deviation from the uniform steady state. We confirm the accuracy of the formula by computer simulations. When the diffusive direction is different between two substances, the directions of stripes in the spatial pattern change smoothly with the magnitude of anisotropy of two substances. When the diffusive direction of the two substances is the same, the stripes are formed either parallel or perpendicular to the common diffusive direction, depending on the relative magnitude of the anisotropy. The transition between these two phases occurs sharply.     

Model of cell signal transduction in a three-dimensional domain

Intracellular signalling molecules form pathways inside the cell. These pathways carry a signal to target proteins which results in cellular responses. We consider a spherical cell with two internal compartments containing localized activating enzymes where as deactivating enzymes are spread uniformly through out the cytosol. Two diffusible signalling molecules are activated at the compartments and later deactivated in the cytosol due to deactivating enzymes. The two signalling molecules are a single link in a cascade reaction and form a self regulated dynamical system involving positive and negative feedback. Using matched asymptotic expansions we obtain approximate solutions of the steady state diffusion equation with a linear decay rate. We obtain three-dimensional concentration profiles for the signalling molecules. We also investigate an extension of the above system which has multiple cascade reactions occurring between multiple signalling molecules. Numerically, we show that the speed of the signal is an increasing function of the number of links in the cascade.     

Glutathionylation-Dependence of Na+-K+-Pump Currents Can Mimic Reduced Subsarcolemmal Na+ Diffusion

The existence of a subsarcolemmal space with restricted diffusion for Na⁺ in cardiac myocytes has been inferred from a transient peak electrogenic Na⁺-K⁺ pump current beyond steady state on reexposure of myocytes to K⁺ after a period of exposure to K⁺-free extracellular solution. The transient peak current is attributed to enhanced electrogenic pumping of Na⁺ that accumulated in the diffusion-restricted space during pump inhibition in K⁺-free extracellular solution. However, there are no known physical barriers that account for such restricted Na⁺ diffusion, and we examined if changes of activity of the Na⁺-K⁺ pump itself cause the transient peak current. Reexposure to K⁺ reproduced a transient current beyond steady state in voltage-clamped ventricular myocytes as reported by others. Persistence of it when the Na⁺ concentration in patch pipette solutions perfusing the intracellular compartment was high and elimination of it with K⁺-free pipette solution could not be reconciled with restricted subsarcolemmal Na⁺ diffusion. The pattern of the transient current early after pump activation was dependent on transmembrane Na⁺- and K⁺ concentration gradients suggesting the currents were related to the conformational poise imposed on the pump. We examined if the currents might be accounted for by changes in glutathionylation of the β₁ Na⁺-K⁺ pump subunit, a reversible oxidative modification that inhibits the pump. Susceptibility of the β₁ subunit to glutathionylation depends on the conformational poise of the Na⁺-K⁺ pump, and glutathionylation with the pump stabilized in conformations equivalent to those expected to be imposed on voltage-clamped myocytes supported this hypothesis. So did elimination of the transient K⁺-induced peak Na⁺-K⁺ pump current when we included glutaredoxin 1 in patch pipette solutions to reverse glutathionylation. We conclude that transient K⁺-induced peak Na⁺-K⁺ pump current reflects the effect of conformation-dependent β₁ pump subunit glutathionylation, not restricted subsarcolemmal diffusion of Na⁺.     

The enhancement of stability of p53 in MTBP induced p53–MDM2 regulatory network

We have modeled an MTBP-MDM2–p53 regulatory network by integrating p53–MDM2 autoregulatory model (Proctor and Gray, 2008) with the effect of a cellular protein MTBP (MDM2 binding protein) which is allowed to bind with MDM2 (Brady et al., 2005). We study this model to investigate the activation of p53 and MDM2 steady state levels induced by MTBP protein under different stress conditions. Our simulation results in three approaches namely deterministic, Chemical Langevin equation and stochastic simulation of Master equation show a clear transition from damped limit cycle oscillation to fixed point oscillation during a certain time period with constant stress condition in the cell. This transition is the signature of transition of p53 and MDM2 levels from activated state to stabilized steady state levels. We present various phase diagrams to show the transition between unstable and stable states of p53 and MDM2 concentration levels and also their possible relations among critical value of the parameters at which the respective protein level reach stable steady states. In the stochastic approach, the dynamics of the proteins become noise induced process depending on the system size. We found that this noise enhances the stability of the p53 steady state level.     

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.     

Characterizing Cell Adhesion by Using Micropipette Aspiration

We have developed a technique to directly quantify cell-substrate adhesion force using micropipette aspiration. The micropipette is positioned perpendicular to the surface of an adherent cell and a constant-rate aspiration pressure is applied. Since the micropipette diameter and the aspiration pressure are our control parameters, we have direct knowledge of the aspiration force, whereas the cell behavior is monitored either in brightfield or interference reflection microscopy. This setup thus allows us to explore a range of geometric parameters, such as projected cell area, adhesion area, or pipette size, as well as dynamical parameters such as the loading rate. We find that cell detachment is a well-defined event occurring at a critical aspiration pressure, and that the detachment force scales with the cell adhesion area (for a given micropipette diameter and loading rate), which defines a critical stress. Taking into account the cell adhesion area, intrinsic parameters of the adhesion bonds, and the loading rate, a minimal model provides an expression for the critical stress that helps rationalize our experimental results.     

Diffusion in lipid bilayers containing barriers

In epithelial cells, a barrier or tight junction restricts the diffusion of lipid probes from the apical to the basolateral side of the outer membrane bilayer. This phenomenon is studied theoretically with the diffusion equation on planar and spherical surfaces. Two models for the tight junction are considered: a penetrable barrier embedded in a monolayer and an impenetrable obstacle in the outer membrane of a bilayer than must be bypassed by flip-flopping between inner and outer membranes. The rate of passing from one side of the cell to the other is calculated for each of these models under steady state conditions. The results are compared with recent fluorescent photobleaching recovery experiments. The theoretical interpretation indicates that it would be difficult to distinguish experimentally between the flip-flop case and the barrier crossing case. Assuming a flip-flop model, large differences in the magnitude of the flip-flop rates of probes are necessary to explain the experimental results as suggested by Dragsten et al. (Dragsten, P. R., R. Blumenthal, and J. S. Handler, 1981, Nature [Lond.], 294:718--722).     

Mathematical model of the primary CD8 T cell immune response: stability analysis of a nonlinear age-structured system

The primary CD8 T cell immune response, due to a first encounter with a pathogen, happens in two phases: an expansion phase, with a fast increase of T cell count, followed by a contraction phase. This contraction phase is followed by the generation of memory cells. These latter are specific of the antigen and will allow a faster and stronger response when encountering the antigen for the second time. We propose a nonlinear mathematical model describing the T CD8 immune response to a primary infection, based on three nonlinear ordinary differential equations and one nonlinear age-structured partial differential equation, describing the evolution of CD8 T cell count and pathogen amount. We discuss in particular the roles and relevance of feedback controls that regulate the response. First we reduce our system to a system with a nonlinear differential equation with a distributed delay. We study the existence of two steady states, and we analyze the asymptotic stability of these steady states. Second we study the system with a discrete delay, and analyze global asymptotic stability of steady states. Finally, we show some simulations that we can obtain from the model and confront them to experimental data.     

Evidence For Cell Generation Controlled Proliferation In the Small Intestinal Crypt

Abstract. In this paper we discuss the hypothesis that cell proliferation is controlled by the number of generations after leaving an 'eternal' stem cell. the theory is based on a simulation of the kinetic behaviour of cells in the intestinal crypts. There is evidence of three, four and five generations of cells which are allowed to enter mitosis in the lower and upper part of the normal intestinal tract, and in some disease states, respectively. We suggest an internal proliferation control: some kind of knowledge that cells carry from generation to generation. It is an open question what sets and changes the generation counter: internal genetic information or external influences such as growth factors or chalones.
The geometric shape of the epithelial tissue in the intestinal tract can be understood as the steady state of a highly dynamic process. Age and death are determined from the beginning; cell-cell interaction or communication is not necessary and can be neglected. Our theory will be illustrated using the intestinal crypts as they are easily accessible, of a simple structure and completely described in the literature.