Michal Přibyl

Discrete Models of Autocrine Cell Communication in Epithelial Layers

Pattern formation in epithelial layers heavily relies on cell communication by secreted ligands. Whereas the experimentally observed signaling patterns can be visualized at single-cell resolution, a biophysical framework for their interpretation is currently lacking. To this end, we develop a family of discrete models of cell communication in epithelial layers. The models are based on the introduction of cell-to-cell coupling coefficients that characterize the spatial range of intercellular signaling by diffusing ligands. We derive the coupling coefficients as functions of geometric, cellular, and molecular parameters of the ligand transport problem. Using these coupling coefficients, we analyze a nonlinear model of positive feedback between ligand release and binding. In particular, we study criteria of existence of the patterns consisting of clusters of a few signaling cells, as well as the onset of signal propagation. We use our model to interpret recent experimental studies of the EGFR/Rhomboid/Spitz module in Drosophila development.

Long-Range Signal Transmission in Autocrine Relays

Intracellular signaling induced by peptide growth factors can stimulate secretion of these molecules into the extracellular medium. In autocrine and paracrine networks, this can establish a positive feedback loop between ligand binding and ligand release. When coupled to intercellular communication by autocrine ligands, this positive feedback can generate constant-speed traveling waves. To demonstrate that, we propose a mechanistic model of autocrine relay systems. The model is relevant to the physiology of epithelial layers and to a number of in vitro experimental formats. Using asymptotic and numerical tools, we find that traveling waves in autocrine relays exist and have a number of unusual properties, such as an optimal ligand binding strength necessary for the maximal speed of propagation. We compare our results to recent observations of autocrine and paracrine systems and discuss the steps toward experimental tests of our predictions.

Parametrical studies of electroosmotic transport characteristics in submicrometer channels.

Spatially two-dimensional nonequilibrium mathematical model describing electroosmotic flow through a submicrometer channel with an electric charge fixed on the channel walls is presented. This system is governed by the hydrodynamic, electrostatic, and mass transport phenomena. The model is based on the coupled mass balances, Poisson, Navier-Stokes, and Nernst-Planck equations. Nonslip boundary conditions are employed. The effect of an imposed electric field on the system behavior is studied by means of a numerical analysis of the model equations. We have obtained the following findings. If the channel width is comparable to the thickness of the electric double layer, the system behaves as an ion-exchange membrane and the dependence of the electric current passing through the channel on the applied voltage is strongly nonlinear. In the case of negatively (positively) charged walls, a narrow region of very low conductivity (so-called ionic gate) is formed in the free electrolyte near the channel entry facing the anode (cathode) side. For a wide channel, the electric current is proportional to the applied voltage and the velocity of electrokinetic flow is linearly proportional to the electric field strength. Complex hydrodynamics (eddy formation and existence of ionic gates) is the most interesting characteristics of the studied system. Hence, current-voltage and velocity-voltage curves and the corresponding spatial distributions of the model variables at selected points are studied and described in detail.

Modeling of hydrogel immobilized enzyme reactors with mass-transport enhancement by electric field

Abstract Effects of electro-transport processes in enzymatic reactors with spatially continuous unstirred reaction media (e.g., gels or polymers) on enzyme reactions are studied by numerical simulations. Two model enzyme reactions are chosen for analysis: (i) with no ionic reaction components, and (ii) with only ionic components and with production of H+ ions. The electrophoretic migration and electro-osmotic flux are considered as mechanisms altering transport rates of reaction components in unstirred reaction medium. Mathematical models of reactor system with hydrophilic membrane (or slab) containing immobilised enzyme with the DC electric field applied in a direction perpendicular to the membrane are constructed. Remarkable increase of the immobilised enzyme productivity was observed when the electric current of proper intensity was applied to the system. This optimum current value depends on substrate concentration, the slab thickness and the rate of enzyme inactivation. Main factor limiting applicability of the electric current to the reaction slab is heating of the slab due to the Joule heat. The electrophoretic migration of H+ ions in the second reaction system prevents local over-acidification, i.e. the averaged reaction yield is higher compared to the system with no electric field applied. An example of experimental results obtained in a laboratory-scale electro-membrane reactor with immobilised penicillin G acylase is also discussed.

Oscillatory motion of water droplets in kerosene above co-planar electrodes in microfluidic chips

We experimentally observed oscillatory motion of water droplets in microfluidic systems with coplanar microelectrodes under imposed DC electric fields. Two-electrode arrangement with no bipolar electrode and eight-electrode arrangement with six bipolar microelectrodes were investigated. Kerosene was used as the continuous phase. We studied the dependences of the oscillation frequency on the electric field intensity and ionic strength of the water phase. We found that the electric field dependence is strongly nonlinear and discussed possible reasons of this phenomenon, e.g., the droplet deformation at electrode edges that affects the charge transfer between the electrode and droplet or the interplay between the Coulomb force on free charge and the dielectrophoretic force. Our experiments further revealed that the oscillation frequency decreases with growing salt concentration in the two-electrode arrangement, but increases in the eight-electrode arrangement, which was attributed to surface tension related processes and electrochemical processes on the bipolar electrodes. Finally, we analyzed the effects of the electric field on the oscillatory motion by means of a simplified mathematical model. It was shown that the electric force imposed on the droplet charge is the key factor to induce the oscillations and the dielectrophoretic force significantly contributes to the momentum transfer at the electrode edges. For the same electric field strength, the model is able to predict the same oscillation frequency as that observed in the experiments.

Transient behavior of an electrolytic diode

The transient behavior of an electrolytic diode system was studied. A gel-like electrolytic diode was incorporated in a capillary microfluidic chip. The microfluidic platform guaranteed a constant composition of solutions on the diode boundaries. The current responses of the electrolytic diode to step-like changes of the imposed DC electric voltage were measured. Some of these transients were accompanied by a short-time overshoot of electric current density. In order to explain this phenomenon, a mathematical model of the electrolytic diode system was developed. Dynamical analysis of the model equations confirmed the existence of the electric current overshoots. Because the results of the experimental and the numerical transient studies were quite similar, we have explained the physical meaning of three selected overshoots by means of an analysis of the reaction-transport processes inside the electrolytic diode system. The transient experiments carried out in this study indicate that our physical concept of the electrolytic diode system presented in previous papers is correct.

Three-phase slug flow in microchips can provide beneficial reaction conditions for enzyme liquid-liquid reactions

Here, we introduce a solution to low stability of a two-phase slug flow with a chemical reaction occurring at the phase interface in a microfluidic reactor where substantial merging of individual reacting slugs results in the loss of uniformity of the flow. We create a three-phase slug flow by introducing a third fluid phase into the originally two-phase liquid-liquid slug flow, which generates small two-phase liquid slugs separated by gas phase. Introduction of the third phase into our system efficiently prevents merging of slugs and provides beneficial reaction conditions, such as uniform flow pattern along the whole reaction capillary, interfacial area with good reproducibility, and intensive water-oil interface renewal. We tested the three-phase flow on an enzyme hydrolysis of soybean oil and compared the reaction conversion with those from unstable two-phase slug flows. We experimentally confirmed that the three-phase slug flow arrangement provides conversions and pressure drops comparable or even better with two-phase liquid-liquid arrangements.

Mathematical modeling of wastewater decolorization in a trickle-bed bioreactor.

This work focuses on mathematical modeling of removal of organic dyes from textile industry waste waters by a white-rot fungus Irpex lacteus in a trickle-bed bioreactor. We developed a mathematical model of biomass and decolorization process dynamics. The model comprises mass balances of glucose and the dye in a fungal biofilm and a liquid film. The biofilm is modeled using a spatially two-dimensional domain. The liquid film is considered as homogeneous in the direction normal to the biofilm surface. The biomass growth, decay and the erosion of the biofilm are taken into account. Using experimental data, we identified values of key model parameters: the dye degradation rate constant, biofilm corrugation factor and liquid velocity. Considering the dye degradation rate constant 1×10⁻⁵ kg m⁻³ s⁻¹, we found optimal values of the corrugation factor 0.853 and 0.59 and values of the liquid velocity 5.23×10⁻³ m s⁻¹ and 6.2×10⁻³ m s⁻¹ at initial dye concentrations 0.09433 kg m⁻³ and 0.05284 kg m⁻³, respectively. A good agreement between the simulated and experimental data using estimated values of the model parameters was achieved. The model can be used to simulate the performance of laboratory scale trickle-bed bioreactor operated in a batch regime or to estimate values of principal parameters of the bioreactor system.

Adaptive mesh simulations of ionic systems in microcapillaries based on the estimation of transport times

Abstract We present results of testing an empirical algorithm of mesh adaptation for modeling of spatially one-dimensional reaction-transport systems with initially separated components and large moving gradients. The algorithm combines newly developed procedures of r -refinement and standard FEMLAB procedures. The developed method uses an expansion of a dense mesh in neighborhoods of localized large gradients and estimates the mesh adaptation interval (based on the evaluation of transport times). The size of the neighborhood is dynamically controlled. The adaptation method has been tested on systems of parabolic–elliptic PDEs with extremely large moving gradients of concentrations and electric potential. We have found that CPU time requirements are comparable to other mesh adaptation solvers. The simplicity of the empirical procedure of the mesh adaptation and its easy implementation in standard dynamic solvers represent main advantages of the proposed method. The studied system, called an “electrolyte diode”, is described by four mass balances of the electrolyte components (parabolic PDEs) and by Poisson equation of electrostatics (an elliptic PDE). Dynamics of formation of open and closed modes of the electrolyte diode are described and current–voltage characteristics are explained. The results obtained with the developed solver agree with the results of steady state analysis. Our tests prove that the proposed algorithm of mesh adaptation can be used in modeling of microfluidics application, e.g., DNA chips, capillary electrophoresis and isoelectrical focusing, where formation of extremely large gradients of electric potential and concentrations of the electrolyte components is expected.

On the origin of bistability in the Stage 2 of the Huang-Ferrell model of the MAPK signaling

Mitogen-activated protein kinases (MAPKs) are important signal transducing enzymes, unique to eukaryotes, that are involved in many pathways of cellular regulation. Successive phosphorylation cascades mediated by MAPKs serve as sensitive switches initiating various cellular processes. Apart from this basic feature, the underlying reaction network is capable of displaying other nonlinear phenomena including bistable steady states and hysteresis as well as periodic oscillations. We show that from the mechanistic point of view, bistability is a consequence of interaction between single and double phosphorylation/dephosphorylation pathways in a Stage 2 subsystem of the Huang-Ferrell model. Within this subsystem we uncover the core subnetwork obtained by systematic reduction relying on the methods of stoichiometric network analysis. For the core model we show that there is either one stable steady state or three steady states of which two are stable and point out the role of interplay between the single and double phosphorylation subnetworks in generating bistability.