Efficient simulations of tubulin-driven axonal growth

This work concerns efficient and reliable numerical simulations of the dynamic behaviour of a moving-boundary model for tubulin-driven axonal growth. The model is nonlinear and consists of a coupled set of a partial differential equation (PDE) and two ordinary differential equations. The PDE is defined on a computational domain with a moving boundary, which is part of the solution. Numerical simulations based on standard explicit time-stepping methods are too time consuming due to the small time steps required for numerical stability. On the other hand standard implicit schemes are too complex due to the nonlinear equations that needs to be solved in each step. Instead, we propose to use the Peaceman–Rachford splitting scheme combined with temporal and spatial scalings of the model. Simulations based on this scheme have shown to be efficient, accurate, and reliable which makes it possible to evaluate the model, e.g. its dependency on biological and physical model parameters. These evaluations show among other things that the initial axon growth is very fast, that the active transport is the dominant reason over diffusion for the growth velocity, and that the polymerization rate in the growth cone does not affect the final axon length.     

The dynamics of phosphodiesterase activation in rods and cones

Phototransduction starts with the activation of a rhodopsin (respectively, coneopsin) molecule, located in the outer segment of rod (respectively, cone) photoreceptors. The subsequent amplification pathway proceeds via the G-protein transducin to the activation of phosphodiesterase (PDE), a G-protein coupled effector enzyme. In this article, we study the dynamics of PDE activation by constructing a Markov model that is based on the underlying chemical reactions including multiple rhodopsin phosphorylations. We derive explicit equations for the mean and the variance of activated PDE. Our analysis reveals that a low rhodopsin lifetime variance is neither necessary nor sufficient to achieve reliable PDE activation. The numerical simulations show that during the rising phase the variability of PDE activation is much lower compared to the recovery phase, and this property depends crucially on the transducin activation rates. Furthermore, we find that the dynamics of the activation process greatly differs depending on whether rhodopsin or PDE deactivation limits the recovery of the photoresponse. Finally, our simulations for cones show that only very few PDEs are activated by an excited photopigment, which might explain why in S-cones no single photon response can be observed.     

An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

In this paper we present a numerical method for the bidomain model, which describes the electrical activity in the heart. The model consists of two partial differential equations (PDEs), which are coupled to systems of ordinary differential equations (ODEs) describing electrochemical reactions in the cardiac cells. Many applications require coupling these equations to a third PDE, describing the electrical fields in the torso surrounding the heart. The resulting system is challenging to solve numerically, because of its complexity and very strict resolution requirements in time and space. We propose a method based on operator splitting and a fully coupled discretization of the three PDEs. Numerical experiments show that for simple simulation cases and fine discretizations, the algorithm is second-order accurate in space and time.     

Chaste: An Open Source C++ Library for Computational Physiology and Biology

Chaste — Cancer, Heart And Soft Tissue Environment — is an open source C++ library for the computational simulation of mathematical models developed for physiology and biology. Code development has been driven by two initial applications: cardiac electrophysiology and cancer development. A large number of cardiac electrophysiology studies have been enabled and performed, including high-performance computational investigations of defibrillation on realistic human cardiac geometries. New models for the initiation and growth of tumours have been developed. In particular, cell-based simulations have provided novel insight into the role of stem cells in the colorectal crypt. Chaste is constantly evolving and is now being applied to a far wider range of problems. The code provides modules for handling common scientific computing components, such as meshes and solvers for ordinary and partial differential equations (ODEs/PDEs). Re-use of these components avoids the need for researchers to ‘re-invent the wheel’ with each new project, accelerating the rate of progress in new applications. Chaste is developed using industrially-derived techniques, in particular test-driven development, to ensure code quality, re-use and reliability. In this article we provide examples that illustrate the types of problems Chaste can be used to solve, which can be run on a desktop computer. We highlight some scientific studies that have used or are using Chaste, and the insights they have provided. The source code, both for specific releases and the development version, is available to download under an open source Berkeley Software Distribution (BSD) licence at http://www.cs.ox.ac.uk/chaste, together with details of a mailing list and links to documentation and tutorials.

This is a PLOS Computational Biology Software Article

     

Numerical Simulation for the Unsteady MHD Flow and Heat Transfer of Couple Stress Fluid over a Rotating Disk

The present work is devoted to study the numerical simulation for unsteady MHD flow and heat transfer of a couple stress fluid over a rotating disk. A similarity transformation is employed to reduce the time dependent system of nonlinear partial differential equations (PDEs) to ordinary differential equations (ODEs). The Runge-Kutta method and shooting technique are employed for finding the numerical solution of the governing system. The influences of governing parameters viz. unsteadiness parameter, couple stress and various physical parameters on velocity, temperature and pressure profiles are analyzed graphically and discussed in detail.     

Influence of the Hall effect on the plasma dynamics in an MTF/MAGO chamber

The role of the Hall effect in experiments on the magnetic implosion of a D-T plasma in a cylindrical MTF/MAGO chamber fed from a helical explosive magnetic generator is investigated. The plasma dynamics is simulated numerically by a 2D code developed for solving the set of MHD equations with account of the Hall effect. In simulations, the generator, the break switch, and other units were replaced with LR circuits. It is shown that taking into account the Hall effect provides better agreement between numerical simulations and experimental data.     

Structural basis for the activity of drugs that inhibit phosphodiesterases

Phosphodiesterases (PDEs) comprise a large family of enzymes that catalyze the hydrolysis of cAMP or cGMP and are implicated in various diseases. We describe the high-resolution crystal structures of the catalytic domains of PDE4B, PDE4D, and PDE5A with ten different inhibitors, including the drug candidates cilomilast and roflumilast, for respiratory diseases. These cocrystal structures reveal a common scheme of inhibitor binding to the PDEs: (i) a hydrophobic clamp formed by highly conserved hydrophobic residues that sandwich the inhibitor in the active site; (ii) hydrogen bonding to an invariant glutamine that controls the orientation of inhibitor binding. A scaffold can be readily identified for any given inhibitor based on the formation of these two types of conserved interactions. These structural insights will enable the design of isoform-selective inhibitors with improved binding affinity and should facilitate the discovery of more potent and selective PDE inhibitors for the treatment of a variety of diseases.     

Parameter estimation for reaction-diffusion models of biological invasions

In this note, we discuss parameter estimation for population models based on partial differential equations (PDEs). Parametric estimation is first considered in the perspective of inverse problems (i.e., when the observation of the solution of a PDE is exactly observed or noise-free). Then, adopting the point of view of statistics, we turn to parametric estimation for PDEs using more realistic noisy measurements. The approach that we describe uses mechanistic-statistical models which combine (1) a PDE-based submodel describing the dynamic under study and (2) a stochastic submodel describing the observation process. This Note is expected to contribute to bridge the gap between modelers using PDEs and population ecologists collecting and analyzing spatio-temporal data.     

Specific localized expression of cGMP PDEs in Purkinje neurons and macrophages

As cGMP hydrolyzing cyclic nucleotide phosphodiesterases (PDEs) have diverse regulatory and catalytic properties, the specific cGMP PDEs a cell expresses will determine the duration and intensity of a cGMP signal. This, in turn, results in different cellular responses between cell types and tissues. Therefore, identifying which cGMP PDEs are expressed in different tissues and cell types could increase our understanding of physiological and pathological processes. The brain is one area where large numbers of diverse cGMP PDEs are expressed in specific regions and cell types. A case in point is differential expression of cGMP PDEs in neuronal cells. For example, we have recently found that PDE5 is expressed in all Purkinje neurons while PDE1B is expressed in only a subset of these neurons. The expression of PDE2 has also been found to be selective for discrete populations of neurons. Another example of selective cGMP PDE expression is seen with cytokine-induced differentiation of monocytes to macrophages. We have recently discovered that monocyte differentiation with the cytokine macrophage colony-stimulating factor (M-CSF) causes an upregulation of PDE2 and a small increase in PDE1B while granulocyte-macrophage colony-stimulating factor (GM-CSF) causes a large increase in PDE1B but a decrease in PDE2. These same cytokines can influence the phenotype of microglial cells and are likely to affect their expression of cGMP PDEs. In this report, we present recent results from our laboratory and review earlier findings illustrating the concept of highly specific expression of cGMP PDEs and discuss how this may be important for understanding brain function and dysfunction.     

Cyclic AMP control measured in two compartments in HEK293 cells: phosphodiesterase K(M) is more important than phosphodiesterase localization

The intracellular second messenger cyclic AMP (cAMP) is degraded by phosphodiesterases (PDE). The knowledge of individual families and subtypes of PDEs is considerable, but how the different PDEs collaborate in the cell to control a cAMP signal is still not fully understood. In order to investigate compartmentalized cAMP signaling, we have generated a membrane-targeted variant of the cAMP Bioluminiscence Resonance Energy Transfer (BRET) sensor CAMYEL and have compared intracellular cAMP measurements with it to measurements with the cytosolic BRET sensor CAMYEL in HEK293 cells. With these sensors we observed a slightly higher cAMP response to adenylyl cyclase activation at the plasma membrane compared to the cytosol, which is in accordance with earlier results from Fluorescence Resonance Energy Transfer (FRET) sensors. We have analyzed PDE activity in fractionated lysates from HEK293 cells using selective PDE inhibitors and have identified PDE3 and PDE10A as the major membrane-bound PDEs and PDE4 as the major cytosolic PDE. Inhibition of membrane-bound or cytosolic PDEs can potentiate the cAMP response to adenylyl cyclase activation, but we see no significant difference between the potentiation of the cAMP response at the plasma membrane and in cytosol when membrane-bound and cytosolic PDEs are inhibited. When different levels of stimulation were tested, we found that PDEs 3 and 10 are mainly responsible for cAMP degradation at low intracellular cAMP concentrations, whereas PDE4 is more important for control of cAMP at higher concentrations.     

A fluid dynamics model of the growth of phototrophic biofilms

A system of nonlinear hyperbolic partial differential equations is derived using mixture theory to model the formation of biofilms. In contrast with most of the existing models, our equations have a finite speed of propagation, without using artificial free boundary conditions. Adapted numerical scheme will be described in detail and several simulations will be presented in one and more space dimensions in the particular case of cyanobacteria biofilms. Besides, the numerical scheme we present is able to deal in a natural and effective way with regions where one of the phases is vanishing.     

cAMP is a ligand for the tandem GAF domain of human phosphodiesterase 10 and cGMP for the tandem GAF domain of phosphodiesterase 11

N-terminal tandem GAF domains are present in 5 out of 11 mammalian phosphodiesterase (PDE) families. The ligand for the GAF domains of PDEs 2, 5, and 6 is cGMP, whereas those for PDEs 10 and 11 remained enigmatic for years. Here we used the cyanobacterial cyaB1 adenylyl cyclase, which has an N-terminal tandem GAF domain closely related to those of the mammalian PDEs, as an assay system to identify the ligands for the human PDEs 10 and 11 GAF domains. We report that a chimera between the PDE10 GAF domain and the cyanobacterial cyclase was 9-fold stimulated by cAMP (EC50= 19.8 microm), whereas cGMP had only low activity. cAMP increased Vmax in a non-cooperative manner and did not affect the Km for ATP of 27 microm. In an analogous chimeric construct with the tandem GAF domain of human PDE11A4, cGMP was identified as an allosteric activator (EC50 = 72.5 microm) that increased Vmax of the cyclase non-cooperatively 4-fold. GAF-B of PDE10 and GAF-A of PDE11A4 contain an invariant NKFDE motif present in all mammalian PDE GAF ensembles. We mutated the aspartates within this motif in both regions and found that intramolecular signaling was considerably reduced or abolished. This was in line with all data concerning GAF domains with an NKFDE motif as far as they have been tested. The data appeared to define those GAF domains as a distinct subclass within the >3100 annotated GAF domains for which we propose a tentative classification scheme.     

A second-order high resolution finite difference scheme for a structured erythropoiesis model subject to malaria infection

We develop a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model describing the within-host dynamics of malaria infection. The model consists of two nonlinear partial differential equations coupled with three nonlinear ordinary differential equations. Convergence of the numerical method to the unique weak solution with bounded total variation is proved. Numerical simulations demonstrating the achievement of the designed accuracy are presented.     

Characterization of a catalytic ligand bridging metal ions in phosphodiesterases 4 and 5 by molecular dynamics simulations and hybrid quantum mechanical/molecular mechanical calculations

Cyclic nucleotide phosphodiesterases (PDEs) constitute a large superfamily of enzymes regulating concentrations of intracellular second messengers cAMP and cGMP through PDE-catalyzed hydrolysis. Although three-dimensional x-ray crystal structures of PDE4 and PDE5 have been reported, it is uncertain whether a critical, second bridging ligand (BL2) in the active site is H2O or HO- because hydrogen atoms cannot be determined by x-ray diffraction. The identity of BL2 is theoretically determined by performing molecular dynamics simulations and hybrid quantum mechanical/molecular mechanical (QM/MM) calculations, for the first time, on the protein structures resolved by x-ray diffraction. The computational results confirm our previous suggestion, which was based on QM calculations on a simplified active site model, that BL2 in PDE4 should be HO-, rather than H2O, serving as the nucleophile to initialize the catalytic hydrolysis of cAMP. The molecular dynamics simulations and QM/MM calculations on PDE5 demonstrate for the first time that the BL2 in PDE5 should also be HO- rather than H2O as proposed in recently published reports on the x-ray crystal structures, which serves as the nucleophile to initialize the PDE5-catalyzed hydrolysis of cGMP. These fundamental structural insights provide a rational basis for future structure-based drug design targeting PDEs.     

Analysis of progress curves by simulations generated by numerical integration.

A highly flexible computer program written in FORTRAN is presented which fits computer-generated simulations to experimental progress-curve data by an iterative non-linear weighted least-squares procedure. This fitting procedure allows kinetic rate constants to be determined from the experimental progress curves. Although the numerical integration of the rate equations by a previously described method [Barshop, Wrenn & Frieden (1983) Anal. Biochem. 130, 134-145] is used here to generate predicted curves, any routine capable of the integration of a set of differential equations can be used. The fitting program described is designed to be widely applicable, easy to learn and convenient to use. The use, behaviour and power of the program is explored by using simulated test data.     

Expression and activity of cGMP-dependent phosphodiesterases is up-regulated by lipopolysaccharide (LPS) in rat peritoneal macrophages

It has been shown that cyclic GMP (cGMP) modulates the inflammatory responses of macrophages, but the underlying molecular mechanisms are still poorly understood. Looking for proteins potentially regulated by cGMP in rat peritoneal macrophages (PMs), in this study we analyzed expression and activity of cGMP-hydrolyzing and cGMP-regulated phosphodiesterases (PDEs). It was found that freshly isolated peritoneal exudate macrophages (PEMs) express enzymes belonging to families PDE1-3, PDE5, PDE10, and PDE11. Analysis of substrate specificity, sensitivity to inhibitors, and subcellular localization showed that PDE2 and PDE3 are the main cGMP-regulated PDE isoforms in PEMs. The profile of PDE expression was altered by maintaining PEMs in culture and treatment with bacterial endotoxin (LPS). After 24 h culture, PDE5 was not present and the levels of PDE2, PDE3, and PDE11 were markedly decreased. However, their expression and activity was recovered after treatment of cultured cells with LPS. A similar pattern of changes was observed for the expression of TNFalpha, but not for guanylyl cyclase A (GC-A). LPS up-regulated PDE expression also in resident peritoneal macrophages (RPMs), although not all PDEs present in PEMs were detected in RPMs. Taken together, our results show that in rat PMs expression of cGMP-dependent PDEs positively correlates with the activation state of cells. Moreover, the fact that most of these PDEs hydrolyze also cAMP indicates that cGMP can play a role of potent regulator of cAMP signaling in macrophages.     

An efficient numerical method for distributed-loop models of the urine concentrating mechanism

In this study we describe an efficient numerical method, based on the semi-Lagrangian (SL) semi-implicit (SI) method and Newton's method, for obtaining steady-state (SS) solutions of equations arising in distributed-loop models of the urine concentrating mechanism. Dynamic formulations of these models contain large systems of coupled hyperbolic partial differential equations (PDEs). The SL method advances the solutions of these PDEs in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The SI approach controls stiffness arising from transtubular transport terms by averaging these terms in time along flow trajectories. An approximate SS solution of a dynamic formulation obtained via the SLSI method can be used as an initial guess for a Newton-type solver, which rapidly converges to a highly accurate numerical approximation to the solution of the ordinary differential equations that arise in the corresponding SS model formulation. In general, it is difficult to specify a priori for a Newton-type solver an initial guess that falls within the radius of convergence; however, the initial guess generated by solving the dynamic formulation via the SLSI method can be made sufficiently close to the SS solution to avoid numerical instability. The combination of the SLSI method and the Newton-type solver generates stable and accurate solutions with substantially reduced computation times, when compared to previously applied dynamic methods.