Instability in a generalized Keller–Segel model

We present a generalized Keller–Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller–Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.     

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

In two space dimensions, the parabolic–parabolic Keller–Segel system shares many properties with the parabolic–elliptic Keller–Segel system. In particular, solutions globally exist in both cases as long as their mass is less than a critical threshold M _c . However, this threshold is not as clear in the parabolic–parabolic case as it is in the parabolic–elliptic case, in which solutions with mass above M _c always blow up. Here we study forward self-similar solutions of the parabolic–parabolic Keller–Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above M _c , which is forbidden in the parabolic–elliptic case.     

A stochastic model for the spatial distribution of species based on an aggregation–repulsion rule

The heterogeneity associated with the spatial distribution of organisms is an awkward problem in ecology because this heterogeneity directly depends on the sampling scale. To specify the scope of the influence of sampling scale on the level of species aggregation, we need data sets that entail excessive sampling costs in situ. To find a solution for this problem, we can use models to simulate patterns of organisms. These models are often very complex models that take into account heterogeneity of habitats and displacement or longevity of studied species. In this article, we introduce a new stochastic model to simulate patterns for one taxon and we want this model to be parsimonious, i.e., with few parameters and able to simulate observed patterns. This model is based on an aggregation–repulsion rule. This aggregation–repulsion rule is defined by two parameters. On a large scale, the number of aggregates present on the pattern is the first parameter. On a smaller scale, the level of aggregation–repulsion among individuals is determined by a probability distribution. These two parameters are estimated from field data set in a robust way so that the simulated patterns reflect the observed heterogeneity. We apply this model to entomological data: four Diptera families, namely the Sciaridae, Phoridae, Cecidomyiidae, and Empididae. The field data for the Phoridae family are used to simulate sampling using different trap sizes. We record changes in the coefficient of variation (C) as a function of the sampling scale, and we can suggest to ecologists emergence traps of 0.6 m², in other words a square 77 × 77 cm trap, to obtain a C value under 20%. Received: February 28, 2000 / Accepted: October 14, 2000     

A parameter-space search algorithm tested on a Hodgkin–Huxley model

We demonstrate a parameter-space search algorithm using a computational model of a single-compartment neuron with conductance-based Hodgkin-Huxley dynamics. To classify bursting (the desired behavior), we use a simple cost function whose inputs are derived from the frequency content of the neural output. Our method involves the repeated use of a stochastic gradient descent-type algorithm to locate parameter values that allow the neural model to produce bursting within a specified tolerance. We demonstrate good results, including those showing that the utility of our algorithm improves as the pre-defined allowable parameter ranges increase and that the initial approach to our method is computationally efficient.     

Combined bioelectrochemical–electrical model of a microbial fuel cell

Several recent studies demonstrated significant charge storage in electrochemical biofilms. Aiming to evaluate the impact of charge storage on microbial fuel cell (MFC) performance, this work presents a combined bioelectrochemical–electrical (CBE) model of an MFC. In addition to charge storage, the CBE model is able to describe fast (ms) and slow (days) nonlinear dynamics of MFCs by merging mass and electron balances with equations describing an equivalent electrical circuit. Parameter estimation was performed using results of MFC operation with intermittent (pulse-width modulated) connection of the external resistance. The model was used to compare different methods of selecting external resistance during MFC operation under varying operating conditions. Owing to the relatively simple structure and fast numerical solution of the model, its application for both reactor design and real-time model-based process control applications are envisioned.     

A mathematical model for embryonic cell division based on a surface “cleavage field”

The process whereby a fertilized egg divides to give rise to an embryo, i.e. the process of cleavage—which can be considered, in some sense, as the early phase of embryonic differentiation—exhibit in many species a precise geometry. Such a geometry may be altered within certain limits, as was done in various classical experiments, and yet normal differentiation may occur. However, since the pattern of cleavage is clearly under genetic control, any model of cleavage should incorporate some device apt to produce a specific geometry. In this paper, a model for embryonic cell division based on a surface “cleavage field” is described. This surface field may be interpreted, for instance, as the surface density of sources of active transport of ions which diffuse into the cell, although other interpretations —such as the surface density of specific binding sites or functional membrane receptors, etc.—are possible. Assumptions relating the geometry of cleavage to the geometry of the level surfaces of the ionic concentration are given together with a discussion of the change in the surface field due to cleavage. Finally, a simplified two-dimensional version of the model is presented which develops interesting patterns of “cleavagerd”, calculated by computer, similar in many ways to those of real threedimensional embryos.     

A phenomenological model for chemico-mechanically induced cell shape changes during migration and cell–cell contacts

A phenomenological model for the evolution of shape transition of cells is considered. These transitions are determined by the emission of growth-factors, as well as mechanical interaction if cells are subjected to hard impingement. The originality of this model necessitates a formal treatment of the mathematical model, as well as the presentation of elementary cases in order to illustrate the consistence of the model. We will also show some small-scale relevant applications.     

A hybrid model of intercellular tension and cell–matrix mechanical interactions in a multicellular geometry

Biomechanics and Modeling in Mechanobiology - Epithelial cells form continuous sheets of cells that exist in tensional homeostasis. Homeostasis is maintained through cell-to-cell junctions that...     

A host–microparasite model with a resistant host

The theory of heterozygote advantage is often used to explain the genetic variation found in natural populations. If a large population randomly mates and the various genotypes have the same growth and death rates, the evolution of the genotypes follows Hardy–Weinberg proportions and polymorphism results. When other environmental stresses, like predators, prey and diseases, are present, polymorphism may or may not occur depending on how the various genotypes are affected by the stress. In this paper, we use a basic host–microparasite model to demonstrate that polymorphism can occur even if one genotype suffers a higher death rate than the others in the absence of the parasite if the heterozygote has resistance or immunity to the parasite.     

A numerical model of cellular blebbing: A volume-conserving,fluid–structure interaction model of the entire cell

In animal cells, blebs are smooth, quasi-hemispherical protrusions of the plasma membrane that form when a section of the membrane detaches from the underlying actin cytoskeleton and is inflated by flowing cytosol. The mechanics behind this common cellular activity are not yet clear. As a first step in the development of a full computational framework, we present a numerical model of overall cell behavior based upon the interaction between a background Newtonian-fluid cytosol and elastic structures modeling the membrane and filaments. The detailed micromechanics of the cytoskeletal network are the subject of future work. Here, the myosin-driven contraction of the actin network is modeled through stressed elastic filaments. Quantitative models of cytoskeletal micromechanics and biochemistry require accurate estimates of local stress and flow conditions. The main contribution of this paper is the development of a computationally efficient fluid–structure interaction model based on operator splitting, to furnish this data. Cytosol volume conservation (as supported by experimental evidence) is enforced through an intermediate energy minimization step. Realistic bleb formation and retraction is observed from this model, offering an alternative formulation to positing complex continuum behavior of the cytoplasm (e.g. poroelastic model of Charras et al., 2008).     

Transport of ginkgolides with different lipophilicities based on an hCMEC/D3 cell monolayer as a blood–brain barrier cell model

Aims

In this report, the transport of ginkgolides with different lipophilicities was investigated using an hCMEC/D3 cell monolayer as a blood–brain barrier (BBB) cell model in vitro in an attempt to explain ginkgolide transport path mediated by lipophilicity.

Main methods

The log P values of ginkgolides were determined by measuring the distribution of the molecule between oil and water. Additionally, the cytotoxicity of ginkgolides on hCMEC/D3 cells was assayed with the MTT method. Ginkgolide contents were determined with an ultra performance liquid chromatograph equipped with an evaporative light scattering detector (ULPC–ELSD) method. Apparent permeability coefficients (P_app) and efflux ratios (P_{appBL → AP}/P_{appAP → BL}) were then calculated to describe the transport characteristics of ginkgolide.

Key findings

The transport of ginkgolide A, ginkgolide B, ginkgolide C, and ginkgolide J across the hCMEC/D3 cell monolayer was non-directional. Additionally, ginkgolide C transport on the cell monolayer was time- and concentration-dependent in the paracellular pathway controlled by cytochalasin D (a tight junction modulator). The transport of ginkgolide N, ginkgolide L, and ginkgolide K across the cell monolayer displayed clear directionality at low ginkgolide concentrations. This behavior indicated that the transport of ginkgolide N, ginkgolide L, and ginkgolide K was influenced by the transcellular pathway containing an efflux protein accompanied by the paracellular pathway for passive diffusion. Additionally, the transport of ginkgolide K was increased significantly by co-culturing with a P-gp inhibitor.

Significance

These findings provide important information for elucidating ginkgolide transport pathways and may be beneficial for the design of ginkgolide molecules with high neuroprotective effects.     

A phenomenological cohesive model for the macroscopic simulation of cell–matrix adhesions

Cell adhesion is crucial for cells to not only physically interact with each other but also sense their microenvironment and respond accordingly. In fact, adherent cells can generate physical forces that are transmitted to the surrounding matrix, regulating the formation of cell–matrix adhesions. The main purpose of this work is to develop a computational model to simulate the dynamics of cell–matrix adhesions through a cohesive formulation within the framework of the finite element method and based on the principles of continuum damage mechanics. This model enables the simulation of the mechanical adhesion between cell and extracellular matrix (ECM) as regulated by local multidirectional forces and thus predicts the onset and growth of the adhesion. In addition, this numerical approach allows the simulation of the cell as a whole, as it models the complete mechanical interaction between cell and ECM. As a result, we can investigate and quantify how different mechanical conditions in the cell (e.g., contractile forces, actin cytoskeletal properties) or in the ECM (e.g., stiffness, external forces) can regulate the dynamics of cell–matrix adhesions.     

Regulating cell–cell junctions from A to Z

Epithelial sheets often present a “cobblestone” appearance, but the mechanisms underlying the dynamics of this arrangement are unclear. In this issue, Choi et al. (2016. J. Cell Biol. http://dx.doi.org/10.1083/jcb.201506115) show that afadin and ZO-1 regulate tension and maintain zonula adherens architecture in response to changes in contractility.The textbook view of epithelial cells is that once such cells adopt a close, hexagonal packing, their “honeycomb” or “cobblestone” arrangement is static. This fixed appearance is misleading, as these cells are more like players in a rugby scrum, locked in a tussle in which the forces exerted by each of the players on the others maintains their seemingly static arrangement, but by a very dynamic force balance. How such balance is maintained in epithelia is a subject of substantial interest. A crucial role is played by F-actin and nonmuscle myosin II isoforms, which are deployed in contractile networks that transiently attach to cell–cell junctions to generate tensile forces along cell–cell boundaries (Lecuit and Yap, 2015). Contractile arrays of actomyosin are regulated by the monomeric G protein Rho, its upstream regulators, including Rho guanine nucleotide exchange factors (Quiros and Nusrat, 2014), and its effectors ROCK/Rho kinase and Shroom3 (Nishimura and Takeichi, 2008), but also by tension-mediated feedback between the myosin network and the junction (Lecuit and Yap, 2015). Cell–cell adhesion, including cadherin-dependent adhesion, also plays a crucial role in this process. As cells engage with one another via interactions of the extracellular domains of their cadherin complexes, they transduce forces to the actomyosin cytoskeleton through catenins. β-Catenin binds to the cytoplasmic domain of classical cadherins and recruits α-catenin, which binds F-actin.Given the dynamic nature of epithelia, the attachment of contractile actomyosin networks to junctions are also subject to regulation. One aspect of epithelial architecture that has received relatively little attention is that a typical epithelial monolayer (Fig. 1 A) displays two main types of cell–cell interfaces: bilateral junctions (BCJs), in which two cells establish a relatively long stretch of contact, and cellular vertices, which represent a confluence of three or more cell edges to form tricellular junctions (TCJs) or multicellular junctions. TCJs are not well understood, but are known to contain unique molecular components (Furuse et al., 2014; Flores-Benitez and Knust, 2015). In this issue, Choi et al. show that the multivalent scaffolding proteins afadin and ZO-1/2 regulate the spacing of and tension along lateral contacts in cultured cells, thereby shedding light on how contractile arrays containing bilateral and tri- or multicellular contact points are regulated in epithelia.Open in a separate windowFigure 1.ZO proteins and afadin regulate junctional tension and organization in cultured cells. (A) Untreated MDCK cells have sinuous cell boundaries, whereas ZO KD cells show extremely straight boundaries. When ZO proteins and afadin are knocked down, cells adopt contact zones of irregular length with other cells, sometimes clustering into foci (asterisks). Images courtesy of Mark Peifer (Choi et al., 2016). (B) A model for actomyosin organization at adherens junctions (adapted from Choi et al., 2016). Contractile actomyosin arrays run parallel to bicellular junctions and are anchored by side-on attachments (pink circles). At TCJs, end-on binding of actin, likely stabilized by afadin, anchors actomyosin filaments. In ZO KD cells, contractile elements and cadherin complexes collapse toward TCJs, and myosin minifilaments adopt a regularly spaced arrangement.Afadin and ZO-1/2 are far from new players at junctions. Afadin binds α-catenin, actin, and other cytoskeletal and junctional proteins and associates with the transmembrane protein nectin, which appears to form an alternative adhesion system at adherens junctions (Mandai et al., 2013). The zonula occludens proteins ZO-1 and ZO-2 are tight junction proteins that bind claudins and are required for tight junction formation (Itoh et al., 1999; Balda and Matter, 2008). In addition, ZO proteins also bind to α-catenin (Itoh et al., 1997), are involved in establishing the zonula adherens (ZA; Ikenouchi et al., 2007), and potentiate cadherin-dependent adhesion in Caenorhabditis elegans (Lockwood et al., 2008) and Drosophila melanogaster (Choi et al., 2011). Knockdown of ZO-1 and ZO-2 (ZO KD) in MDCK cells has previously been shown (Fanning et al., 2012) to lead to dramatic alterations of the ZA: F-actin and myosin IIs assemble into striking apical arrays at the ZA, spaced at regular intervals. In addition, the normally sinuous boundaries between cells give way to very straight borders (Fig. 1 A).Using superresolution microscopy, diffraction-limited junctional laser ablation, cell morphometry, kinetic analysis, and a whole-monolayer approach to contractility, Choi et al. (2016) now extend this story. To test whether contractility is increased after ZO KD, the authors first measured the recoil after laser ablation of ZO KD cells; an increase in recoil velocity indicated that the straight junctional boundaries between ZO-depleted cells are under tension. Imaging analysis of BCJs showed that the increase in contractility in ZO KD cells is associated with a strikingly dynamic behavior of the BCJs. Individual BCJs were found to undergo periods of shortening and elongation, whereas neighboring BCJs underwent compensatory, opposite changes in length. These changes in contractility have effects on the entire tissue sheet as well: whereas control cell sheets remained flat when detached from the substratum, ZO KD cells contracted into a cup-like shape. This constriction was blocked by the myosin inhibitor blebbistatin. Overall, these experiments indicated that ZO proteins regulate myosin assembly and contractility across the cellular sheet.To dissect the protein network mediating increased contractility in ZO KD cells, Choi et al. (2016) examined the role of ROCK and found that ROCK inhibitors abolished the straight BCJs, which became curvilinear. Additionally, Shroom3, which is known to recruit ROCK (Nishimura and Takeichi, 2008), was cytoplasmic in control cells but junctional in ZO KD cells. Transient Shroom3 overexpression led to ROCK recruitment to the ZA and drove formation of an actomyosin network similar to that in ZO KD cells. Conversely, Shroom3 knockdown resulted in loss of the actomyosin arrays in ZO KD cells. Collectively, these data indicated that Shroom3 is an effector of increased apical contractility in ZO KD cells.The researchers used ZO KD cells to test how tissue integrity is maintained despite elevated contractibility and how junctions are remodeled to maintain integrity when increased tension is present. Afadin is a good candidate: the Drosophila homologue of afadin, Canoe, plays roles in convergent extension and collective cell migration; in its absence, actomyosin networks at the apex of constricting epithelial cells in the embryo contract in a catastrophic, uncontrolled fashion (Sawyer et al., 2009), suggesting a potential role for afadin in the maintenance of tissue integrity during morphogenetic movements. Choi et al. (2016) therefore turned their attention to afadin. ZO KD cells have significantly more afadin at their adherens junctions and TCJs, a pattern reminiscent of the normal distribution of Canoe in Drosophila (Sawyer et al., 2009). Knocking down afadin by shRNA in ZO KD cells led to further defects in cell–cell boundary maintenance. In addition to the taut appearance of bicellular borders, cell boundary length became much more irregular, with occasional foci of highly contracted cells (Fig. 1 A). Velocimetry analysis and live-cell imaging indicated that loss of both ZO proteins and afadin led to large-scale cell movements within the monolayer not seen after ZO KD alone.New imaging techniques used by Choi et al. (2016) revealed further details about the changes in actomyosin arrays in ZO KD cells. Superresolution imaging of myosin light chain kinase staining via structured illumination showed that myosin II assembles into arrays of myosin minifilaments spaced ∼415 nm apart along bicellular contacts. Superresolution and transmission electron microscopy also revealed reorganization of F-actin and E-cadherin at TCJs in ZO KD cells. Lateral F-actin bundles appeared to terminate end-on at TCJs at sites where E-cadherin was present. ZO KD therefore induces assembly of a remarkably ordered actomyosin array along BCJs, and these arrays appear to be separate contractile units that anchor end-on at the ZA. Moreover, based on staining for vinculin and a specific epitope in αE-catenin that serve as markers for regions under high tension (Yonemura et al., 2010), the end-on attachments of actin cables to the ZA at TCJs experience significant tensile stress. Strikingly, although vinculin and αE-catenin accumulation at TCJs was relatively uniform after ZO KD, their distribution was more heterogeneous after ZO/afadin KD. Differences in staining paralleled differences in cell border length and correlated with the level of tension measured at BCJs after laser cutting, suggesting that afadin contributes to the ability of cells to distribute forces at TCJ/multicellular junctions throughout the monolayer. Lastly, the researchers investigated whether internal cues downstream of ZO KD are sufficient for myosin recruitment or whether such recruitment depends on mechanical cues exerted by neighboring cells. They designed an assay mixing small islands of wild-type cells surrounded by ZO KD cells (or vice versa) and found that the development of the contractile array at the ZA depends on the contractility of neighboring cells; however, afadin recruitment to the ZA was less dependent on the sustained contractility of neighboring cells. Taking these data together, Choi et al. (2016) propose that cells respond to elevated contractility by increasing junctional afadin; because combined ZO/afadin knockdown dramatically alters cell shape and barrier function in response to elevated contractility, afadin acts as a robust scaffold that maintains ZA architecture most crucially at TCJs.Although many aspects of the model proposed by Choi et al. (2016) remain to be tested, their data suggest new features regarding the detailed assembly of actomyosin contractile arrays in confluent cells (Fig. 1 B). In control cells, actomyosin arrays presumably extend parallel to individual BCJs. Choi et al. (2016) propose that these actomyosin bundles act as separate contractile units that terminate near TCJs, allowing the generation of tension along BCJs. In ZO KD cells, excessive assembly of actomyosin filaments, perhaps exacerbated by the tendency of F-actin/myosin minifilament arrays to self-assemble, somehow leads to regularly spaced actomyosin arrays, and perhaps collapse of cadherin complexes and other components toward TCJs. There is a precedent for such lateral collapse of cadherin-dependent attachments: it is a prominent feature of cadherin complexes at sites of high tension in the epidermis of the C. elegans embryo (Choi et al., 2015). If the new model of Choi et al. (2016) is correct, then the foci seen in ZO KD/afadin KD cells may be similar to what happens in a game of tug of war when one team stops pulling. If some end-on attachments (assisted by afadin) fail, filaments might be expected to collapse along BCJs as the other, still tethered end of a set of filaments contracts toward the remaining attachment at the opposite cell vertex.Several other interesting questions remain. First, what is the relationship of the striking, regularly spaced bipolar myosin II minifilaments that form in ZO KD cells to myosin arrays in normal cells? It is clear that untreated cells have junctional actomyosin networks, but not with this strict periodicity. One possibility is that this spacing is simply an epiphenomenon; when not appropriately anchored along junctions, actomyosin networks may self-organize as they are known to do in other systems, such as in the contractile ring and in migrating cells (Srivastava et al., 2015; Fenix et al., 2016). More optimistically, the spacing may represent an intensified version of processes that operate in normal cells at bicellular and multicellular contact sites. If so, components of the model of Choi et al. (2016) will require further investigation. For example, the organization of F-actin along BCJs remains unclear, as are the proteins that mediate the side-on binding envisioned in this model. It is also uncertain whether proteins assist bundling of filaments and what role dynamic growth and shrinkage of actin filaments plays in end-on binding. In some contexts, junctions are capable of seeding polymerization of F-actin (Brieher and Yap, 2013), and it may be that actin dynamics are important in the processes studied here.A second question has to do with the community events within monolayers that Choi et al. (2016) describe. The neighbor effects on ZA morphology that they document are intriguing, as are the long-range accelerated movements of cells lacking both ZO proteins and afadin. Collective properties of monolayers are only beginning to be explored; connecting these properties with subcellullar events is an exciting future challenge. Whatever the answers to these new questions, the work of Choi et al. (2016) refines our understanding of the roles of key scaffolding proteins in organizing and anchoring junctions in epithelia.     

ProPose: a docking engine based on a fully configurable protein–ligand interaction model

Virtual high-throughput screening of molecular databases and in particular high-throughput protein–ligand docking are both common methodologies that identify and enrich hits in the early stages of the drug design process. Current protein–ligand docking algorithms often implement a program-specific model for protein–ligand interaction geometries. However, in order to create a platform for arbitrary queries in molecular databases, a new program is desirable that allows more manual control of the modeling of molecular interactions.For that reason, ProPose, an advanced incremental construction docking engine, is presented here that implements a fast and fully configurable molecular interaction and scoring model. This program uses user-defined, discrete, pharmacophore-like representations of molecular interactions that are transformed on-the-fly into a continuous potential energy surface, allowing for the incorporation of target specific interaction mechanisms into docking protocols in a straightforward manner. A torsion angle library, based on semi-empirical quantum chemistry calculations, is used to provide minimum energy torsion angles for the incremental construction algorithm. Docking results of a diverse set of protein–ligand complexes from the Protein Data Bank demonstrate the feasibility of this new approach.As a result, the seamless integration of pharmacophore-like interaction types into the docking and scoring scheme implemented in ProPose opens new opportunities for efficient, receptor-specific screening protocols. Figure ProPose — a fully configurable protein-ligand docking program — transforms pharmacophores into a smooth potential energy surface.This revised version was published online in October 2004 with corrections to the Graphical Abstract.     

T cell–tumor cell: a fatal interaction?

Fas (Apo-1/CD95) is a cell-surface protein that is responsible for initiating a cascade of proteases (caspases) culminating in apoptotic cell death in a variety of cell types. The function of the Fas/FasL system in the dampening of immune responses to infectious agents through the autocrine deletion of activated T cells has been well documented. More recently, it has been proposed that tumor cells express FasL, presumably to avoid immune detection. In this review, we focus on the role of the interaction of Fas and FasL in the modulation of antitumor responses. We critically examine the evidence that FasL is expressed by tumor cells and explore alternative explanations for the observed phenomena in vitro and in vivo. By reviewing data that we have generated in our laboratory as well as reports from the literature, we will argue that the Fas/FasL system is a generalized mechanism used in an autocrine fashion to regulate cell survival and expansion in response to environmental and cellular cues. We propose that FasL expression by tumor cells, when present, is indicative of a perturbed balance in the control of proliferation while “immune privilege” is established by “suicide” of activated antitumor T cells, a form of activation-induced cell death. Received: 5 May 1998 / Accepted: 20 May 1998     

Effects of curvature and cell–cell interaction on cell adhesion in microvessels

It has been found that both circulating blood cells and tumor cells are more easily adherent to curved microvessels than straight ones. This motivated us to investigate numerically the effect of the curvature of the curved vessel on cell adhesion. In this study, the fluid dynamics was carried out by the lattice Boltzmann method (LBM), and the cell dynamics was governed by the Newton’s law of translation and rotation. The adhesive dynamics model involved the effect of receptor-ligand bonds between circulating cells and endothelial cells (ECs). It is found that the curved vessel would increase the simultaneous bond number, and the probability of cell adhesion is increased consequently. The interaction between traveling cells would also affect the cell adhesion significantly. For two-cell case, the simultaneous bond number of the rear cell is increased significantly, and the curvature of microvessel further enhances the probability of cell adhesion.     

A multicomponent reaction–diffusion model of a heterogeneously distributed immobilized enzyme

A physical model was derived for the synthesis of the antibiotic cephalexin with an industrial immobilized penicillin G acylase, called Assemblase. In reactions catalyzed by Assemblase, less product and more by-product are formed in comparison with a free-enzyme catalyzed reaction. The model incorporates reaction with a heterogeneous enzyme distribution, electrostatically coupled transport, and pH-dependent dissociation behavior of reactants and is used to obtain insight in the complex interplay between these individual processes leading to the suboptimal conversion. The model was successfully validated with synthesis experiments for conditions ranging from heavily diffusion limited to hardly diffusion limited, including substrate concentrations from 50 to 600 mM, temperatures between 273 and 303 K, and pH values between 6 and 9. During the conversion of the substrates into cephalexin, severe pH gradients inside the biocatalytic particle, which were previously measured by others, were predicted. Physical insight in such intraparticle process dynamics may give important clues for future biocatalyst design. The modular construction of the model may also facilitate its use for other bioconversions with other biocatalysts.