In vertebrates, supply of oxygen and nutrients to tissues is carried out by the blood vascular system through capillary networks. Capillary patterns are closely mimicked by endothelial cells cultured on Matrigel, a preparation of basement membrane proteins. On the Matrigel surface, single randomly dispersed endothelial cells self-organize into vascular networks. The network is characterized by a typical length scale, which is independent on the initial mean density of deposed cells N over a wide range of values of N. We give a detailed description of a mathematical model of the process which has proven able to reproduce several qualitative and quantitative features of in vitro vascularization experiments. The model is basically a multidimensional Burgers\' equation coupled to an equation modeling the diffusion of a chemoattractant factor. Starting from sparse initial data, mimicking the initial conditions realized in laboratory experiments, the solutions to the model equations develop characteristic network structures, similar to observed ones, whose average size is related to the finite range of chemoattractant diffusion.

• Thematic program: Mathematical Biology (2004)

• Thematic program: Mathematical Biology (2004)

The spreading of metastases through angiogenesis has been recently modeled by some authors in a deterministic way. A stochastic version, which may take into account the so-called \"dormancy\" phenomenon, is presented here. The system of growing and spreading tumors is represented like a gas of independent and proliferating random walks in the discrete space of sizes, with absorption in 0. The small (microscopic) sizes play here a critical role, and qualitative differences with respect to the deterministic model are pointed out.

• Thematic program: Mathematical Biology (2004)

• Thematic program: Mathematical Biology (2004)

• Thematic program: Mathematical Biology (2004)

• Thematic program: Mathematical Biology (2004)

Cellular interactions with the extracellular matrix and migration therein are fundamental to tissue inflammation, wound repair, and cancer progression. Using static and time-resolved bright field, confocal, and multiphoton microscopy, we have reconstructed the live-cell morphology, position, and dynamics in fibroblast of tumor and immune cells in 3D tissue culture models in vitro as well as by intravital microscopy in vivo. 5D semiquantitative reconstruction in tissue reconstructs was used to show (i) cytoskeletal dynamics of filamentous actin in invading B16 melanoma cells, (ii) the time-resolved subcellular topography of proteolytic cleavage of the extracellular matrix structures by invading cancer cells, and (iii) calcium transients and redistribution of lipid rafts in immune cells scanning antigen-presenting cells (APC). Thereby, live-cell imaging in vitro and in vivo has provided new concepts and tools to monitor molecular cell dynamics and tissue patterning, including proteolytic and nonproteolytic pathways to tissue remodeling and regeneration, single and collective cancer cell invasion into connective tissues migration and the migratory tissue scanning by T cells resulting in serial immune cell interactions with APC and the formation of a dynamic immunological immune synapse.

• Thematic program: Mathematical Biology (2004)

Experiments of in vitro vasculogenesis show that when endothelial cells are spread on a matrix gel, they self-organize into quite regular geometrical patterns. These structures mimic in two dimensions the capillary networks forming the blood vascular system in vertebrates. Patterns arise just when the initial number of cells $\\bar n$ is in a specific range and, inside this range, the typical spatial scale of the structures is nearly independent on $\\bar n$. The morphogenesis process occurs along three distinct stages: in the early 3--6 hours cells walk on the matrigel, later they adhere to the substrate and finally eject philopodes that are stretched up to a stabilization of the new vasculature. Mathematical models of this process develop according to two mainstreams. Murray and coworkers (see for instance [1]) focus on the strain process and state that the formation of the lacunae in the cell density just as a consequence of the tension exerted by the cells hanging on the matrigel. More recently it has been noticed that pure mechanical effects cannot predict the measured invariance of the chord length: the strain field can only amplify lacunae that are dictated by the initial conditions and no characteristic lengths can arise. One can instead properly account for this experimental fact when noting that cells migrate before adhering, a random walk biased by a chemotactic signalling [2]. It is then possible to describe the system of the cells as a fluid, including chemoattraction as a bulk force. A characteristic length naturally arises and can be expressed in terms of physical parameters that to be measured independently. Aim of this talk is to provide a new model, providing a unified view of the morphogenetic process. When including both cell migration and matrigel stretch it possible to predict both the characteristic length scale and the thinning effect due to the tension field. ;; [1] Manoussaki, D., Lubkin, S.R., Vernon, R.B., Murray, J.D. A mechanical model for the formation of vascular networks in vitro. Acta Biotheoretica 44:271--282 (1996). ; [2] D. Ambrosi, A. Gamba and G. Serini. Cell directional and chemotaxis in vascular morphogenesis, Bulletin of Mathematical Biology, 66 (2004).

• Thematic program: Mathematical Biology (2004)

A Chapman-Enskog expansion is used to derive hyperbolic models for chemosensitive movements as a hydrodynamic limit of a velocity-jump process. On the one hand, it connects parabolic and hyperbolic chemotaxis models since the former arise as diffusion limits of a similar velocity-jump process. On the other hand, this approach provides a unified framework which includes previous models obtained by ad hoc methods or methods of moments. Numerical methods with different orders of accuracy are proposed to approximate these hyperbolic models. First and second order well-balanced finite volume schemes are presented. This approach provides an exact conservation of the steady state solutions. Then, a high order finite difference weighted essentially non-oscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme in order to exactly preserve steady states and to retain high order accuracy. Numerical simulations are also performed and are motivated by recent experiments with human endothelial cells on matrigel. Their movements lead to the formation of networks that are interpreted as the beginning of a vasculature. These structures cannot be explained by parabolic models but are recovered by numerical experiments on hyperbolic models.

• Thematic program: Mathematical Biology (2004)

Cells communicate via a variety of signalling mechanisms, stimulating responses ranging from chemotactic migration to mitosis. In this talk, I will introduce a mathematical model that incorporates the microscopic aspects of signalling, from the extracellular space via the cell membrane to an internal signal. This model will be used to understand two types of cellular signalling. In the first, the problem of chemical relays will be considered. Here, I will use the model to understand how different local modes of signalling affect the macroscopic patterns of signal spread. In the second example I will use the model to understand how morphogen gradients can be robustly set up in a population of embryonic cells.

• Thematic program: Mathematical Biology (2004)

• Thematic program: Mathematical Biology (2004)

Directed migration of amoeboid cells is involved in processes such as embryonic development, wound healing, and the metastasis of cancer. The motile machinery of cells involved in these types of processes is controlled by a diffusible signal released into the surrounding tissue from sources that depend on the particular process in question. It has also been shown experimentally that mechanical stresses which are transmitted to and from the cell play an important role in cell motility and have an effect on intracellular signalling processes. Because of the importance of intracellular stresses, we focus on the role of mechanics in amoeboid cell motility. We present a phenomenological model of cellular mechanics in which the protrusion and retraction typically associated with amoeboid cell \'crawling\' are incorporated through a localized multiplicative decomposition of the deformation gradient into active and passive parts. The active part of the deformation gradient is meant to incorporate the role of actin dynamics in cell motility. Experiments have shown that the passive response of the cell is viscoelastic, and we incorporate this rheology through an appropriately chosen constitutive equation. Finite element numerical simulations will be shown, and we will also discuss numerical issues associated with solving a nonlinear model of this magnitude.

• Thematic program: Mathematical Biology (2004)

While computational models for cell migration on two-dimensional substrata have described how various molecular and cellular properties and physicochemical processes are integrated to accomplish cell locomotion, the same issues, along with certain new ones, might contribute differently to a model for migration within three-dimensional matrices. To address this more complicated situation, we have developed a computational model for cell migration in three-dimensional matrices using a force-based dynamics approach. This model determines an overall locomotion velocity vector, comprising speed and direction, for individual cells based on internally-generated forces transmitted into external traction forces and considering a time-scale during which attachment and detachment events occur. Key parameters characterize cell and matrix properties, including cell/matrix adhesion, mechanical properties of the matrix, and proteolytic matrix degradation; certain underlying molecular properties are incorporated explicitly or implicitly. Model predictions agree well with experimental results for the limiting case of migration on two-dimensional substrata as well as recent experiments in three-dimensional natural tissues and synthetic gels.

• Thematic program: Mathematical Biology (2004)

Actin polymerizes in cells and generates movement and deformations. This phenomenon of force generation by monomer assembly is currently restricted to biological systems. To probe the basis of polymerization-driven force production, we use biomimetic experimental systems that allow for versatile handling, where parameters like the size, and the deformability of the propelled object can be controlled. With this set-up, we are able to measure the force generated as a function of the velocity of propulsion. The main questions are now : how do we use biomimetic systems to better understand cell motility? and Is movement induced by polymerization more general in science, and not only confined to cytoskeletal proteins?

• Thematic program: Mathematical Biology (2004)

The overall trend in theoretical studies of self-propelled cell motion is to break down the underlying molecular biology and physics of the main cell components responsible for motility. To gain quantitative insight into the interactions between these components, we adapt a two-phase reactive flow model for the active motion of a single cell on a surface, incorporating simple sub-models for adhesion and membrane morphology. We exploit the disparate length and time scales characterising the physically widely-relevant thin-cell limit to simplify dramatically the governing equations. Incorporating simple sub-models for protrusive and retractive force generation at the contact line, the strong adhesion limit can result in novel multi-valued contact-line laws describing the motion of the outer cell periphery. This can lead to some intriguing types of behaviour, ranging from periodic contraction and expansion (pulsation) to steady propagation at a constant speed and an unsteady combination of pulsation and propagation. These have all been observed in practice. The resulting contact-line behaviour is highly sensitive to environmental signals and such a formulation may accordingly provide a useful \'minimal\' modelling framework for investigation of chemotactic effects at the cell scale.

• Thematic program: Mathematical Biology (2004)

• Thematic program: Mathematical Biology (2004)

A great deal of progress has made toward understanding how microscopic aspects of cell movement can be incorporated into macroscopic chemotaxis equations for simple swimmers such as E. coli, but the problem is far more difficult for crawling cells such as leukocytes. In this talk we will identify some essential behavioral aspects that macroscopic equations for chemotaxis of amoeboid cells must reflect, and discuss progress on the micro- to macro problem for such cells.

• Thematic program: Mathematical Biology (2004)

• Thematic program: Mathematical Biology (2004)

The way cells evaluate chemical signals in order to reorient themselves in their actual chemical environment and the way they interact with each other during chemotactic movement has a strong effect on the macroscopic, namely cell population level. During the talk the interplay between microscopic effects and macroscopic structure formation will be disussed.

• Thematic program: Mathematical Biology (2004)

The ability of eukaryotic cells to sense spatial gradients of chemoattractant factors and to move towards their increasing concentrations underlie the development of complex organisms and of life in general. Cells exposed to shallow gradients in chemoattractant concentration respond with strongly asymmetric accumulation of several factors, including the phosphoinositides PI(3,4,5)P3 and PI(3,4)P2, and the PI 3-kinase (PI3K) and phosphatase (PTEN). This early amplifying stage is believed to trigger effector pathways leading to cell movement. Although many factors implied in directional sensing are known, the mechanism itself is still rather mysterious. Here we consider the possibility that the main features of directional sensing observed in the experiments are the consequence of a phase ordering process driven by the distribution of the external signal. We develop a computational model based on these principle, which naturally realizes large amplification of shallow chemical gradients, selective localization of chemical factors, macroscopic response timescales, ability to respond over a wide range of stimulus concentration and spontaneous polarization. The proposed mechanism is robust with respect to variations of the system parameters.

• Thematic program: Mathematical Biology (2004)

In the literture on chemotaxis modeling the existence of blow-up solutions has been a major focus. If the finite volume of individuls is introduced into the model, then solutions will no longer blow-up but exist globally. In addition they show interesting pattern fromations. Painter and I classified the volume effects into (i) \"volume filling\" if the finite cell size is incorporated, (ii) \"quorum sensing\" for cells that release a repulsive chemical signal, and (iii) \"finite sampling radius\" to model the measurement of a chemical signals on the cells outer membrane.; While the volume-filling model has been discussed in a paper by Hillen and Painter in 2000, I will use this talk to focus on the quorum-sensing effect and the finite sampling radius. I will show reslts on global in time existence and pattern formation. Moreover, I will pose some open problems.

• Thematic program: Mathematical Biology (2004)

Abstract: Crawling of animal cells is based on three coupled mechanisms: protrusion of the leading edge, graded adhesion to the surface, and contraction of the cytoskeleton. It is also regulated by a rapid turnover of actin. In order to elucidate basic principles of mechanochemistry of the cell movements, we develop a computational model of migrating fish keratocyte cells. These flat, simple shaped cells move rapidly, smoothly and persistently. They are arguably the simplest cell motility modeling system. The model consists of few `modules\': (i) polymerization ratchet model of protrusion, (ii) dynamic actin-myosin contraction model, (iii) dendritic-nucleation actin turnover model. We combine these models and simulate the cell as a 2-D domain with a free boundary using finite elements. The simulations reproduce the observed shapes and movements of the keratocyte cells and predict F-actin and G-actin densities. Comparing the computational and experimental results allows us to suggest the dynamic principles of spatio-temporal organization of cell movements and forces.

• Thematic program: Mathematical Biology (2004)

The actin cytoskeleton and its associated proteins produce the force for locomotion, and adhesion molecules transmit this force to the substrate to move the cell. The connection between the cytoskeleton and the substrate, formed by integrins, a major family of adhesion molecules, is dynamic and regulated. The link is formed at the front of the cell; integrins then release from both the substrate and the cytoskeleton toward the rear as the cell moves. Modulation of integrin-mediated adhesion to substrates occurs through three mechanisms: changes in adhesion molecule expression levels, adjustment of affinity (by conformational changes in the integrin molecule) and changes in avidity (by integrin rearrangement). Rearrangement of integrins can activate adhesion independent of affinity changes. Since the cytoskeleton controls integrin rearrangement, regulation of Integrin-cytoskeleton connections also has consequences for integrin-substrate adhesion, by changing avidity. This mechanism is effective both under static conditions and under flow, as occurs in the bloodstream. Under flow, integrins cooperate with another class of adhesion molecules, the selectins, in avidity adjustment. The regulation of this process is complex, and only beginning to be understood.

• Thematic program: Mathematical Biology (2004)

Chemotaxing cells, such as Dictyostelium and mammalian neutrophils, sense shallow chemoattractant gradients and respond with highly polarized changes in cell morphology and motility. Uniform chemoattractant stimulation induces the transient translocations of several second messengers, including PI3K, PTEN, and PI(3,4,5)P3. In contrast, static spatial chemoattractant gradients elicit the persistent, amplified localization of these molecules. We have proposed a model in which the response to chemoattractant is regulated by a balance of a local excitation and a global inhibition (LEGI), both which are controlled by receptor occupancy. The LEGI model can account for both the transient and spatial responses to chemoattractants, but alone does not amplify the external gradient. In this talk we show how parallel LEGI mechanisms induce an amplified PI(3,4,5)P3 response that agrees quantitatively with experimentally obtained PH-GFP distributions.

• Thematic program: Mathematical Biology (2004)

Chemotaxis is the cell movement induced by chemical substances. It has been studied in a mathematical way since \'50s but experienced a boost after the seminal work of Keller and Segel (late \'60s). In this talk we show the basic features of chemotaxis and present a short introduction to the main models in the literature. First we present the Keller-Segel model and subsequently the so called Othmer-Dunbar-Alt model (kinetic models). We show the relevance of studying two different scales of description by pointing out questions that have different and similar answers in each scales. In particular we try to shed light on the question of blow up. We also study the drift-diffusion limit of the kinetic models, showing precise mathematical conditions such that the limit of kinetic models is the Keller-Segel model.

• Thematic program: Mathematical Biology (2004)

We investigate the spontaneous behaviors of the actomyosin cell cortex when the control of its contractility by microtubules is suppressed. Under these conditions and in the absence of substrate adhesions, the cell cortex spontaneously breaks and a membrane bulge devoid of detectable actin and myosin is expelled through the hole. A constriction ring at the base of the bulge oscillates from one side of the cell to the other. The movement is accompanied by sequential redistribution of actin and myosin to the membrane. We observe this oscillatory behavior also in cell fragments of various sizes, providing a simplified, nucleus-free system for biophysical studies (1). We conclude that it reveals an intrinsic behavior of the actomyosin cortex. Our observations suggest a mechanism based on active gel dynamics and inspired by our study of symmetry breaking in actin gels growing around beads (2). We further propose that bleb formation and cortical flows are direct consequences of cortical contractility and analyze the implications of these spontaneous behaviors in cell locomotion. ;; (1) Paluch E, Piel M, Prost J, Bornens M, Sykes C. 2005. Cortical actomyosin breakage triggers shape oscillations in cells and cell fragments. Biophys. J. 89:724-733.; (2) van der Gucht J, Paluch E, Plastino J, Sykes C. 2005. Stress release drives symmetry breaking for actin-based movement. Proc. Natl. Acad. Sci. USA 102:7847-7852.

• Thematic program: Mathematical Biology (2004)

Actin polymerization and network formation are key processes in cell motility. Numerous actin binding proteins controlling the dynamic properties of actin networks have been studied and models such as the dendritic nucleation scheme have been proposed for the functional integration of at least a minimal set of such regulatory proteins. However, a complete understanding of actin network dynamics is still lacking. Even at the actin-filament level, the dynamics of the distribution of filament lengths and nucleotide profiles is still not fully understood. In this talk we will describe recent work on the evolution of the distribution of filament lengths and nucleotide profiles of actin filaments. The distributional dynamics of actin filaments are investigated in the framework of both deterministic and stochastic chemical kinetics. For the latter we develop a master equation for the biochemical processes involved at the individual filament level and simulate the dynamics by generating numerical realizations using a Monte Carlo scheme. We will also discuss work aimed at integrating microscopic models of actin dynamics into cell-level descriptions of motility.

• Thematic program: Mathematical Biology (2004)

The quantification of cell migration parameters has become an important issue for clinicians, especially in the context of leukocyte migration. Experimental setups used to test the effect of newly developed drugs include so-called direct assays, where individual cells are automatically tracked and monitored, as well as indirect assays, where the behavior of a whole population of cells can be characterized. Starting from a well-established mathematical model describing the behavior of a cell population in the presence of an attracting chemical, we present a mathematical method to identify certain parameters in this model from experimental data. We test the accuracy of our method and apply it to data obtained from leukocyte experiments. Finally, we give on outlook on possible further applications of the method.

• Thematic program: Mathematical Biology (2004)

We describe the meshwork of Actin-filaments in Keratocyte-lammelipodia using a locally two-directional model on the basis of the basic physical properties of its constituents. In this sense our modeling approach is to start from \"first principles\". In this work we use an ansatz for rotationally symmetric solutions. Going to the continuous limit with respect to the distribution of filaments we obtain the Lagrangian and its associated variational equations. We assume that the dynamics of the network are driven by a constant rate of polymerisation and depolymerisation and we (numerically) compute the evolution of the network using a quasi-stationary approach. The simulations reproduce several features of the Actin-Network dynamics found experimentally: treadmilling, lateral flow and a characteristic distribution of angles between filaments.

• Thematic program: Mathematical Biology (2004)

Dissipative particle dynamics (DPD) is a mesoscopic simulation technique that is being increasingly used for studying soft matter systems. While similar to Molecular Dynamics, its use of soft potentials and coarse-grained particles allow it to be used to study system sizes many orders of magnitude larger than can be achieved with Molecular Dynamics. Here we use DPD to study the process of actin monomers self-assembling into filaments. We are able to measure filament properties such as mean length quantitatively, and are working to quantify the filament stiffness and length diffusion. The frequencies of monomer addition and loss at the two ends of a filament are tunable parameters in the model allowing us to explore a wide variety of growth conditions.

• Thematic program: Mathematical Biology (2004)

We study the mechanism of cell protrusion as the first step in normal cell migration and as the earliest pathological event visible in metastasis. We approach this question by reverse engineering the mechano-chemical pathways underlying cell morphodynamic outputs. Our first task is to reconstruct the spatiotemporal distribution of forces generated by cytoskeleton assembly, contraction, and engagement with adhesion complexes. One way of accessing intracellular force distributions is by probing actin network deformation and modelling of the inverse relationship between force exertion and resulting deformation based on assumptions of the cytoskeleton material properties. For this purpose, we have developed quantitative Fluorescent Speckle Microscopy (qFSM) which provides high-resolution spatiotemporal data of both the actin cytoskeleton deformation and dynamic material properties. Using qFSM, we discovered that the protrusion of epithelial cells is mediated by two kinetically, kinematically, and molecularly distinct, yet spatially overlapping actin networks. The first network, referred to as the lamellipodium, exhibits rapid polymerization and depolymerization over 1 – 2 microns, driven by Arp2/3 and cofilin function. The fast retrograde flow of this network is independent of myosin II activity. In contrast, the second network, referred to as the lamella, assembles independently of Arp2/3 function and retrograde flow is powered by myosin II contraction. This discovery complicates the analysis of the cell protrusion mechanism substantially. Local protrusion efficiency is not only determined by the balance of at least three force generating machineries but is modulated by transient structural integration of lamellipodium and lamella in a heterogeneous cytoskeleton architecture. To address this second issue first, we have developed statistical methods that exploit the intrinsic spatiotemporal heterogeneity of actin dynamic and protrusion events in unstimulated wound healing responses to pinpoint the precise sequence and timing of dynamic events that lead to cell edge movements. This data will now serve as input for numerical models of the protrusion mechanism. To summarize the most important aspects of our measurements, we discovered that bursts of protrusion precede bursts of Arp2/3-mediated F-actin assembly, suggesting that nucleation of a dendritically branched F-actin network is not the origin of cell edge advancement. The time lag disappears in cells where Arp2/3-function is inhibited directly or indirectly via overexpression of tropomyosin. In combination with measurements of small Rho GTPase activity this data suggests that cell protrusion is initiated by Arp2/3-independent expansion of the lamella, leading to the assembly of actin filaments free of tropomyosin decoration. These support then the formation of lamellipodium filaments which branch off of lamella filaments and transiently disintegrate due to the severing activity of cofilin. Together, these and other data discussed in this presentation have lead to the formulation of a working model of the structural and functional integration of lamellipodium and lamella in a mechano-chemical regulatory network, which now awaits verification by rigorous dynamic systems analysis in a mathematical framework.

• Thematic program: Mathematical Biology (2004)

• Thematic program: Mathematical Biology (2004)

• Thematic program: Mathematical Biology (2004)

Motility of isolated cultured tissue cells (e.g. human keratinocytes) on 2-dimensional substrates is characterized by pulsative protrusion and retraction activity of lamellipodia, accompanied by steady retrograde flow of the actin cytoplasm from the tip towards the cell body. While non-polarized cells or cell fragments (of stable disc-like shape in 2-d projection) usually perform on-spot motility, chemical or mechanical stimulation induces transition to a stable polarized shape with a protruding front and a retracting rear end, leading to ongoing forward locomotion with mean migration speed depending on substrate adhesiveness. These ubiquitous phenomena are well reproduced by a simple one-dimensional hyperbolic-elliptic differential equation system, modeling the transiently cross-linked actin-myosin network as a contractile viscous two-phase fluid, where the solvent pouring through the network can induce hydrodynamic pressure gradients according to Darcy’s law. The model assumes that such hydrodynamic pressures together with thermodynamic swelling pressures (in the range of a few Pascal) are sufficient to push the plasma membrane forward, thus giving space for new actin assembly at the lamella tip. On the other hand, at places where sufficiently many actin filaments are bound to membrane proteins (e.g. integrin patches), the contractile and viscous tension in the actin network is able to withhold the lamella tip and retract it backwards with the retrograde actin flow. An amplified model including diffusion of free membrane proteins (integrins, cdc42 etc.), stochastic binding of F-actin and induced configuration changes of bound protein complexes, reproduces the dynamical appearance and motility of cortical actin spots, as they are frequently observed, also e.g. in budding yeast. A further model extension takes into account additional binding kinetics of integrin complexes to adhesion sites of an underlying fixed substrate. Distributions of free and bound adhesion complexes underneath the cell depend on local concentration and stress (!) within the actin filament network. On the other hand, dynamics of contractile actin flow and the resulting force exerted to the substrate - thus also the induced cell migration speed - depend on the spatial distribution of adhesion complexes. This dynamic mechano-chemical feedback system with tension-dependent dissociation kinetics of adhesion complexes, has the potential to explain basic questions of spontaneous and evoked cell polarization, principal mechanisms of cytoplasm dynamics, its coupling to formation and rupture of adhesion complexes as well as its resulting effects on cell migration.

• Thematic program: Mathematical Biology (2004)

The synaptic weight between a pre and a postsynaptic neuron depends in part on the number of postsynaptic receptors. On the surface of neurons, receptors traffic by random motion in and out from a synaptic microstructure called the Postsynaptic Density (PSD). In the PSD, receptors can be stabilized at the membrane when they bind to scaffolding proteins. In this talk, we will present recent mathemtical computations of the mean time it takes for a receptor to escape a micro-domains through small holes. A Markovian approach is then used to estimate the mean number of receptor at the PSD. Because dendritic spine, the locus of the post-synaptic excitatoty connections, are highly motile structures, we propose that this fast motility can regulate the number of receptors and thus the synaptic weight.

• Thematic program: Mathematical Biology (2004)