A basic model for the penetration of healthy tissue by bacteria is presented. There, the bacteria produce enzymes which react with and destroy the tissue, providing space and nutrients for the bacteria. The model problem is given by a coupled system of a parabolic and an ordinary differential equation and a typically large parameter governs the degradation rate. In this talk we present an analysis of travelling wave solutions. Existence of travelling waves can be shown for a continuum of speeds. A crucial question is the identification of the minimal speed. Depending on the value of the degradation parameter we find two different principles which determine this speed.
(Joint work with D. Hilhorst, Paris-Sud, and J. R. King, Nottingham)
We consider the so called Moran process with frequency dependent fitness given by a certain pay-off matrix. For finite populations, we show that the final state must be homogeneous, and show how to compute the fixation probabilities. Next, we consider the infinite population limit, and discuss the appropriate scalings for the drift-diffusion limit. In this case, a degenerated parabolic PDE is formally obtained that, in the special case of frequency independent fitness, recovers the celebrated Kimura equation in population genetics. We then show that the corresponding initial value problem is well posed and that the discrete model converges to the PDE model as the population size goes to infinity. We also study some game-theoretic aspects of the dynamics and characterize the best strategies, in an appropriate sense.
This is a Joint work with Max O. Souza
The problem of assessing the significance of rare phenomena involving scoring schemes is an example in which probability theory has been quite useful for bio-molecular data analysis. A key reason for the success of this line of research is the ability to focus on questions and models that retain generality and relevance for applications while introducing enough structure to be of theoretical interest and beauty. I shall explore this interplay while reviewing few contributions made in this direction. For example, gapless local alignment is linked to asymptotic of large exceedances in random sequences which is closely related to queuing theory and sequential statistics. Under somewhat different assumptions it leads instead to an asymptotic of waiting times that are highly relevant for information theory. The assessment of significance of approximate local matching for 3D protein structures results with asymptotic theory for maxima of partial sums indexed by geometrical structures. Finally, theoretical considerations of local optimality yield for a certain parameter regime both logarithmic growth of the gapped local alignment score and a bound on its p-value.
Swarms of birds (as cranes or starlets) as well as confluent layers of tissue cells (as keratinocytes) show characteristic patterns of cohort formation and movement, which resemble the dynamics of compressible fluids with quite smooth free boundaries. Derivation and simulation of neighbor interaction models implemented into standard stochastic ODE systems for multi-particle motion reproduce a wide spectrum of observed phenomena and dynamic behavior as, for example, nonlinear waves and phase transitions between different types of swarm formation. Moreover, for such nonlinear local interaction systems we consider, at least in the 1-dimensional case, suitable continuum limits yielding hyperbolic equations of generalized Navier-Stokes type.
(joint work with Michael Eichler)
Different concepts based on graphical models are presented in order to discuss causality relations between the components of multivariate time series.
One concept is based on the partial spectral coherence of two components given the remaining components of the series leading to an undirected graph. Another concept is based on the notion of Granger causality leading to directed graphs. We present results on the relation between Markov properties of the processes and path properties of the graph.
Inference for time series graphs is done by parametric and nonparametric methods in combination with model selection or testing procedures. We discuss inference with AR-models, the partial spectral coherence and a partial correlation function. As an example we consider multivariate autoregressive processes.
The methods are applied to air pollution data, EEG data and spike trains from a network of neurons.
Vision is an important sense for most species throughout the animal kingdom and even in the dimmest habitats animals have functional eyes. Different eye types are not adaptations to ecological conditions alone but eye design is also constrained by the physical nature of light. Vision in dim light is photon limited and the superposition eye in insects has evolved to utilise as much photons reaching the eye as possible. Some nocturnal insects, however, have apposition eyes, an eye type not optimal for vision at low luminances.
Spatial summation and temporal integration can enable a diurnal eye to improve vision in dim light. The aim of this study is to model and examine neural adaptations of receptive fields of interneurons under dim-light conditions. The model is based on natural image data. It is shown that receptive fields get spatially enlarged as images get noisier. Furthermore the effect of motion on the time response of photoreceptors is investigated. Noisier images require a prolonged integration time whereas motion causes a shortening of integration time. It is shown that spatial pooling and temporal integration deteriorate signal-to-noise ratio at high spatial and temporal frequencies in order to improve signal-to-noise ratio at low frequencies.
The spatial part of the receptive field is modeled by a semilinear neural net (Backpropagation algorithm). The temporal properties are modeled with a log-normal function and parameter optimisation is done by means of a direction set method (Powell's algorithm).
In cytometric measurements of fluorescence color marked molecular elements (e.g. genes in the nucleus) by microscopic imaging, in standard cases 2D projections of the whole object or an appropriate section are obtained. In addition, the object has usually been distorted by biochemical treatments. Therefore, the problem of reconstructing the object or computing the postition of the marked region leads to several mathematical questions. A convenient measure in the microscopic images are distances of two points. Interestingly, for many types of given point densities (e.g. by splines) on the ball, the resulting (projected) distance distributions can be derived analytically. The principle question, namely the inverse problem of determining the point densities (e.g. the localisation of chromosomal domains in the nucleus) from the statistical distribution of the (projected) distances, is far from being solved.
A fundamental problem in the field of animal development is to understand how well-defined cellular patterns can emerge in the presence of fluctuations. A well-established means of tissue patterning is given by morphogens. These are signaling molecules that spread from a restricted source into an adjacent target tissue forming a concentration gradient. The fate of cells in the target tissue is determined by the local concentration of such morphogens. In the presence of fluctuations, it is an important question how precise the positional information encoded in a morphogen gradient can be. Here, we first give a brief introduction to morphogen transport. We then investigate the precision of the gradient of the morphogen Dpp in the Drosophila wing disk both experimentally and theoretically. We measure the normalized fluctuations of the Dpp gradient as a function of the distance to the source. We find that these fluctuations grow monotonously for large distances to the source, while close to the source they can decrease. Our theoretical analysis reveals that cell-to-cell variability in the target tissue can generate the observed behavior of the fluctuations. This suggests that the concentration fluctuations in the gradient reflect the random components of intercellular signaling and transport.
We investigate nonlinear dynamics near an unstable constant equilibrium in the two classical models. Given any general perturbation of magnitude $\delta$, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of $ln(1/\delta)$. Our result can be interpreted as a rigourous mathematical characterization for early pattern formation in the Keller-Segel model and Turing instability.
We have recently introduced the concept of network entropy [1] as a global proxy for the resilience of a network against random perturbations. At the example of two protein interaction networks (yeast and C.elegans), I will illustrate that these ideas can be used to rank proteins according to their structural importance in the network and that this ranking scheme correlates with functional importance (lethality).
I will illustrate how the entropic characterisation of networks can be turned into a novel model of network evolution which is based on a biological selection principle.
[1] L. Demetrius & T. Manke, Robustness and network evolution - an entropic principle Physica A 346 (2005) 682-696.
In contrast to classical engineering materials, living tissues show the fascinating ability to adapt their internal microstructure to a given mechanical environment. Although this functional adaptation of biological tissues has been known for more than a century, it was only within the past two decades that continuum biomechanics has attracted a considerably growing attention. This talk focuses on the historical development of continuum biomechanics and discusses its fundamental differences from classical, traditional continuum mechanics. Computational simulations of functional adaptation, self healing, microstructural alignment and self organization illustrate the potential of modern biomechanical models for growth and remodeling in hard and soft tissues.
Bioconvection patterns are observed in shallow suspensions of randomly, but on average upwardly, swimming micro-organisms which are a little denser than water. The basic mechanism is analogous to that of Rayleigh-Benard convection, in which an overturning instability develops when the upper regions of fluid become denser than the lower regions. The reason for the upswimming however depends on the species of micro-organism: certain biflagellate algae are bottom-heavy, and therefore experience a gravitational torque when they are not vertical; certain oxytactic bacteria swim up oxygen gradients that they generate by their consumption of oxygen. Rational continuum models can be formulated and analysed in each of these cases, as long as the cell volume fraction $n$ is low enough for hydrodynamic or other cell-cell interactions to be neglected ($n$ \leq 0.1%). The key mathematical step is the calculation of the probability density function for the cells' swimming velocity from a suitable Fokker-Planck equation, when that is justifiable. Both examples will be discussed from this point of view.Another sort of pattern-formation ("whorls and jets") is observed in very concentrated, very shallow cultures of swimming bacteria on agar plates. Here cell-cell interactions are crucial, but it is not clear how to derive an appropriate macroscopic model that is consistent with the laws of mechanics at the cellular level. A recent attempt (J Lega & T Passot, Phys Rev E, 67, 2003) succeeds in generating patterns on the correct scale, but appears not to be rationally justified. Here we examine the deterministic swimming of model organisms which interact hydrodynamically but do not exhibit intrinsic randomness except in their initial positions and orientations. A micro-organism is modelled as a squirming, inertia-free sphere with prescribed tangential surface velocity. Pairwise interactions have been computed using the boundary element method, supplemented by lubrication theory, and the results stored in a database. The movement of 27 identical squirmers is computed by the Stokesian Dynamics method, with the help of the database of interactions (the restriction to pairwise interactions requires that the suspension be semi-dilute, with particle volume fraction less than about 0.1). It is found that the spreading is correctly described as a diffusive process a sufficiently long time after the motion is initiated, although all cell movements are deterministic. The effective translational and rotational diffusivities depend strongly on volume fraction and mode of squirming.
Somites arise as the result of a complex process that takes places in the early vertebrate embryo: a seemingly uniform field of cells is organised into discrete blocks via a mechanism which is tightly regulated both in space and time. Further differentiation of the cells within these somitic segments leads to the formation of the vertebrae, ribs and other associated features of the vertebrate musculature.
Various experimental results have shown the existence of a wavefront of gene signalling along the vertebrate embryo, which, coupled with a segmentation clock, is able to gate the cells into blocks that will later go on to form somites. I will use a signalling based approach, with cues from the wavefront and the segmentation clock, to mathematically model somite formation and show that the model can reproduce the results seen in vivo when progression of the wavefront is disturbed.
We consider the travelling patterns which appear in different biological processes (for example: during cells aggregation, in tumour growth, in a fibrillating heart). We focus on a mathematical model for the propagation of the signal during the aggregation of some microorganisms. We show the existence of travelling pulse solutions using singular perturbation methods. A biological interpretation of the results will be given.
It is quite common and useful to describe physical systems at different levels of resolution: micro-, meso- and macroscopic scales are naturally introduced in their modelling. The macroscopic biosystems are usually made of several small components, so that a similar procedure might be used. Here it is emphasized the role of the growth (multiplication) process which produces a "dynamic" transition between the microscopic initial size to the macroscopic one. Examples of this nontrivial interplay between the micro- and the macroscopic scales will be given, in the context of tumour growth modelling and population dynamics.
A one-dimensional driven two-species model with parallel sublattice update and open boundaries is considered. Although the microscopic many-body dynamics is symmetric with respect to the two species and interactions are short-ranged, there is a region in parameter space with broken symmetry in the steady state.
The symmetry breaking can be shown to be the result of an amplification mechanism of fluctuations. In contrast to previously considered models, this leads to a proof for spontaneous symmetry breaking which is valid throughout the whole region in parameter space with a symmetry broken steady state.
In the field of population dynamics, some models of competing species are described by reaction-diffusion systems. Among them, there is a system which is taken into account the population pressure created by the competitors. It is called a cross-diffusion and competition system in mathematical biology. This system, especially, the global structures of equilibrium solutions have been investigated in nonlinear PDE communities. From the biological motivation, we discuss that such a cross-diffusion system can be approximated by a kind normal diffusion system, taking some singular limit.
This is a joint work with H. Ninomiya (Ryukoku Univ.) and M. Iida(Iwate Univ.).
The crawling movement of several types of cells is based on the dynamics of a part of the cytoskeleton, the lamellipodial network of actin polymers. It involves polymerization and depolymerization, branching, crosslinking, the creation and removal of adhesion sites, interaction with myosin motors, the reaction to stresses produced by deformation and by the cell membrane, the transport of monomers, and probably a few other mechanisms. Our aim is to create a framework for the mathematical modelling and simulation of these effects in order to quantify their contribution to cell movement.
A first step in this program will be presented, where a two-dimensional actin network is modelled as a grid of two families of locally parallel filaments (microscopic model), and a continuum (macroscopic) model is derived by a homogenization limit. First numerical simulations of a dynamically stable rotationally symmetric rest state of a lamellipodial fragment (which can be created in the laboratory) will also be shown.
Growing cell cultures often exhibit branch patterns. The cultures are usually modelled as free boundary problems. Here, we review these models and investigate the appearance of branch patterns using methods from asymptotic analysis. Two extreme cases - large kinetics and small kinetics - are considered. For large kinetics the models reduce to the well-studied Stefan problem, which is known to exhibit a diffusive-instability and hence shows branch patterns. Considering small kinetics, small perturbations are smoothed out resulting in regular shapes of the cultures. The branching phenomenon in general can be understood as somewhere between these two extremes.
The non-linear dynamic response of polymers is dominated by the non-equilibrium phenomenon of tension propagation, which is e.g. pertinent to DNA micro-manipulations and mechanical signal transduction through the cytoskeleton. I will sketch how the surprisingly rich deterministic response emerges from the underlying stochastic differential equations via a scale separation between conformational and tensile dynamics.
The phenomenon of collective motion is of wide interest and therefore a lot of scientific research has been done on that in the recent years. Especially in biological and social systems which seem to be far from equilibrium most of the time, coherent motion of groups of individuals can be observed very often. Schools of fish, flocks of birds, groups of ants, systems of microorganism like bacteria or cellular slime molds etc. show cooperative behavior and collective moving modes of a large group of these species. The types of motion lead from translational directed motion over rotational motions up to more complex cooperative moving patterns of particles. After an survey of existing biologically and physically motivated models the theory of Active Brownian particles is going to be introduced in more detail.
A discrete stochastic model of a spatially extended cell population with local interactions is presented, which is capable to describe adhesive behavior as well as persistence of cell movement but which is nevertheless analytically tractable. The biological motivation comes from Steinberg's Differential Adhesion Hypothesis, which states that cell sorting is caused by random cell movement in combination with differential cell-cell adhesion. The long-time behavior of this interacting particle system is analyzed and its relation with statistical mechanics is discussed. It is shown that the evolution of the particle system can be interpreted as cell sorting. The role of cell persistence in cell sorting is indicated and it is discussed how persistence influences the systems behavior.
In this talk, the following degenerate parabolic system modelling chemotaxis is considered. $(KS): u_t = \nabla (\nabla u^m - u \nabla v ), x in R^N, t>0, \tau v_t = \Delta v - v + u, x in R^N, t>0, u(x,0) = u_0(x), \tau v(x,0) = \tau v_0(x), x \in \R^N,$ where $m>1, \tau=0 or 1$, and $N \ge 1$. We discuss the existence of a global weak solution of $(KS)$ under some appropriate conditions on m without any restriction on the size of the initial data.Specifically, it is discussed that a solution (u,v) of (KS) exists globally in time either (i) $2 \le m$ for large initial data or (ii)$1 < m \le 2-\frac{2}{N}$ for small initial data. In the case of (ii), the decay properties of the solution ($u,v$) are also presented.
Actin networks are continuously reorganized in cells that rapidly change their shape. Applying total internal reflection fluorescence (TIRF) microscopy at acquisition rates of 10 to 20 Hz, we measured an average growth rate of 3 m x sec-1 for filamentous actin structures throughout the entire substrate-attached cortex of Dictyostelium cells. New filaments often proceed along pre-existing ones, resulting in bundle formation concurrent with filament growth. In cells that orientate in a gradient of chemoattractant, prominent actin assemblies enriched in the Arp2/3 complex are inserted into the network, primarily at the base of filopods that point into the direction of the gradient. The Arp2/3 complex promotes actin nucleation and branching of the filaments. This complex is inhibited by coronin, a WD40-repeat protein. Accordingly, coronin is recruited to a zone behind the leading edge or to other sites where actin assembly ceases. We propose that high turnover rates of actin filaments confer the plasticity to the cell cortex that is required for rapid accommodation to external stimuli.Actin networks are continuously reorganized in cells that rapidly change their shape. Applying total internal reflection fluorescence (TIRF) microscopy at acquisition rates of 10 to 20 Hz, we measured an average growth rate of 3 m x sec-1 for filamentous actin structures throughout the entire substrate-attached cortex of Dictyostelium cells. New filaments often proceed along pre-existing ones, resulting in bundle formation concurrent with filament growth. In cells that orientate in a gradient of chemoattractant, prominent actin assemblies enriched in the Arp2/3 complex are inserted into the network, primarily at the base of filopods that point into the direction of the gradient. The Arp2/3 complex promotes actin nucleation and branching of the filaments. This complex is inhibited by coronin, a WD40-repeat protein. Accordingly, coronin is recruited to a zone behind the leading edge or to other sites where actin assembly ceases. We propose that high turnover rates of actin filaments confer the plasticity to the cell cortex that is required for rapid accommodation to external stimuli.
Enzymes are the specific catalysts of almost any physiological reaction. Studies on the simplest types of enzymes and enzyme model systems show that such minimal systems are able to show nonlinear dynamic behaviour. Their dynamics (as well as their bifurcation between different states) is investigated under homogeneous conditions. In these cases, it can be shown that the source of the nonlinearities lies in the kinetics of the reactions underlying the enzyme system. Oscillatory dynamics are not a mere side effect of the kinetics, but are also used to exert biological functions. Last, but not least, a series of experiments shows that nonlinear dynamics is also possible in enzyme model systems where the enzyme is bound to a membrane, such that transport processes may play an important role in inducing dynamic behaviours.
Dynamical systems can be considered on different levels of precision. For studying the qualitative behaviour of a system, it can be useful to choose a very coarse-grained description: The state space is divided into a small number of subregions, each coded by a symbol. In a discrete-time dynamical system, sequences of successive states of the system are turned into symbol sequences.
Symbolic dynamics studies dynamical systems on the base of the symbol sequences obtained for a suitable partition of the state space. It is also applied to the qualitative analysis of empirical time series. We focus on symbolic dynamics describing the ordinal structure of a system. Here we follow an idea of Bandt and Pompe, who have introduced the ordinal viewpoint into time series analysis. In particular, we describe EEG-based methods for detecting, visualizing and analyzing temporal and spatial qualitative changes of brain states related to epileptic activity.
We present a result, which relates the stability of a population, as defined by the rate of decay of fluctuations induced by demographic stochasticity, with evolutionary entropy.
This entropy is a measure of the variability in the age of reproducing individuals in the population. Its definition is based on concepts originally developed in statistical mechanics and the proof of our result uses the theory of large deviations in dynamical systems.
In this talk we will mainly focus on the biological interpretation of our mathematical result. In addition we discuss a possible generalization of our ideas to metabolic networks, which can be mathematically described by graphs.
Over the last 25 years, coalescent theory became more and more important for ancestral population genetics. The coalescent is a stochastic process, which approximates the ancestral tree of a sample of n individuals, provided the total population size is sufficiently large. Corresponding first convergence theorems go back to Kingman (1982). The Kingman coalescent has the particular property that only binary mergers of ancestral lineages appear with positive probability. For a huge class of population models, the Kingman coalescent arises in the limit for large total population size, similar to the normal distribution in the central limit theorem. This robustness justifies the relevance of the coalescent for the applied sciences.
This talk gives an introduction to coalescent theory and characterizes the class of population models in the domain of attraction of the Kingman coalescent. At the same time, the approaches used in the proofs yield a classification of all coalescent processes which appear in the weak limit when the population size tends to infinity. In general, these coalescent processes allow for simultaneous multiple collisions of ancestral lines. The talk closes with a summary on ongoing research in this field and emphasizes the interdisciplinary importance of the coalescent for biology, computer science, mathematics and medicine.
In response to starvation Myxococcus xanthus cells initiate a multicellular developmental program that culminates in the formation of spore-filled fruiting bodies. Fruiting body formation involves two morphogenetic processes, aggregation and sporulation, which are coordinated in time and space: aggregation of cells precedes sporulation and cells do not sporulate until they have accumulated inside the fruiting bodies. The starting point in the formation of fruiting bodies is a nearly symmetric distribution of cells in a homogeneous mat, and the endpoint is the asymmetric distribution of cells inside the multicellular fruiting bodies. The mechanism underlying this redistribution of cells is changes in organised cell movements from swarming in non-starving cells to aggregation in starving cells. Key questions in understanding fruiting body formation are how the aggregation process is accomplished and how aggregation and sporulation are coordinated. The intercellular C-signal has a key role in inducing and choreographing the aggregation process and in coordinating aggregation and sporulation.
To understand how the C-signal acts at the molecular level, we have analysed the mechanism of the C-signal. The C-signal is cell surface-associated and signal transmission occurs by a contact-dependent mechanism. C-signal accumulates during development. C-signal induces aggregation and sporulation at specific thresholds: at intermediate levels aggregation is induced and at high levels sporulation is induced. An ordered increase in the level of C-signalling in combination with the specific thresholds ensures the correct temporal order of aggregation and sporulation. The contact-dependent signal transmission mechanism allows the spatial coordination of aggregation and sporulation by coupling cell position and signalling levels.
To understand how the C-signal induces aggregation, we have analysed the effect of the C-signal on cell behaviour using fluorescence time-lapse video microscopy. The C-signal induced motility response includes increases in transient gliding speeds and in the duration of gliding intervals and decreases in stop and reversal frequencies. This response results in a switch in cell behaviour from an oscillatory to a unidirectional type in which the net-distance travelled by a cell per min is increased. We have proposed that the C-signal dependent regulation of the reversal frequency is essential to the aggregation process. We have proposed a qualitative model for aggregation. In this model, C-signal transmission is a local event, which occurs between two cells and without reference to the global pattern, and the result is a global organization of cells. This pattern formation mechanism does not require a diffusible substance or other actions at a distance.
Edge-reinforced random walk was introduced by Diaconis in the late 1980s. Diaconis asked for which values of d the edge-reinforced random walk on $Z^d$ is recurrent. This question is stillopen for all $d \ge 2$. In this talk, I will present recent results for edge-reinforced random walk on ladders. These include recurrence results and limit theorems. The analysis is based on a representation of the edge-reinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a marginal of a multi-component Gibbsian process. A transfer operator technique and entropy estimates from statistical mechanics are used to analyze this Gibbsian process.This is joint work with Franz Merkl.
Stochastic particle systems far from equilibrium are frequently applied in Physics or Biology and show a great variety of collective phenomena already in one dimension. The zero range process (ZRP) is an important model of this class, closely related to driven diffusive systems modeling e.g. biopolymerization or transport across membranes. We study the ZRP on a periodic lattice with rates inducing an effective attraction between particles. If the particle density exceeds a critical value, the system phase separates into a homogeneous background and a condensate, where the excess particles accumulate. We proof this by showing the equivalence of the canonical and the grand-canonical stationary measure. We also show that for large systems the condensed phase typically consists only of a single, randomly located site. Using heuristic arguments supported by Monte Carlo simulations, we study the dynamics of the condensation and its dependence on space dimension and symmetry of the jump rates. The results are extended to condensation in two-component systems, showing a particularly rich critical behaviour.
New results on homoclinic bifurcation from certain generic codimension-1 and -2 heteroclinic cycles between an equilibrium and a periodic orbit are presented. We show implications for spectral stability of associated travelling waves in spatially onedimensional reaction-diffusion systems, and new results concerning the absolute spectrum of periodic wave trains. These are used to partially explain the phenomenon of 'tracefiring', which is the bifurcation of a stable excitation pulse to a self-replicating pulse-chain. Here, secondary pulses periodically grow out of the wake of the rear pulse and travel in the same direction as the chain. The aforemention codimension-1 case serves as an organizing center for some of the spatial patterns involved. We present this phenomenon in the Oregonator model of the light-sensitive BZ reaction. For this case, the primary pulse's instability is explained through its interaction with a small amplitude periodic wave train. This relies on the above codimension-2 and spectral structure results, as well as numerical computations of spectra for the pulse and periodic wave train.
In the first part, we discuss the theory of model reductions under quasistationarity hypotheses. We briefly repeat the singular perturbation approach before addressing the method of intrinsic low-dimensional manifolds.
The second part of the talk is devoted to model reduction problems arising in chemical engineering and biochemistry. The examples stem from reactive separation and from enzyme reactions.
Individuals in streams and rivers are constantly subject to predominantly unidirectional flow in their environment. Some individuals, such as fishes, have mechanisms to actively swim against the water current. Many other species, such as stream insects, lack appropriate mechanisms, and their movement in the water column is determined largely by the flow. The question of how these populations can persist in upper stream reaches is known as the ``drift paradox''. In this talk, we model a population of stream insects using an integrodifferential equation for non-local dispersal, and analyze conditions for persistence and for upstream spread of the population. This work is closely related to models for the plug-flow reactor and biofilms.
In this talk, we will discuss a rigorous numerical method for the study and verification of global dynamics for gradient systems. The procedure involved relies on first verifying the structure of the set of stationary solutions, as often depicted in a bifurcation diagram produced via continuation methods. This includes proving the existence and uniqueness of computed branches of the diagram as well as showing the nonexistence of additional stationary solutions. Topological information in the form of the Conley index is then used to build a model for the set of bounded solutions that form an attractor for the system. As illustration, this method is used to produce a (semi-) conjugacy between an attractor for the Swift-Hohenberg equation and a constructed model system.
We introduce three new examples of kinetic models for chemotaxis, where a kinetic equation for the phase-space density is coupled to a parabolic or elliptic equation for the chemo-attractant, in two or three dimensions. We prove that these models have global-in-time existence and rigorously converge, in the drift-diffusion limit to the Keller-Segel model. Furthermore, the cell density is uniformly-in-time bounded. This implies, in particular, that the limit model also has global existence of solutions.
Physiologic or morphogenetic systems rarely optimize natural explicit cost functions, i.e. based on thermodynamic or energetic criteria built from their underlying physico-chemical framework. Despite this absence, we can model these systems with simple dynamical equations (dealing with polynomial ODEs of Lienard type or rational fraction ODEs of Hill or MWC type). These models well explain the asymptotic behavior experimentally observed and offer more, i.e. an explanation of attractors in terms of singular trajectories of gradient, or hamiltonian or of mixed gradient-hamiltonian systems. This last description gives an a posteriori interpretation of the dynamics using energy cost functions based on observable variables. Examples will be studied concerning the cardio-respiratory system, the hippocampic system and the plant morphogenesis.
Fish demonstrate a remarkable diversity in coloration and patterning. These patterns result from the spatial organisation of specific pigment cells within the skin. The development of these patterns is a particularly elegant example of morphogenesis: the emergence of form in the growing embryo. In certain species, including members of the marine angelfish Pomacanthus the pattern evolves as the fish matures through larval, juvenile and adult stages. In the first part of this talk, I will introduce a continuous mathematical model based on cell chemotaxis that replicates the growing patterns in these species.
In the second part of the talk I will explore the formation of the larval pigment pattern in the Zebrafish, Danio rerio. Here, we will use a discrete-continuous model to understand how the pattern is set-up, from the emergence of pigment cells from the neural crest to their establishment in the skin. The modelling of cells as discrete objects allows the incorporation of specific biological mechanisms. We shall use the model to make a number of hypotheses on the mechanistic basis of patterning.
Among biological rhythms, those with a circadian (close to 24h) period are conspicuous by their ubiquity and by the key role they play in allowing organisms to adapt to their periodically changing environment. Thanks to genetic and biochemical advances on the molecular bases of circadian rhythms in a variety of organisms, mathematical models closely related to experimental observations can be considered for the regulatory mechanisms of circadian clocks. In the best-studied organism Drosophila, circadian rhythms originate from the autoregulatory negative feedback exerted by a complex between the PER and TIM proteins on the expression of their genes. A model of this genetic regulatory network predicts the occurrence of sustained circadian oscillations in continuous darkness. Numerical integration of the differential equations governing the time evolution of the model show that the oscillations correspond to the evolution toward a limit cycle. When incorporating the effect of light, the model accounts for phase shifting of the rhythm by light pulses and for entrainment by light-dark cycles. The model can also explain the long-term suppression of circadian rhythms by a single pulse of light, and predicts the occurrence of autonomous chaotic behavior. Stochastic simulations show how circadian oscillations are affected by molecular noise. An extension of the model to circadian rhythms in mammals permits an investigation of the dynamical bases of certain physiological disorders of the sleep-wake cycle in humans.
In technical chemistry and biotechnology the construction of predictive models has become an essential step in process design and product optimization. Accurate modelling of the reactions in a reactor requires detailed knowledge about the processes involved. However, if, f.e., the development of new products and new production techniques is concerned, this knowledge often is not available. It thus is a typical situation that one has to work with a selection of possible models and the main tasks of early development is to discriminate these models.
Model discrimination means the ranking of models wrt. their ability to reproduce certain experimental data for the process under investigation. In this talk we present a new statistical approach to model discrimination that ranks models wrt. the probability with which they reproduce the data. The talk will shortly review some other prominent approaches in the field, introduces the new approach, discusses its statistical background, presents numerical techniques for its implementation and illustrates the application to realistic examples from biokinetics.
Developmental biology of multi-cellular animals comprises the progressive transformation of the fertilized egg into an adult of approximately a trillion cells in case of the mouse. During this process, organs and tissues derive from individual progenitor pools, and in each of these a characteristic cocktail of morphogenetic mechanisms, like proliferation of cell clones, cell migration and mixing, and programmed cell death (apoptosis) leads to characteristic spatial patterns in the mature organs. Some of these events can be examined retrospectively by genetic cell labelling techniques in artificial chimaeras; e.g. by marking a fraction of cells in an early embryonic stage followed by retrospective analysis of the derived mosaic; these techniques have been applied to the mouse for many years (cf. [1]). We have used green fluorescent transgenic mice [2] to label cells for parental origin in combination with unlabelled mice to produce mouse chimeras [3]. Tissues of these contain green fluorescent labelled cell patches, which morphology and distribution are historical records of underlying developmental processes and enable to develop models of organ development. Examples for the clonal cell distribution patterns in the heart, skeletal [3] and smooth muscle tissue, and the pancreas will be presented.
1. West, J. D. (1999) Curr Top Dev Biol 44, 1-20. 2. Jockusch, H., Voigt, S., Eberhard, D. (2003), J Histochem Cytochem, 3, 401- 4. 3. Eberhard, D. and Jockusch, H. (2004) Developmental Dynamics. In press.
A simple ODE model of anticancer drug efficacy and healthy tissue toxicity in a chronobiological setting will be presented, its parameters being identified after animal experimental data. The use of optimal control (nonlinear conjugate gradient) is advocated, yielding a best infusion profile, with numerical results, and showing in this frame the advantage of time scheduled regimens over constant infusion[joint work with Claude Basdevant].
These results are dependent on tumour growth kinetics and pharmacodynamic modelling, for which possible extensions will be discussed. Cell cycle kinetics modelling at the level of an aged-structured population of cells will also be evoked in the perspective of its control by natural circadian rhythms and external action of anticancer drugs [joint work with Benoît Perthame].
The modelling of cancer provides an enormous mathematical challenge because of its inherent multi-scale nature. For example, in vascular tumours, nutrient is transported by the vascular system, which operates on a tissue level. However, it effects processes occuring on a molecular level. Molecular and intra-cellular events in turn effect the vascular network and therefore the nutrient dynamics. Our modelling approach is to model, using partial differential equations, processes on the tissue level and couple these to the intercellular events (modelled by ordinary differential equations) via cells modelled as automaton units. Thusfar, within this framework we have modelled structural adaptation at the vessel level and we have modelled the cell cycle in order to account for the effects of p27 during hypoxia. These preliminary results will be presented.
Many physical models are based on the idea that several non-linear effects act simultaneously and (hence) the vector fields enter the model equations in an additive way (as in reaction diffusion equations). The additivity can be justified by Trotter formulas or the fractional step approach where the simultaneous action is replaced by a periodic action with short period.
Here different actions are coupled diffusively, i.e., the transition between different vector fields is governed by Poisson processes. While for linear vector fields the Trotter and the Poisson approach lead to the same limiting equation (for short periods or large coupling rates), for non-linear vector fields the limiting equations are quite different.
Away from the limiting case, for moderate coupling rates, it may be quite difficult to extract the behavior of the coupled system from the behavior of its components. The problem may even be meaningless for arbitrary systems. However, there are several useful scenarios:
- Coupling closely related dynamics (caricature of coupled map lattice).
- Coupling transport equations to reaction equations (equivalent damped wave equations, stability of stationary states, travelling front problem, linear determinacy).
- Coupling spatial spread to an infection dynamics (justification of epidemic contact distributions in terms of spatial processes).
- Coupling an arbitrary dynamics to a quiescent state. Although from the modeling point of view this extension should act similar to a delay, the qualitative features are quite different. A general stability theorem can be shown.
(joint work with Thomas Hillen and Mark Lewis)
Contact tracing is a quite interesting method to fight infectious diseases: if an infected person is discovered, this person is asked about the (potentially) infectious contacts. The corresponding individuals are then examined, and in this way it is possible to find further infected persons.
Though the common belief and the common feeling is that this measure to fight diseases is quite effective (also with monetary aspects in mind). However, there are almost no monitoring structures or possible to derive an objective idea about the efficacy of contact tracing.
We propose a (stochastic, individual-based) model for contact tracing. The central aspect of this model is the graph that is generated by infected persons: the nodes are the infected persons, a directed edge goes from infector to infectee. Contact tracing takes place along the edges of this graph. First, we derive (in an heuristic way) an ordinary differential equation. Then, we derive results about the structure of the stochastic graph. In an outlook, we try to sketch ideas how to use these models to set up monitoring tools using these results.
Physiologically structured population models try to do justice to the mechanisms at the individual level when investigating phenomena at the population level. So models first of all pertain to the individual level and are then "lifted", by bookkeeping, to the population level. In line with the folklore of applied mathematics, this leads to partial differential equations (but with non-local boundary conditions).
In this lecture it will be argued that there is an attractive alternative to a pde formulation. This alternative is in terms of cumulative quantities and (Stieltjes) integral equations. And "attractive" means that unbounded operators, with hard to determine domains, are avoided completely.
The lecture is based on joint work with Mats Gyllenberg and Hans Metz.
Collaboration with: Hitesh Mistry, Sandrine Etienne-Manneville, John Billingham
A time-lapse record of cell motion is examined after a cut is made in a monolayer of astrocytes. Cell-flux and cell-density are extracted over periods of time in sections of the record. The data suggests that flux depends more on cell-density than on cell-density gradient, with the average speed of cell motion being approximately constant up to a maximum density. Numerical solutions of a simple model, with flux a function of density and a small amount of diffusion, are qualitatively consistent with the experiment. Curiously, depending on the initial data, the model is inadequate without some mechanism for initiating motion. Diffusion serves such a purpose but the results then depend quite significantly on the amount of diffusion.
We study the classical model for chemotaxis, the so-called Keller-Segel model, which is a drift-diffusion equation for the cell density coupled with an elliptic equation describing the evolution of the chemoattractant. We consider the case of small diffusivity and investigate the limit as the diffusion coefficient goes to zero. Considering a model where the drift term vanishes at high cell densities leads to a nonlinear equation which allows the formation of shocks in the limit. Moreover, we look at the long term behaviour of solutions.
This seminar proposes a mechanism to explain allometric relations between metabolic rate and body size in organisms . The model postulates that energy transduction in biological organisms is constrained by two classes of dynamical processes : the first process has its origin in Quantum mechanics and the constraints which the coupling of electron transport and proton translocation impose on metabolic activity . The second derives from evolutionary dynamics and the constraints which environmental factors impose on metabolic rate . This model applies to unicells , plants and animals and it is thus distinguished from competing models whose range of application is highly restricted .
We have a system of non-linear partial differential equations, each equation representing a different phase of the cell cycle (G1, S, G2, M). Our independent variables are x, relative DNA content, and t, time in hours. In the equation for G1-phase, a non-local term represents newly divided cells from M-phase and gives rise to steady DNA distributions as seen experimentally in cell lines unperturbed by cancer treatment.
We can use this system as a starting point for modelling cancer treatment by changing appropriate model parameters. In particular we consider paclitaxel, an anticancer drug that halts cell division and induces cell death in tumor cell populations. The model enables us to quantify the rate of cell death and the subsequent rate of DNA degradation. These rates are currently unable to be obtained experimentally.
Using an individual-based model, I will show how the clustered growth of bacteria in biofilms enables the evolution of a 'primitive' altruistic strategy that reduces growth rate in favour of yield, thereby increasing the fitness of the cluster at the cost of decreased fitness for the individual.
This perspective on life in biofilms can for the first time explain biofilm structure and characteristics as promoting the origin and maintenance of the altruistic strategy.
Chemostat models and experiments mislead microbiologists into believing that growth rate but not yield decide the outcome of competition.
The aim of this work is to show under which conditions a receptor-based model can produce and regulate patterns. Such model is applied to the pattern formation and regulation in a fresh water polyp, hydra.
The model is based on the idea that both head and foot formation could be controlled by receptor-ligand binding. Positional value is determined by the density of bound receptors. The model is defined in the form of reaction-diffusion equations coupled with ordinary differential equations.
The objective is to check what minimal processes are sufficient to produce patterns in the framework of a diffusion-driven (Turing-type) instability. Three-variable (describing the dynamics of ligands, free and bound receptors) and four-variable models (including also an enzyme cleaving the ligand) are analysed and compared. The minimal three-variable model takes into consideration the density of free receptors, bound receptors and ligands. In such model patterns can evolve only if self-enhancement of free receptors, i.e. a positive feedback loop between the production of new free receptors and their present density, is assumed.
The final pattern strongly depends on initial conditions. In the four-variable model a diffusion-driven instability occurs without the assumption that free receptors stimulate their own synthesis. It is shown that gradient in the density of bound receptors occurs if there is also a second diffusible substance, an enzyme, which degrades ligands. The four-variable model is able to capture some results from cutting experiments and reflects {\it{de novo}} pattern formation from dissociated cells.
The results of grafting experiments suggest that model should involve a memory-based relation. It is shown that the model is able to capture results from experiments if the dynamics of production of ligands and enzyme are described by the system of ordinary differential equations showing hysteresis.
Rod theory has the application in the modelling of DNA and some bacterial fibers in the mechanical aspect. In this talk, we relate a geometric functional of curves to the simplest Kirchhoff elastic energy of rods so that the generalized elastic curves are the centerlines of elastic rods. Under a mild assumption, we derive the long time existence of the total-length preserving evolution of the generalized elastic curves.
Based on first principles, we derive a general model to describe the spatio-temporal dynamics of two morphogens. The diffusive part of the model incorporates the dynamics, growth and curvature of one- and two-dimensional domains embedded in R^3.
Our generalized diffusion process includes spatio-temporal varying diffusion coefficients, advection and dilution terms. We present specific examples by analyzing a third order activator-inhibitor mechanism for the kinetic part. We carry out illustrative numerical simulations on two-dimensional growing domains having different geometies. Comparisons with former results on fixed domains show the crucial role of growth and curvature on pattern selection. Evidence is given that both effects might be biologically relevant in explaining the selection of some observed patterns and in changing or enhancing their stability.
We consider a simple model arising in modeling angiogenesis and more specifically the development of capillary blood vessels due to an exogenous chemoattractive signal (solid tumors for instance). It is given as coupled system of parabolic equations through a nonlinear transport term. We show that, by opposition to some classical chemotaxis model, this system admits a positive energy. This allows us to develop an existence theory for weak solutions. We also show that, in two dimensions, this system admits a family of self-similar waves.
The spatial-temporal organisation of growing epithelial cell populations in vitro is studied by a mathematical model that is designed to allow a quantitative comparison with experiments. For this purpose a three-dimensional single-cell based, lattice-free model is developed. The model is able to explicitly account for the conservation of the cell volume, for cell-cell and cell-substrate interaction, and for the visco elastic properties of the individual cells. Each model input variable is an experimentally measurable quantity hence the model parameters can vary within small parameter ranges only.
The effect of changes of cell-kinetic parameters and parameters that characterize cell-cell and cell-matrix interactions is investigated in order to identify those parameters that determine the macroscopic growth kinetics of the cell populations. As demonstrated the model also allows the evaluation of extra-cellular matrix modifications including modifications of viscoelastic properties, and of the spatial structure and composition of the matrix. The results represent the first step towards models of complex biological tissues as single- or multilayered epithelia. Examples are the formation of the epithelial barrier of skin and of the oral mucosa, and the macroscopic self-organisation processes in the intestinal mucosa.
Helper T lymphocytes control the immune reponse by secreting specific cytokines. To analyse the molecular mechanisms that determine which cytokine pattern is expressed in naive T cells and subsequently memorized in differentiated Th1 and Th2 cells, we have developed mathematical models of the processing of antigenic stimulation by phosphorylation cascades and of the transcriptional control mechanisms determining cytokine memory. The model results are discussed in relation to experimental data.
In the first part of the talk, I consider an equation that describes orientational aggregation of actin filaments by a jump process.
I show that a delta peak is a stable stationary solution if filaments attract each other. Then I study the bifurcation behaviour of the model. In particular, I show that when there is a generic bifurcation from the homogeneous distribution (i.e., exactly one of the eigenmodes changes stability), there is an analytic branch of stationary solutions emanating from the bifurcation point whose power series expansion can be computed explicitly. Moreover I show that our model can exhibit a wide variety of types of dynamical behaviour, for example travelling waves, periodic solutions not of travelling wave type, and solutions combining characteristics of both.
In the second part of the talk, I present two models for the cortex formation of actin filaments near the inner membrane of a vesicle. The first model is a partial differential equation that contains anisotropic spatial diffusion and reorientation of filaments. In the second model a system of partial differential equation mimics the dynamic processes that take place when actin polymerization is started at the boundary of the vesicle. Numerical simulations show that both models can lead to aggregation.
Actin is an abundant cellular protein that self-assembles into filaments with lengths on the order of several microns. Because actin filaments are linear chain polymers, their mechanics are well described by polymer physics.
The assembly process is well-characterized, but coupling of energy dissipative reactions to polymerization complicates the picture. Actin filaments in solution form complex networks in which interactions between polymer chains become important. Interesting concentration-dependent phenomena occur in these systems, which are currently being investigated via a simple experimental in vitro model. The concentration regime in which a partial phase transition to a nematic liquid crystalline state occurs is focused upon. As predicted by Onsager, isotropic and anisotropic domains coexist in this region. Larger scale spatial structure, which is not predicted by theory, has also been observed. This structure is suggestive of pattern formation.
The dissipative biochemical reactions responsible for the phenomenon know as treadmilling guarantee that the actin networks under consideration are not in thermodynamic equilibrium. Preliminary results suggest that the presence of molecules involved in energy dissipative processes is correlated with the observed large scale structural variation. Therefore, investigation of the role of nonequilibrium processes in the creation of large scale ordering in this in vitro system are being pursued via advanced microscopy techniques. Using such methods, including polarization, confocal and multi-photon fluorescence, the structural variation can be quantitatively characterized. In this way it can be determined if the system can be truly classified as pattern forming.
Using the information-geometric framework, it is possible to naturally define the complexity of a composite system as its deviation from the set of non-complex systems. Here, a system is considered to be non-complex if it is just the superposition of its constituent elements. The aim of my presentation is to give some analytical results on complex systems and to illustrate them by computer simulations. Applying my approach to the field of neural networks, it leads to a generalized version of Linsker's infomax principle.
Chemotaxis is a process in which bacteria, or, more generally, cells, change their state of movement, reacting to the presence of a chemical substance, approaching chemically favorable environments and avoiding unfavorable ones.
In this talk, we rigorously prove that a kinetic model for chemotaxis introduced by Othmer, Dunbar and Alt has as its drift-diffusion limit the celebrated Keller-Segel model. Furthermore, we show that finite-time-blow-up, which generally occurs in the Keller-Segel model, does not occur in the kinetic model under certain biologically relevant assumptions on the turning kernel.
We present a model for the evolution of networks of occupied sites on undirected regular graphs. At every iteration step in a parallel update I randomly chosen empty sites are occupied and occupied sites having degree outside of a given interval (textl,textu) are set empty. Depending on the influx I and the values of both lower threshold and upper threshold of the degree different kinds of behaviour can be observed. In certain regimes stable long-living patterns appear. We distinguish two types of pattern: static patterns arising on graphs with low connectivity and dynamic patterns found on high connectivity graphs. Increasing I patterns become unstable and transitions between almost stable patterns, interrupted by disordered phases, occur. For still larger I the lifetime of occupied sites becomes very small and network structures are dominated by randomness. We develop methods to analyze nature and dynamics of these network patterns, give a statistical description of defects and fluctuations around them, and elucidate transitions between different patterns. Results and methods presented can be applied to a variety of problems in different fields and a broad class of graphs. Aiming chiefly at the modeling of functional networks of idiotypic networks are discussed.
The talk is devoted to study of electrosensory system of paddlefish. We first characterize spontaneous oscillatory properties of electroreceptors. We show that a single electroreceptor is a complex system composed from two distinct stochastic self-sustained oscillators. The coupling of these oscillators results in specific quasiperiodic firing pattern of electroreceptor afferents (sensory neurons). Application of periodic electric stimuli shows the phenomenon of synchronization, while stimulation with noise reveals noise-induced transitions from tonic to bursting behavior.
Matrix models for stage-structured populations have been very useful and successful to describa and predict the dymanics, steadt states, cycles,... of populations. Integrodifference equations are a tool to include space explicitly into time discrete population models.
We first present some analytical results on stage-structured integrodifference equations which relate directly to the study of minimal habitat size and species persistence in fragmented habitat. We then discuss the so-called dispersal success approximation which reduces the complexity of the model and makes it more easily applicable to experimental data. Finally, we present some numerical simulations for a population divided into two stages: juveniles and adults.
The purpose of this talk is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 70's and subsequently established experimentally by him and his coworkers. In the simplest version of this model, an avascular tumor secretes a tumor growth factor, (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network.
In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a sub case. A third mechanism, receptor inhibition, can be incorporated into this model although we do not discuss this mechanism in this talk.
Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of the various molecular species in porous media.
One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor.
The structural response and internal organization of eukaryotic cells is governed by its cytoskeleton - a highly-organized polymer network extending throughout the cell. Composed of three types of protein filaments - actin filaments, microtubules and intermediate filaments - the cytoskeleton is directly responsible for intra- and intercellular movements as well as cellular shape changes. Hence, the cytoskeleton is a dynamic material, whose architecture and composition reflect cell function and state. This high correlation of cytoskeletal architecture to basic cell functions such as mitosis and transport, has led researchers in the pathology of several diseases, including cancer, to conclude that cytoskeletal changes are key, perhaps even diagnostic, in the progression of these diseases. This, in turn, implies that the change in cell elasticity or structural response due to cytoskeletal change is an important parameter to characterize cells.
There have been many experimetal techniques to characterize eukaryotic cell elasticity. There have also been polymer rheology experiments for the structural characterization of in vitro polymer networks. Our work focusses on relating these two fields, by understanding the results of cell deformation experiments, based on the polymer physics of the cytoskeleton. We have performed detailed theoretical investigations of the structural properties of the actin cytoskeleton and its contribution to cell strength. The ultimate aim of this work is to quantify the contribution of each individual cytoskeletal polymer network to the entire structural response of the cell.
Cooperative ordinary differential equations are of importance in the context of monotonicity and comparison theorems. The special qualitative properties of cooperative systems were elucidated by M.W. Hirsch some decades ago. The usual notion of cooperativity is based on the standard ordering in real n-space defined by the cone of positive vectors. In this talk, an extension to general order cones will be presented, and it will be shown that the usual monotonicity and comparison theorems hold. We also exhibit classes of relevant examples that are cooperative with respect to a nonstandard ordering.
This lecture introduces a special kind of problem solving: Instead of integrating all knowledge about a given problem (in the case of drug development for example 3d-structures of the targets/proteins and/or physico-chemical parameters) the real problem is transformed to an abstract problem which is solved by abstract methods and at last retransformed to a real world solution. This procedure enables algorithms to organize into unknown and unusual solution space. The first part of the lecture introduces methods - used at the Max Planck Institute of Biophysical Chemistry in Goettingen - of algorithmic self organization to study molecular evolution. The second part gives an example of a rapid de novo molecular optimization of an thrombin inhibitor using the principles introduced in the first part.
Various psychophysical findings suggest the existence of a common framework of distinguished temporal intervals in human perception and cognition. The corresponding experimental paradigms include the study of periodic and simple sequential stimuli in different modalities, e.g. by determining perceptual thresholds or examining the influence of temporal parameters on the resulting percepts, the study of latencies in simple and complex recognition as well as the investigation of human time perception and duration discrimination. The determined time intervals exhibit a structure of nearly rational or integer size relations. In agreement with Geisslers (1985, 1992) taxonomic model this regularity can be expressed by the representation of these intervals as multiples of a quantum of about 4.5 msec. In my talk I will present a neurophysiological principle of discrete mental timing which relates to the concept of synfire chains (Abeles et al.,1993). In this scheme, interaction of precisely timed neural delays with oscillations in neural feature maps constrains the delays to simple rational size relations. On such a basis, various psychophysical findings on discrete timing, including quantization, are linked to recent neurophysiological results.
Cultured cells on a multielectrode array show spontaneous activity which changes characteristically in the presence of certain substances. We present some methods to describe these substance-specific patterns of spiking neurons.
The dynamics of the growth of cell colonies and tumours has been characterised by means of scaling techniques, which allow for very accurate descriptions of cell contour variation as a function of time . Scaling techniques used to analyse the fractal nature of cell colonies growing in vitro, and that of tumours developing in vivo, showed them to exhibit exactly the same type of dynamics, irrespective of phenotype, genotype or malignant status. This dynamics corresponds to the Molecular Beam Epitaxy universality class, which implies a linear growth rate ( in strong contrast with the commonly assumed exponential - type growth ), a growing border , and interface cell migration. The main mechanism responsible for tumour progression, as for any cell proliferation process, seems to be surface cell diffusion on the tumour border.
This result has a number of important consequences concerning solid tumours .To begin with, the effectiveness of chemotherapy appears to be somehow dependent on the specific surface of tumours .A second implication is that the most malignant cells are always located at the tumour border . These remarks may have profound implications for a better understanding of the tumour kinetics, the progression of malignant phenotypes at different stages of development, and the host - tumour relationship.
Studies of genetic differentiation of populations are of critical importance for understanding evolutionary divergence and speciation. The causes of differentiation are generally thought to be natural selection, random genetic drift and restricted migration. However, there is little empirical evidence showing how and to what extent these forces drive population differentiation. We analyze the relative impact of these processes on levels and patterns of DNA sequence variation and differentiation in Drosophila ananassae, a highly subdivided species whose geographic center is in Southeast Asia. In particular, we test the hypothesis that selection for locally favored alleles (i.e. local adaptation) is primarily responsible for population differentiation in D. ananassae. The second part of the talk will about the detection of footprints of natural selection ("selective sweeps") based on a population genomics approach. A newly developed likelihood ratio test will be used to identify selective sweeps along chromosomes. This test allows us to estimate the strength and frequency of selected substitutions as well as the "target sites" of selection in the genome.
More than 90% of our visual information is catched by a very small region of the retina, occupying less than 1% of their total surface area, and being called the fovea centralis. This name was given because in the adult Primate retina, at this place the ''inner'' retinal layers (containing second-and third-order neurons) are shifted aside, leaving the photoreceptor cells - as the light perceptive cells - directly exposed to the light which enters the eye. It has been speculated that this organization is important for vision; in humans suffering from albinism, this regional specialization is not developed. Interestingly enough, this specialization develops mostly during the early postnatal period (up to the 4th year in humans). At birth, this area is even thicker than the rest of the retina. The mechanisms underlying this ''foveation'' are largely unknown. The hypothesis will presented that one of the retinal layers (between the photoreceptor cells and the inner layers) is a ''locus minoris resistentiae'', and allows a ''sliding like that of the tectonic plates'' between the cells in the photoreceptor layer (centripetally) and those of the inner layers (centrifugally). Biological data are available (and further data will be obtained); what we would greatly appreciate is any help in modeling the process of foveation.
We show that stochastic resonance can be interpreted as stochastic phase synchronization. Introducing a phase desccription to stochastic bistable dynamics we investigate conditions where strong phase relations are obeyed between an inputting signal and the stochastic output of the bistable dynamics. By considering dichotomic input and output signals we are able to analytically prove the occurrence of noise-induced frequency and phase locking. Regions of stochastic synchronization form Arnold tongues in the amplitude/noise intensity space.
In the second part we apply concepts of stochastic synchronization to behavioural experiments on paddlefish. It was shown experimentally that juvenile paddlefish makes use of stochastic resonance during prey detection. We show an increase of the average phase locking period for subthreshold signals acting on the paddlefish rostrum in the neighbourhood of swarms yielding a source of external electrical noise.
Literature:J. A. Freund, A. Neiman and L. Schimansky-Geier, Europhys. Lett. 50 8-14 (2000). J. A. Freund, J. Kienert, L. Schimansky-Geier, B. Beisner, A. Neiman, D. Russel, T. Yakusheva and F. Moss, Phys. Rev. E 63, 031910 1-11 (2001); Journal of Theoret. Biology, accepted for publication.
Proteasomes are the key proteases in cytosolic protein degradation. Some of their degradation products, peptides of 8-11 amino acids in length, are presented on the cell surface bound to major histocompatibility complex (MHC) class I molecules. These, in turn, can be recognized by T lymphozytes screening the body for virus-infected cells.
Prediction of proteasomal cleavage specificity would facilitate the search for CTL epitopes by narrowing down the number of potential peptides selected by MHC class I binding prediction.
Network-based model proteasomes are developed and trained by an evolutionary algorithm with the experimental cleavage data of yeast and human 20 S proteasomes with an array of affinities for a window of ten flanking amino acids.
The affinity parameters of the model, which decide for or against cleavage, correspond with the cleavage motifs determined experimentally. The model proteasomes reproduce and predict proteasomal cleavages, positions and quantitative cleavage strength, with high degree of accuracy.
The prediction algorithms can be used via an Internet site.
All situations considered in population dynamics have a common problem that could be characterized as the "structure" distribution problem. The objects forming a population, like polymers, cell structures, single cells, cells in a tissue, animals or people can be modelled with different degrees of internal structure, like age, energy content, metabolic activity, or health status. On the other side the individual's environment has in general itself a structure, bacteria in the soil live in a porous medium, oceanic populations must cope with turbulent diffusion and oceanic currents, human populations are mixed by various processes affecting for example the spread of diseases, and trees experience self-induced vertical solar energy gradients. It is clear that any realistic structuring of a population gives rise to arbitrary complex models. It is important to find consistent frameworks which allow a sufficient projection of the real-world problem into mathematical equations and furthermore are sufficiently tractable by analytical and numerical methods. In the talk I will give some concrete (mostly ecological and biotechnological) examples how such a modelling program can be followed. Here the mathematical restriction will be to use deterministic models, i.e. PDE and integral equations.
Strong variations of the G+C content in genomes of warm-blooded vertebrates ("isochores") have been observed a long time ago, but any explanation of their possible origin has remained out of reach. Here we present one possible mechanism that could explain the dynamic origin of isochores by using mutation patterns derived from the recently published human genome. The key element of the dynamic model is the strong G+C dependence of mutation rates. We show that the observed mutation patterns imply an unstable mutational equilibrium, and that deviations from equilibrium are amplified exponentially. In this way G+C rich regions become even richer, whereas G+C poor region become poorer.
An important basis of neuron plasticity is the intracellular calcium concentration in response to specific stimulation protocols. It turns out that the transmission of action potentials at synapses undergo durable changes in dependence of the stimulation - an effect called long-term-potentiation and long-term-depression.
The intracellular calcium dynamics are investigated in a theoretical model in dependence of different stimulations. The model is based on the experimental knowledge about single membrane proteine characteristics. The calcium answer is shown for pyramidal neurons and is quantitatively compared to experimental observation. The relevance of different calcium sources for the neuron plasticity is discussed.
Allergic hypersensitivity of type I for hymenoptera venoms is the most frequent reason for acute IgE-mediated anaphylactic reactions. The subcutaneous injection of increasing doses of purified allergen followed by long term administration of an adequate maintenance dose over a period of 3 to 5 years called venom immunotherapy has been proven to be a very effective treatment achieving protection in about 96% of allergic patients. Even though the principle of hyposensitisation has been introduced already 90 years ago, the underlying immunoregulatory mechanisms of venom immunotherapy remain poorly understood. Recent studies suggest a shift in cytokine production from a Th2 to a Th1 cytokine profile during therapy. In this paper a mathematical model for T-cell regulation and a model for mast cell/basophil desensitisation are presented and analysed to explain the mechanism of conventional, rush and ultra-rush venom immunotherapy.
Hematopoiesis is a tissue with high proliferative activity being responsible for peripheral blood cell production. The normal steady-state production can temporarily be impaired during the administration of anti-cancer therapies. In particular, the deficiency of neutrophil granulocytes (neutropenia) in the peripheral blood is of clinical importance, since patients are prone to potentially life-threatening infections. G-CSF is a hematopoietic growth factor which specifically stimulates granulopoietic cell production. Therefore, G-CSF is widely used in clinical practice to attenuate chemotherapy induced neutropenia. However, the timing of G-CSF administration is crucial for optimal attenuation.
In this talk, I will present a mathematical model of human granulopoiesis which is able to describe the blood neutrophil dynamics during chemotherapy and G-CSF administration. A set of concatenated compartments is used to describe the population kinetics of hematopoietic stem cells, granuloid precursors and mature blood neutrophils. The system is regulated by several feed-back loops. The model was used to systematically explore the impact of different G-CSF administration schedules on neutrophil population dynamics in order to identify optimal schedulings.
In fish schools individuals adapt their orientation of movement and their speed to that of their neighbors. A model for this behavior is derived in two steps in one space dimension based on reaction transport equations.
First, the adaptation of orientation is modeled under the assumption of constant speed. The model is investigated analytically and numerically. Then speed adaptation is modeled leading to a hyperbolic/elliptic system, for which existence of solutions is shown.
Analysing the principles of collective cell motion and interaction is of primary importance in order to understand not only life cyles of bacteria or social amoebae but also the organization of tissues, wound healing or organogenesis. Cellular automata offer a modelling perspective to typical examples of cellular pattern formation based on direct cell-cell interactions.
We have characterized basic, in particular density- and orientation-dependent, interactions. Density-dependent interaction provides a model of differential adhesion while orientation-dependent interaction yields a model of collective cell motion or swarming. Orientation reversal induced by cellular collisions can explain periodic rippling patterns in myxobacteria if, additionally, refractoriness of the reversal mechanism is assumed.
Cellular automata can be used as simulation tool of spatio-temporal pattern formation but also allow for straightforward analysis. In particular, Fourier analysis permits to deduce important orientational and spatial aspects of simulation outcomes.
The development of a primary solid tumour (e.g., a carcinoma) begins with a single normal cell becoming transformed as a result of mutations in certain key genes, this leades to uncontrolled proliferation. An individual tumour cell has the potential, over successive divisions, to develop into a cluster (or nodule) of tumour cells consisting of approximately $10^{6}$ cells. This avascular tumour cannot grow any further, owing to its dependence on diffusion as the only means of receiving nutrients and removing waste products. For any further development to occur the tumour must initiate angiogenesis - the recruitment of blood vessels. After the tumour has become vascularised via the angiogenic network of vessels, it now has the potential to grow further and invade the surrounding tissue. There is now also the possibility of tumour cells finding their way into the circulation and being deposited in distant sites in the body, resulting in metastasis.In this talk we present two types of mathematical model which describe the invasion of host tissue by tumour cells. In the models, we focus on three key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix, ECM) and matrix-degradative enzymes (MDE) associated with the tumour cells. The first model focusses on the macro-scale structure (cell population level) and considers the tumour as a single mass. The mathematical model consists of a system of partial differential equations describing the production and/or activation of degradative enzymes by the tumour cells, the degradation of the matrix and the migratory response of the tumour cells. Numerical simulations are presented in one and two space dimensions and compared qualitatively with experimental and clinical observations. The second type of model focusses on the micro-scale (individual cell) level and uses a discrete technique developed in previous models of angiogenesis. This technique enables one to model migration and invasion at the level of discrete cells whilst still allowing the chemicals (e.g. MDE, ECM) to remain continuous. Hence it is possible to include micro-scale processes both at the cellular, such as proliferation, cell/cell adhesion and sub cellular such as cell mutation properties. This in turn allows us to examine the effects of such micro-scale changes upon the overall tumour geometry and subsequently the potential for metastatic spread.
Almost half a century ago Charles Elton (1958) warned of the increasing frequency of foreign species introduction, and of the inevitable biological dislocations that follow. Today, the number and type of invading organisms is growing --- understanding and monitoring the process of alien species spread is an important applied ecological problem. A key element of this process is prediction of spread rate for the invader. It was thought for many years that this issue of spread rate was essentially resolved by analysis of an equation derived by Fisher (1937) for invading genotypes. It is now clear that the Fisher spread model does not hold for many relevant biological situations. In particular, the model tacitly ignores rare, long distance dispersal events that initiate secondary invasion foci, far ahead of the bulk of invasion. These events can be shown to drive the invasion process at much higher speeds than previously thought, speeds which may continue to increase as the invasion progresses. The resulting spatial pattern of spread is patchy, with the patches linked historically via long-distance dispersal.
In my talk I will discuss these and other issues related to the effects of environmental variability on spread rates. I will propose a nonparametric method for estimating spread rate which makes no assumptions about the underlying distribution of dispersal distances, and will discuss the role of mathematical models in explaining dynamics of several well-studied biological invasions, including the house finch in eastern North America, and the historical spread of trees in response to climate change.
An organism which got infected by a pathogen will respond with a very complex immune reaction aimed to destroy the invasor and neutralize its toxicity. Among other things, so called B cells will start producing antibodies which possess a high ability to bind to that individual pathogen. At the same time, the immune system will try to further improve the quality of the produced antibodies. It does that by creating small compartments in the lymphoid tissue, called germinal centers. There, B cells proliferate rapidly while the part of the genome which encodes the antibodies shows a very high mutation rate. After some time, a large number of B cells, each one enconding antibodies with different properties, is available. The cells capable of producing the best fitting antibodies are selected out of that bunch and released into the blood stream. Thus, a few days after the infection, the body is able to produce antibodies with higher affinity towards the given pathogen than the original ones.
Although many details of the germinal center function are well known, the mechanisms governing the dynamics of the B cells and the selection processes within the germinal center are still unclear. The introduction of mathematical models might shed some light into that questions. In this talk, an introduction into the subject will be given and some possible mechanisms and computer simulation results will be presented.
We investigate a model where idiotypes (characterizing B-lymphocytes and antibodies of an immune system) and anti-idiotypes are represented by complementary bitstrings of a given length d allowing for a number of mismatches (matching rules). In this model, the vertices of the hypercube in dimension d represent the potential repertoire of idiotypes. A random set of (with probability p) occupied vertices corresponds to the expressed repertoire of idiotypes at a given moment. Vertices of this set linked by the above matching rules build random clusters.
We give a structural and statistical characterisation of these clusters -- or in other words -- of the architecture of the idiotypic network. Increasing the probability p one finds at a critical p a percolation transition where for the first time a large connected graph occures with probability one. Increasing p further, there is a second transition above which the repertoire is complete in the sense that any newly introduced idiotype finds a complementary anti-idiotype. We introduce structural characteristics such as the mass distributions and the fragmentation rate for random clusters, and determine the scaling behaviour of the cluster size distribution near the percolation transition including finite size corrections. We find that slightly above the percolation transition the large connected cluster (the central part of the idiotypic network) consists typically of one highly connected part and a number of weakly connected constituents and coexists with a number of small isolated clusters. This is in accordance with the picture of a central and a peripheral part of the idiotypic network and gives some support to idealized architectures of the central part used in recent dynamical mean field models.
The experimental observation of calcium oscillations in living cells led to the development of several mathematical models which describe such oscillations. Besides simple rather periodic oscillations, complex nonperiodic oscillations have also been observed experimentally. These complex oscillations have not been explained by the existing models so far. We therefore developed a new model which is able to display simple and complex calcium oscillations in time and space. We describe the dynamics of this system in detail.
The neuromuscular synapse provides a unidirectional flux of signals (action potentials) from the nerve to the muscle. In the terminology of cybernetics this junctional structure is an important nonlinear biological transfer element.
A mathematical description of its operation could be of interest for a better understanding of experimentally observed transmission characteristics and of possible pharmacological influences. Therefore a model of neuromuscular transmission had been proposed which considers morphological structure and interrelated chemical and electrical mechanisms. In this interrelation after-effects cut a great figure. The result of this modelling is a system of linear and nonlinear first order differential equations. By an analytical investigation of derived differential equations detailed information about the global properties of the solutions is obtained. The special time-dependent solutions computer simulated with biologically relevant parameters are in good agreement with experimentally determined transient responses of the neuromuscular synapse.
Since the introduction of coalescent theory by J.F.C. Kingman in 1982, the mainstream of theoretical population genetics changed from a prospective to a retrospective point of view. The focus is now the reconstruction of aspects of evolutionary history from a molecular population sample. Coalescent theory describes the evolution of such a sample backwards in time, and provides a probabilistic description of the events that occured in the genealogy since the most recent common ancestor of the sample.
In this talk I will give a short introduction to the coalescent process and its use in modelling the evolution of a sample of a population. However, the main part of this talk will be dealing with the challenging task of infering demographic aspects of the population the sample stems from.
After exposure of tissues to radiation or growth factors or during tumor formation often an increased cell division rate is experimentally found. In tissue sheets as in intestinal crypts or the skin this is often accompanied by the folding of the sheet.
We suggest a generic mechanism which provides a potential explanation for this phenomenon.
Our studies partly base on a stochastic single cell Monte Carlo model and partly on a coarse grained analytic approach. For simplicity and in order to obtain a clear picture of the underlying folding principle we focus on one-dimensional cell chains in two-dimensional space.
The basic physical model assumptions are the existence of attractive nearest-neighbor interactions between cells to maintain the integrity of the one-layered tissue, a bending energy that models the polarity of the cells in a sheet and cell division that takes into account potential size changes of the sheet.
The effect of cell division on the tissue geometry and growth law is studied for different tissue geometries, bending rigidities (which measure the resistance of a tissue sheet against bending), cell division rates and division algorithms (daughter cells of the same size of mother cells or smaller than mother cells; the latter case corresponds to the situation found during blastula formation).
We find that as a tissue domain grows above a certain size the bending energy becomes too small to smooth local undulations that are stochastically created by local fluctuations in the growth of the layer hence the layer roughens.
If this occurs before cell deformations or compressions become so strong that cell division is hindered, the cell number increases exponentially (if the cycle time is large or the bending rigidity small) otherwise the growth law changes to sub-exponential growth before the folding occurs.
It has been observed in many species that the release of a chemical signal can trigger individuals to change their movement behavior such that a swarm or an aggregation is formed (chemotaxis). To model this effect I will start with a transport equation for the population, which comes from a velocity jump process. An appropriate scaling analysis allows to relate this model to the well known parabolic Keller-Segel equations for chemotaxis. I will give some very general assumptions and show an approximation result. It turns out that the diffusion limit in general is non isotropic and necessary and sufficient conditions for isotropy will be derived. Finally I will give several examples, which lead to appropriate models for specific populations.
(Joint work with H. Othmer, Minneapolis.)
