## Seminars 2006-2007

Dr. Michael Freitas will present a plan to create a core group of researchers in clinical proteomics at OSU. There would be three major components: clinical sample collection and processing; mass spectrometry; and bio-informatics. The talk will cover the resources now in place as well as those still needed, especially in the area of bio-informatics.

A brief overview of the the visual system, focusing on the primary visual cortex. In particular, we will derive an activity-based developmental model of ocular dominance column formation in primary visual cortex that takes into account cortical growth. The resulting evolution equation for the densities of feedforward afferents from the two eyes exhibits a sequence of pattern forming instabilities as the size of the cortex increases. We use linear stability analysis to investigate the nature of the transitions between successive patterns in the sequence. We show that these transitions involve the splitting of existing ocular dominance columns, such that the mean width of an OD column is approximately preserved during the course of development. This is consistent with recent experimental observations of postnatal growth in cat. Additionally, I will briefly discuss the joint development of cytochrome oxidase blobs with ocular dominance columns, another problem I have worked on.

Joint work with Paul Bressloff (Dept. of Mathematics, University of Utah).

In this talk a biofilm model with growth and detachment is discussed, hence, making the model interesting in applications. The crucial feature of our model is that cells are able to enter an adapted resistant state when challenged with antimicrobials (adaptation). The model is presented in both a qualitative and quantitative manner. Existence and uniqueness of solutions can be shown as well as the existence and non-uniqueness of steady-state solutions. Another question of interest is the effective dosing of biocide, i.e., exploring dosing strategies that could minimize the number of living cells or biofilm thickness. Constant and periodic dosing regimes were modelled numerically. One of our main results is that on and off dosing is significantly better than the other dosing types. The model contributes to a better understanding of one of the resistant mechanisms in biofilms.

I will talk about basic neurobiology needed to set up the model equations. Then I will provide sufficient conditions to guarantee the existence and stability of traveling waves (both homoclinic orbits and heteroclinic orbits) for each equation or each system. In particular, I will provide speed analysis (estimates, limits, etc), speed index function and stability index function and their relationship.

Ever since Hodgkin and Huxley developed their revolutionary mathematical model of neuronal activity, the field of computational neuroscience has been growing rapidly. Today, computer simulations are used extensively to model systems as small as individual ion channels and as large as networks consisting of 10,000 morphologically accurate model cells.

A drawback of simulations performed on detailed cell models is that the most commonly used numerical schemes are implicit, meaning that the voltage update is global in scope. This approach does not allow focusing of computational effort on those regions of the cell that are most active. The result is an unnecessary slow-down in neural simulations when activity is localized to a small region of the cell.

I will present a predictor-corrector numerical method we developed that decouples all of the branches from each other. This allows the algorithm to detect which regions of the cell are active and to focus computational effort on those regions while saving computations in other regions of the cell that are at rest. As a result, the computational cost of a simulation scales with activity, not with the physical size of the system. I will show several simulations that illustrate this idea, including reproductions of recent experimental results from our lab.

Articular cartilage is the primary load-bearing soft tissue in joints such as the knee, shoulder and hip. This tissue can be idealized as a continuum mixture of interstitial water and a solid extracellular matrix comprised of collagen fibers and negatively charged proteoglycan macromolecules. Degradation of cartilage extracellular matrix leads to osteoarthritis, a painful condition that is predominantly associated with aging. The extracellular matrix is maintained by a sparse population of cells (chondrocytes) that are encapsulated by a thin, stiff layer called the pericellular matrix. Since mature cartilage is avascular and aneural, mechanical variables in the vicinity of the chondrocytes strongly influence cell metabolic activity and, ultimately, the health of the tissue. In this talk, I will present an overview of biphasic (solid-fluid) and triphasic (solid-fluid-ion) continuum mixture models of mechanical and chemical loading in articular cartilage. Analytical and numerical solutions of the associated interface problems will be presented. Results will be discussed in the context of assessing the impact of pericellular matrix properties, and alterations with osteoarthritis, on the cellular microenvironment in cartilage.

This talk is motivated by the study of 1D fluid models for blood flow in the vascular system. In our work, we consider blood modeled as an incompressible shear-thinning generalized Newtonian fluid in a straight rigid and impermeable vessel with circular cross-section of constant radius. To study this problem, we use an approach based on the Cosserat theory (also called director theory) related to fluid dynamics which reduces the exact three-dimensional equations to a system depending only on time and on a single spatial variable. From this new system we obtain the unsteady relationship between mean pressure gradient and volume flow rate over a finite section of the tube for the specific case of power law model and also the correspondent equation for the wall shear stress which enters directly in the formulation as a dependent variable.

We study a Lotka-Volterra reaction-diffusion-advection model for two competing species in a heterogeneous environment. The species are assumed to be identical except their dispersal strategies: one disperses by random diffusion only, the other by both random diffusion and advection along environmental gradient. When the two competitors have the same diffusion rates and the strength of the advection is relatively weak in comparision to that of the random dispersal, we show that the competitor that moves toward more favorable environments has the competitive advantage, provided that the underlying spatial domain is convex, and the competitive advantage can be reversed for certain non-convex habitats. When the advection is strong relative to the dispersal, we show that both species can invade when they are rare, and the two competitors can coexist stably. The biological explanation is that for sufficiently strong advection, the "smarter" competitor will move toward more favorable environments and is concentrated at the place with maximum resources. This leaves enough room for the other species to survive since it can live upon regions with less resources.

In this talk, we will introduce numerical approaches to find the optimal shape and topology for elliptic eigenvalue problems in an inhomogeneous medium by using shape derivatives and topological derivatives in combination with level set methods. The common numerical approach for these problems is to start with an initial guess for the shape and then gradually evolve it, until it morphs into the optimal shape. One of the difficulties is that the topology of the optimal shape is unknown. The level set approach based on shape derivatives has been well known for its ability to handle topology changes but it may get stuck at shapes with fewer holes than the optimal geometry. By incorporating topological derivatives into the level set method one can provide an alternative way to create holes efficiently and escape from the shapes with fewer holes. We will show results in applications including resonant frequency control and photonic devices design.

Biofilms are ubiquitous biological ecosystems consisting of large numbers of microbial organisms contained in a self-secreted polymeric matrix. Biofilm structure plays an important role in biofilm function and control. It is thus important to determine physical properties of biofilms regarded as materials. Measurements indicate that biofilms can be characterized as viscoelastic fluids. Observations demonstrate the tendency for material failure via sloughing of large chunks. In this talk I will discuss some efforts to understand physical properties of biofilms as materials and to use this knowledge to construct models capable of describing long-time behavior of biofilms.

Multicellular tumor spheroids (MCTS) have been used as a model system because of their remarkable ability of reproducing the properties of tumors in vivo. MCTS are made of three layers with different mechanical properties, i.e. proliferating outer layer, quiescent middle zone, and necrotic zone. Helmlinger et al (1997) were able to measure the residual stress generated by tumor growth in agarose gel. These results showed that tumor growth can be regulated by stress and that mechanical properties of extracellular matrix (ECM), such as stiffness, can inhibit the tumor growth in vitro. These authors also found that MCTS grew in ellipsoidal shape rather than spherical shape when grown in a long cylinder, which indicates that stress was a controling factor in MCTS growth. The residual stress caused by uncontrolled cell proliferation is believed as possible cause of localized blood vessel collapse in the tumor, thereby causing malfunction of vital organs.

For the proliferating zone, I consider force balance equation to get the evolution of cell movement where each cell is considered as a growing viscoelastic ellipse with two major axes. The discretely modeled cells can divide, push each other, and find the right path to move. These cells keep dividing as long as they get the necessary nutrients. As cells proliferate, cells in the center do not have enough nutrient and start to die, a process called necrosis. Cells in outer part of spheroid continue to proliferate, which produces residual stresses. Increased stresses surrounding the spheroid then inhibit MCTS growth and increase the cell density in the proliferating zone. By considering that the gel, quiescent zone, and necrotic region are viscoelastic materials, I use continuum model in these regions and couple the continuum model to the discrete cell model. Reaction-diffusion equations for nutrients are considered to describe the evolution of concentration of nutrients.

I discuss the stress effect on tumor growth and growth behavior of active tumor cells. I will also discuss a possible mechanism for reduction of cell volume. My future work includes the incorporation of a shedding effect, which is when tumor cells detach from the primary tumor and shed into the suspension medium. I will discuss how I plan to incorporate this effect, which is an important issue in cancer such as certain brain cancers. Another aspect of my future work is the inclusion of internal cell dynamics. I will discuss this as well.

The goal of tissue engineering is to direct the generation and remodeling of tissues often for potential clinical applications. This talk will explore the use of mechanobiology - the study of the biological effects of mechanical forces - in tissue engineering using three examples; mechanically induced growth and remodeling of blood vessels cultured outside of the body, capillary morphogenesis in 3D substrates with various mechanical properties, and the in vitro differentiation of precursor cells to insulin producing pancreatic cells. Efforts by my laboratory and others to use mathematical modeling to better understand these systems will be noted but the focus of the talk will be the underlying biological process.

Spatial heterogeneity, habitat connectivity, and rates of movement can have large impacts on whether a disease, such as rabies, persists or becomes extinct. In this talk, we consider equilibrium properties for a pair of related frequency-dependent SIS epidemic models in which the movement of susceptible and infected individuals in discrete or continuous space is represented by a system of differential equations, or a coupled pair of reaction-diffusion equations, respectively.

In both models, local differences in disease transmission and recovery rates characterize whether regions are low-risk or high-risk, and these differences collectively determine whether the spatial domain (or habitat) is low-risk or high-risk. We relate the basic reproduction number (R_0) to the speed with which infected individuals move within the habitat. For low-risk habitats, the disease-free equilibrium (DFE) is stable provided that the rate at which infected individuals move lies above a threshold value. For high-risk habitats, the DFE is always unstable. When the DFE is unstable, a unique endemic equilibrium (EE) exists. This EE tends to a spatially inhomogeneous DFE as the rate at which susceptible individuals move becomes very small. The limiting DFE is positive on all low-risk regions and can also be positive on some high-risk regions. Sufficient conditions for the limiting DFE to be positive or zero on high-risk regions are given, and these conditions are illustrated using numerical examples.

This work is in collaboration with Linda Allen (Texas Tech), Ben Bolker (Florida), and Yuan Lou (Ohio State).

Conductance-based models for hypothalamic neuronal thermosensitivity

The question as to how the ratio of horizontal to vertical transmission depends on the coefficient of horizontal transmission is investigated in host-parasite models with one or two parasite strains. In an apparent paradox, this ratio decreases as the coefficient is increased provided that the ratio is taken at the equilibrium at which both host and parasite persist. Moreover, a completely vertically transmitted parasite strain that would go extinct on its own can coexist with a more harmful horizontally transmitted strain by protecting the host against it. Several stability results are presented for the coexistence equilibrium (host and two parasite strains). Under standard incidence, undamped oscillations may occur.

It is not widely appreciated that mitochondria are important participants in intracellular Ca2+ signaling. In cells expressing highly dynamic Ca2+ signaling behavior, mitochondria can shape the spatiotemporal patterns. In addition, mitochondria have been shown to act as an independent network of oscillators. These phenomena are gaining attention as evidence points to mitochondrial dysfunction as a potential factor in diseases such as neurodegeneration. I will review the work of others and present some of our own results regarding mitochondria and Ca2+ signaling. In particular I will discuss efforts to model dynamic phenomena involving mitochondria, and I will sketch out areas which might benefit from additional theoretical attention.

There is growing interest in understanding and controlling the spread of diseases through realistically structured host populations. We investigate how network structures, ranging from circulant, through small-world networks, to random networks, and vaccination strategy and effort interact to influence the proportion of the population infected, the size and timing of the epidemic peak, and the duration of the epidemic.

We found these three factors, and their higher-order interactions, significantly influenced epidemic development and extent. Increasing vaccination effort (from 0 - 90%) decreased the number of hosts infected while increasing network randomness worked to increase disease spread. On average, vaccinating hosts based on degree (hubs) resulted in the smallest epidemics while vaccinating hosts with the highest clustering coefficient resulted in the largest epidemics. In a targeted test of five vaccination strategies on a small-world network (probability of rewiring edges = 5%) with 10% vaccination effort we found that vaccinating hosts preferentially with high-clustering coefficients (similar to some real-world strategies) resulted in twice the number of hosts infected as random vaccinations and nearly a 30-fold higher number of cases than our strategy targeting hubs (highest degree hosts). Our model suggests how vaccinations might be implemented to minimize the extent of an epidemic (e.g., duration and total number infected) as well as the timing and number of hosts infected at a given time over a wide range of structured host networks.

The basal ganglia are a group of subcortical nuclei involved in the generation of voluntary movement, cognition and emotion. Dysfunction of the basal ganglia is associated with movement disorders such as Parkinson's disease and Huntington's chorea. Structures within the basal ganglia have been the target of recent therapeutic surgical procedures including deep brain stimulation(DBS). The basal ganglia display complex firing patterns which differ between normal and pathological states. Neither the origins of these firing patterns nor the neuronal mechanisms that underlie the patterns are understood. Conventional theories of basal ganglia are based on the average firing rate of the neurons and ignore the importance of temporal dynamics; they do not explain where tremors come from or why DBS would alleviate the symptoms. In this lecture, I will describe recent progress on the development of more realistic, biophysically based models and review various firing patterns which emerge in those models. I will also discuss geometric dynamical methods for analyzing the irregular activity patterns.

Experiments have demonstrated that the projection neurons (PNs) within a mammal's olfactory bulb or insect's antennal lobe (AL) produce complex firing patterns in response to an odor. The firing patterns may consist of epochs in which a subset of PNs fire synchronously. At each subsequent epoch, PNs drop in and drop out of the ensemble, giving rise to "dynamic clustering". I will present a biologically motivated model of the AL that produces dynamic clustering, as well as other complex features of AL activity patterns. For example, the model exhibits of form of spatial decorrelation in which the temporal representations of similar odors evolve to distinct patterns. Using singular perturbation methods, we reduce the analysis of the model to a discrete system. The discrete system allows us to systematically study how properties of the attractors depend on parameters including network architecture.

Tissue engineering might be characterised as 'the science of spare parts'. Although currently in its infancy, its long-term aim is to grow functional tissues and organs to replace those which have become defective through age, trauma or disease. A promising method for growing liver tissue in vitro involves the culture of hepatocytes (the cell type making up around 80% of the liver in vivo) as multicellular aggregates. At present, however, the aggregation process is not well understood. In this talk, I shall describe the development of simple mathematical models of the initial stages of aggregation, focusing on the effects of the hepatocytes' interactions with stellate cells (with which they are sometimes co-cultured), and adhesion to the extracellular matrix on which they are seeded.

Dynamic control of the actin network in eukaryotic cells plays an essential role in their movement, but to date our understanding of how the network properties are controlled in space and time is still rudimentary. For example, how the cell maintains the pools of monomeric actin needed for a rapid response to signals, how the filament length distribution is controlled, and how the actin network properties are modulated by various bundling and severing proteins to produce the mechanical response is not known. In this talk we focus on the development and analysis of mathematical models which enable us to investigate the temporal evolution of the filament length distribution and the effect of the nucleotide composition on the dynamics of actin filaments in vitro. We discuss recent results on the relevant time scales for establishment of a time-invariant length distribution. We find that there are very long-lived intermediate length distributions that are not exponential. Also, we set up a master equation for the biochemical processes appearing at the actin-filament level and simulate the corresponding dynamics by generating numerical realizations through a Monte Carlo scheme. Statistical analysis of ensembles of generated realizations provides the moments of the various distributions of interest. Various challenges in this direction concerning the complexity of the Monte Carlo scheme are addressed and an analysis of the statistically-derived moments in the framework of simplified analytic models and correlated random walks is discussed.

A multilocus analysis of frequency-dependent selection on a quantitative trait.

Knowledge of haplotypes is useful for understanding block structure in the genome and disease risk associations. Direct measurement of haplotypes in the absence of family data is presently impractical, and hence, several methods have been developed for reconstructing haplotypes from population data. We have developed a new population-based method using a Bayesian Hidden Markov Model (HMM) for the source of the ancestral haplotype segments. In our Bayesian model, a higher order Markov model is used as the prior for ancestral haplotypes, to account for linkage disequilibrium. Our model includes parameters for the genotyping error rate, the mutation rate, and the recombination rate at each position. Computation is done by Markov Chain Monte Carlo (MCMC) using the Forward-Backward algorithm to efficiently sum over all possible state sequences of the HMM. We have used the model to reconstruct the haplotypes of 129 children at a region on chromosome 5 in the data set of Daly et al. [2001] (for which true haplotypes are obtained based on parental genotypes), and of 30 children at selected regions in the CEU and YRI data of the HAPMAP project. The results are quite close to the family-based reconstructions and comparable to the state-of-the-art PHASE program [Stephens et al., 2001, Stephens and Donnelly, 2003, Stephens and Scheet, 2005]. Our haplotype reconstruction method does not require division of the markers into small blocks of loci. The recombination rates inferred from our model can help to predict haplotype block boundaries, and estimate recombination hotspots.

We investigate the propagation of the traveling wave fronts in a one-dimensional integrate-and-fire network of synaptically coupled neurons for the case of one, two and multiple spike waves. We use an integro-differential equation characterizing the evolution of the firing times as a function of spatial position to determine the relationship between the speed of the propagating wave and its acceleration. We use the evolution equation to show that for a network of neurons with exponential synaptic connectivity and instantly rising, then exponentially decaying synapses, the evolution of the propagation is fully determined by the instantaneous speed of the traveling wave front. In this case the history of the firing map determines the initial speed of the transient propagation; the acceleration however depends only on the instantaneous speed, thus greatly simplifying the understanding of the network dynamics. This allows for a clear understanding of the conditions required for propagation failure, as well as of the mechanisms by which sustained transient propagation evolves towards the stable constant-speed traveling wave solution. Expanding the equation for the two and multiple wave cases yields further insight on the mechanisms by which sustained transient propagation evolves towards the stable constant-speed traveling wave solutions. In addition, we show that the wave speed and interspike intervals of the asymptotically stable state depend mainly on the interaction between a few successive wavefronts. It then follows that a unique asymptotical solution is selected from an infinite number of theoretically possible solutions and that this solution is independent of the initial conditions in the neural network.

The secondary forest succession is caused by the interaction between pioneer and climax tree species. Such interaction is modeled by a reaction diffusion system. The succession can be described by traveling wave solution connecting the initial and final stages of the succession.

Under some mild conditions, we show the existence, uniqueness and stability of the traveling wave solutions and their asymptotics in the model system. We use method of positive invariant region, method of super and lower-solutions, sliding domain method, asymptotic analysis as well as spectral analysis in the proofs.

The implications of the results are very interesting: the existence of the traveling wave solution implies that the succession is wave-like; the wave has certain particular shape and maintains its speed during the propagation. The asymptotics of the traveling wave solutions reveal that the rate the pioneer tree species leaving the site is proportional to that of the climax tree species entering the site in the final stage of the succession. The uniqueness the wave solution shows that the same wave can also be observed at another time and location. The stability of the traveling wave solution in the weighted Banach spaces tells us that the succession is delicate, in the sense that any big disturbance can disrupt the process.

Another interesting finding is the failure of the K-Selection due to the Allee effect in the climax tree species.

Mesencephalic dopamine neurons ordinarily will not fire faster than about 10/s in response to somatic current injections. However, in response to dendritic excitation, much higher rates are briefly attained. In analysis of a simplified biophysical model, we have suggested a way such transient high-frequency firing may be evoked. The neuron is represented as a number of electrically coupled compartments with different natural frequencies, which correspond to the soma and parts of the dendrite. The model is highly reduced: all the diversity of the compartments that describes real dendritic geometry is substituted by a pair of compartments: the slowest, somatic and the fastest, the most distal dendritic one. In the absence of any synaptic stimulation, oscillatory pattern in this model is controlled by the somatic compartment, and, therefore, has a very low frequency. We consider and compare the influence of N-methyl-D-aspartate (NMDA) and alpha-amino-5-hydroxy-3-methyl-4-isoxazole propionic acid (AMPA) receptor activation in the dendritic compartment. The main result is that activation of the dendritic NMDA receptors evokes oscillations at a much higher frequency. Dendritic AMPA activation, by contrast, cannot increase the frequency significantly. We show that the elevated frequency emerges due to the domination by the dendritic compartment during the application of NMDA. We consider further details of the mechanisms of such domination in the present model as well as in a model comprised of a higher number of compartments.

Molecular phylogenetics is concerned with inferring evolutionary relationships from biological sequences (for example, DNA or proteins). The probabilistic models of sequence evolution that underly statistical approaches in this field exhibit a rich mathematical structure.

After an introduction to the inference problem and an overview of phylogenetic substitution models, this talk will focus on the important statistical issue of identifiability of phylogenetic models. Identifiability is needed for valid inference of model parameters --- whether tree topologies, edge lengths, or substitution rates --- but current understanding still lags well behind common practice.

With the availability of large-scale, high-density single-nucleotide polymorphism (SNP) markers and information on haplotype structures and frequencies, a great challenge is how to take advantage of haplotype information in the association mapping of complex diseases or complex traits in case-control studies. We developed a novel approach for association mapping based on directly mining haplotypes (i.e., phased genotype pairs) produced from case-control data or case-parent data via a density-based clustering algorithm, which can be applied to whole-genome screens as well as candidate-gene studies in small genomic regions. The method directly explores the sharing of haplotype segments in affected individuals that are rarely present in normal individuals. The measure of sharing between two haplotypes is defined by a new similarity metric that combines the length of the shared segments and the number of common alleles around any marker position of the haplotypes, which is robust against recent mutations/genotype errors and recombination events. The effectiveness of the approach is demonstrated by using both simulated datasets and real datasets. The results show that the algorithm is accurate for different population models and for different disease models, even for genes with small effects, and it outperforms some recently developed methods. We recently extend the method to QTL mapping.

In many biological, chemical and physical systems modeled mathematically as bifurcation problems, the bifurcation parameter may vary naturally and slowly with time or the parameter may be slowly varied by the experimenter. Mathematically, these are called slow passage or ramp problems . Of particular interest is when a parameter passes slowly through a Hopf bifurcation and the system's response changes from a slowly varying steady state to slowly varying oscillations. The interesting phenomena is that the transition may not occur until the parameter is considerably beyond the value predicted from a static bifurcation analysis, no matter how slow the parameter is varied, and the delay in onset is dependent on the initial state of the system (memory effect). Previous studies have focussed on linear or constant speed ramps. In this talk I will introduce the problem of slow accelerating and de-accelerating ramps, obtain new results using numerical and asymptotic methods, and apply the results to problems in nerve membrane accommodation and neuronal bursting.

Schistosomiasis remains a major public health problem in many developing countries and represents a class of infectious diseases with strong environmental links. Environmental effects on the transmission of schistosomiasis are well recognized, but the role of specific factors like climate and agricultural practices in modulating transmission is seldom characterized quantitatively. To understand the effects of these factors in the transmission and to explore how the understanding could help to inform better control strategies, a mathematical model was developed for the site-specific characterization of schistosomiasis transmission in irrigated agricultural environments in western China. The model incorporated impact of microclimatic condition on infective stages, the ecology of intermediate snail host, and spatial and temporal heterogeneities associated with human exposure. The model was then calibrated using field data from intervention studies in three villages and simulated to predict the effects of alternative control options. Both the results of these interventions and earlier epidemiological findings confirm the central role of environmental factors, particularly those relating to snail habitat, site and timing of exposure and agricultural and sanitation practices. Moreover, the findings indicate the inadequacy of current chemotherapy-mollusciciding strategies alone to achieve sustainable interruption of transmission in some endemic areas. More generally, the analysis suggests a village-specific index of transmission potential and how this potential is modulated by time-varying factors, including climatological variables, seasonal water-contact patterns, and irrigation practices, which altogether provide a framework for evaluating the likelihood of sustained schistosomiasis transmission and suggest an approach to quantifying the role of environmental factors for other environmentally-mediated infectious diseases.

Angiogenesis is one of the many important processes that occurs in both normal development such as placental growth and embryonic development as well as in abnormal growth such as in the rapid growth of malignant tumors. The purpose of this talk is to give a sense of some of the underlying biochemical and mathematical ideas that have been coupled together to model this process. Most of what we have to say here has already appeared in the mathematical biology literature. However, we have added some new modifications that have yet to appear but that may offer new insight into modeling this complex phenomenon.

A mathematical analysis of SELEX (Systematic Evolution of Ligands by Exponential Enrichment)

Gene regulation is currently at the center of attention of bio-medical community. The genomes of many species are known and the next big challenge of systems biology is to understand complex interactions between genes and their products in the context of a wide variety of pre- and post- transcriptional control and modification.

In this endeavour the concept of a "network" plays a central role. The biological data is often recorded and transmitted in literature and a list of proteins, genes or ligands which either up- or down- regulate the production, expression or concentration of other agents. This leads to a common picture of the network as a graph with nodes that correspond to the agents and signed arrows, that describe the interactions between them.

There are recent attempts to model the interaction at each node of the graph as a logical gate, where inputs are transformed to an output by a set of logic rules.

We discuss the situations in which both network and logical gate models of gene regulation are incomplete. While these widely used models may lead to important insights, one should be aware of pitfalls and limitation of such simplified view of gene regulation.

When modelling the spatial spread of infections, several paradigms are available. In densely populated areas, partial differential equations are very appropriate. In more sparsely populated zones, or for problems involving the spread between administrative units such as Counties, Provinces or Countries, metapopulations become an interesting alternative. Such models consist in a large number of differential equations coupled within multi-digraphs. I will discuss the general context of diseases in metapopulations, and present elements of a theorization of the domain.