Lorentz Center - Mathematical Modeling and Analysis of Biological Networks NDNS+ workshop from 29 Jan 2007 through 2 Feb 2007
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    Mathematical Modeling and Analysis of Biological Networks
    NDNS+ workshop
    from 29 Jan 2007 through 2 Feb 2007

Co-receptor Modulation of T Cell Antigen Receptor Avidity

Co-receptor Modulation of T Cell Antigen Receptor Avidity

Hugo van den Berg (Warwick)


The cellular immune system faces the problem of mounting an effective immune response against pathologically altered body cells, while sparing healthy, unaltered cells. When the immune system fails to solve this problem, autoimmune disease ensues; but such failure is relatively rare. On a widespread (but naive) concept of T cell recognition as an all-or-none lock-and-key-type phenomenon, a host of difficulties is encountered, suggesting that the immune system ought to be very unsuccessful in solving the problem of appropriately directed recognition. However, these difficulties vanish when the naive concept is replaced by a biophysically more realistic theory which treats recognition as continuous rather than discrete. The continuous quantity expressing the strength of recognition has been termed "functional avidity". Recent experimental evidence suggests that the co-receptor CD8 plays a crucial role in modulating this avidity, and in focussing reactivity on the salient (i.e. disease-associated) antigens. I review models used to analyse the relative contributions made by the various mechanisms through which avidity modulation is effected.



Mapping global sensitivity: a new approach to sensitivity analysis, parameter reduction and experimental optimisation for cellular network dynamics

David Rand (Warwick)


A key challenge of systems biology is to develop analytical tools to understand and predict the behaviour of regulatory, signalling and metabolic networks and to connect models of the networks to data. This is difficult because the dynamical systems that arise are highly nonlinear, have high-dimensional state spaces and depend on large numbers of parameters. I develop a more global approach to sensitivity analysis that studies the variation of the whole solution rather that focusing on just one output variable. Such an approach is practicable for cellular networks because such systems have a local geometric rigidity. This leads to new approaches to sensitivity analysis, parameter reduction and experimental optimisation. A key result is that all the sensitivities of such a complex dynamical system can be represented in terms of a single graphical object. Moreover, the approach provides a framework to predict which experimental protocols and perturbations best reveal which aspects of the system and to provide a cost-benefit analysis of the different possibilities for data collection.



Flexible house hunting strategies in social insects

Bob Planqué (Amsterdam)


The study of decentralized decision making in social insects and other groups of social animals has revealed a number of prominent mechanisms such as positive feedback, inhibition of behaviours and response thresholds. One of the prime examples in which many of these behaviours are employed to form collective decisions is house hunting by colonies of ants. When their old nest is destroyed, scouts go looking for potential new nests, recruit other ants to these nests through a process called tandem-running, and switch from recruitment to carrying by monitoring if a quorum of ants has been reached inside a new nest. Using these different behaviours allows the ants to efficiently trade speed for accuracy when deciding which nest to emigrate to. One of the behaviours commonly observed during colony emigrations has sparked much speculation, and does not fit the above emigration paradigm: reverse tandem running from the new to the old nest. Although ants are usually regarded as simple automatons obeying innate rules, close scrutiny reveals that they are capable of rich individual behaviour, including learning and even teaching others. In this talk I will highlight the challenges to model such behaviour in order to understand collective decision making in these ants. Then, using a number of models, we will explore different hypotheses that might explain the role of reverse tandem running.



Ancestral processes with selection: branching and Moran models

Ellen Baake (Bielefeld, Kloostermanhoogleraar 2006)


We consider two versions of stochastic population models with mutation and selection.

The first approach relies on a multitype branching process; here, individuals reproduce and mutate independently of each other, without restriction on population size. We analyze the equilibrium behaviour of such models, both in the forward and in the backward direction of time; the backward point of view emerges if the ancestry of individuals chosen randomly from the present population is traced back into the past.

The second approach is the Moran model with selection. Here, the population has constant size N. Individuals reproduce (at rates depending on their types), the offspring inherits the parent's type, and replaces a randomly chosen individual (to keep population size constant). Independently of the reproduction process, individuals can change type, i.e., mutate. As in the branching model, we consider the ancestral lines of single individuals chosen from the  equilibrium population. We use analytical results of Fearnhead (2002) to determine the explicit properties, and parameter dependence, of the ancestral distribution of types, and its relationship with the stationary distribution in forward time.


This is joint work with Robert Bialowons.



Structural principles of protein-protein recognition

Joël Janin (Paris)


The Protein Data Bank (PDB) is a rich source of information on the interaction between biological  macromolecules. We examine X-ray structures in the PDB to derive rules that govern macromolecular recognition and identify properties that distinguish between specific, functional and non-specific, biologically insignificant, interactions. We select four types of macromolecular assemblies: (i) transient protein-protein complexes, (ii) homodimeric proteins, (iii) packing contacts in protein crystals, (iv) icosahedral virus capsids. Crystal packing contacts illustrate non-specific interactions. They are generally much less extensive that the specific contacts that occur in the three other types. A few of the interfaces created by crystal packing contacts are comparable in size to those of protein-protein complexes, but they bury fewer atoms and are less tightly packed than in homodimers or protein-protein complexes (1). Virus capsids contain interfaces with a wide range of sizes. The larger ones resemble the interfaces in homodimers, the smaller ones are more like crystal packing contacts (2). Whereas a close-packed interface creates a buried core that is more hydrophobic than the protein surface and differs in its amino acid composition, crystal contacts have no core and a composition similar to the surface. All protein-protein interfaces include water molecules that make extensive packing and polar interactions. Specific interfaces have about one water per 100 Å2 of buried protein surface, crystal packing interfaces are more highly hydrated with one water per 60 Å2 (3). These rules have implications on the mechanism of formation of macromolecular assemblies, which we illustrate by analyzing a viral capsid.


(1) Bahadur et al (2004) J. Mol. Biol. 336:943-955.

(2) Bahadur



Plasticity and dynamics of neuronal networks involved in cognition

Arjen Brussaard (Amsterdam)


Neuronal networks involved in cognition in humans, are located in two brain areas also known as the hippocampus and the prefrontal cortex. The functions performed by the neuronal microcircuits in these brain areas are to a large extent dependent on the anatomical and physiological properties of the various synaptic pathways connecting neurons. Neural microcircuits across various species and brain regions are similar in terms of their repertoire of neurotransmitters, their synaptic kinetics, and their plasticity. In the CNCR, we study these networks using a variety of technical approaches, including electrophysiology techniques, functional imaging and neuroinformatics approaches. The rationale behind our program is that information processing in neuronal networks under various (patho-) physiological conditions primarily depends on the efficacy of synaptic transmission. In turn alterations in synaptic transmission may be triggered at a molecular level in the synapse, by modulation of the presynaptic output and/or by modulation (short term) or modification (long term) of the postsynaptic sensitivity to the neurotransmitter being released. Under normal conditions, synaptic modulation and plasticity may occur in an input specific fashion and as such may play a crucial role in determining the neuronal wiring diagram of neuronal networks. During adult life synaptic plasticity is thought to keep interconnected neurons within a proper dynamic range at which normal alterations in neuronal networks are still feasible, but at the same time thereby preventing pathophysiological conditions of activity. I will discuss a number of these issues and argue that an intimate collaboration between experimental neuroscience and statistical analysis is entering a new time era.



From Cognition to Dynamics

Frank van der Velde (Leiden)


The brain has many levels of organization, ranging from genes to neuronal systems. Each of these levels contributes to cognition, although the processes that generate cognition (e.g., language, vision) occur in full at the level of neuronal systems. Two complementary approaches can be distinguished in the study to understand how the brain generates cognition. The first one begins at lower levels of organization and tries to understand how the processes at those levels influence the higher levels of organization, up to the level of cognition. The second approach begins with an analysis of cognitive processes in terms of neuronal systems, and tries to understand how the processes in these systems are influenced by lower levels of organization. In this talk, I will follow the second approach, using the example of visual cognition. In particular, I aim to show how the dynamical, computational and cognitive constraints found at different levels of organization can be combined in the study of neurocognition.



Use of nonlinear and complex signal analysis to characterize functional networks in the brain

Cees Stam (Amsterdam)


The brain can be considered a complex network of interacting specialized subunits. One of the key problems in neuroscience is to understand how interactions take place between the elements of this network, and how local specialization and global integration of information processing are reconciled. The modern theory of nonlinear dynamical systems (Stam et al., 2005; Stam, 2006), and, more recently, the modern theory of complex networks (Bassett and Bullmore, 2006), have provided useful concepts and signal analysis tools which can be used to study various aspects of functional integration in the brain.

First, statistical interdependencies between time series of brain activity (fMRI, EEG and MEG) provide information about functional interactions between the brain regions producing these signals. These interdependencies can be studied with classic linear techniques such as correlations or coherence. Recently, nonlinear measures of synchronization, based either on phase synchronization of generalized synchronization have become popular. We will discuss one of these techniques, the synchronization likelihood, in more detail, and illustrate its application to MEG recordings in neurological disorders such as Alzheimer's disease, Parkinson's disease, multiple sclerosis and brain tumours. In all these disorders, synchronization analysis reveals characteristic changes in functional interactions between various brain regions.

Second, the discovery of small-world networks in 1998 and scale-free networks in 1999 has lead to a new field of complex network analysis. This approach allows to characterize the nature of both anatomical as well as functional connectivity networks, and to determine their properties with measures such as the clustering coefficient, the path length and the degree distribution. We will show how graph analysis can be applied to matrices of synchronization strengths in multi channel EEG and MEG data, and discuss what conclusions can be drawn about large scale brain networks in health and disease. More specifically, we will consider the hypothesis that normal brain networks are small-world systems, and that various types of pathology give rise to a more random architecture, possibly lowering the threshold for seizures.




Bassett DS, Bullmore E. Small-world brain networks. The neuroscientist 2006; 12: 512-523.


Stam CJ. Nonlinear dynamical analysis of EEG and MEG: review of an emerging field. Clin Neurophysiol 2005; 116: 2266-2301. 


Stam CJ. Nonlinear brain dynamics. Nova Science Publishers, New York, 2006.



Mathematical modelling of Magnetoencephalographic data

Fetsje Bijma (Amsterdam)


Neural brain activity generates very weak magnetic fields outside the head. A rather new technique called Magnetoencephalography (MEG) is a sensitive technique used for measuring these fields outside the head. MEG has a high temporal resolution: measurements are sampled at a few kHz. One of the goals of MEG is to determine which brain areas were active during the measurement and to what extent. Due to the high noise level in the data, this is not straightforward. Usually, a certain area of interest in the brain is activated by an external stimulus (e.g. a picture) and the responses to this stimulus are recorded several times. The so obtained recorded signals are averaged, and so called “source localisation” is performed on this average measurement. Different models for incorporating the spatiotemporal noise covariance in this parameter estimation problem are discussed. These models are also interesting for determining spatial and temporal characteristics of the background brain activity. Specifically, finding different processes in the background activity is interesting.



Activity-dependent homeostasis: a modeling approach in Purkinje neurons

E. de Schutter (Antwerpen)


The electrical activity of a neuron is strongly dependent on the ionic channels present in its membrane. Modifying the maximal conductances from these channels can have dramatic impact on neuron behavior. But the effect of such modifications can also be cancelled out by compensatory mechanisms between different channels. Robustness of neural activity to channel alterations, also called functional homeostasis, has been observed in several experimental conditions. We have investigated this in a large compartmental Purkinje cell model by trying  to reproduce up to small details its complex neuronal electrical activity using dissimilar sets of ionic currents. This model exhibits different modes of activity depending upon the current that is injected in it: it can be silent, spiking or bursting. By running an evolution strategy with a fitness function based on phase-plane analysis we obtained 20 very different models of this cell. All these models produced very similar outputs to current injections, including tiny details of the complex firing pattern. Overall, the 24 maximal conductances of the channels of these models, that are our free parameters, exhibit large differences with the one of the data. This demonstrates that the precise reproduction of electrical activity can be obtained from a wide range of parameter values. We have tested several hypothesis that can explain this diversity of conductances: low influence of  some parameters, weak compensations between parameters, linear correlations between them, summation of the impact of each ionic current from different locations, etc. But none of them is sufficient to explain the observed diversity. We have then analyzed in more detail the phase-space and found that regions of good models surround our 20 solutions and often -but not always- link them together. But around every good individual, a small difference in only one of its parameters can lead to very bad models. The hypervolume delimitating good points resembles a foam and is restricted to the hyperplanes defined by averages of our original solutions. Our method is efficient in finding good models in this complex landscape. It has been suggested that cytoplasmic calcium concentrations could be the homeostatic sensor.  In the Purkinje cell model this was not the case as predicted calcium concentrations vary widely, except in the spiny dendrite where calcium plays an important role in synaptic plasticity.



Pattern formation in bacterial biofilms: From cell to continuum

Hans Othmer (Minneapolis)


The bacterium Proteus mirabilis can swarm over a hard surface and form spectacular concentric ring patterns. During pattern formation the colony front is observed to move towards the periphery of the Petri dish either continuously or periodically, due to collective movement of elongated, hyper-flagellated swarmer cells at the leading edge. The formation of the rings was thought to arise from periodic colony expansion. However, recent experimental results show that swimmer cells stream inward toward the inoculation site, and form a number of complex patterns, including radial and spiral streams, rings and traveling trains. In this talk we will discuss a hybrid cell-based model which incorporates a simplified single cell signal transduction model with both the adaptation and excitation components. By assuming that swimmer cells respond to a chemoattractant that they produce, we predict the formation of radial streams as a result of the modulation of the local attractant concentration by the cells.  We will also discuss how to derive macroscopic evolution equations from an individual-based description of the tactic response of cells that use a “run-and-tumble'' strategy and for the more complex type of behavioral response characteristic of crawling cells, which detect a signal, extract directional information from a scalar concentration field, and change their motile behavior accordingly.



Control and realization of biochemical systems

Jan H. van Schuppen (Amsterdam)


Control and system theoretic problems for biochemical systems are motivated by the quest for understanding the biology of cells and by biotechnology.

Problems to be discussed include:

(1) modeling of metabolic networks as biochemical systems;

(2) system reduction of biochemical systems;

(3) realization of biochemical systems; and

(4) modeling of a protein network.



Modeling Polar Auxin Transport

Bert Peletier (Leiden)


Auxin is a growth hormone in plants, endowed with a unique and rapid transport mechanism. Discovered by Went in 1928, this mechanism is only gradually being understood. In spite of significant advances in moleucular biological insights, many questions still remain, in particular with regard to the dynamics. In this talk we report on the development of a mathematical model based on recent experiments done at the {\em Plant Biodynamics Laboratory}  by Kees Libbenga, Kees Boot and Pauline van Spronsen.



Qualitative simulation of the carbon starvation response in Escherichia coli

Hidde de Jong (Rhône-Alpes)


The adaptation of living organisms to their environment is controlled at the molecular level by large and complex networks of genes, mRNAs, proteins, metabolites, and their mutual interactions. In order to understand the overall behavior of an organism, we must complement molecular biology with the dynamic analysis of cellular interaction networks, by constructing mathematical models derived from experimental data, and using simulation tools to predict the behavior of the system under a variety of conditions. Following this strategy we have started the analysis of the network of global transcription regulators controlling the adaptation of the bacterium Escherichia coli to environmental stress conditions. Even though E. coli is one of the best studied organisms, it is currently little understood how a stress signal is sensed and propagated throughout the network of global regulators, so as to enable the cell to respond in an adequate way. Using a qualitative method that is able to overcome the current lack of quantitative data on kinetic parameters and molecular concentrations, we have modeled the carbon starvation response network and simulated the response of E. coli cells to carbon deprivation. This has allowed us to identify essential features of the transition between exponential and stationary phase and to make new predictions on the qualitative system behavior following a carbon upshift. The model predictions have been tested experimentally by means of gene reporter systems.



Models of Population Structure in Genetic Association Studies

David Balding (Londen)


One of the simplest population genetic models to go beyond the basic notion of a random-mating population, is that of the finite island subpopulation model, in which there are K randomly-mating subpopulations (islands) that occasionally exchange migrants.  The problem of "population stratification" causing confounding in genetic association studies is usually formulated and tackled in the context of the island model.  I will discuss a broader view of the problem which leads to superior solutions, illustrated via simulation results and some analyses of plant and human data sets.



Effects of deleterious mutations on the evolution of reproductive modes

Patsy Haccou (Leiden)


The prevalence of sexual reproduction is still one of the great mysteries of evolutionary biology, since asexual populations have a twofold fitness advantage over their sexual counterparts. Thus, whenever the two reproductive strategies compete, the elimination of the sexual mode of reproduction is expected, unless there are factors that counterbalance its disadvantages. Nevertheless, most eukaryotes reproduce sexually. Several theories have been developed to explain this.

One of these is that in large populations segregation and recombination may lead to a lower load of deleterious mutations, if there are synergistic interactions between such alleles. In the absence of such epistasis, recombination has no effect on expected viability of females in sexual populations with equal male and female ploidy. Furthermore, their expected mutational load is then equal to that of clonally reproducing females. However, even without synergy, sexual reproduction does reduce the mutation load if there is a stronger selection against deleterious mutations in males than in females. We examined the effect of several forms of sexual and asexual reproduction on mutation load in the absence of epistasis and found that some forms of asexual reproduction lead to a lower mutation load, and thus a higher viability, than diplodiploid sexual reproduction and, furthermore, that haplodiploidy (a mode of sexual reproduction where females are diploid and males haploid) strongly reduces mutation load.

The lion¹s share of the discussion concerning competition between sexual and asexual modes of reproduction, is, however, based on comparison of population growth rates. In fact, most of the reasoning in theoretical evolutionary biology is based on such comparisons. The general idea is that when an initially small mutant population is able to grow to large numbers in a large, stable population of residents, it will take over and replace the resident population. This argument, however, disregards the fact that, although the probability of such a take-over may be positive, it can still be extremely small, thus making it a very rare event indeed, even on an evolutionary scale.

Furthermore, a comparison of population growth rates does not necessarily give correct predictions concerning invasion probabilities. For instance, since haplodiploid populations have a higher viability than diplodiploid ones, they might be expected to be more robust against invasion by asexual reproductive modes. In fact, the opposite is true.

Finally, effects of initial conditions have been ignored. The initial mutation load of a mutant female that starts to reproduce asexually is determined by the distribution of mutation loads in the resident population. Invasion chances are considerably affected by such initial conditions.



Likelihood-based inference for $\Lambda$-coalescents

Matthias Birkner (Berlijn)


$\Lambda$-coalescents are generalisations of Kingman's coalescent which allow for simultaneous coalescence of several ancestral lineages, and are being discussed as a model for genealogies for species with highly variable offspring numbers. We extend results of Griffiths and Tavare, which allow to estimate the likelihood of sequence observations using a Monte Carlo approach, to this setting, and illustrate our method using simulated and real datasets.



A stochastic model of genetic hitchhiking

Anton Wakolbinger (Frankfurt)


Occasionally in evolution a strongly advantageous allele enters the stage at some genetic locus and rapidly  becomes fixed in the population. Such a "selective sweep" leads to a reduction of genetic diversity at neutral loci in the neighbourhood of the selective one: an allele carried by the founder of the sweep at some neigbouring locus spreads into the population along with the founder's offspring, counteracted only by recombination, which now and then tears the two loci apart. This "hitchhiking effect" was first addressed and modeled by Maynard-Smith and Haigh in 1974. In a pioneering analysis of such "selective sweeps", Kaplan, Hudson and Langley (1989) modelled the increase in the frequency of the advantageous gene by a deterministic logistic curve and described the genealogy at a neighbouring neutral locus as a structured coalescent.

The model presented in the lecture replaces the structured coalescent by a marked Yule tree and models the freqency of the advantageous gene by a random diffusion process rather than a deterministic curve. This allows a refined description of the important early phase of the sweep. Simulation confirms that the new hitchhiking model with its "random driver" performs well in competition with the older logistic model.


The talk is based on the two papers

A. Etheridge, P. Pfaffelhuber and A. Wakolbinger, An approximate sampling formula under genetic hitchhikng,  Ann. Appl. Probab. 16 (2006), 685-729.

P. Pfaffelhuber, B. Haubold and A. Wakolbinger, Approximate genealogies under genetic hitchhiking, Genetics 174 (2006),  1995-2008.